thermally induced shape-memory effects in polymers: quantification and related modeling approaches

Thermally Induced Shape-Memory Effects in Polymers: Quantification and

Related Modeling Approaches

Matthias Heuchel, Tilman Sauter, Karl Kratz, Andreas Lendlein

Institute of Biomaterial Science and Berlin Brandenburg Centre for Regenerative Therapies, Helmholtz-Zentrum Geesthacht,

14513 Teltow, Germany

Correspondence to: A. Lendlein (E-mail: [email protected])

Received 2 November 2012; accepted 18 December 2012; published online 6 February 2013

DOI: 10.1002/polb.23251

ABSTRACT: Thermo-sensitive polymers, which are capable to

exhibit a dual-, triple-, or multi-shape effect or a temperature-

memory effect (TME), characterized by a controlled shape

change in a predefined way, are of current technological in-

terest for designing and realization of actively moving intelli-

gent devices. Here, the methods for the quantitative

characterization of shape-memory effects in polymers and

recently developed thermomechanical modeling approaches

for the simulation of dual-, triple-, and multi-shape polymers

as well as materials that exhibit a TME are discussed and

some application oriented models are presented. Standardized

methods for comprehensive quantification of the different

effects and reliable modeling approaches form the basis for a

successful translation of the extraordinary achievements of

fundamental research into technological applications. VC 2013

Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys.

2013, 51, 621–637

KEYWORDS: shape-memory effect; temperature-memory effect;

stimuli-sensitive polymers; modeling; mechanical properties

INTRODUCTION A prerequisite of polymers that exhibit ther-mally induced shape-memory effects (SMEs) is their elasticdeformability in combination with their thermosensitivity. Ingeneral, shape-memory polymers (SMPs) are materials thatcan be deformed and fixed into various temporary shapes,which are stable until specific response temperatures areexceeded. In contrast to other shape-changing polymers,which deform solely as long as a stimulus is applied, therecovery of the original permanent shape is actuated oncethe SMPs’ response temperature is exceeded.1 A greatadvantage of SMPs is their ability to perform (complex)active movements, when the SME is initiated. So far, the ma-jority of SMPs are dual-shape polymers (DSPs) that changefrom a temporary shape (A) to a memorized original shape(B) when activated.1–12 Besides, such dual-shape materialsrecently polymers with a triple- or multiple-shape capabilityhave been introduced3,13–20 as well as materials exhibiting apronounced temperature-memory effect (TME).21–24

A key structural element of SMPs are switching domainsrelated to a thermal transition, for example, a glass or melt-ing transition (Ttrans ¼ Tg or Tm), which act as reversiblecrosslinks and are responsible for fixation of the temporaryshape. In SMPs, the segments forming the switching domainsare connected to netpoints, which determine the permanentshape. They can be either of chemical (polymer networks) orphysical nature (thermoplastics). Alternatively, the SME hasalso been shown for liquid crystalline elastomers, for which

the thermal transition is characterized by a clearing tempera-ture (Ttrans ¼ Tcl) where the mesogenic units change from anematic/smectic into an isotropic phase.25

In polymers, a specific thermomechanical treatment namedshape-memory creation procedure (SMCP) or programinghas to be applied for implementation of the shape-memoryfunctionality. A conventional dual-shape creation procedureconsists of deforming the polymer at temperatures aboveTtrans, followed by cooling to temperatures below Ttrans whilekeeping the external load, whereby the reversible crosslinkssolidify, and finally, the external stress is removed to obtainthe fixed temporary shape. When such programed SMPs areheated again to temperatures above Ttrans, the original shapeis recovered under stress-free conditions, whereas underconstant strain recovery a characteristic recovery-stress isobtained. The driving force for the recovery process is theentropy gained by the switching chain segments, when mov-ing from an oriented (programed) conformation to a randomcoil-like (recovered) conformation.

On the macroscopic level, the shape-memory properties ofpolymers are typically quantified by the extent of fixing theexternally applied deformation em in the temporary shape(shape fixity ratio Rf) and the percentage of recovering theoriginal shape (shape recovery ratio Rr). One of the mostcommon test procedures for examination of SMPs are cyclicthermomechanical tensile tests, which allow on the one handside a precise control of the applied programing parameters

VC 2013 Wiley Periodicals, Inc.

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and on the other hand provide a complete dataset describingthe materials’ stress-temperature-strain behavior with time.Therefore, additional characteristic measures such as theswitching temperature Tsw from the obtained strain-tempera-ture plot under stress-free recovery conditions and the tem-perature at recovery-stress maximum Tr,max or the tempera-ture at the inflection point Tr,inf of the stress-temperaturecurve under constant strain conditions can be determined.26

Based on the numerous available experimental data in recentyears, various qualitative and quantitative modelingapproaches have been developed for description and predic-tion of polymers’ thermomechanical as well as shape-mem-ory properties. While the short-term objective of the theoret-ical approaches is a description of experimental data, theultimate goal is a true prediction of the macroscopic shape-memory behavior for a particular polymer and application.Thereby, the model description should be capable to

reproduce the influence of different essential programing pa-rameters such as the deformation temperature Tdeform or theapplied deformation em on the shape-memory properties,whereby also morphological model aspects representing thepolymers’ molecular structure have to be considered. Cur-rently, two different modeling approaches can be distin-guished: the application of existing linear viscoelastic modelsconsisting of coupled spring, dashpot, and frictional elementsfor modeling of SMPs.27–36 Such models were applied topolymer networks as well as thermoplastic materials. Morerecently, models have been proposed that consider in greaterdetail the nature of the specific thermal transition, Ttrans ¼Tg or Tm.

37–49 First summaries of modeling approaches forSMPs with thermal actuation were published recently.50,51

In this article, a brief overview about the characterization ofSMPs via cyclic thermomechanical tensile tests and therelated thermomechanical modeling approaches is given.

Matthias Heuchel studied chemistry and got his Ph.D. in Physical Chemistry at University of

Leipzig. 1997-99 he was Marie-Curie-Fellow at the Department of Chemical Engineering,

University of Edinburgh. In 2000 he joined the Polymer Physics group of GKSS and is

currently Senior Researcher in the Department of Polymer Engineering of the Institute of

Biomaterial Science at the Helmholtz-Zentrum Geesthacht in Teltow, Germany. His research

interests range from atomistic computer simulation of polymers to thermomechanical

modeling of shape-memory polymers.

Tilman Sauter studied Mechanical Engineering at Technical University of Munich and

Medical Engineering at the Royal Institute of Technology Stockholm. He obtained his

degrees in 2010 and currently works for his PhD project on the shape-memory properties of

polymeric scaffolds for biomedical applications in the Department of Polymer Engineering

of the Institute of Biomaterial Science at the Helmholtz-Zentrum Geesthacht in Teltow,

Germany.

Karl Kratz received his diploma in Chemistry and doctor’s degree from the University

Bielefeld. At first he worked a scientific project manager and director resources in the

industry on the development of degradable shape-memory polymers. Currently he holds the

position of a senior research associate and is head of the Polymer Engineering Department

of the Institute of Biomaterial Science at the Helmholtz-Zentrum Geesthacht. His primary

research interests are design, development and characterization of biomaterials and stimuli-

sensitive polymers.

