Research ArticleModeling Based Characterization of ThermorheologicalProperties of Polyurethane ESTANEƒ

Ehsan Ghobadi1 Rakulan Sivanesapillai1 Jana Musialak2 and Holger Steeb12

1 Institute of Mechanics (CE) University of Stuttgart Pfaffenwaldring 7 70569 Stuttgart Germany2Institute of Mechanics Faculty of Civil and Environmental Engineering Ruhr-University BochumUniversitatsstraszlige 150 44801 Bochum Germany

Correspondence should be addressed to Ehsan Ghobadi ehsanghobadimechbauuni-stuttgartde

Received 19 March 2016 Revised 7 June 2016 Accepted 22 June 2016

Academic Editor Kai Chen

Copyright copy 2016 Ehsan Ghobadi et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Shape-Memory Polymers (SMPs) have the ability to be deformed and memorize this deformation until an external activationstimulus (eg heat) is applied Therefore they have attracted great interest in many areas especially for applications wherereconfigurable structures are required (eg Shape-Memory (SM) stents or micro air vehicles) Nevertheless prior to technicalapplication the effective thermomechanical behavior of SMPs must be thoroughly understood In the current contributionan assessment of thermorheological properties of the commercially available polyurethane system ESTANE is presentedThermorheological properties were investigated using Dynamic Mechanical Thermal Analysis (DMTA) and complementaryuniaxial stress relaxation experiments Upon material parameter optimization a finite viscoelastic and incompressible materialmodel was used to model experimentally observed viscoelastic properties

1 Introduction

Shape-Changing Materials (SCMs) together with Shape-Memory Materials (SMMs) are two kinds of smart materialswith the ability to vary their shape upon being exposed toactivation stimuli [1] Typical stimuli for activating SMMs(or Shape-Memory Polymers SMPs) consist of tempera-ture (including both cooling and heating either directly orindirectly eg inductive or resistive heating) chemicalsand light [1 2] In contrast to SCMs (or Shape-ChangingPolymers SCPs) which can preserve a deformed shape onlyas long as an external stimulus is applied SMMs (or SMPs)can be deformed and fixed to several different forms andrecover their reference shape when the activation stimulus istriggered

Depending on environmental and boundary conditionsfor example temperature and the applied stimulus a poly-mer could be either SMP or SCP [3] In the case of heat asthe activation stimulus the SMP is called thermally inducedSMP The capability of a material to temporarily preservea deformed shape and recover the undeformed referenceshape upon triggering using an external activation stimulus

is known as the Shape-Memory-Effect (SME) In fact thereare two types of thermoresponsive SME The first typeis responsive with respect to heating which is an almostintrinsic feature of all polymers and the other one is coolingresponsive which is only limited to a couple of speciallydesigned polymeric materials [3] Compared to other smartmaterials several properties of SMPs for example theirglass transition temperature (119879119892) or dynamic modulus canbe easily tailored to meet specific requirements Thereforethey have drawn much attention and are intensively used indifferent applications [4]

Essential requisites for a polymer to exhibit the SME arean appropriate polymer structure and morphology togetherwith the use of appropriate processing and programmingmethods [2 4] In the so-called uniaxial Shape-MemoryCreation Procedure (SMCP) [5 6] the polymer is heatedto a temperature (119879prog) which is normally higher thanits transition temperature (119879119892 or melting temperature 119879119898)Afterwards it is deformed in a stress-controlled or strain-controlled process to a prescribed stretch ratio 1205821 in loadingdirection e1 Subsequently the polymer is cooled down to atemperature below the transition temperature (119879low) in order

Hindawi Publishing CorporationInternational Journal of Polymer ScienceVolume 2016 Article ID 7514974 11 pageshttpdxdoiorg10115520167514974

2 International Journal of Polymer Science

to solidify the switching domains resulting in mechanicalfixation of the deformed shape upon stress unloading Foractivation of the SME the programmed shape should beexcited for example by heating to a temperature above thetransition temperature (119879sw) [7]

Two important characteristics for quantification of SMEare shape fixity ratio (119877119891) and shape recovery ratio (119877119903)which are usually quantified by cyclic thermomechanicalinvestigations under uniaxial (biaxial) tension or compres-sion or even bending experiments Macroscopically shapefixation is the extent up to which a deformation can be fixedafter programming and shape recovery is the recoverability ofthe sample after being heated [8]

Characterization of SME requires investigations on ther-momechanical and rheological behavior of the polymericsystem through various modeling and experimental studies[9ndash13] Generally two different phenomenological modelingapproaches can be used to model the mechanism of SMEThe first model is called Standard-Linear-Solid (SLS) [14ndash16] which requires phenomenological constitutive relationsfor the temperature-dependence of the material propertiesIn recent years such models have been well established andintensively discussed in literature compare [17ndash21] Althoughthese models provide insight into the shape recovery behav-ior they have limited predictive capabilities and their appli-cations are mostly limited to one-dimensional problemsTherefore recent works have shifted towards more physicaldescriptions of the underlying (Shape-Memory) SM mecha-nisms [22] A further set of modeling approaches explicitlyaccounts for the underlying morphology of the polymerThesemodels are known as phase transition or frozen volumefraction approaches [23ndash25] and describe the transition ofthe dominant microscopic deformation mechanism fromentropy-driven conformational chain rearrangements at hightemperatures tomolecular deformations corresponding to aninternal energy change at low temperaturesHere amorphouspolymers are considered as two-phase materials composedof frozen and active parts such that the total strain could berepresented as the sum of two contributions from respectivevolume fractions [26] Although these models deliver goodresults with comparatively small number of parameters stilllimited perception about the underlying physics could beachieved Therefore for design and evaluation of mechanicalcomponents with complex geometries under realistic load-ing and environmental conditions prior to any modelingactivity comprehensive experimental studies on thermal andmechanical properties over a wide temperature time andstrain range are required [13] To this end an extensive ther-momechanical analysis of a SM-polyurethane commerciallyavailable under the name ESTANE is first performed Byapplying the theory of finite viscoelasticity and based on therheological results achieved here the constitutive viscoelasticbehavior of ESTANE is represented by a generalizedMaxwell-Zener model

2 Material and Methods

21 Material In the present contribution aliphatic SM-poly-urethane ESTANE ETE75DT3 NAT022 granulates (Lubrizol

OveleWesterlo Belgium) have been processed using an injec-tion molding machine (Arburg Allrounder 270M 500-210Lossburg Germany) Sample preparation and preprocessingsteps have been described in detail in [27]

For quantitative characterization of the polymer twocomplementary sets of experiments have been performedThe Dynamic Mechanical Thermal Analysis (DMTA) andquasistatic stress relaxation experiments

22 Thermorheological Characterization To determine thetime-temperature-dependent viscoelastic properties DMTAtests in torsion mode were performed using rectangularsamples with dimensions of 119882 times 119867 times 119871 2mm times 10mmtimes 50mm Here DMTA experiments were performed witha stress-controlled rheometer with integrated Peltier-basedtemperature chamber (Anton Paar Physica MCR 301 plusCTD 180 Graz Austria) The isothermal scans were carriedout at specially chosen temperature set points in the rangeof 20 to 90∘C For frequencies from 001 up to 100 radsdotsminus1harmonic twist rotations were applied with maximum ampli-tudes of 02 while resulting torques as well as phase lagswere monitored Subsequently RheoPlus software [28] wasused to calculate the corresponding magnitude of complexshear modulus |119866lowast| storage modulus 1198661015840 loss modulus11986610158401015840 and loss factor tan 120575 which are related to each other

according to the following relations [29]1003816100381610038161003816119866lowast1003816100381610038161003816 =radic11986610158402 + 119866101584010158402

tan 120575 = 11986610158401015840

1198661015840

(1)

The dynamic shear modulus 119866lowast = 1198661015840 + 11989411986610158401015840 is the appro-priate constitutive measure to evaluate stresses relaxationduring strain-controlled experiments Equivalently in stress-controlled experiments the complex shear compliance 119869lowastdetermines strain retardation where

1003816100381610038161003816119869lowast1003816100381610038161003816 =

1

|119866lowast|

1198691015840=1003816100381610038161003816119869lowast1003816100381610038161003816 cos 120575

11986910158401015840=1003816100381610038161003816119869lowast1003816100381610038161003816 sin 120575

(2)

Here 1198691015840 represents the storage compliance and 11986910158401015840 the losscompliance Finally for adequate thermorheological char-acterization of ESTANE the (complex) dynamic viscositywas computed The in-phase viscosity 1205781015840 and out-of-phaseviscosity 12057810158401015840 as well as complex viscosity |120578lowast| are related to theloss and storage moduli in terms of

1205781015840=11986610158401015840

120596

12057810158401015840=1198661015840

120596

1003816100381610038161003816120578lowast1003816100381610038161003816 =

1003816100381610038161003816119866lowast1003816100381610038161003816

120596

(3)

where 120596 is the frequency

International Journal of Polymer Science 3

23 Stress Relaxation Experiments For quasistatic relaxationexperiments ESTANE plates were punched to dumbbellspecimens of type 5A according to DIN EN ISO 527-2 (119882 times119867 times 119871 2mm times 4mm times 75mm) Uniaxial stretching andrelaxation experiments were performed on a spindle-driventesting machine (Schenck Trebel RM50 New York USA)with a stretching and compression capacity of plusmn500N Themachine was equipped with a heating chamber of type Fre-senberger TK18400100 (Wipperfurth Germany) attachedto a PID temperature controller (Eurotherm 3508 LimburgGermany) so that temperatures in the range of minus100 to 400∘Ccould be adjusted and kept constant with a precision of plusmn1∘C

For stress relaxation experiments in displacement drivencontrol mode all specimens were pneumatically fixed alongtheir stretching axis and heated up with a rate of 3∘Csdotminminus1 toa temperature119879prog which was equilibrated for at least 10minAfterwards the specimens were deformed with a strain rateof 0005 sminus1 from their initial length 119871 to a new length 119897corresponding to a local uniaxial strain of about 120576 = (119897 minus119871)119871 = 10 This uniaxial stretch 120582 = 119897119871 = 11 was keptconstant for at least 60min At different temperatures 119879progranging from 10∘C up to 80∘C the decay of stress over timewasmonitored Considering a linear viscoelastic behavior forESTANE the total stress will ultimately reach an equilibriumstress following the decay of multiple superimposed relax-ation processes as long as no viscous flow occurs It is worthmentioning that all experiments were repeated two times toimprove confidence

24Thermal Expansion Experiments Todetermine theCoef-ficient of Thermal Expansion (CTE) two different experi-ments were conducted The first procedure was performedwith the same spindle-driven machine as used for therelaxation experiments Here the samples were heated whilea force controlled loop kept a tensile force of approximately05N constant using the feedback signals of a load cell thatwas attached to the moving upper clamp and an actuatorTheupper clamp displacement during heating was tracked andassuming a homogeneous deformation subsequently usedto determine the sample deformation The correspondingtemperature (from 10 to 140∘C) versus stretch data was usedto compute the Coefficients of Thermal Expansion