Andreas Lendlein is the Director of the Institute of Biomaterial Science at Helmholtz-Zentrum

Geesthacht in Teltow, Germany and Member of the Board of Directors of the Berlin-

Brandenburg Centre for Regenerative Therapies. He is a professor at the University Potsdam

and Honorary Professor at Freie Universit€at Berlin as well as Member of the Medical Faculty

of Charit�e University Medicine Berlin. He completed his Habilitation in Macromolecular

Chemistry in 2002 at RWTH Aachen University and received his doctoral degree in Materials

Science from the Swiss Federal Institute of Technology (ETH) in Zurich. His current research

interests include (multi)functional polymer-based materials, biomaterials and their interac-

tion with biological environments as well as the development of medical devices and

controlled drug delivery systems especially for regenerative therapies.

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While in the first section, the cyclic testing of DSPs and themost relevant concepts for constitutive theoretical predictionmodels are introduced in particular, in the next paragraph,the respective characterization and modeling approacheswere extended to polymers, which are capable of a triple- ormulti-shape effect and temperature-memory polymers. Thefollowing section addresses selected application-orientedthermomechanical tests and theoretical models includingfinite element (FE) approaches. Finally, the main challengesand future perspectives in cyclic thermomechanical testingand modeling of SMPs are discussed.

CYCLIC THERMOMECHANICAL TESTING AND MODELING

OF DUAL SHAPE-MEMORY POLYMERS

Cyclic Thermomechanical Testing of Dual-ShapePolymersCyclic thermomechanical uniaxial tensile, bending, or com-pression tests are widely used to characterize SMPs, becausea complete dataset of stress-temperature-strain over time canbe generated. Such as modeling of stress–strain relationshipsis possible by fitting a theoretical model to existing datasets;in the same way, cyclic thermomechanical experiments serveas basis to evaluate and to fit theoretical models. As comparedwith a stress–strain relationship, where a constant strain rateis applied until a desired elongation is reached, which mightbe regarded as a single-step procedure, the thermomechanicalevaluation of DSPs is a multistep procedure involving the cre-ation of a temporary shape (called SMCP) and the recovery ofthe original shape under different conditions.

The first step during SMCP is the elongation of the sample to theprograming strain em at a temperature Tdeform, which is abovethe transition temperature Ttrans of the switching segment [seeFig. 1(a)]. At em, the sample will be hold for a certain time toallow the sample to equilibrate. Subsequently, the sample will becooled to Tlow, which is below Ttrans, either under constant strain(em) or constant stress conditions (rm ¼ const.) resulting in amaximum deformation e(rm) ¼ el. During the cooling procedure,the formerly flexible switching segment will be fixed during thecrystallization of the polymer chains, if Ttrans ¼ Tm, or during thetransition from the rubbery into the glassy state, if Ttrans ¼ Tg.After Tlow has been reached, the sample is unloaded and thetemporary shape eu is obtained. The degree of how well theapplied deformation em or el can be maintained is called theshape-fixity ratio Rf. Rf is calculated according to eq 1, while N isthe number of repeated measurement cycles, if the sample iscooled under strain controlled conditions, or according to eq 2,if the sample is cooled under stress controlled conditions.

Strain controlled : RfðNÞ ¼euðNÞem

(1)

Stress controlled : RfðNÞ ¼euðNÞ

el(2)

The recovery process can be investigated under stress-free(r ¼ 0 MPa) conditions to observe the shape recovery dueto entropy elastic behavior of the switching segment aboveTtrans, when the polymer is heated to Thigh � Ttrans. The tem-

perature, at which the shape change occurs, is called theswitching temperature Tsw. The recovered shape is deter-mined as ep and the shape-recovery ratio Rr, which definesthe degree of the shape recovery, can be obtained after oneor more cycles, where each cycle is composed of the SMCPfollowed by the recovery procedure, and is defined in eq 3,if the programing was conducted under strain controlledconditions, or under stress controlled conditions (eq 4).

Strain controlled : RrðNÞ ¼em � epðNÞ

em � epðN � 1Þ (3)

Stress controlled : RrðNÞ ¼el � epðNÞ

el � epðN � 1Þ (4)

The stress-temperature-strain diagram in Figure 1 shows asimplified recovery process, where the sample fully recoversits original shape to ep ¼ e0. Typically, ep remains higherthan e0 due to irrecoverable strain. Alternatively, the samplecan be reheated under constant strain (e ¼ const.) conditionsto measure the evolution of the stress at increasing tempera-tures. Here, the characteristic values that can be obtainedfrom the stress–temperature curve for thermoplastic materi-als are the peak recovery stress rmax and the respective tem-perature Tr,max. Above Tr,max, the recovery stress is reducedagain due to softening of the polymer. In contrast to thermo-plastic materials, the stress increases continuously forpolymer networks and rmax is maintained even at highertemperatures. The characteristic temperature Tr,inf is thusdetermined by the inflection point of the stress–temperaturecurve.26

Modeling of Dual-Shape PolymersA schematic representation of a stress-temperature-straindiagram obtained in a cyclic thermomechanical tensile test

FIGURE 1 (a) Schematic illustration of a r-T-e diagram obtained

in cyclic thermomechanical tensile tests consisting of a heating-

cooling-heating shape-memory creation procedure (SMCP) and

recovery under stress-free conditions with characteristic states:

(1) start and end point of the cycle (0, Thigh, e0 ¼ ep); (2) after end

stretching (r(em), Thigh, em); (3) after cooling at constant strain

(rm, Tlow, em); (4) after unloading before the recovery step starts

(0, Tlow, eu). (b) Schematic representation of the states of the test

cycle by the standard linear solid (SLS) model.



of a thermoplastic DSP is shown in Figure 1(a), whereby inFigure 1(b) a simple viscoelastic model consisting of an elas-tic spring and a Maxwell model connected in parallel at dif-ferent characteristic steps of the test cycle is illustrated. Thismodel is called the standard linear solid (SLS).52 The elasticspring with a Young’s modulus of Eeq represents the equilib-rium response on stress by the permanent netpoints (hardsegments). The nonequilibrium branch is presented byspring and dashpot in series. The dashpot has a viscosity gor a characteristic relaxation time s ¼ g /Er , which dependstrongly on temperature, representing the temperature de-pendent chain mobility of the switching segments (reversiblecrosslinks). At first, the viscosity of the dashpot is very lowat Thigh, as depicted in state 1 in Figure 1. At Thigh, the defor-mation takes place so that the energy is stored mainly in theleft spring of the equilibrium branch (see state 2 in Fig. 1),and the spring with modulus Er of the Maxwell unit isalmost undeformed compared with the initial state. Duringcooling to Tlow, which is below Ttrans, the viscosity or relaxa-tion time of the dashpot increases by orders of magnitude.In state 3, the external load/stress has been removed at Tlow.Because of the high viscosity, the Maxwell unit behaves likean elastic spring under constant stress. The stress free state(r ¼ 0) generates a redistribution of stored elastic energy inboth springs combined with only a very small deformationem � eu, because the modulus of the spring in the Maxwellunit of this model is in general much higher than the one inthe equilibrium branch Er � Eeq. So, the deformation of theDSP at Thigh is fixed. For the next step of the cycle, it is im-portant to note that the deformation of the spring in theMaxwell unit is compressive, whereas the sample itself isstill elongated. During heating, the sample from Tlow to tem-perature above Ttrans, the viscosity in the dashpot is dramati-cally decreased, and the force, created through the spring ofthe Maxwell unit moves at Tsw the viscous extension back tozero, and the original shape (state 1) is recovered.

For the understanding of the mechanism of the SME two fea-tures of the dashpot play an important role: (i) the viscousstrain, which is frozen and locked at Tlow, and (ii) the tre-mendous dependence of viscosity on temperature, whichallows above Ttrans to unlock viscous strain and to bringback the original shape.

The whole thermomechanical cycle has been at first theoreti-cally described and modeled by Tobushi et al.34,35 The modelextended the SLS model with a slip element to account fordissipated energy due to internal friction, which results inan incomplete recovery. The model expressed the fact thatduring stress free recovery at a specific temperature, a cer-tain part of the extension remained as irrecoverable strain. Acomparison of experimental and calculated data35 for a seg-mented polyurethane composed of amorphous ether-basedsoft segments consisting with a Tg around 45 �C, which con-tained adipic acid and bisphenol A moieties extended withethylene or propylene oxide and crystallizable hard segmentssynthesized from 4,40-diphenylmethane diisocyanate (MDI)and 1,4-butanediol (BD) are shown in Figure 2. For a maxi-

mum extension of em ¼ 20%, all steps of the cycle: (1)stretching, (2) cooling, (3) unloading are qualitatively welldescribed, as well as including the e(T)-curve (not shown)during the stress-free recovery (4).

This model is a first example of a general approach to calcu-late thermomechanical properties of DSPs based on its visco-elastic properties. Other rheological models consisting ofspring, dashpot, and frictional elements were applied toSMPs, for example, by Lin and Chen.31 These models werefurther developed and stepwise adapted to the specific na-ture of the thermal transition in the polymer, either a glasstransition with Ttrans ¼ Tg

40,41,43,53,54 among others or amelting transition Ttrans ¼ Tm.

33,37,38 The models for SMPswith crystallizable switching segments followed a mechanicalapproach, in which the stress–strain behavior is described asa combination of spring or dashpot units,33 or the observedexpansion and contraction is predicted based on so-called‘‘constitutive equations’’ for a rubbery phase, a semicrystal-line phase, the crystallization and melting process, which areadjusted to predict the conditions of a specific SMP.37

The current capability of viscoelastic models is demonstratedon the 3D finite deformation constitutive model for amor-phous SMPs recently developed by Westbrook et al.,55 whichis shown in Figure 3.

The authors investigated an acrylate-based covalently cross-linked polymer network (Tg ¼ 42 �C). Isothermal uniaxialcompression experiments were conducted at strain rates of0.01/s and 0.1/s. To explore the rubbery and glassy behav-ior, the isothermal uniaxial compression tests were per-formed at various temperatures between 0 and 100 �C in 10K increments. Further cyclic experiments, both under con-strained and free conditions during the recovery step werecarried out. For constrained recovery, the programing tookplace at Tdeform ¼ Thigh ¼ 60 �C, the unloading (fixing) tookplace at Tlow ¼ 10 �C, and the recovery again at Thigh ¼ 60�C. During constrained recovery, a stress overshoot peak foran initial compression deformation is observed on thestress–temperature graph [Fig. 4(a)]. At stress-free recovery

FIGURE 2 Modeling of a cyclic thermomechanical test of a seg-

mented polyurethane with a maximum extension of em ¼ 20%

using an extended standard linear solid (SLS) model with addi-

tional slip element due to account for internal friction (Reprinted

from Ref. 35, VC 2001, with permission from Elsevier).


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experiments of samples programed at Tdeform ¼ 40 �C andunloaded at Tlow ¼ 10 �C, the recovery temperature was var-ied from Thigh ¼ 30, 35, 40, to 50 �C. As it becomes obviousfrom Figure 4(b), the recovery behavior is strongly depend-ent on Thigh.

These experiments were modeled with a viscoelastic modelwhere every material point moves in space under conditions,which can be represented in 1D by the rheological model pre-

sented in Figure 3, where a thermal expansion element isarranged in series with mechanical elements. The mechanicalelements consist of an equilibrium branch composed of an elas-tic spring with a Young’s modulus of Eeq, a glassy nonequili-brium branch represented by a Maxwell element based on anelastic spring and a dashpot characterized by a relaxation timesg ¼ gg/Eg, which are placed in series, and several further Max-well elements, presenting the rubbery nonequilibrium branches.This model represents an extension of the simple SLS model.The glassy branch describes the segmental relaxation of thepolymer. This most fundamental relaxation process defines themonomer friction coefficient and it is on the bottom of all otherand longer time scale chain relaxation processes. The modelconsiders a finite number of n different relaxation processes inthe rubbery (or molten) state. For the thermal expansion, amultiparameter theory has been used,56 which describes thevolume departure of the polymer in the nonequilibrium situa-tion below the glass transition. The elastic behavior of the DSPabove Tg is modeled by a hyperelastic material model typicallyfor rubbers, that is, the Arruda-Boyce eight chain model.57 Forthe nonequilibrium viscoelastic branches, one can assume thatall branches follow the same viscous flow rules, but with differ-ent relaxation times. The relaxation times are temperature de-pendent. According to the thermo-rheological simplicity princi-ple, a time-temperature superposition shift factor aT(T) can bedetermined, which relates all n þ 1 relaxation times

siðTÞ ¼ aTðTÞsi0 (5)

with i ¼ 0,…,n to a reference relaxation time si0, where the shiftfactor is equal to 1. For temperatures below Tg, the shift factor canbe determined from an Arrhenius-type behavior,58 and at temper-atures close to or above Tg, the WLF equation59 can be used.

FIGURE 3 1D rheological representation of the constitutive

model for amorphous SMPs55 with thermal expansion element

in series with parallel mechanical elements consisting of an

equilibrium branch (spring) and nonequilibrium branches of

Maxwell elements for one glassy and m rubbery nonequili-

brium branches of different relaxation times (Reprinted from

Ref. 55, VC 2011, with permission from Elsevier).

FIGURE 4 Modeling of experimental recovery data of an acrylate-based covalently crosslinked polymer with the constitutive

model, presented in Figure 3. (a) Comparison between experiment and simulation for the stress response during constant strain

recovery. (b) Comparison between experiment and simulation for the strain recovery during isothermal heating at different tem-

peratures TH2 under stress-free condition. Samples were programed at Thigh ¼ TH1 ¼ 40 �C (Reprinted from Ref. 55, Figures 2 and

3, VC 2011, with permission from Elsevier).



The authors implemented the constitutive model into a FEsoftware package and fitted at first model parameters forseveral key features: (i) the overall behavior of the isother-mal uniaxial compression results at a strain rate of 0.01/s,(ii) the location and magnitude of the stress overshoot peakin the constrained recovery experiments, and (iii) the overallbehavior in stress-free recovery experiments at all Thighvalues. It turned out that the number of nonequilibrium rub-bery branches could be assessed to n ¼ 2. The model wasthen capable to predict the strain rate dependency in theisothermal uniaxial compression experiments. For the con-strained recovery conditions [see Fig. 4(a)], the model wasable to capture magnitude and temperature location of thestress overshoot peak during heating. However, the overallbehavior above 35 �C during cooling or heating and the loca-tion of the temperature at which the stress overshoot is stillnot well depicted. For the free recovery after a shape-mem-ory cycle, the experimental results are well described [seeFig. 4(b)]. For Thigh � Tg, the stress-free recovery behavior(both the recovery rate mrec and the temperature intervalDTrec, where the transition from the temporary to the origi-nal shape starts (Tstart) and ends (Tend)) is equivalent exclud-ing the thermal expansion for the case where Thigh > Tdeform.