In addition to this the same rheometer which was usedfor DMTA experiments was applied as an alternative methodfor stress-controlled CTE-measurement In equivalence tothe above approach a tensile force of 05N was controlled tobe constant while heating the samples from 10 to 140∘C Incontrast to the uniaxial spindle-driven testing machine theadvantage of the rheometer apparatus is the use of a high-resolution load cell with an accuracy of 0005N

3 Results and Discussion

31 Characterization

311 Determination of Thermal Expansion A subtle dis-continuity in the CTE which otherwise exhibits a lineartemperature dependency is observed for both rheometer

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

1000

1005

1010

1015

1020

1025

1030

Rheometer

Rheometer

Spindle

Spindle

120582(mdash

)

T (∘C)

T (∘C)

F(N

)

10

05

00

RheometerSpindle

120572r = 2549 middot 10minus4 Kminus1

120572r = 2842 middot 10minus4 Kminus1

120572g = 1876 middot 10minus4 Kminus1

120572g = 1350 middot 10minus4 Kminus1

Figure 1 Thermal expansion during heating while applying con-stant tensile force of 05N

and spindle-driven testing procedures The correspondingdata points and best fitting linear regressions are depictedin Figure 1 The tensile force which was controlled duringheating is plotted in the inset to Figure 1 Pronouncedfluctuations in the applied tensile force around 119879119892 suggestthat quantitative measurement of CTE is impeded for theless accurate spindle-driven testing procedure The forcecontrolling mechanism was more accurate during rheometertesting which we attribute to higher accuracy of the loadsensor The magnitudes of elongation are furthermore foundto be distinctly small such that data acquisition errors need tobe consideredThe corresponding magnitudes of uncertaintyare illustrated by means of error bars and found to be com-paratively high at low temperatures The subscripts ldquo119892rdquo andldquo119903rdquo indicate the regions of glass-like and rubbery behaviorrespectively Below 119879119892 the CTE is less pronounced whichproduces a kink in the 120582-T curve The discontinuity of linearthermal expansion which we attribute to glass transition isfound at 119879119892 asymp 55

∘C This coincides with previously reportedresults in [27 30]

32 Dynamic Thermomechanical Experiments Aim of the(cyclicharmonic) amplitude sweeps is to determine a linearregime where effective properties are independent of the(shear strain) amplitude Therefore DMTA amplitude sweeptests where the strain amplitude increases incrementally withfixed frequency are used to determine the linear viscoelasticregion (LVR) Within the LVR the viscoelastic response ofthe polymer is independent of the amplitude of deformationsuch that constitutive nonlinearities and plasticity effects canbe excluded In Figure 2 the variations of storage and lossmoduli with strain amplitude 1205740 at different temperatures (040 and 80∘C) are depicted In the low temperature regime(119879 = 0∘C Figure 2 black dots) the mechanical response(1198661015840) is dominated by the hard segment phase It could beobserved that 1198661015840 and 11986610158401015840 is hardly amplitude dependentwithin 003 lt 120574 lt 1and no hint for any plastic behavior isobserved In the high temperature regime (T = 80∘C Figure 2

4 International Journal of Polymer Science

104

103

102

101

104

103

102

101

001 01 1 001 01 1

G998400

(MPa

)

1205740 ()

G998400998400

(MPa

)

G998400 G998400998400

T = 0∘C

T = 80∘CT = 40∘C

Figure 2 Amplitude sweep test at different temperatures

blue diamonds) the mechanical properties are determinedby the soft segments In this graph the linear viscoelasticregion can be identified as a pronounced plateau For shearamplitudes within the range 003 lt 120574 lt 1 the storageand loss moduli remain almost constant This is a fairlybroad LVRwhen compared to previously reported LVRs [29]For shear amplitudes smaller than 003 the decrease inthe experimental signal-to-noise ratio results in an artificialdeviation of moduli from the LVR plateau

Upon identifying the appropriate strain amplitude rangestorage and loss moduli have been measured using tempera-ture sweeps as shown in Figure 3(a) As can be seen DMTA intemperature sweep mode gives intriguing information aboutstructural properties of ESTANE It is shown that the storagemodulus 1198661015840 decreases gradually with temperature until theglass transition region is reached The plotted behavior istypical for physically cross-linked block copolymers [31]The inflection point of the 1198661015840-119879-graph determines 119879119892 of thesampleThis point shifts to a higher temperature as frequencyincreases 119879119892 of SM-ESTANE was measured to be approxi-mately 55∘C which is in agreement with previously reportedresults [30] Interestingly the temperature sweep tests of SM-ESTANE scanned at different frequencies indicates that thereduction of 1198661015840 is less pronounced at higher frequenciesthan at lower ones Moreover the storage modulus decreasesuntil it reaches a minimum This is more considerable forsamples under lower tested frequencies This minimumlocated at temperatures higher than 119879119892 is characteristicfor another relaxation inside the polymer chains due tomolecular heterogeneities of the specimen Samples underhigher frequencies have less or not enough time to showthis relaxation An inverse temperature sweep test with thesame frequency proves that this observation is almost notrepeatable

The loss modulus 11986610158401015840 shows a different behavior 11986610158401015840-119879-plot has also a nonmonotonic form The curves increasewith increasing temperature until 119879119892 is attained and decreaseslowly at higher temperatures Moreover at higher frequen-cies the loss modulus takes higher values The maximum of

11986610158401015840-119879-graph is interpreted as 119879119892 of the sample Likewise a

minimum at 119879 gt 119879119892 characteristic for heterogeneities of thesample is perceptible

Moreover in addition to storage and loss moduli storageand loss compliance are plotted in Figure 3(b) As can be seenthe loss and storage compliance show a nonuniform behaviorand obviously are in an almost opposite form in comparisonwith the loss and storage moduli The storage compliance1198691015840 increases first gradually with respect to temperature untilit reaches a maximum The plotted loss compliance as afunction of temperature for different frequencies has alsoa monotonic shape It decreases with temperature until 119879119892is reached and then increases with higher slopes up to thesecond relaxation temperature

In addition to this the results of dynamic and out-of-phase viscosities as a function of temperature are plotted inFigure 4 Again a nonmonotonic behavior can be observedfor these properties and a stronger frequency dependencethan that of moduli and compliance is obvious It can beconcluded that at higher frequencies the material showsmore internal energy dominated glassy behavior whereby atdecreasing frequency a transition to a nearly entropic rubberystate is surveyed This is because of the fact that at higherfrequencies polymer chains absorb energy much better thanat lower frequencies

As shown in Figure 5 the temperature dependency ofstorage and loss moduli exhibit weak sensitivity with respectto heating rate 120585 which was varied in the range from 025 to2∘Csdotminminus1 during temperature sweeps Below 119879119892 the storagemodulus does not noticeably change with temperature and asmooth decreasing trend is observed ESTANE demonstratestherefore a linear elastic behavior in this region In the regionaround119879119892 a harsh fall of storage modulus is observed In thisrange a pronounced difference can be observed for differentheating rates For higher heating rates the storage modulusexhibits a global minimum at around 119879119892 This could beattributed to the fact that for higher heating rates the kineticenergy of polymer chains increases noticeably resulting in abigger free volume inside the polymer matrix This volumeexpansion obtains the needed energy and space for chainmovements in a defined time scale leading to an extremesubsidence of energy However after this range the polymerchains resist to stresses due to physical cross-links and chainentanglements Finally one can conclude that the changes ofmechanical properties dependmore strongly upon frequencyor strain amplitudes rather than heating rates Additionallythe results of loss factor (damping coefficient) are shown inFigure 6 The peak values and the temperatures regardingthis maximum point are very well detectable A small shifttowards higher temperatures for tan 120575 is seen for higherheating rates A similar behavior is also recorded in otherthermal analyses such as TMA or DSC [32]

Rearrangement of chain sequences in polymeric systemsleading to molecular relaxations accelerates at higher ratesand elevated temperatures This direct equivalence betweentime and temperature leads to the fact that the time overwhich these relaxations take place can be simply reiteratedby conducting the experiment at higher temperatures andshifting the resulting data to lower frequencies This treating

International Journal of Polymer Science 5

1000

100

10

1000

100

10

0 20 40 60 80 100120

G998400

(MPa

)

G998400 G998400998400

G998400998400

(MPa

)

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

(a) Variation of storage and loss shear modulus with temperature

0 20 40 60 80 100120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

J998400(1

MPa

)

J998400998400(1

MPa

)

J998400 J998400998400

10minus1

10minus2

(b) Variation of storage and loss compliance with temperature

Figure 3 Temperature sweep tests at different frequencies

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus1

10minus2

100

101

102

103

10minus1

10minus2

100

101

102

103

120578998400 120578998400998400

120578998400

(MPa

middots)

120578998400998400

(MPa

middots)

Figure 4 Temperature sweep tests at different frequencies variationof dynamic and out-of-phase viscosities with temperature

of the results of dynamic experiments is called the Time-Temperature Superposition Principle [33] and can be usedto produce a corresponding master curve of a polymer withwhich the material properties can be estimated for very longand not measurable time scales [29] The horizontal shiftfactor values (119886119879) for linking the individual experimental dataand producing the master curve can be obtained through theWilliams-Landel-Ferry (WLF) equation [29]

log 119886119879 = minus1198621 (119879 minus 119879119903)

1198622 + (119879 minus 119879119903) (4)

where 1198621 and 1198622 are empirical constants and 119879119903 denotes thereference temperature

Figure 7(a) demonstrates the experimentally obtaineddata of 1198661015840 at different isothermal temperatures varying from20∘C to 70∘C in the range of unreduced angular frequencies

103

102

101

103

102

101

G998400

(MPa

)

G998400998400

(MPa

)

G998400 G998400998400

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

Figure 5 Temperature sweep tests with different heating ratesvariation of storage and loss shear modulus with temperature

120596 from 01 to 100 radsdotsminus1 Complementary to this obtainedmaster curves of 1198661015840 and 11986610158401015840 are shown in Figure 7(b) Here119879119903 is 50

∘Cand1198621 and1198622 are found to be 8 and 18 respectivelyAs can be seen the shift factor values of WLF equation andthe chosen temperature increments lead to acceptable mastercurves from experimentally acquired data revealing thatESTANE may be considered as thermorheologically simpleHowever because of segmental and global heterogeneityconsidering ESTANE as thermorheological complex [34 35]would lead to even better master curves

A very interesting inference which can be concludedfrom the aspect of the resulting master curve is the feature ofthe polymer system to show the SME In the fact of the viscousnature of the system which is essential to the shape-storageand recovery one can deduce that the SME results only fromthe time-temperature-dependent property of the system and

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 International Journal of Polymer Science

to solidify the switching domains resulting in mechanicalfixation of the deformed shape upon stress unloading Foractivation of the SME the programmed shape should beexcited for example by heating to a temperature above thetransition temperature (119879sw) [7]

Two important characteristics for quantification of SMEare shape fixity ratio (119877119891) and shape recovery ratio (119877119903)which are usually quantified by cyclic thermomechanicalinvestigations under uniaxial (biaxial) tension or compres-sion or even bending experiments Macroscopically shapefixation is the extent up to which a deformation can be fixedafter programming and shape recovery is the recoverability ofthe sample after being heated [8]