The established model allows, for example, to study theinfluence of the heating rate on the recovery behavior up tothe question of how fast the SMP can recover its memorizedshape, if it could be heated instantaneously, or to simulate amagnetosensitive SMP composite to determine the effect ofchanging the particle size and concentration on the freerecovery behavior.

The just discussed model represents implicitly also a tool tostudy the influence of the programing parameters Tdeform andem on the recovery behavior of the SMP due to the incorpo-rated temperature dependency of the relaxation times. Itshould also be mentioned that this constitutive model needs23 parameters. The model of Srivastava et al.60 is currentlythe most elaborated with 45 parameters. The effort in the pa-rameter description, together with the often mathematicallychallenging formulation of these models, is at present a disad-vantage for a wider application of these kind of models in theSMP research. From this point of view, simple modeling con-cepts, which only focus on the essential aspects of the shape-memory behavior, are of special interest.

An example is the approach of Bonner et al.61 to predict therecovery time of SMPs depending on programing parameters.A simple Kelvin–Voigt model (elastic spring in parallel with adashpot) is used to analyze data of a transient stress dip test,where a sample is stretched at constant rate up to a certainmaximum length, then the stretching direction is very quicklyreversed for a short time, and finally, the strain developmentas function of time is observed and the recovery stress rR ismeasured. It was found that rR is a linear function of k2 � 1/k where k is the draw ratio, further that rR decreases withincreasing deformation temperature, and increases withincreasing strain rate. It turned further out that the ratio ofthe recovery stress to the total stress is a good approximationinvariant to the three programing variables Tdeform, em, and

strain rate _e. The shape recovery of the SMP occurs when thespring component of the Kelvin–Voigt model or the storedstress in the material exerts a force capable of deforming theviscous component or the dashpot element of the model,which typically occurs at the temperature of shape recovery.

Into the same direction of a simplified model with a few char-acteristic parameters aims also the recently modified SLSmodel.36 The simple analytical solution allows to predict thenonisothermal free recovery of an amorphous DSP by onlyeight calibration parameters obtained by relatively quickdynamic mechanical thermal analysis (DMTA) measurements.

For the second set of general modeling approaches to calcu-late thermomechanical properties, the polymers’ morphologyis the starting point. Covalently crosslinked amorphous DSPsare described as a two-phase material composed of a glassyand rubbery phase. Liu et al.42 developed a constitutivemodel where the total strain e can be represented as sum e¼ /fef þ (1 � /f)ea of two strain contributions from the fro-zen phase and the active phase. The strain in the frozen(glassy) phase ef can be determined from three contribu-tions, that is, the (stored) entropic stain, the internal ener-getic strain, and the thermal strain. In the active phase, thestrain deformation ea consists of two parts: the externalstress induced entropic strain and the thermal strain.

The essential idea of this type of models is shown in Figure5(a). In the glassy state at Tlow, the major phase of a poly-mer is the frozen phase composed of frozen bonds, whereconformational motions of chain segments are locked. Byheating over a deformation temperature Tdeform, the volumefraction of the frozen phase is reduced, and at the sametime, the fraction of the active phase with free conforma-tional motion is increasing till the polymer is at Thigh in apractically full rubbery state. This process is assumed asfully reversible.

From a thermodynamic point of view, the frozen volume fraction/f is an internal state variable of the system and it is assumedthat /f depends only on the temperature (T). Based on experi-mental results (a free strain recovery response during heating,where the sample has experienced predeformation epre andstrain-storage process), an analytical phenomenological function

/fðTÞ ¼ 1� 1

1þ cfðThigh � TÞn (6)

can be determined. The two numerical parameters cf and ncan be fitted from the experimental strain ratio data. Veryrecently, Kazakeviciute-Makovska et al.62 analyzed Liu’s law(6), and other /f(T) relations obtained by different research-ers43,44,47,48,53 and proposed a modified general functionbased on reduced temperature valuesn ¼ T=Tdeform; nhigh ¼ Thigh=Tdeform; nlow ¼ Tlow=Tdeform (7)

with two parameters a and b, and following shape

/fðnÞ ¼1

1þ expðbðn� ð1þ aÞÞ ; (8)


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where a can be positive or negative, and b > 0. The influenceof these parameters on the T-dependency of the frozen volumefraction is shown in Figure 5(b). Parameter a shifts the transi-tion curve relative to Tdeform, and b determines the width of thetransition. The parameters a and b for the generalized evolutionlaw for the frozen volume fraction may be obtained in a sys-tematic way from stress–strain experiments. A simple approach

has been used by Volk et al.,47 who assumed that the frozenvolume fraction function [see Fig. 6(b)] is given by the normal-ized stress free recovery profile presented in Figure 6(a), thatis, /f(T) ¼ e (T)/em. Based on the generalized evolution law ofthe frozen volume fraction /f, it was possible to model severalexperimental sets of thermomechanical tests,63,64 including thealready discussed experiments of Tobushi et al.34

FIGURE 6 Approximation of frozen volume fraction for a polystyrene-co-butadiene copolymer network as normalization of the

extension recovery /f(T) ¼ e (T)/em. (a) Stress free strain recovery. (b) Frozen volume fraction fitted by normalized extension data

(Reprinted from Ref. 47, with permission from IOP Publishing Ltd.).

FIGURE 5 Sketch of the two-phase model. (a) Polymer phase at T ¼ Tlow << Ttrans (left) at T ¼ Ttrans (middle), and T ¼ Thigh >>

Ttrans (right). (b) Curves of the frozen volume fraction /f for normalized temperatures (see eq 7) presented through the generalized

function (8) with two parameters a and b (Reprinted from Ref. 62, VC Springer Verlag 2012, with kind permission from Springer Sci-

ence and Business Media).



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The two-phase approach was confirmed by Wang et al.48 Basedon this theory, Chen and Lagoudas40,53 reported an approach todescribe the thermomechanical properties of SMPs under largedeformations, where general constitutive functions of neo-Hoo-kean type for nonlinear thermo-elastic materials are used forthe active and frozen phases. This model was applied to analyzeexperimental cyclic tests for a polystyrene-based SMP net-work.47,64 Figure 7 presents the comparison of experiment andmodeling for different maximum extension values.

The model was calibrated by calculating values for modelparameters (coefficient of thermal expansion, shear moduli,and the frozen volume fraction) using the data of thethermomechanical test with em ¼ 25%. Based on these data,the model allows to predict in qualitative agreement the fullr-T-e tensile tests for small as well as large em of 10%, 50%,and 100%. The model failed to capture the irrecoverableextension at the end of the recovery step. The model wasalso used to predict the stress free and constrained recoverycurves for a polyurethane SMP, which were found to be ingood agreement with experimental data.65

Despite its successes, the two-phase approach has a stronglimitation. As Diani et al. discuss in ref. 66, the volume frac-tion of each phase in such models must be given as a func-tion of temperature /f ¼ /f(T). This function is obtainedfrom fitting experimental shape-memory data at a certain

heating rate. Therefore, the resulting models can fit these ex-perimental data but cannot predict the shape-memorybehavior of the material under different heating rates orheating profiles.

Another important aspect was highlighted by Nguyenet al.43 with an own model development based on the none-quilibrium character of the glassy state. The authorspointed out that the concept of two mixed phases (frozen/active) is not in true agreement with the physical processesof the glass transition and thus results in ‘‘nonphysical’’ pa-rameters, such as the volume fractions of the glassy andrubbery phases. Instead, it would be necessary to consider,as primary molecular mechanism, the time-dependentstructural and stress relaxation of the glass forming poly-mer material. The model developed under this relaxationconcept seems to reproduce the stress-free strain–tempera-ture response, the temperature and strain-rate dependentstress–strain response, and important features of the tem-perature dependence of the shape memory response well.67

The modeling approach allowed recently to study the effectof physical aging (up to 180 days) on the stress free recov-ery of (meth)acrylate-based polymer networks.68 Anotherapplication of this stress-relaxation concept was the model-ing of experiments where a SMP network was programedby cold compression.69

FIGURE 7 r-T-e profile for 10%, 50%, and 100% extension experiments of a polystyrene-co-butadiene copolymer network and model pre-

dictions: (a) 10% extension, (b) 50% extension, and (c) 100% extension (Reprinted from Ref. 47, with permission from IOP Publishing Ltd.).


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http://dx.doi.org/10.1088/0964-1726/19/7/075005

http://dx.doi.org/10.1088/0964-1726/19/7/075005

http://dx.doi.org/10.1088/0964-1726/19/7/075005

As extension of the discussed ‘‘two-phase’’ models, Kim et al.70

developed especially for shape-memory polyurethanes(SMPUs) a constitutive model consisting of three phases.SMPUs consist of hard and switching domains. The switchingdomains can form a variable state ranging from a rubbery(active) to a rigid (frozen) phase depending on temperature.Hard domains act as fixed (permanent) netpoints. The three-phase model divides the SMP into one hard segment phasethat governs the viscoelastic behavior and two switching seg-mented phases (frozen and active phase) that undergo a re-versible phase transformation as function of temperature. Athree-element viscoelastic model (two springs and a dash-pot) was used to present the hard segment phase. For thetwo switching segment phases, a Mooney-Rivlin solid model(hyperelastic spring) was used. The two hyperelastic springsfor the frozen and active switching segment phases werelinked in the model as series or parallel. The stresses in eachphase were calculated using hyperelastic and viscoelasticequations, and combined into the total stress. The materialproperties for the constitutive equations were determined bydifferential scanning calorimetry and a series of uniaxial ten-sile stress–strain tests at three temperature conditions. Sim-ulations of cyclic thermomechanical tests gave good agree-ment with experimental data and showed a certaindependence on the assumed linking model (parallel orseries).

CYCLIC THERMOMECHANICAL TESTING OF TRIPLE- OR

MULTI-SHAPE AND TEMPERATURE-MEMORY POLYMERS

AND RELATED MODELING APPROACHES

Programing Procedures for Triple-Shape PolymersPolymers containing at least two distinct domains having sig-nificantly different thermal transition temperatures (Ttrans,Band Ttrans,A) are able to memorize two temporary shapes and,hence, are called triple-shape polymers (TSPs).13,71 When sucha TSP is heated to Thigh, which is above the two transition tem-peratures Ttrans,B and Ttrans,A, both switching domains are in therubbery elastic state and it can be easily deformed, for example,by stretching to em ¼ 50%. Cooling to Tmid, which is belowTtrans,B but above Ttrans,A, results in the fixation of domain B bycrystallization or vitrification (depending on the nature ofTtrans,B, which can be a Tg or Tm), whereas domain A is still inthe flexible state. Hence, cooling to Tmid affects only domain B,but not domain A, and unloading of the sample at Tmid resultsin the temporary shape eB. In a next step, the sample isstretched by an additional proportion to 100% at Tmid andcooled to Tlow, which is below Ttrans,A. During this step, only do-main A is affected, because the cooling step only passes onethermal transition, which is Ttrans,A (either Tg or Tm). Unloadingat Tlow results in the second temporary shape eA. To fix the tem-porary shape eB, different programing procedures have beendescribed.3 The stress-free recovery of a crosslinked multiphasepolymer network is shown in Figure 8 for different eB and eA.

13

As extension of the dual SME, the shape-fixity ratios Rf(C !B) and Rf(B ! A) for TSPs quantifying the differencebetween two shapes together with the overall fixity Rf(C !A) are described in eqs 9–11

RfðC! BÞ ¼ eB � eCeloadB � eC

(9)

RfðB! AÞ ¼ eA � eBeloadA � eB

(10)

RfðC! AÞ ¼ eA � eCeloadA � eC

(11)

Recovery of TSPs is generally conducted under stress-freeconditions (r ¼ 0 MPa) to observe the shape recovery. Thesample is heated from Tlow to Thigh to observe a change inshape. The heating rate should be chosen as constant toreally investigate whether the material shows a triple-shapeeffect.13,71,72 Upon heating, the sample contracts to therecovered shape B at erecB when exceeding the switching tem-perature from shape A to B (Tsw(A ! B)), at which the do-main B is still in the glassy or semicrystalline state. The finalshape C at erecC is recovered when the sample reaches Tsw(B! C).

The recovery of the two memorized shapes in TSPs is calcu-lated in analogy to DSPs. The shape-recovery ratios can becalculated for the recovery of the first recovery step Rr(A !B), the second change in shape Rr(B ! C), and the totalshape recovery Rr(A ! C) according to the eqs 12–14.

RrðA! BÞ ¼ eA � erecB

eA � eB(12)

RrðB! CÞ ¼ eB � erecC

eB � eC(13)

RrðA! CÞ ¼ eA � erecC

eA � eC(14)

Different material design concepts such as covalently cross-linked multiphase polymer networks13,15,71–73 or

FIGURE 8 TSP: Recovery of shapes B and C by heating a cross-

linked multiphase polymer network from Tlow to Thigh. A constant

heating rate of 1 K min�1 was applied. By different combinations

of eB and eA, different recoveries can be achieved: solid line, eB ¼50% and eA ¼ 100%; dashed line, eB ¼ 30% and eA ¼ 100%; dotted

line, eB ¼ 50% and eA ¼ 120% (Taken from Ref. 13, VC 2006 National

Academy of Sciences, USA, reproduced by permission).



thermoplastic polymers containing two thermal transitionscan be used as switching domains or copolymers with abroad thermal transition18,74,75 as well as multimaterialapproaches17,76 (which, in general, combine two dual-shapematerials) have been reported for the realization of TSPs.