Characterization of SME requires investigations on ther-momechanical and rheological behavior of the polymericsystem through various modeling and experimental studies[9ndash13] Generally two different phenomenological modelingapproaches can be used to model the mechanism of SMEThe first model is called Standard-Linear-Solid (SLS) [14ndash16] which requires phenomenological constitutive relationsfor the temperature-dependence of the material propertiesIn recent years such models have been well established andintensively discussed in literature compare [17ndash21] Althoughthese models provide insight into the shape recovery behav-ior they have limited predictive capabilities and their appli-cations are mostly limited to one-dimensional problemsTherefore recent works have shifted towards more physicaldescriptions of the underlying (Shape-Memory) SM mecha-nisms [22] A further set of modeling approaches explicitlyaccounts for the underlying morphology of the polymerThesemodels are known as phase transition or frozen volumefraction approaches [23ndash25] and describe the transition ofthe dominant microscopic deformation mechanism fromentropy-driven conformational chain rearrangements at hightemperatures tomolecular deformations corresponding to aninternal energy change at low temperaturesHere amorphouspolymers are considered as two-phase materials composedof frozen and active parts such that the total strain could berepresented as the sum of two contributions from respectivevolume fractions [26] Although these models deliver goodresults with comparatively small number of parameters stilllimited perception about the underlying physics could beachieved Therefore for design and evaluation of mechanicalcomponents with complex geometries under realistic load-ing and environmental conditions prior to any modelingactivity comprehensive experimental studies on thermal andmechanical properties over a wide temperature time andstrain range are required [13] To this end an extensive ther-momechanical analysis of a SM-polyurethane commerciallyavailable under the name ESTANE is first performed Byapplying the theory of finite viscoelasticity and based on therheological results achieved here the constitutive viscoelasticbehavior of ESTANE is represented by a generalizedMaxwell-Zener model

2 Material and Methods

21 Material In the present contribution aliphatic SM-poly-urethane ESTANE ETE75DT3 NAT022 granulates (Lubrizol

OveleWesterlo Belgium) have been processed using an injec-tion molding machine (Arburg Allrounder 270M 500-210Lossburg Germany) Sample preparation and preprocessingsteps have been described in detail in [27]

For quantitative characterization of the polymer twocomplementary sets of experiments have been performedThe Dynamic Mechanical Thermal Analysis (DMTA) andquasistatic stress relaxation experiments

22 Thermorheological Characterization To determine thetime-temperature-dependent viscoelastic properties DMTAtests in torsion mode were performed using rectangularsamples with dimensions of 119882 times 119867 times 119871 2mm times 10mmtimes 50mm Here DMTA experiments were performed witha stress-controlled rheometer with integrated Peltier-basedtemperature chamber (Anton Paar Physica MCR 301 plusCTD 180 Graz Austria) The isothermal scans were carriedout at specially chosen temperature set points in the rangeof 20 to 90∘C For frequencies from 001 up to 100 radsdotsminus1harmonic twist rotations were applied with maximum ampli-tudes of 02 while resulting torques as well as phase lagswere monitored Subsequently RheoPlus software [28] wasused to calculate the corresponding magnitude of complexshear modulus |119866lowast| storage modulus 1198661015840 loss modulus11986610158401015840 and loss factor tan 120575 which are related to each other

according to the following relations [29]1003816100381610038161003816119866lowast1003816100381610038161003816 =radic11986610158402 + 119866101584010158402

tan 120575 = 11986610158401015840

1198661015840

(1)

The dynamic shear modulus 119866lowast = 1198661015840 + 11989411986610158401015840 is the appro-priate constitutive measure to evaluate stresses relaxationduring strain-controlled experiments Equivalently in stress-controlled experiments the complex shear compliance 119869lowastdetermines strain retardation where

1003816100381610038161003816119869lowast1003816100381610038161003816 =

1

|119866lowast|

1198691015840=1003816100381610038161003816119869lowast1003816100381610038161003816 cos 120575

11986910158401015840=1003816100381610038161003816119869lowast1003816100381610038161003816 sin 120575

(2)

Here 1198691015840 represents the storage compliance and 11986910158401015840 the losscompliance Finally for adequate thermorheological char-acterization of ESTANE the (complex) dynamic viscositywas computed The in-phase viscosity 1205781015840 and out-of-phaseviscosity 12057810158401015840 as well as complex viscosity |120578lowast| are related to theloss and storage moduli in terms of

1205781015840=11986610158401015840

120596

12057810158401015840=1198661015840

120596

1003816100381610038161003816120578lowast1003816100381610038161003816 =

1003816100381610038161003816119866lowast1003816100381610038161003816

120596

(3)

where 120596 is the frequency

International Journal of Polymer Science 3

23 Stress Relaxation Experiments For quasistatic relaxationexperiments ESTANE plates were punched to dumbbellspecimens of type 5A according to DIN EN ISO 527-2 (119882 times119867 times 119871 2mm times 4mm times 75mm) Uniaxial stretching andrelaxation experiments were performed on a spindle-driventesting machine (Schenck Trebel RM50 New York USA)with a stretching and compression capacity of plusmn500N Themachine was equipped with a heating chamber of type Fre-senberger TK18400100 (Wipperfurth Germany) attachedto a PID temperature controller (Eurotherm 3508 LimburgGermany) so that temperatures in the range of minus100 to 400∘Ccould be adjusted and kept constant with a precision of plusmn1∘C

For stress relaxation experiments in displacement drivencontrol mode all specimens were pneumatically fixed alongtheir stretching axis and heated up with a rate of 3∘Csdotminminus1 toa temperature119879prog which was equilibrated for at least 10minAfterwards the specimens were deformed with a strain rateof 0005 sminus1 from their initial length 119871 to a new length 119897corresponding to a local uniaxial strain of about 120576 = (119897 minus119871)119871 = 10 This uniaxial stretch 120582 = 119897119871 = 11 was keptconstant for at least 60min At different temperatures 119879progranging from 10∘C up to 80∘C the decay of stress over timewasmonitored Considering a linear viscoelastic behavior forESTANE the total stress will ultimately reach an equilibriumstress following the decay of multiple superimposed relax-ation processes as long as no viscous flow occurs It is worthmentioning that all experiments were repeated two times toimprove confidence

24Thermal Expansion Experiments Todetermine theCoef-ficient of Thermal Expansion (CTE) two different experi-ments were conducted The first procedure was performedwith the same spindle-driven machine as used for therelaxation experiments Here the samples were heated whilea force controlled loop kept a tensile force of approximately05N constant using the feedback signals of a load cell thatwas attached to the moving upper clamp and an actuatorTheupper clamp displacement during heating was tracked andassuming a homogeneous deformation subsequently usedto determine the sample deformation The correspondingtemperature (from 10 to 140∘C) versus stretch data was usedto compute the Coefficients of Thermal Expansion

In addition to this the same rheometer which was usedfor DMTA experiments was applied as an alternative methodfor stress-controlled CTE-measurement In equivalence tothe above approach a tensile force of 05N was controlled tobe constant while heating the samples from 10 to 140∘C Incontrast to the uniaxial spindle-driven testing machine theadvantage of the rheometer apparatus is the use of a high-resolution load cell with an accuracy of 0005N

3 Results and Discussion

31 Characterization

311 Determination of Thermal Expansion A subtle dis-continuity in the CTE which otherwise exhibits a lineartemperature dependency is observed for both rheometer

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

1000

1005

1010

1015

1020

1025

1030

Rheometer

Rheometer

Spindle

Spindle

120582(mdash

)

T (∘C)

T (∘C)

F(N

)

10

05

00

RheometerSpindle

120572r = 2549 middot 10minus4 Kminus1

120572r = 2842 middot 10minus4 Kminus1

120572g = 1876 middot 10minus4 Kminus1

120572g = 1350 middot 10minus4 Kminus1

Figure 1 Thermal expansion during heating while applying con-stant tensile force of 05N

and spindle-driven testing procedures The correspondingdata points and best fitting linear regressions are depictedin Figure 1 The tensile force which was controlled duringheating is plotted in the inset to Figure 1 Pronouncedfluctuations in the applied tensile force around 119879119892 suggestthat quantitative measurement of CTE is impeded for theless accurate spindle-driven testing procedure The forcecontrolling mechanism was more accurate during rheometertesting which we attribute to higher accuracy of the loadsensor The magnitudes of elongation are furthermore foundto be distinctly small such that data acquisition errors need tobe consideredThe corresponding magnitudes of uncertaintyare illustrated by means of error bars and found to be com-paratively high at low temperatures The subscripts ldquo119892rdquo andldquo119903rdquo indicate the regions of glass-like and rubbery behaviorrespectively Below 119879119892 the CTE is less pronounced whichproduces a kink in the 120582-T curve The discontinuity of linearthermal expansion which we attribute to glass transition isfound at 119879119892 asymp 55

∘C This coincides with previously reportedresults in [27 30]

32 Dynamic Thermomechanical Experiments Aim of the(cyclicharmonic) amplitude sweeps is to determine a linearregime where effective properties are independent of the(shear strain) amplitude Therefore DMTA amplitude sweeptests where the strain amplitude increases incrementally withfixed frequency are used to determine the linear viscoelasticregion (LVR) Within the LVR the viscoelastic response ofthe polymer is independent of the amplitude of deformationsuch that constitutive nonlinearities and plasticity effects canbe excluded In Figure 2 the variations of storage and lossmoduli with strain amplitude 1205740 at different temperatures (040 and 80∘C) are depicted In the low temperature regime(119879 = 0∘C Figure 2 black dots) the mechanical response(1198661015840) is dominated by the hard segment phase It could beobserved that 1198661015840 and 11986610158401015840 is hardly amplitude dependentwithin 003 lt 120574 lt 1and no hint for any plastic behavior isobserved In the high temperature regime (T = 80∘C Figure 2

4 International Journal of Polymer Science

104

103

102

101

104

103

102

101

001 01 1 001 01 1

G998400

(MPa

)

1205740 ()

G998400998400

(MPa

)

G998400 G998400998400

T = 0∘C

T = 80∘CT = 40∘C

Figure 2 Amplitude sweep test at different temperatures

blue diamonds) the mechanical properties are determinedby the soft segments In this graph the linear viscoelasticregion can be identified as a pronounced plateau For shearamplitudes within the range 003 lt 120574 lt 1 the storageand loss moduli remain almost constant This is a fairlybroad LVRwhen compared to previously reported LVRs [29]For shear amplitudes smaller than 003 the decrease inthe experimental signal-to-noise ratio results in an artificialdeviation of moduli from the LVR plateau