Programing Procedures for Multi-Shape PolymersPolymers that exhibit a broad thermal glass transition, suchas perfluorosulfonic acid ionomer (PFSA) with a DTg from 55to 130 �C, are also able to memorize and recover more thantwo predefined shapes. A multishape creation procedureinvolves to heat the sample to Thigh, where all domains (i ¼1,…,n) are in the rubbery phase, then to stretch the sampleto the programing strain of shape S1, eS1, and finally to coolbelow the highest transition temperature Ttrans,1 to the de-formation temperature Td1, where S1 is fixed and the samplecan be unloaded. In the next step, the process of stretchingat Td(i), cooling to Td(iþ1) and unloading is repeated until thedesired number of temporary shapes is obtained and Tlow,where all n domains are in the glassy state. The shape-fixityrate, which is a measure of how well the temporary shapescan be maintained, is increased the lower the transition ofthe domain, because more domains are already in the glassystate. Hence, these multishape polymers can, depending onthe programing procedure, show a dual-, triple-, and multi-SME as illustrated in Figure 9. Recovery of such a multishapeprogramed sample is achieved by heating the sample fromTlow to Thigh under stress-free conditions, as shown inFigure 10(b). In contrast to the TSP example shown in Fig-ure 8, the recovery of the different shapes was realized witha discontinuous heating regime.77 The broad DTg of PFSA

can be treated as ensemble of sharp transitions, wherebyeach particular sharp transition acts as a controlling unitwith a specific Ttrans,i. This current point of view is theresult of recent model considerations. First, Sun andHuang78 proposed that such SMPs consist of ‘‘tiny units ofa dual part system’’ composed of an elastic spring as elasticpart, and in parallel to this a thermoresponsive elementrepresenting the so-called transition part. In heating andcooling processes, the thermoresponsive element showsbetween the temperature values Ts and Tf a linear functionof ‘‘hard portion in the transition part.’’ At T ¼ Ts, the hardportion is 100% and at T ¼ Tf the hard portion is 0%. Theauthors argued that under these idealized conditions it ispossible during the programing to store stepwise contribu-tions of elastic energy at certain elongations when the SMPis stepwise cooled from Tf to Ts. It becomes comprehensi-ble that the stored elastic energy may be released stepwisein a free recovery during stepwise heating from Ts to Tfand creates the (entropic) driving force for this multi-SME.The basis is a single broad thermal transition, which canbe assumed as integral distribution of an infinite numberof ‘‘tiny units’’ showing all a separate sharp transition. Sowithin the transition range, a portion of the transition partis glassy, whereas the rest is in a rubbery state. Based onthe ideas of Sun and Huang very recently, the first quanti-tative model to predict the thermomechanical cycle of amultishape memory effect has been developed by Yuet al.77 and applied to the experimental PFSA data. Theauthors used the already introduced finite deformationmodel of Westbrook et al.55 (see Fig. 3). In the model,

FIGURE 9 Temporal development of temperature T, stress r (above), and strain e (below) in cyclic thermomechanical tests of dif-

ferent SMCP for perfluorosulfonic acid ionomer (PFSA) with a DTg from 55 to 130 �C in comparison with a theoretical modeling

approach (eqs 15–17). (a) Dual-shape cycle. (b) Triple-shape cycle. (c) Multishape cycle (Reproduced from Ref. 77, with permission

of The Royal Society of Chemistry).


POLYMER SCIENCE


http://dx.doi.org/10.1039/C2SM25292A




individual nonequilibrium branches represent differentrelaxation modes of polymer chains with different relaxa-tion times. As the temperature increases in a staged man-ner, for a given temperature, different numbers of branches(or relaxation modes) became shape-memory active or inac-tive. The main equations of the model are the relations forthe total stress r (t), for the elastic strain eei (t), and the vis-cous strain evi (t) (the strain in the dashpot) in the ith indi-vidual nonequilibrium branch.

rðtÞ ¼ EeqeðtÞ þXni¼0

Ei

Z t

0

deðsÞdt

exp � t � ssiðTÞ

� �ds (15)

eei ðtÞ ¼Z t

0

deðsÞdt

exp � t � s

siðTÞ

� �ds (16)

evi ðtÞ ¼ eðtÞ � eei ðtÞ (17)

For i ¼ 0, si ¼ sg, and E0 ¼ Eg. As discussed before, the relaxa-tion times in the individual branches are strongly temperaturedependent (see eq 5). Following the SMCP, eq 15 was solvedfor total strain e (t) under the condition of the temperatureand time dependent total stress. The individual elastic and vis-cous strain values are calculated with eqs 16 and 17. Duringstress free recovery (r ¼ 0), eq 15 was solved for the total re-covery strain e (t) with the left hand being zero.

As already illustrated in Figure 3, each domain i can betreated as a Maxwell element with the same spring constantbut different damping constants, or retardation times, withinthe rubbery phase. With the model presented by eqs 15–17,all dual-, triple-, and multi-shape recoveries can be wellpredicted.

Programing Procedures for Temperature Memory Effectsin PolymersIf a multishape polymer such as PFSA is programed into aDSP at a certain Tdeform, which is within DTg and above a

certain number of j transition temperatures, where j is belowa maximum values n (1 � j � n) Ttrans,j, j domains will befixed into the temporary shape, once cooled to Tlow, whereasall n � j domains with a higher transition temperature willnot be affected by the programing procedure. When heatedfrom Tlow to Thigh, the entropy elastic recovery will be com-pleted above Ttrans,j; hence, the response temperature (e.g.,Tsw or Tr,max) of the sample will correspond to the deforma-tion temperature Tdeform. This so-called TME describes poly-mers with the capability to remember the temperature atwhich they were deformed and by reversing this deforma-tion upon reheating above the deformation temperature.Therefore, both Tsw and Tr,max can be related to Tdeform.

22

The dependency of Tsw and Tr,max toward Tdeform is pre-sented in Figure 11. The relationship of the deformation andthe recovery temperature, which is shown experimentally inFigure 11(a), can be understood by application of the modelpresented in Figure 3. In analogy, a similar behavior wasfound for this model when applied to PFSA with n ¼ 7domains, as illustrated in Figure 10(a), which in principlewould implicate seven different Tsw.

77 These so-called tem-perature-memory polymers might exhibit a broad thermalTg,

18,23,24,79 a broad thermal melting transition Tm22,29 as

well as a combination of two thermal transitions.21 It shouldbe emphasized again that in contrast to a multishape mem-ory programing procedure, which requires the application ofmultiple deformations at various temperatures within a sin-gle programing protocol, a TME is characterized by severalsubsequent dual-shape test cycles, where different deforma-tion temperatures are applied in subsequent cycles.

FURTHER APPLICATION-ORIENTED MODELS

Cyclic thermomechanical tests are the basic experiments fordetermination of the polymers’ thermomechanical propertiesrequired as key datasets for constitutive viscoelastic and vis-coplastic models as presented in the previous sections.

FIGURE 10 Evolution of viscous strain evi (t) (the strain in the dashpot) in the glassy (G) and the seven nonequilibrium rubbery

branches of the 1D rheological model presented in Figure 3 to describe the recovery after different SMCPs presented in Figure 9.

(a) Dual-shape recovery of the single Maxwell elements. The superposition results in the free recovery presented in Figure 9(a). (b)

Multishape cycle presented in Figure 9(b) (Reproduced from Ref. 77, with permission of The Royal Society of Chemistry).






Application oriented testing and modeling, however, requiresthe investigation of additional parameters, for example, aque-ous physiological environment, as well as different thermo-mechanical settings, for example, the time-dependent shape-recovery at a constant recovery temperature. For example,the TME might be used to generate several temporaryshapes at different deformation temperatures and to simu-late their recovery behavior by application of a constant re-covery temperature. Additionally, device components havingmore complex geometries such as cardiovascular stents needrather structural-dependent shape-memory and shape-recov-ery modeling. This is achieved by the FE analysis, which,besides investigation of the recovery kinetics in constantenvironments, will be presented in the following.