Upon identifying the appropriate strain amplitude rangestorage and loss moduli have been measured using tempera-ture sweeps as shown in Figure 3(a) As can be seen DMTA intemperature sweep mode gives intriguing information aboutstructural properties of ESTANE It is shown that the storagemodulus 1198661015840 decreases gradually with temperature until theglass transition region is reached The plotted behavior istypical for physically cross-linked block copolymers [31]The inflection point of the 1198661015840-119879-graph determines 119879119892 of thesampleThis point shifts to a higher temperature as frequencyincreases 119879119892 of SM-ESTANE was measured to be approxi-mately 55∘C which is in agreement with previously reportedresults [30] Interestingly the temperature sweep tests of SM-ESTANE scanned at different frequencies indicates that thereduction of 1198661015840 is less pronounced at higher frequenciesthan at lower ones Moreover the storage modulus decreasesuntil it reaches a minimum This is more considerable forsamples under lower tested frequencies This minimumlocated at temperatures higher than 119879119892 is characteristicfor another relaxation inside the polymer chains due tomolecular heterogeneities of the specimen Samples underhigher frequencies have less or not enough time to showthis relaxation An inverse temperature sweep test with thesame frequency proves that this observation is almost notrepeatable

The loss modulus 11986610158401015840 shows a different behavior 11986610158401015840-119879-plot has also a nonmonotonic form The curves increasewith increasing temperature until 119879119892 is attained and decreaseslowly at higher temperatures Moreover at higher frequen-cies the loss modulus takes higher values The maximum of

11986610158401015840-119879-graph is interpreted as 119879119892 of the sample Likewise a

minimum at 119879 gt 119879119892 characteristic for heterogeneities of thesample is perceptible

Moreover in addition to storage and loss moduli storageand loss compliance are plotted in Figure 3(b) As can be seenthe loss and storage compliance show a nonuniform behaviorand obviously are in an almost opposite form in comparisonwith the loss and storage moduli The storage compliance1198691015840 increases first gradually with respect to temperature untilit reaches a maximum The plotted loss compliance as afunction of temperature for different frequencies has alsoa monotonic shape It decreases with temperature until 119879119892is reached and then increases with higher slopes up to thesecond relaxation temperature

In addition to this the results of dynamic and out-of-phase viscosities as a function of temperature are plotted inFigure 4 Again a nonmonotonic behavior can be observedfor these properties and a stronger frequency dependencethan that of moduli and compliance is obvious It can beconcluded that at higher frequencies the material showsmore internal energy dominated glassy behavior whereby atdecreasing frequency a transition to a nearly entropic rubberystate is surveyed This is because of the fact that at higherfrequencies polymer chains absorb energy much better thanat lower frequencies

As shown in Figure 5 the temperature dependency ofstorage and loss moduli exhibit weak sensitivity with respectto heating rate 120585 which was varied in the range from 025 to2∘Csdotminminus1 during temperature sweeps Below 119879119892 the storagemodulus does not noticeably change with temperature and asmooth decreasing trend is observed ESTANE demonstratestherefore a linear elastic behavior in this region In the regionaround119879119892 a harsh fall of storage modulus is observed In thisrange a pronounced difference can be observed for differentheating rates For higher heating rates the storage modulusexhibits a global minimum at around 119879119892 This could beattributed to the fact that for higher heating rates the kineticenergy of polymer chains increases noticeably resulting in abigger free volume inside the polymer matrix This volumeexpansion obtains the needed energy and space for chainmovements in a defined time scale leading to an extremesubsidence of energy However after this range the polymerchains resist to stresses due to physical cross-links and chainentanglements Finally one can conclude that the changes ofmechanical properties dependmore strongly upon frequencyor strain amplitudes rather than heating rates Additionallythe results of loss factor (damping coefficient) are shown inFigure 6 The peak values and the temperatures regardingthis maximum point are very well detectable A small shifttowards higher temperatures for tan 120575 is seen for higherheating rates A similar behavior is also recorded in otherthermal analyses such as TMA or DSC [32]

Rearrangement of chain sequences in polymeric systemsleading to molecular relaxations accelerates at higher ratesand elevated temperatures This direct equivalence betweentime and temperature leads to the fact that the time overwhich these relaxations take place can be simply reiteratedby conducting the experiment at higher temperatures andshifting the resulting data to lower frequencies This treating

International Journal of Polymer Science 5

1000

100

10

1000

100

10

0 20 40 60 80 100120

G998400

(MPa

)

G998400 G998400998400

G998400998400

(MPa

)

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

(a) Variation of storage and loss shear modulus with temperature

0 20 40 60 80 100120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

J998400(1

MPa

)

J998400998400(1

MPa

)

J998400 J998400998400

10minus1

10minus2

(b) Variation of storage and loss compliance with temperature

Figure 3 Temperature sweep tests at different frequencies

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus1

10minus2

100

101

102

103

10minus1

10minus2

100

101

102

103

120578998400 120578998400998400

120578998400

(MPa

middots)

120578998400998400

(MPa

middots)

Figure 4 Temperature sweep tests at different frequencies variationof dynamic and out-of-phase viscosities with temperature

of the results of dynamic experiments is called the Time-Temperature Superposition Principle [33] and can be usedto produce a corresponding master curve of a polymer withwhich the material properties can be estimated for very longand not measurable time scales [29] The horizontal shiftfactor values (119886119879) for linking the individual experimental dataand producing the master curve can be obtained through theWilliams-Landel-Ferry (WLF) equation [29]

log 119886119879 = minus1198621 (119879 minus 119879119903)

1198622 + (119879 minus 119879119903) (4)

where 1198621 and 1198622 are empirical constants and 119879119903 denotes thereference temperature

Figure 7(a) demonstrates the experimentally obtaineddata of 1198661015840 at different isothermal temperatures varying from20∘C to 70∘C in the range of unreduced angular frequencies

103

102

101

103

102

101

G998400

(MPa

)

G998400998400

(MPa

)

G998400 G998400998400

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

Figure 5 Temperature sweep tests with different heating ratesvariation of storage and loss shear modulus with temperature

120596 from 01 to 100 radsdotsminus1 Complementary to this obtainedmaster curves of 1198661015840 and 11986610158401015840 are shown in Figure 7(b) Here119879119903 is 50

∘Cand1198621 and1198622 are found to be 8 and 18 respectivelyAs can be seen the shift factor values of WLF equation andthe chosen temperature increments lead to acceptable mastercurves from experimentally acquired data revealing thatESTANE may be considered as thermorheologically simpleHowever because of segmental and global heterogeneityconsidering ESTANE as thermorheological complex [34 35]would lead to even better master curves

A very interesting inference which can be concludedfrom the aspect of the resulting master curve is the feature ofthe polymer system to show the SME In the fact of the viscousnature of the system which is essential to the shape-storageand recovery one can deduce that the SME results only fromthe time-temperature-dependent property of the system and

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Nano

materials

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Journal ofNanomaterials

International Journal of Polymer Science 3

23 Stress Relaxation Experiments For quasistatic relaxationexperiments ESTANE plates were punched to dumbbellspecimens of type 5A according to DIN EN ISO 527-2 (119882 times119867 times 119871 2mm times 4mm times 75mm) Uniaxial stretching andrelaxation experiments were performed on a spindle-driventesting machine (Schenck Trebel RM50 New York USA)with a stretching and compression capacity of plusmn500N Themachine was equipped with a heating chamber of type Fre-senberger TK18400100 (Wipperfurth Germany) attachedto a PID temperature controller (Eurotherm 3508 LimburgGermany) so that temperatures in the range of minus100 to 400∘Ccould be adjusted and kept constant with a precision of plusmn1∘C

For stress relaxation experiments in displacement drivencontrol mode all specimens were pneumatically fixed alongtheir stretching axis and heated up with a rate of 3∘Csdotminminus1 toa temperature119879prog which was equilibrated for at least 10minAfterwards the specimens were deformed with a strain rateof 0005 sminus1 from their initial length 119871 to a new length 119897corresponding to a local uniaxial strain of about 120576 = (119897 minus119871)119871 = 10 This uniaxial stretch 120582 = 119897119871 = 11 was keptconstant for at least 60min At different temperatures 119879progranging from 10∘C up to 80∘C the decay of stress over timewasmonitored Considering a linear viscoelastic behavior forESTANE the total stress will ultimately reach an equilibriumstress following the decay of multiple superimposed relax-ation processes as long as no viscous flow occurs It is worthmentioning that all experiments were repeated two times toimprove confidence

24Thermal Expansion Experiments Todetermine theCoef-ficient of Thermal Expansion (CTE) two different experi-ments were conducted The first procedure was performedwith the same spindle-driven machine as used for therelaxation experiments Here the samples were heated whilea force controlled loop kept a tensile force of approximately05N constant using the feedback signals of a load cell thatwas attached to the moving upper clamp and an actuatorTheupper clamp displacement during heating was tracked andassuming a homogeneous deformation subsequently usedto determine the sample deformation The correspondingtemperature (from 10 to 140∘C) versus stretch data was usedto compute the Coefficients of Thermal Expansion

In addition to this the same rheometer which was usedfor DMTA experiments was applied as an alternative methodfor stress-controlled CTE-measurement In equivalence tothe above approach a tensile force of 05N was controlled tobe constant while heating the samples from 10 to 140∘C Incontrast to the uniaxial spindle-driven testing machine theadvantage of the rheometer apparatus is the use of a high-resolution load cell with an accuracy of 0005N

3 Results and Discussion

31 Characterization

311 Determination of Thermal Expansion A subtle dis-continuity in the CTE which otherwise exhibits a lineartemperature dependency is observed for both rheometer

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

1000

1005

1010

1015

1020

1025

1030

Rheometer

Rheometer

Spindle

Spindle

120582(mdash

)

T (∘C)

T (∘C)

F(N

)

10

05

00

RheometerSpindle

120572r = 2549 middot 10minus4 Kminus1

120572r = 2842 middot 10minus4 Kminus1

120572g = 1876 middot 10minus4 Kminus1

120572g = 1350 middot 10minus4 Kminus1

Figure 1 Thermal expansion during heating while applying con-stant tensile force of 05N

and spindle-driven testing procedures The correspondingdata points and best fitting linear regressions are depictedin Figure 1 The tensile force which was controlled duringheating is plotted in the inset to Figure 1 Pronouncedfluctuations in the applied tensile force around 119879119892 suggestthat quantitative measurement of CTE is impeded for theless accurate spindle-driven testing procedure The forcecontrolling mechanism was more accurate during rheometertesting which we attribute to higher accuracy of the loadsensor The magnitudes of elongation are furthermore foundto be distinctly small such that data acquisition errors need tobe consideredThe corresponding magnitudes of uncertaintyare illustrated by means of error bars and found to be com-paratively high at low temperatures The subscripts ldquo119892rdquo andldquo119903rdquo indicate the regions of glass-like and rubbery behaviorrespectively Below 119879119892 the CTE is less pronounced whichproduces a kink in the 120582-T curve The discontinuity of linearthermal expansion which we attribute to glass transition isfound at 119879119892 asymp 55

∘C This coincides with previously reportedresults in [27 30]

32 Dynamic Thermomechanical Experiments Aim of the(cyclicharmonic) amplitude sweeps is to determine a linearregime where effective properties are independent of the(shear strain) amplitude Therefore DMTA amplitude sweeptests where the strain amplitude increases incrementally withfixed frequency are used to determine the linear viscoelasticregion (LVR) Within the LVR the viscoelastic response ofthe polymer is independent of the amplitude of deformationsuch that constitutive nonlinearities and plasticity effects canbe excluded In Figure 2 the variations of storage and lossmoduli with strain amplitude 1205740 at different temperatures (040 and 80∘C) are depicted In the low temperature regime(119879 = 0∘C Figure 2 black dots) the mechanical response(1198661015840) is dominated by the hard segment phase It could beobserved that 1198661015840 and 11986610158401015840 is hardly amplitude dependentwithin 003 lt 120574 lt 1and no hint for any plastic behavior isobserved In the high temperature regime (T = 80∘C Figure 2

4 International Journal of Polymer Science

104

103

102

101

104

103

102

101

001 01 1 001 01 1

G998400

(MPa

)

1205740 ()

G998400998400

(MPa

)

G998400 G998400998400

T = 0∘C

T = 80∘CT = 40∘C

Figure 2 Amplitude sweep test at different temperatures

blue diamonds) the mechanical properties are determinedby the soft segments In this graph the linear viscoelasticregion can be identified as a pronounced plateau For shearamplitudes within the range 003 lt 120574 lt 1 the storageand loss moduli remain almost constant This is a fairlybroad LVRwhen compared to previously reported LVRs [29]For shear amplitudes smaller than 003 the decrease inthe experimental signal-to-noise ratio results in an artificialdeviation of moduli from the LVR plateau

Upon identifying the appropriate strain amplitude rangestorage and loss moduli have been measured using tempera-ture sweeps as shown in Figure 3(a) As can be seen DMTA intemperature sweep mode gives intriguing information aboutstructural properties of ESTANE It is shown that the storagemodulus 1198661015840 decreases gradually with temperature until theglass transition region is reached The plotted behavior istypical for physically cross-linked block copolymers [31]The inflection point of the 1198661015840-119879-graph determines 119879119892 of thesampleThis point shifts to a higher temperature as frequencyincreases 119879119892 of SM-ESTANE was measured to be approxi-mately 55∘C which is in agreement with previously reportedresults [30] Interestingly the temperature sweep tests of SM-ESTANE scanned at different frequencies indicates that thereduction of 1198661015840 is less pronounced at higher frequenciesthan at lower ones Moreover the storage modulus decreasesuntil it reaches a minimum This is more considerable forsamples under lower tested frequencies This minimumlocated at temperatures higher than 119879119892 is characteristicfor another relaxation inside the polymer chains due tomolecular heterogeneities of the specimen Samples underhigher frequencies have less or not enough time to showthis relaxation An inverse temperature sweep test with thesame frequency proves that this observation is almost notrepeatable

The loss modulus 11986610158401015840 shows a different behavior 11986610158401015840-119879-plot has also a nonmonotonic form The curves increasewith increasing temperature until 119879119892 is attained and decreaseslowly at higher temperatures Moreover at higher frequen-cies the loss modulus takes higher values The maximum of

11986610158401015840-119879-graph is interpreted as 119879119892 of the sample Likewise a

minimum at 119879 gt 119879119892 characteristic for heterogeneities of thesample is perceptible

Moreover in addition to storage and loss moduli storageand loss compliance are plotted in Figure 3(b) As can be seenthe loss and storage compliance show a nonuniform behaviorand obviously are in an almost opposite form in comparisonwith the loss and storage moduli The storage compliance1198691015840 increases first gradually with respect to temperature untilit reaches a maximum The plotted loss compliance as afunction of temperature for different frequencies has alsoa monotonic shape It decreases with temperature until 119879119892is reached and then increases with higher slopes up to thesecond relaxation temperature

In addition to this the results of dynamic and out-of-phase viscosities as a function of temperature are plotted inFigure 4 Again a nonmonotonic behavior can be observedfor these properties and a stronger frequency dependencethan that of moduli and compliance is obvious It can beconcluded that at higher frequencies the material showsmore internal energy dominated glassy behavior whereby atdecreasing frequency a transition to a nearly entropic rubberystate is surveyed This is because of the fact that at higherfrequencies polymer chains absorb energy much better thanat lower frequencies

As shown in Figure 5 the temperature dependency ofstorage and loss moduli exhibit weak sensitivity with respectto heating rate 120585 which was varied in the range from 025 to2∘Csdotminminus1 during temperature sweeps Below 119879119892 the storagemodulus does not noticeably change with temperature and asmooth decreasing trend is observed ESTANE demonstratestherefore a linear elastic behavior in this region In the regionaround119879119892 a harsh fall of storage modulus is observed In thisrange a pronounced difference can be observed for differentheating rates For higher heating rates the storage modulusexhibits a global minimum at around 119879119892 This could beattributed to the fact that for higher heating rates the kineticenergy of polymer chains increases noticeably resulting in abigger free volume inside the polymer matrix This volumeexpansion obtains the needed energy and space for chainmovements in a defined time scale leading to an extremesubsidence of energy However after this range the polymerchains resist to stresses due to physical cross-links and chainentanglements Finally one can conclude that the changes ofmechanical properties dependmore strongly upon frequencyor strain amplitudes rather than heating rates Additionallythe results of loss factor (damping coefficient) are shown inFigure 6 The peak values and the temperatures regardingthis maximum point are very well detectable A small shifttowards higher temperatures for tan 120575 is seen for higherheating rates A similar behavior is also recorded in otherthermal analyses such as TMA or DSC [32]

Rearrangement of chain sequences in polymeric systemsleading to molecular relaxations accelerates at higher ratesand elevated temperatures This direct equivalence betweentime and temperature leads to the fact that the time overwhich these relaxations take place can be simply reiteratedby conducting the experiment at higher temperatures andshifting the resulting data to lower frequencies This treating

International Journal of Polymer Science 5

1000

100

10

1000

100

10

0 20 40 60 80 100120

G998400

(MPa

)

G998400 G998400998400

G998400998400

(MPa

)

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

(a) Variation of storage and loss shear modulus with temperature

0 20 40 60 80 100120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

J998400(1

MPa

)

J998400998400(1

MPa

)

J998400 J998400998400

10minus1

10minus2

(b) Variation of storage and loss compliance with temperature

Figure 3 Temperature sweep tests at different frequencies

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus1

10minus2

100

101

102

103

10minus1

10minus2

100

101

102

103

120578998400 120578998400998400

120578998400

(MPa

middots)

120578998400998400

(MPa

middots)

Figure 4 Temperature sweep tests at different frequencies variationof dynamic and out-of-phase viscosities with temperature

of the results of dynamic experiments is called the Time-Temperature Superposition Principle [33] and can be usedto produce a corresponding master curve of a polymer withwhich the material properties can be estimated for very longand not measurable time scales [29] The horizontal shiftfactor values (119886119879) for linking the individual experimental dataand producing the master curve can be obtained through theWilliams-Landel-Ferry (WLF) equation [29]

log 119886119879 = minus1198621 (119879 minus 119879119903)

1198622 + (119879 minus 119879119903) (4)

where 1198621 and 1198622 are empirical constants and 119879119903 denotes thereference temperature

Figure 7(a) demonstrates the experimentally obtaineddata of 1198661015840 at different isothermal temperatures varying from20∘C to 70∘C in the range of unreduced angular frequencies

103

102

101

103

102

101

G998400

(MPa

)

G998400998400

(MPa

)

G998400 G998400998400

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

Figure 5 Temperature sweep tests with different heating ratesvariation of storage and loss shear modulus with temperature

120596 from 01 to 100 radsdotsminus1 Complementary to this obtainedmaster curves of 1198661015840 and 11986610158401015840 are shown in Figure 7(b) Here119879119903 is 50

∘Cand1198621 and1198622 are found to be 8 and 18 respectivelyAs can be seen the shift factor values of WLF equation andthe chosen temperature increments lead to acceptable mastercurves from experimentally acquired data revealing thatESTANE may be considered as thermorheologically simpleHowever because of segmental and global heterogeneityconsidering ESTANE as thermorheological complex [34 35]would lead to even better master curves

A very interesting inference which can be concludedfrom the aspect of the resulting master curve is the feature ofthe polymer system to show the SME In the fact of the viscousnature of the system which is essential to the shape-storageand recovery one can deduce that the SME results only fromthe time-temperature-dependent property of the system and

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 International Journal of Polymer Science

104

103

102

101

104

103

102

101

001 01 1 001 01 1

G998400

(MPa

)

1205740 ()

G998400998400

(MPa

)

G998400 G998400998400

T = 0∘C

T = 80∘CT = 40∘C

Figure 2 Amplitude sweep test at different temperatures

blue diamonds) the mechanical properties are determinedby the soft segments In this graph the linear viscoelasticregion can be identified as a pronounced plateau For shearamplitudes within the range 003 lt 120574 lt 1 the storageand loss moduli remain almost constant This is a fairlybroad LVRwhen compared to previously reported LVRs [29]For shear amplitudes smaller than 003 the decrease inthe experimental signal-to-noise ratio results in an artificialdeviation of moduli from the LVR plateau

Upon identifying the appropriate strain amplitude rangestorage and loss moduli have been measured using tempera-ture sweeps as shown in Figure 3(a) As can be seen DMTA intemperature sweep mode gives intriguing information aboutstructural properties of ESTANE It is shown that the storagemodulus 1198661015840 decreases gradually with temperature until theglass transition region is reached The plotted behavior istypical for physically cross-linked block copolymers [31]The inflection point of the 1198661015840-119879-graph determines 119879119892 of thesampleThis point shifts to a higher temperature as frequencyincreases 119879119892 of SM-ESTANE was measured to be approxi-mately 55∘C which is in agreement with previously reportedresults [30] Interestingly the temperature sweep tests of SM-ESTANE scanned at different frequencies indicates that thereduction of 1198661015840 is less pronounced at higher frequenciesthan at lower ones Moreover the storage modulus decreasesuntil it reaches a minimum This is more considerable forsamples under lower tested frequencies This minimumlocated at temperatures higher than 119879119892 is characteristicfor another relaxation inside the polymer chains due tomolecular heterogeneities of the specimen Samples underhigher frequencies have less or not enough time to showthis relaxation An inverse temperature sweep test with thesame frequency proves that this observation is almost notrepeatable

The loss modulus 11986610158401015840 shows a different behavior 11986610158401015840-119879-plot has also a nonmonotonic form The curves increasewith increasing temperature until 119879119892 is attained and decreaseslowly at higher temperatures Moreover at higher frequen-cies the loss modulus takes higher values The maximum of

11986610158401015840-119879-graph is interpreted as 119879119892 of the sample Likewise a

minimum at 119879 gt 119879119892 characteristic for heterogeneities of thesample is perceptible

Moreover in addition to storage and loss moduli storageand loss compliance are plotted in Figure 3(b) As can be seenthe loss and storage compliance show a nonuniform behaviorand obviously are in an almost opposite form in comparisonwith the loss and storage moduli The storage compliance1198691015840 increases first gradually with respect to temperature untilit reaches a maximum The plotted loss compliance as afunction of temperature for different frequencies has alsoa monotonic shape It decreases with temperature until 119879119892is reached and then increases with higher slopes up to thesecond relaxation temperature