Modeling of Shape Recovery KineticsSMP-based materials intended for regenerative biomedicalapplications require, besides their biocompatibility anddegradability, mainly an open porous structure for the cellsto infiltrate and to regenerate functional tissue while the ma-terial is degraded over time. Furthermore, the nutrition andwaste exchange as well as cell–cell crosstalks can be main-tained by a porous structure, which is called scaffold. Typi-cally, autologous cells are seeded in vitro onto the scaffolduntil they have proliferated and migrated throughout thescaffold, while the cell-scaffold is then implanted in thedefect site of the patient. These cells, however, need anactive environment such as mechanical strains to proliferatewell and to prevent dedifferentiation. In this context,recently, mechanically active scaffolds based on SMPs wereintroduced, which can undergo autonomous, controlledshape changes (e.g., pore geometry) under physiological con-ditions.80 Such scaffolds are intended to serve as model scaf-folds for investigating the mechanical stimulation of mecha-nosensitive cells such as mesenchymal stem cells in vitro or

in vivo. The feasibility of actively moving scaffolds was dem-onstrated using model scaffolds prepared from radio-opaquecopolyetherurethane composites (PEUC) with a broad mixedglass transition in the range from 20 to 90 �C, where origi-nally square-shaped pores were temporarily fixed in anexpanded circular shape at different Tdeform.

80 It could bedemonstrated that the kinetics of the shape change obtainedunder physiological conditions could be adjusted by variationof Tdeform in the range from 40 �C to 60 �C between 1 and 6h. In further work, it was investigated how the variation ofphysical parameters applied during programing such as Tde-form and em influence the recovery behavior of this copolye-therurethane (PEU).54 A theoretical model was developed,which was able to describe the different shape recoverykinetics observed for radiopaque PEU composites based 3Dsubstrates under isothermal conditions (37 �C in water),which could be adjusted by variation of the deformationtemperature (Tdeform) applied during the programing step.80

Systematic stress relaxation experiments were carried outwith a tensile tester at seven different deformation tempera-tures (Tdeform ¼ 0, 25, 37, 50, 60, 70, and 80 �C) for strainvalues em ¼ 100%, 150%, 200%, or 250%. Figure 12(a) dis-plays the typical stress relaxation curves for the largestapplied strain of em ¼ 250% at four Tdeform. Figure 12(b)shows the stress r0 at the beginning of the relaxation pro-cess for different strain values em..

At a certain Tdeform value, the initial stress increases with strainem, and at constant elongation, the initial stress becomes drasti-cally smaller with increasing Tdeform. The applied model con-sisted of one spring and two Maxwell units in parallel. It couldbe understood as simplification of the model shown in Figure3 without thermal expansion component and only two Maxwellbranches. The fits of the relaxation curves for four elongationsat Tdeform ¼ 25 �C with the applied model are shown in theupper graph of Figure 12(b). The model describes well the

FIGURE 11 Temperature-memory properties of a polymer network having a broad melting transition temperature. (a) Strain-tem-

perature recovery curves for the same polymer, which was stretched at different Tprog (solid line: 0 �C; dashed line: 25 �C; dotted

line: 50 �C; dash-dotted line: 75 �C; dash-double dotted line: 100 �C). (b) Relation of the response temperatures Tsw and Tr,max on

Tdeform. Filled squares: Tsw determined in five subsequent cycles with increasing Tdeform ¼ 0, 25, 50, 75, 100 �C; open squares: five

subsequent cycles with decreasing Tdeform ¼ 100, 75, 50, 25, 0 �C. Filled circles Tr,max determined in five subsequent cycles with

increasing Tdeform ¼ 0, 25, 50, 75, 100 �C; open circles: five subsequent cycles with decreasing Tdeform ¼ 100, 75, 50, 25, 0 �C (em ¼75%, Tlow ¼ �10 �C, and Thigh ¼ 110 �C). The dashed trend line represents Tsw or Tr,max ¼ Tdeform (Taken from Ref. 22, VC Wiley-

VCH Verlag GmbH & Co. KGaA, Weinheim, reproduced with permission).


POLYMER SCIENCE


experimental curves. The viscous element in the ‘‘fast’’ Maxwellunit has a relaxation time in the range of 102 s, the ‘‘slow’’ unitof about 103 s. At constant strain [see Fig. 12(b) below], the sit-uation is more complex. The initial strong drop of stress-valueschanges with temperature, but also the final values of therelaxed stress after long observation times depend not propor-tionally on Tdeform. The information from the stress relaxationexperiments of PEU was applied to model the isothermal re-covery of PEUC scaffolds at 37 �C. It was found that the ‘‘fast’’relaxation can be neglected, and the viscoelastic properties ofPEU can be described by the simple SLS model. The calculatedrecovery ratio according to this model agreed well with the ex-perimental data (Fig. 13). It can be seen that the model is qual-itatively capable to present the Tdeform influence on an isother-mal recovery calculation (fastest recovery if Tdeform is close toThigh ¼ 37 �C).

Modeling Extension Toward Real DevicesA current task is to combine the developed 1D deformationmodels with more sophisticated model approaches todescribe quantitatively experimental data. These future cal-culations may be realized by implementation of a thermome-chanical model into a FE analysis. This combination is cur-rently a trend in SMP modeling investigations, because itallows to model the shape-memory properties of real devi-ces. Examples are the already mentioned stent structure,46

or of a flat diamond-lattice–shaped specimen.60 An instruc-

tive example is the simulation of a series of shape-memorytorsion tests of a flat sheet prepared from an covalentlycrosslinked epoxy polymer network with Ttrans ¼ Tg.

66 Thetime–temperature dependency of the thermoelastic behaviorof the polymer network was tested by detailed DMTA meas-urements using dynamic frequency sweeps at 0.2% strain

FIGURE 12 Modeling of relaxation experiments for an actively moving scaffold. (a) Relaxation experiments for PEU. (a-a0) em ¼250% and Tdeform ¼ 25, 37, 60, and 80 �C. (a-b0) Stress at the beginning of the relaxation process for different em and Tdeform. (b)

Modeling of experimental stress relaxation curves for PEU with a simplified model of Figure 3 consisting out of a spring and two

Maxwell units. (b-a0) Tdeform ¼ 25 �C, em ¼ 100%, 150%, 200%, and 250%. (b-b0) At em ¼ 250% and Tdeform ¼ 25, 37, 60, and 80 �C

(Reprinted from Ref. 54, VC 2010, with permission from Elsevier).

FIGURE 13 Fit of experimental free recovery ratio Rapp values

at Thigh¼ 37 �C for PEU based scaffolds, which were pro-

gramed at different Tdeform-values (Reprinted from Ref. 54,

VC 2010, with permission from Elsevier).



for frequencies ranging from 0.01 to 63 Hz, and at tempera-tures between 40 �C and 60 �C. With the time–temperaturesuperposition principle, a storage modulus master curve wasconstructed, and respective parameters of a generalized Max-well model out of 12 branches were fitted to this curve. To-gether with time-temperature parameters determined withthe WLF equation and coefficients of linear thermal expan-sion for glassy and rubbery state from thermal mechanicalanalysis experiments, it was possible to predict the torsionshape memory response of the epoxy material in very goodagreement with the experimental data (Figure 14b, c). In aFE calculation, a sheet of 90 � 10 � 1.6 mm3 was presentedby a network of 50 � 10 � 4 ¼ 2000 of 8-node linear brickhybrid elements (Figure 14a).

The combined approach should help to design SMP deviceswith large deformation requirements, such as stents anddeployable structures, based on its prediction results of theshape memory response of polymers with varying composi-tion, structure, and geometry, and under varying thermome-chanical cycling conditions. The presented approach seemstransferrable to other shape memory polymers, which are

intrinsically viscoelastic materials with time–temperature-de-pendent properties.