In addition to this the results of dynamic and out-of-phase viscosities as a function of temperature are plotted inFigure 4 Again a nonmonotonic behavior can be observedfor these properties and a stronger frequency dependencethan that of moduli and compliance is obvious It can beconcluded that at higher frequencies the material showsmore internal energy dominated glassy behavior whereby atdecreasing frequency a transition to a nearly entropic rubberystate is surveyed This is because of the fact that at higherfrequencies polymer chains absorb energy much better thanat lower frequencies

As shown in Figure 5 the temperature dependency ofstorage and loss moduli exhibit weak sensitivity with respectto heating rate 120585 which was varied in the range from 025 to2∘Csdotminminus1 during temperature sweeps Below 119879119892 the storagemodulus does not noticeably change with temperature and asmooth decreasing trend is observed ESTANE demonstratestherefore a linear elastic behavior in this region In the regionaround119879119892 a harsh fall of storage modulus is observed In thisrange a pronounced difference can be observed for differentheating rates For higher heating rates the storage modulusexhibits a global minimum at around 119879119892 This could beattributed to the fact that for higher heating rates the kineticenergy of polymer chains increases noticeably resulting in abigger free volume inside the polymer matrix This volumeexpansion obtains the needed energy and space for chainmovements in a defined time scale leading to an extremesubsidence of energy However after this range the polymerchains resist to stresses due to physical cross-links and chainentanglements Finally one can conclude that the changes ofmechanical properties dependmore strongly upon frequencyor strain amplitudes rather than heating rates Additionallythe results of loss factor (damping coefficient) are shown inFigure 6 The peak values and the temperatures regardingthis maximum point are very well detectable A small shifttowards higher temperatures for tan 120575 is seen for higherheating rates A similar behavior is also recorded in otherthermal analyses such as TMA or DSC [32]

Rearrangement of chain sequences in polymeric systemsleading to molecular relaxations accelerates at higher ratesand elevated temperatures This direct equivalence betweentime and temperature leads to the fact that the time overwhich these relaxations take place can be simply reiteratedby conducting the experiment at higher temperatures andshifting the resulting data to lower frequencies This treating

International Journal of Polymer Science 5

1000

100

10

1000

100

10

0 20 40 60 80 100120

G998400

(MPa

)

G998400 G998400998400

G998400998400

(MPa

)

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

(a) Variation of storage and loss shear modulus with temperature

0 20 40 60 80 100120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

J998400(1

MPa

)

J998400998400(1

MPa

)

J998400 J998400998400

10minus1

10minus2

(b) Variation of storage and loss compliance with temperature

Figure 3 Temperature sweep tests at different frequencies

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus1

10minus2

100

101

102

103

10minus1

10minus2

100

101

102

103

120578998400 120578998400998400

120578998400

(MPa

middots)

120578998400998400

(MPa

middots)

Figure 4 Temperature sweep tests at different frequencies variationof dynamic and out-of-phase viscosities with temperature

of the results of dynamic experiments is called the Time-Temperature Superposition Principle [33] and can be usedto produce a corresponding master curve of a polymer withwhich the material properties can be estimated for very longand not measurable time scales [29] The horizontal shiftfactor values (119886119879) for linking the individual experimental dataand producing the master curve can be obtained through theWilliams-Landel-Ferry (WLF) equation [29]

log 119886119879 = minus1198621 (119879 minus 119879119903)

1198622 + (119879 minus 119879119903) (4)

where 1198621 and 1198622 are empirical constants and 119879119903 denotes thereference temperature

Figure 7(a) demonstrates the experimentally obtaineddata of 1198661015840 at different isothermal temperatures varying from20∘C to 70∘C in the range of unreduced angular frequencies

103

102

101

103

102

101

G998400

(MPa

)

G998400998400

(MPa

)

G998400 G998400998400

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

Figure 5 Temperature sweep tests with different heating ratesvariation of storage and loss shear modulus with temperature

120596 from 01 to 100 radsdotsminus1 Complementary to this obtainedmaster curves of 1198661015840 and 11986610158401015840 are shown in Figure 7(b) Here119879119903 is 50

∘Cand1198621 and1198622 are found to be 8 and 18 respectivelyAs can be seen the shift factor values of WLF equation andthe chosen temperature increments lead to acceptable mastercurves from experimentally acquired data revealing thatESTANE may be considered as thermorheologically simpleHowever because of segmental and global heterogeneityconsidering ESTANE as thermorheological complex [34 35]would lead to even better master curves

A very interesting inference which can be concludedfrom the aspect of the resulting master curve is the feature ofthe polymer system to show the SME In the fact of the viscousnature of the system which is essential to the shape-storageand recovery one can deduce that the SME results only fromthe time-temperature-dependent property of the system and

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

International Journal of Polymer Science 5

1000

100

10

1000

100

10

0 20 40 60 80 100120

G998400

(MPa

)

G998400 G998400998400

G998400998400

(MPa

)

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

(a) Variation of storage and loss shear modulus with temperature

0 20 40 60 80 100120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

J998400(1

MPa

)

J998400998400(1

MPa

)

J998400 J998400998400

10minus1

10minus2

(b) Variation of storage and loss compliance with temperature

Figure 3 Temperature sweep tests at different frequencies

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Hz)011

10100

10minus1

10minus2

100

101

102

103

10minus1

10minus2

100

101

102

103

120578998400 120578998400998400

120578998400

(MPa

middots)

120578998400998400

(MPa

middots)

Figure 4 Temperature sweep tests at different frequencies variationof dynamic and out-of-phase viscosities with temperature

of the results of dynamic experiments is called the Time-Temperature Superposition Principle [33] and can be usedto produce a corresponding master curve of a polymer withwhich the material properties can be estimated for very longand not measurable time scales [29] The horizontal shiftfactor values (119886119879) for linking the individual experimental dataand producing the master curve can be obtained through theWilliams-Landel-Ferry (WLF) equation [29]

log 119886119879 = minus1198621 (119879 minus 119879119903)

1198622 + (119879 minus 119879119903) (4)

where 1198621 and 1198622 are empirical constants and 119879119903 denotes thereference temperature

Figure 7(a) demonstrates the experimentally obtaineddata of 1198661015840 at different isothermal temperatures varying from20∘C to 70∘C in the range of unreduced angular frequencies

103

102

101

103

102

101

G998400

(MPa

)

G998400998400

(MPa

)

G998400 G998400998400

0 20 40 60 80 100 120

T (∘C)0 20 40 60 80 100120140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

Figure 5 Temperature sweep tests with different heating ratesvariation of storage and loss shear modulus with temperature

120596 from 01 to 100 radsdotsminus1 Complementary to this obtainedmaster curves of 1198661015840 and 11986610158401015840 are shown in Figure 7(b) Here119879119903 is 50

∘Cand1198621 and1198622 are found to be 8 and 18 respectivelyAs can be seen the shift factor values of WLF equation andthe chosen temperature increments lead to acceptable mastercurves from experimentally acquired data revealing thatESTANE may be considered as thermorheologically simpleHowever because of segmental and global heterogeneityconsidering ESTANE as thermorheological complex [34 35]would lead to even better master curves

A very interesting inference which can be concludedfrom the aspect of the resulting master curve is the feature ofthe polymer system to show the SME In the fact of the viscousnature of the system which is essential to the shape-storageand recovery one can deduce that the SME results only fromthe time-temperature-dependent property of the system and

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 International Journal of Polymer Science

0 20 40 60 80 100 120 140

T (∘C)

(Kmiddotminminus1)120585 = 2120585 = 1

120585 = 05120585 = 025

tan 120575

(mdash)

00

01

02

03

04

05

06

07

08

Figure 6 Temperature sweep tests with different heating ratesvariation of loss factor with temperature

its linear viscoelasticity [17] By decreasing the temperaturethe deformation is stored as viscous strain by a large increasein the viscosity or equivalently in the relaxation times (cfSection 321) Additionally the shape recovery would beachieved through viscous strain release owing to a decreasein viscosity by temperature elevation Therefore the shapefixation regardless of the cooling history can be estimateddirectly from themaster curve through119877119891 = 1minus119866119890119866119892 where119866119890 is the rubbery (at small frequencies) and 119866119892 is the glassy(at high frequency) modulus

321 Constitutive Model A suitable constitutive frame-work for thermorheologically simple polymers is generalizedMaxwell-Zener model [33] that is composed of a singleneo-Hookean equilibrium element aligned in parallel to 119899nonequilibrium Maxwell elements These nonequilibriumthermoviscoelastic elements describe distinct relaxation pro-cesses of the polymer [36] In attempt to accurately representthe continuous viscoelastic spectrum as measured usingDMTA 119899 = 19 discrete nonequilibrium Maxwell elementsare taken into account The constitutive behavior of the 119895thMaxwell element is defined by the shear modulus 119866119895 andthe reduced relaxation time 120591119895

119877and together they are referred

to as Prony series coefficients Both material parameters areoptimized using the storage modulus master curve 1198661015840(120596 sdot120572119879) for the reference temperature 119879119903 as shown in Figure 7We account for temperature dependency of the discreterelaxation processes by scaling 120591119895 with the temperature shiftfactors 119886119879 according to the Time-Temperature SuperpositionPrinciple (TTSP) As outlined in Section 32 the shift factors119886119879 have been found to obey the WLF relationship within thetested temperature range For the present thermorheologi-cally simplematerial temperature dependency of the discreterelaxation processes takes the form 120591119895(119879) = 120591119895

119877120572119879(119879) The

elastic shear modulus 119866eq of the neo-Hookean equilibrium

Table 1 Generalized Maxwell model relaxation times 120591119895119877

andassociated shear moduli 119866119895 for ESTANE

120591119895

119877[s] 119866

119895 [MPa]1000 times 10minus9 8495 times 101

5995 times 10minus9 7954 times 101

3594 times 10minus8 7414 times 101

2154 times 10minus7 6874 times 101

1291 times 10minus6 6334 times 101

7743 times 10minus6 5794 times 101

4641 times 10minus5 5254 times 101

2782 times 10minus4 4713 times 101

1668 times 10minus3 4168 times 101

1000 times 10minus2 3602 times 101

5995 times 10minus2 2932 times 101

3594 times 10minus1 2106 times 101

2154 times 100 1399 times 101

1291 times 101 9998 times 100

7743 times 101 8412 times 100

4641 times 102 7613 times 100

2782 times 103 6331 times 100

1668 times 104 4651 times 100

1000 times 105 2823 times 100

element is evaluated in the limit lim120596rarr01198661015840(120596 sdot 120572119879) as 119866eq =

29MPa Subsequently the Prony coefficients 119866119895 and 120591119895119877are

optimized using the Tikhonov regularization method [37]such that the discrete representations of storage and lossmoduli

1198661015840(120596 119879) = 119866eq +

119899

sum

119895=1

1198661198951205962(120591119895(119879))2

1 + 1205962 (120591119895 (119879))2

11986610158401015840(120596 119879) =

119899

sum

119895=1

119866119895 120596120591

119895(119879)

1 + 1205962 (120591119895 (119879))2

(5)

adequately represent experimentally measured DMTA dataThe best fitting Prony coefficients are listed in Table 1The fitted storage and loss moduli are plotted in Figure 7and found to accurately represent the dynamic materialproperties within the frequency and temperature range ofinterest

322 Modeling Parameter Identification In this section thematerial parameter optimization techniques are discussedAccording to the rheological properties presented here thetheory of finite viscoelasticity is used [38] By consideringhomogeneous uniaxial deformations the resulting constitu-tivemodel is reduced to a scalar equation entitled constitutivedriver Consistent with [39 40] the total deformation gra-dient F = 120597x120597X = 119865119894119895119894 otimes 119895 can be decomposed into anelastic and inelastic part F = F119890 sdot F119894 For uniaxial tension

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

International Journal of Polymer Science 7

203040

506070

102

01 1 10 100

G998400

(MPa

)

120596 (radmiddotsminus1)

14121086420

minus2minus4minus6

20 30 40 50 60 70 80

lna T

(mdash)

T (∘C)

(a)

102

101

100 101 102 103 104 105 106 10710minus3 10minus2 10minus1

G998400 G

998400998400(M

Pa)

G998400

G998400998400G998400-fitG998400998400-fit

120596 middot aT (radmiddotsminus1)

(b)

Figure 7 (a) Storage modulus as a function of angular frequency at different isothermal conditions obtained from DMTA experiments and(b) master curves of storage and loss moduli shifted to a reference temperature of 119879

119903= 50∘CThe solid line in the insert shows the calculated

shift factors obtained fromWLF approximation

of an incompressible material in e2 direction assuming anisotropic thermal expansion one can write

F =[[[[[

[

1

radic120582

0 0

0 120582 0

0 01

radic120582

]]]]]

]

(6)

where 120582 is the previously introduced stretch Similarly thestretch ratio of the 119895th Maxwell unit 120582119895 can be decomposedinto 120582119895 = 120582119895

119890120582119895

119894 A neo-Hookean strain energy function is

taken into account for both the equilibrium and nonequilib-rium stresses The resulting constitutive equation reads

T = Teq + Tneq = 119866eqB +119899

sum

119895=1

119866119895B119895119890minus 119901I (7)

where T B = F sdot F119879 and B119895119890= F119890 sdot F119879119890 are the Cauchy stress

tensor the total left Cauchy-Green tensor and the elastic leftCauchy-Green tensor of the 119895th Maxwell unit respectivelyThe volumetric Lagrange parameter 119901 in constitutive equa-tion (7) which accounts for the incompressibility constraintand thus interpreted as a pressure can be evaluated from theboundary condition Since the material is stretched in just

one direction the orthogonal directions are stress free Thismeans 11987911 = 11987933 = 0 which results in

119879 = 119866eq (1205822minus1

120582) +

119899

sum

119895=1

119866119895 [

[

1205822

(120582119895

119894)2minus120582119895

119894

120582

]

]

(8)

Using the relation above the inelastic component 120582119895119894can be

replaced by120582119895120582119895119890The evolution of the inelastic rightCauchy-

Green tensorC119895119894 which takes the role of the internal variables

reads in general [40]

C119895119894=2

120591119895 (119879)[C minus 1

3tr (C sdot [C119895

119894]minus1

)C119895119894] (9)

with the right Cauchy-Green tensor C = F119879 sdot F and theinelastic right Cauchy-Green tensor of the 119895th Maxwell unitC119895119894= (F119895119894)119879sdot F119895

For linearization of this equation we use the method des-cribed in [40]

C119895119894=2

120591119895 (119879)[C minus C119895

119894] (10)

An alternative approach is constituted by Shutov et al propos-ing a simple iteration scheme without linearization by animplicit integration procedure and a simple explicit updateformula for unimodular deformations [41] In view of the

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 International Journal of Polymer Science

400500400300200100

300

200

100

00 1000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

10∘C-simulation10∘C-experiment30∘C-simulation

30∘C-experiment40∘C-simulation40∘C-experiment

10 20 30 40

20

15

10

5

20

15

10

5

01000 2000 3000 4000 5000 6000

120590(M

Pa)

t (s)

T (∘C)

50∘C-simulation50∘C-experiment60∘C-simulation

70∘C-simulation70∘C-experiment80∘C-simulation80∘C-experiment

50 60 70 80

60∘C-experiment

120590m

(MPa

)

120590m

(MPa

)

Figure 8 Temporal development of loading and stress relaxation experiments and their associated simulations under different isothermalconditions

scalar constitutive driver the inelastic stretch component 120582119895119894

takes the role of the internal variable The correspondinglinear evolution equation reads

119895

119894=

1

2120591119895 (119879)[1205822

120582119895

119894

minus 120582119895

119894] (11)

The linearized evolution in (10) and (11) is only applicableto boundary value problems with small deformation ratesFor these large deviations with respect to the point oflinearization C119895

119894= I are not expected Equation (11) can be

solved by the implicit backward Euler procedure in time

asymp120582119895

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

Δ119905

=1

2120591119895 (119879)[1205822(119905 + Δ119905)

120582119895

119894(119905 + Δ119905)

minus 120582119895

119894(119905 + Δ119905)]

(12)

Equation (12) is reformulated into a root-finding problemfor the updated inelastic stretch ratio of the 119895th Maxwellelement which is solved using a Newton-Raphson schemeThe corresponding residual form in the 119896th iteration reads

R119896= 120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905)

minusΔ119905

2120591119895 (119879)[

1205822(119905 + Δ119905)

120582119895119896

119894(119905 + Δ119905) minus 120582

119895

119894(119905 + Δ119905)

]

(13)

Every Newton-Raphson iteration needs a new tangent

H119896=

120597R

120582119895119896

119894(119905 + Δ119905)

= 1 +Δ119905

2120591119895 (119879)

[

[

1 +(120582 (119905 + Δ119905))

2

(120582119895

119894(119905 + Δ119905))

2]

]

(14)

It is worth stating that the root-finding algorithm terminateswhen the residual function (R119896) reaches the threshold valueof 108

323 Finite Deformation Modeling Rheology-Based Asshown before Prony coefficients as well as theWLF constantshave been determined for small deformations from frequencysweep DMTA experiments The parameter identificationis based on these results Here for parameter validationrelaxation experiments described in Section 23 were usedand compared to modeling results as obtained from theconstitutive approach discussed in Section 32

The stress relaxation experiments with ESTANE havebeen conducted at different temperatures (119879prog = 10 30 4050 60 70 and 80∘C) The typical stress relaxation curvesand the appending simulated data of relaxations are depictedin Figure 8 once for temperature below 119879119892 and once aboveit Obviously a pronounced peak maximum at 119905 = 10 scan be surveyed belonging to the final extension of 10The correspondingmaximum tensile stresses are called initialstresses 120590119898 and are illustrated in the inset graphs of Figure 8As can be seen larger inelastic stresses evolving duringdeformation decrease with increasing temperature due to thereduction of cross-linking densities According to Eyringrsquos

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

International Journal of Polymer Science 9

Table 2 Fitting results obtained by the analysis of the relaxation according to the model description of (15) for ESTANE

119879prog 1198661

1198662

1205911

1205912

119866eq Corel Coeff R2

10 054 010 456 89350 036 097530 070 001 801 88081 012 098340 051 011 1695 129661 010 098550 024 018 10230 88342 052 098760 017 021 38690 772480 059 0987

theory [33] this can be related to activation volume which isthe volume of the polymer segment involved in the polymerflow [42] At temperatures below 119879119892 polymer chains are ina kind of frozen-in state where insufficient thermal energyhinders the rotational and transitional movement of wholechain segments However Haward [42] has declared thatthe activation volume has no physical significance despitehaving the dimension of volume By increasing temperaturethe thermal energy increases towards the barriers for theconsiderate segmental movement which results in smallerstress levels during deformation

According to the results illustrated in Figure 8 theexperimental initial stresses 120590119898 are higher than simulationwhichmight be associated with the nonoptimalmaster-curvegeneration However for all temperatures a fast relaxationprocess dominates the initial period of relaxations and thena slow process is characteristic for the further relaxationsMoreover for temperatures below 119879119892 the amount of exper-imental and simulated relaxation stresses converges to eachother very fast and good agreement can be observed Attemperatures above glass transition aside from 119879 = 60∘C theconformity deteriorates slightly The emerging discrepanciesbetween experimental and simulation results at 119879prog = 50∘Care believed to be based on the fact that this temperatureis located in the glass transition range where main chainsegmental motions are activated (120572-relaxation) As long as119879119892exceeds better fits could be again achieved

On the other hand another interesting aspect is thekinetics of stress relaxations Figure 9 shows for five selectedtemperatures the temporal change of stresses as relativevalues with respect to 120590119898 at the beginning of the relaxationTherefore the model relaxation process starts with an initialvalue of 1 By trying to find a suitable mathematical fit itwas clear that a simpler generalized Maxwell-Zener modelwith only five independent parameters (119866eq 1198661 1198662 120591

1 1205912) is

thoroughly sufficientTherefore the experimental data pointswere fitted to a five-parameter model with a numerical leasterror square method The modified Maxwell-Zener modelconsists of two Maxwell units and a spring in parallel whichrepresents the long-time elastic behavior of the model Thismodel has the following function

119866 (119905) =120590 (119905)

120576= 119866eq + 119866

1 exp( 1199051205911) + 119866

2 exp ( 1199051205912) (15)

The solid lines in Figure 9 are the obtained fitting curvesand the calculated fitting parametersThese fitting parametersare listed in Table 2 As can be seen from Figure 9 therelaxation process can be fitted very well with this model and

00

01

02

03

04

05

06

07

08

09

10

11

0 1000 2000 3000 4000 5000 6000

t (s)

10∘C30∘C

50∘C70∘C

40∘C

120590(t)

120590m

Figure 9 Modeling of relaxation curves for ESTANE with themodified Maxwell-Zener model

the following five unknown parameters at each temperaturecan be estimated with a good accuracy the relaxed stress (orthe equilibrium stress) 119866eq the elastic moduli of springs 1198661

and 1198662 and relaxation times of dashpots 1205911 and 1205912According to the results listed in Table 2 and the fitting

curves in Figure 9 it can be established that the mainrelaxation times 1205911 increase with increasing 119879prog meaningthat the first unit of the model relaxes faster than the otherone Since 1198661 is the dominant spring modulus in the modeland because it decreases with temperature 1205911 increases pro-portional to the temperature Finally at temperatures below119879119892 119866eq decreases with increasing temperature as presentedin [43] and then increases to higher values for temperatureshigher than glass transition

4 Conclusion

Based on the Dynamic Mechanical Thermal Analysis andquasistatic experiments the thermorheological properties oflittle known ESTANE have been investigated According tothe results presented here it was possible to estimate itsCoefficient of Thermal Expansion above and below glasstransition temperature Consistent with large force fluctua-tions in tension machine during CTE-determination it is