A further example for combining a general 3D constitutivemodel on the micromechanical level with FE-simulation ofsimple devices is the work of Baghani et al.81 Consideringthe small strain regime, the total strain is additively decom-posed into four parts: (i) a SMP-related strain consisting outof glassy and rubbery parts, (ii) a strain contribution fromhard segments (similar to ref. 70), and contributions from ir-reversible (iii) and thermal strains (iv). The general modelwas verified using three different experimental datasets.First, the experimental strain and stress recovery curves ofthe epoxy-based polymer network used by Liu et al.42 weresimulated (11 material parameters), ignoring time-dependenteffects and the hard segment contribution. As second exam-ple, data of a SMP composite (hollow glass microballoonsdispersed in a epoxy-based SMP matrix) were used,82 whichwere analyzed before49 with the simplified two-phase con-cept.42 With 19 material parameters, the overall trend inshape recovery for both regimes (stress free and constantstrain) could be simulated in good agreement with the

FIGURE 14 Simulation of a series of shape-memory torsion tests of a flat sheet prepared from an epoxy network. (a) Finite ele-

ment (FE) simulation of the torsion shape recovery of the sample representative mesh when heated at 1 K min�1 a 360� deforma-

tion. (b) Model simulation results and experimental data of the shape recovery process for the epoxy under various heating rates.

(c) Model prediction of the isothermal shape recovery as a function of time at 42 �C (Reprinted from Ref. 66, VC 2012, with permis-

sion from Elsevier).


POLYMER SCIENCE


experimental data. Also the experiments of Volk et al.64 couldbe well represented using 15 material parameters. Based onthe verified micromechanical models with the material pa-rameters of the Liu experiment, but assuming a hard seg-ment volume fraction of 0.33, that it is possible to solveboundary value problems, for example, for a 3D beam and amedical stent. The beam had a length of 100 mm, 12.5 mmwidth, and 10 mm height and was under fully clampedboundary conditions at one end. For certain stress, strainrate, and holding times, a full thermomechanical test cyclewas calculated over 1000 min, including a stress-free recov-ery. Once FE calculations performed successfully for simplestructures as the beam, also more complicated devices canbe simulated, as was demonstrated for a medical SMP stent.

The pioneering studies for the FE-based simulation of thethermomechanical cycle of three-dimensional SMP struc-tures, including a realistic SMP stent, were carried out byReese et al.46 and Srivastava et al.60 By incorporation of thetwo-phase model of Liu et al.42 into a FE calculation, Reeseet al.46 were able to calculate the deformations of a realisticstent structure consisting of a SMP during a thermomechani-cal cycle. At the same time, Srivastava et al.60 conductedlarge strain compression experiments for a tert-butyl acry-late (tBA) based SMP network, calibrated with these data thematerial parameter of a constitutive model considering largedeformations, and used FE modeling to simulate the behav-ior of a cylindrical stent. The stent was made by photopoly-

merization of tBA with the crosslinking agent poly(ethyleneglycol) dimethacrylate with diamond shaped perforationswhen it is inserted in an artery. The stent was modeled as atube made from a nonlinear elastic material. Figure 15(a)shows the initial undeformed stent. Figure 15(b) shows thestent after radial compression at Thigh ¼ 60 �C, and Figure15(c) shows snapshots of the stent inside the artery duringshape-recovery at 22, 42, and 60 �C. The aforementionedexamples show the capability of FE calculations, once consti-tutive models of the specific SMP material have been estab-lished, to be a useful and appropriate tool for design, analy-sis, and optimization of 3D structures made of SMPs. Despitethe promising results regarding real device applications, acurrent disadvantage for FE simulations of 3D-structures isthe lack of 3D constitutive approaches. So far, only a few 3Dmodels are available, which are capable to assess largestrains.41,46,55,60,81

CONCLUDING REMARKS

SMEs of polymers are the result of the polymer network’sstructure/morphology in combination with the application ofspecific programing procedures (SMCPs). While in the past,the focus was mainly concentrated on the synthesis of newSMPs for variation of the shape-memory properties towardthe requirements of the desired application; recently, theinfluence of the variation of physical parameters duringSMCP or recovery on the SME of polymers is target of inten-sive research efforts. An impressive example for the influ-ence of programing parameters on the SME is the TME,where a polymer material is capable to memorize the defor-mation temperature, whereby the SME occurs upon reheat-ing above the deformation temperature, and in this way, thepolymers’ response temperature can be easily adjusted. Inthis context, the applied processing technologies for pro-graming of SMPs turned out as a versatile toolbox for adjust-ing the shape-memory properties of polymers by solelyphysical methods, which should strongly support the transla-tion of the scientific achievements into (industrial) applica-tions of SMPs as one and the same material can be adjustedto the requirements of different applications.

Along with the development of more complex programingprocedures for tailoring the materials’ shape-memory capa-bility at the same time highly sophisticated testing protocolssuch as cyclic thermomechanical tests have been developedallowing a comprehensive quantitative characterization ofthe polymers’ thermomechanical behavior with time duringprograming and shape recovery. The obtained completestress-temperature-strain datasets were the basis for the de-velopment of various qualitative and quantitative modelingapproaches for description and prediction of shape-memoryproperties. Also a substantial progress could be achieved inthe theoretical approaches for the description of experimen-tal thermomechanical data. While the first constitutive modelapproaches allowed only to describe some main features ofSMPs, the recently introduced advanced thermomechanicalconstitutive models are able to represent the materialsbehavior (e.g., the stress–temperature-strain development

FIGURE 15 Finite element (FE) simulation of a cylindrical stent

made from tBA/poly(ethylene glycol) dimethacrylate with dia-

mond-shaped perforations. For clarity, the mesh has been mir-

rored along relevant symmetry planes to show the full stent

and artery. (a) Undeformed original stent. (b) Deformed stent.

(c) Shape recovery of the stent inside the artery at different

temperatures (Reprinted from Ref. 60, VC 2010, with permission

from Elsevier).



with time) in a very accurate way and further implementa-tion of FE models allows the simulation of particular SMP-based devices (e.g., stents).

To achieve significant progress in this area, it will be neces-sary to close the currently existing gap between the complextheoretical model descriptions containing numerous parame-ters, which require an extensive characterization of the poly-mers, and real application-related issues. In this context, onthe one hand, pragmatic and simple validated models withfew parameters, for example, obtained by standard thermo-mechanical tests such as DMTA,36 have to be provided bytheorists, which can be applied by engineers for design anddevelopment of shape memory polymers, and on the otherhand, more sophisticated 3D constitutive models arerequired as basis for reasonable FE studies on SMP-baseddevices.

Further research activities on the theoretical approaches arenecessary, which investigate, for example, the influence ofholding times during programing as well as the relevantinfluence parameters in the real application such as con-straints from the surrounding environment or fluctuations intemperature or humidity. Finally, for the design of novelSMPs, it would be very helpful, if multiscale modelingapproaches can be developed, which combine the predictionof the molecular behavior (e.g., influence of small moleculessuch as additives or water) with the macroscopic directedmotion of a SMP-based device.

ACKNOWLEDGMENTS

The authors thank S. Benner and K. Schm€alzlin for her adminis-trative assistance. T. Sauter is grateful to the Berlin-Branden-burg School for Regenerative Therapies (DFG-GSC 203) for afellowship.

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