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 International Journal of Polymer Science

believed that the measured CTE from the rheometer test ismore reliable Furthermore it could be shown that the linearviscoelastic region of ESTANE lay in the range of 003 to1 which is wider than normally measured with DMTA Asmaintained by temperature sweep tests the glass transitiontemperature of ESTANE could be measured as the minimumof the 1198661015840-119879-graph and is equal to approximately 55∘C

Additionally it could be shown that the dynamic andout-of-phase viscosities depend strongly on frequency andESTANE shows a typical more internal energy dominatedbehavior at lower frequencies In one step further basedon Time-Temperature Superposition Principle the long-timebehavior of ESTANE could be described A generalizedMaxwell model consisting of 19 Maxwell units was thenapplied tomodel the viscoelastic behavior of ESTANEMore-over relaxation experiments were performed for validatingthe optimized Prony coefficients It was observed that attemperatures below 119879119892 the experimental data obtained inrelaxation experiments coincide better with simulations Inone step further it was revealed that the relaxation results intwo steps one fast and one slow Since the slower processdecays only after some minutes the relaxation model couldbe simplified to a five-parameter model

Finally in the line of this work a better understandingabout the relaxation kinetics of ESTANE could be achievedBased on this work we anticipate that these results mightprovide a necessary knowledge base for further progresses inthe development of Shape-Memory Polymers and motivatefurther studies addressing the particular parameters influenc-ing the kinetics of relaxations

Competing Interests

The authors declare no competing interests

Acknowledgments

Ehsan Ghobadi and Holger Steeb are grateful to GermanScience Foundation for the financial support with theDFG Priority Program SPP 1713 ldquoChemomechanicsrdquo (GrantProject no STE 9698-1)The authors thankRKazakeviciute-Makovska A Marquardt F Varnik and G Eggeler Ruhr-University Bochum for sample preparation injection mold-ing and valuable discussions

References

[1] M Behl J Zotzmann and A Lendlein ldquoShape-memory poly-mers and shape-changing polymersrdquo in Shape-Memory Poly-mers vol 226 ofAdvances in Polymer Science pp 1ndash40 Springer2010

[2] L SunWM Huang Z Ding et al ldquoStimulus-responsive shapememory materials a reviewrdquo Materials and Design vol 33 no1 pp 577ndash640 2012

[3] W M Huang Y Zhao C C Wang et al ldquoThermochemo-responsive shape memory effect in polymers a sketch ofworking mechanisms fundamentals and optimizationrdquo Journalof Polymer Research vol 19 no 9 pp 9952ndash9986 2012

[4] P T Mather X Luo and I A Rousseau ldquoShape memorypolymer researchrdquoAnnual Review ofMaterials Research vol 39no 1 pp 445ndash471 2009

[5] M Heuchel T Sauter K Kratz and A Lendlein ldquoThermallyinduced shape-memory effects in polymers quantification andrelated modeling approachesrdquo Journal of Polymer Science PartB Polymer Physics vol 51 no 8 pp 621ndash637 2013

[6] T Sauter M Heuchel K Kratz and K A Lendlein ldquoQuantify-ing the shape-memory effect of polymers by cyclic thermome-chanical testsrdquo Polymer Reviews vol 53 no 1 pp 6ndash40 2013

[7] Y Liu H Du L Liu and J Leng ldquoShape memory polymersand their composites in aerospace applications a reviewrdquo SmartMaterials and Structures vol 23 no 2 Article ID 023001 2014

[8] W Yan L Fang U Noechel K Kratz and A LendleinldquoInfluence of deformation temperature on structural variationand shape-memory effect of a thermoplastic semi-crystallinemultiblock copolymerrdquo Express Polymer Letters vol 9 no 7 pp624ndash635 2015

[9] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoInfluenceof the addition of water to amorphous switching domainson the simulated shape-memory properties of poly(L-lactide)rdquoPolymer vol 54 no 16 pp 4204ndash4211 2013

[10] E Ghobadi M Heuchel K Kratz and A Lendlein ldquoAtomisticsimulation of the shape-memory effect in dry and waterswollen poly[(rac-lactide)-co-glycolide] and copolyester ure-thanes thereofrdquoMacromolecular Chemistry and Physics vol 215no 1 pp 65ndash75 2014

[11] E Ghobadi R Sivanesapillai J Musialak and H SteebldquoThermo-rheological characterization of polyetherurethaneparameter optimization and validationrdquo in Constitutive Modelsfor Rubbers IX B Marvalova and I Petrikova Eds vol 1chapter 24 pp 157ndash163 CRC Press 2015

[12] X Guo L Liu Y Liu B Zhou and J Leng ldquoConstitutive modelfor a stress- and thermal-induced phase transition in a shapememory polymerrdquo Smart Materials and Structures vol 23 no10 Article ID 105019 2014

[13] B Atli F Gandhi and G Karst ldquoThermomechanical charac-terization of shape memory polymersrdquo Journal of IntelligentMaterial Systems and Structures vol 20 no 1 pp 87ndash95 2009

[14] H Tobushi T Hashimoto S Hayashi and E Yamada ldquoTher-momechanical constitutive modeling in shape memory poly-mer of polyurethane seriesrdquo Journal of Intelligent MaterialSystems and Structures vol 8 no 8 pp 711ndash718 1997

[15] H Tobushi K Okumura S Hayashi and N Ito ldquoThermo-mechanical constitutive model of shape memory polymerrdquoMechanics of Materials vol 33 no 10 pp 545ndash554 2001

[16] S Wineman and K R Rajgapol Mechanical Response of Poly-mers An Introduction CambridgeUniversity Press CambridgeUK 2000

[17] K K Westbrook P H Kao F Castro Y Ding and H J Qi ldquoA3D finite deformation constitutive model for amorphous shapememory polymers a multi-branch modeling approach fornonequilibrium relaxation processesrdquo Mechanics of Materialsvol 43 no 12 pp 853ndash869 2011

[18] R Xiao C M Yakacki J Guo C P Frick and T D NguyenldquoA predictive parameter for the shape memory behavior ofthermoplastic polymersrdquo Journal of Polymer Science Part BPolymer Physics vol 54 no 14 pp 1405ndash1414 2016

[19] S Alexander R Xiao and T DNguyen ldquoModeling the thermo-viscoelastic properties and recovery behavior of shape memorypolymer compositesrdquo Journal of Applied Mechanics Transac-tions ASME vol 81 no 4 Article ID 041003-1 2014

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

International Journal of Polymer Science 11

[20] K Yu and H J Qi ldquoTemperature memory effect in amorphousshape memory polymersrdquo Soft Matter vol 10 no 47 pp 9423ndash9432 2014

[21] Q Ge X Luo C B Iversen et al ldquoA finite deformation ther-momechanical constitutive model for triple shape polymericcomposites based on dual thermal transitionsrdquo Journal ofApplied Mechanics vol 51 no 15-16 pp 2777ndash2790 2014

[22] T D Nguyen ldquoModeling shape-memory behavior of polymersrdquoPolymer Reviews vol 53 no 1 pp 130ndash152 2013

[23] Y Liu K Gall M L Dunn A R Greenberg and J DianildquoThermomechanics of shapememory polymers uniaxial exper-iments and constitutive modelingrdquo International Journal ofPlasticity vol 22 no 2 pp 279ndash313 2006

[24] T D Nguyen H J Qi F Castro and K N Long ldquoA ther-moviscoelastic model for amorphous shape memory polymersincorporating structural and stress relaxationrdquo Journal of theMechanics and Physics of Solids vol 56 no 9 pp 2792ndash28142008

[25] B LVolkDC Lagoudas andY-CChen ldquoAnalysis of the finitedeformation response of shape memory polymers II 1D cali-bration and numerical implementation of a finite deformationthermoelastic modelrdquo Smart Materials and Structures vol 19no 7 Article ID 075006 2010

[26] R Kazakeviciute-Makovska H Steeb and A O Aydin ldquoOn theevolution law for the frozen fraction in linear theories of shapememory polymersrdquo Archive of Applied Mechanics vol 82 no 8pp 1103ndash1115 2012

[27] S Mogharebi R Kazakeviciute-Makovska H Steeb G Eggelerand K Neuking ldquoOn the cyclic material stability of shapememory polymerrdquo Materialwissenschaft und Werkstofftechnikvol 44 no 6 pp 521ndash526 2013

[28] Anton Paar Germany GmbH RheoPlus Software SoftwareVersion 30x 2006

[29] K P Menrad Dynamic Mechanical Analysis A Practical Intro-duction CRC Press Boca Raton Fla USA 2008

[30] R Kazakeviciute-Makovska A Ozlem Ozarmut and H SteebldquoCharacterization of shape memory polymer estane by meansof dynamic mechanical thermal analysis techniquerdquo SmartMaterials Research vol 2014 Article ID 250258 9 pages 2014

[31] C Liu H Qin and P T Mather ldquoReview of progress in shape-memory polymersrdquo Journal of Materials Chemistry vol 17 no16 pp 1543ndash1558 2007

[32] J D Menczel and R B Prime Thermal Analysis of PolymersFundamentals and Applications JohnWiley amp Sons 1st edition2009

[33] L H Sperling Introduction to Physical Polymer Science WileyNew York NY USA 4th edition 2006

[34] S S Es-Haghi A A Yousefi and A Oromiehie ldquoThermorheo-logical complexity of poly(methyl methacrylate)poly(vinyli-dene fluoride) miscible polymer blend terminal and segmentallevelsrdquo Journal of Polymer Science Part B Polymer Physics vol45 no 20 pp 2860ndash2870 2007

[35] N A Hardikar S Bobba and R Jha ldquoApplicability of time tem-perature superposition principle to an immiscible blend ofpolyphenyleneoxide and polyamiderdquo Journal of Polymer Engi-neering vol 31 no 2-3 pp 223ndash236 2011

[36] J J Aklonis W J Mac Knight and M Shen Introduction toPolymer Viscoelasticity A Practical Introduction John Wiley ampSons New York NY USA 1972

[37] JWeese ldquoA regularizationmethod for nonlinear ill-posed prob-lemsrdquoComputer Physics Communications vol 77 no 3 pp 429ndash440 1993

[38] P Haupt Continuum Mechanics and Theory of MaterialsSpringer Berlin Germany 1999

[39] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering John Wiley amp Sons Chichester UK2000

[40] M Johlitz H Steeb S Diebels A Chatzouridou J Batal andW Possart ldquoExperimental and theoretical investigation of non-linear viscoelastic polyurethane systemsrdquo Journal of MaterialsScience vol 42 no 23 pp 9894ndash9904 2007

[41] A V Shutov R Landgraf and J Ihlemann ldquoAn explicitsolution for implicit time stepping in multiplicative finite strainviscoelasticityrdquo Computer Methods in Applied Mechanics andEngineering vol 265 pp 213ndash225 2013

[42] R N HawardThe Physics of Glassy Polymers Springer ScienceNew York NY USA 2nd edition 2012

[43] M Heuchel J Cui K Kratz H Kosmella and A LendleinldquoRelaxation based modeling of tunable shape recovery kineticsobserved under isothermal conditions for amorphous shape-memory polymersrdquo Polymer vol 51 no 26 pp 6212ndash6218 2010

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials