A finite deformation constitutive model for shape memory polymers based on Hencky strain

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  • Mechanics of Materials 73 (2014) 110Contents lists available at ScienceDirect

    Mechanics of Materials

    journal homepage: www.elsevier .com/locate /mechmatA finite deformation constitutive model for shape memorypolymers based on Hencky strainhttp://dx.doi.org/10.1016/j.mechmat.2013.11.0110167-6636/ 2014 Elsevier Ltd. All rights reserved.

    Corresponding author at: Department of Mechanical Engineering,Sharif University of Technology, Tehran, Iran. Tel.: +98 21 6616 5546; fax:+98 21 6600 0021.

    E-mail address: naghdabd@sharif.edu (R. Naghdabadi).M. Baghani a, J. Arghavani b, R. Naghdabadi b,c,a School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, IranbDepartment of Mechanical Engineering, Sharif University of Technology, Tehran, Iranc Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran

    a r t i c l e i n f o a b s t r a c tArticle history:Received 12 March 2013Received in revised form 6 October 2013Available online 13 March 2014

    Keywords:Shape memory polymerFinite deformationHencky strain tensorContinuum thermodynamicsFinite elementIn many engineering applications, shape memory polymers (SMPs) usually undergo arbi-trary thermomechanical loadings at finite deformation. Thus, development of 3D constitu-tive models for SMPs within the finite deformation regime has attracted a great deal ofinterest. In this paper, based on the classical framework of thermodynamics of irreversibleprocesses, employing the logarithmic (or Hencky) strain as a more physical measure ofstrain, a 3D large-strain macromechanical model is presented. In the constitutive modeldevelopment, we adopt a multiplicative decomposition of the deformation gradient intoelastic and stored parts. In addition, employing the averaging scheme, the logarithmic elas-tic strain tensor is decomposed into the rubbery and glassy parts. The evolution equationsfor internal variables are introduced for both cooling and heating processes. The time-dis-crete form of the proposed model in the implicit form is also presented. Comparing the pre-dicted results with experimental data reported in the literature, the model is validated.Finally, using the finite element method, two boundary value problems e.g., a 3D beamand a medical stent made of SMPs are numerically simulated.

    2014 Elsevier Ltd. All rights reserved.1. Introduction

    Shape memory materials opened unexplored horizonsin the materials science and engineering, giving access tonumerous unconventional functions in different materialsclass: metals, ceramics and polymers (Schmidt et al.,2011) These materials are able to return from a temporaryshape to their permanent shape. The shape memory effect(SME) is typically induced by external stimuli such as heat,electricity or magnetism (Lendlein and Langer, 2002;Baghani et al., 2012a). This gives shape memory materialsan increasing potential for application in sensors,actuators, smart devices (e.g., medical stents) and mediarecorders (Reese et al., 2010). The mechanisms controllingthe thermally-induced SME in shape memory polymers arevery different from those operating in shape memory al-loys or ceramics, resulting in differences in the materialbehavior (Diani et al., 2006).

    Many attempts have been made to experimentallycharacterize the behavior of SMPs (see e.g., Liu et al.,2004, 2006; Atli et al., 2009; Kim et al., 2010; Volk et al.,2010b,a; Prima et al., 2010a,b,c among others), but a con-siderable part of the research has also been devoted tothe description of the SMPs performance through thedevelopment of constitutive models.

    Most SMPs constitutive models have been developedfor thermally-activated ones. They usually tend to followeither a phase transition approach (see Liu et al., 2006; Bar-ot et al., 2008; Chen and Lagoudas, 2008a; Qi et al., 2008;

    http://crossmark.crossref.org/dialog/?doi=10.1016/j.mechmat.2013.11.011&domain=pdfhttp://dx.doi.org/10.1016/j.mechmat.2013.11.011mailto:naghdabd@sharif.eduhttp://dx.doi.org/10.1016/j.mechmat.2013.11.011http://www.sciencedirect.com/science/journal/01676636http://www.elsevier.com/locate/mechmat

  • Fig. 1. Stressstrain-temperature diagram showing the thermomechan-ical behavior of a pre-tensioned SMP under different strain or stressrecovery conditions.

    2 M. Baghani et al. /Mechanics of Materials 73 (2014) 110Kim et al., 2010; Reese et al., 2010; Xu and Li, 2010;Baghani et al., 2012b among others) or a thermoviscoelas-tic approach (see Diani et al., 2006; Nguyen et al., 2008,2010; Westbrook et al., 2011; Ghosh and Srinivasa, 2011,2013 among others). The phase transition approach pos-sesses a distinct advantage in that the concepts can beapplied to an extensive variety of SMPs with different tran-sition mechanisms (Nguyen et al., 2010).

    Based on the phase transition concept, Liu et al. (2006)proposed a macroscopic constitutive model for SMPs in thesmall strain regime. In this model, the strain is decomposedinto thermal, elastic, and a storedpart. The stored strain isde-fined to capture the strain storage/release phenomenon.However, themodel does not present themathematical evo-lution equations upon the heating process (Chen and Lagou-das, 2008a,b). Based on this model, Chen and Lagoudas(2008a,b) proposed a constitutivemodel to capture the char-acteristics of SMPs behavior in large strain loadings. Ghoshand Srinivasa (2011) proposed a one-dimensional constitu-tivemodel for SMPs in the range of small strains. Theirmodelwas able to simulate the response of the material duringheating and cooling cycles and the sensitive dependence ofthe response on thermal expansion. Recently, based on theframework of continuum thermodynamics, Baghani et al.(2012c) have proposed a phenomenological model, in whichevolution laws have been introduced for stored strain notonly in the coolingprocess but also in theheating process un-der 3D loadings. In anotherwork, Baghani et al. (2012b) haveextended the model to the time-dependent regime. Theyhave also added a hard segment in the constitutive modelto predict the behavior of the SMP-based foams.

    In recent years, much attention has been devoted to thedevelopment of constitutive models for large deformationsof solids. The logarithmic strain (natural or Hencky strain),introduced by Hencky (1928), is a favored measure ofstrain due to its remarkable properties at finite deforma-tions in solid mechanics. An important feature is thatamong all finite strain measures, only the spherical andthe deviatoric parts of the Hencky strain can additivelyseparate the volumetric and isochoric deformations fromthe total deformation (Xiao et al., 2004).

    Furthermore, the Hencky strain has certain intrinsicproperties that shows its important position among allstrain measures. For instance, the Eulerian logarithmicstrain is the unique strainmeasure that its corotational rate(associatedwith the so-called logarithmic spin) is the strainrate tensor (Xiao et al., 2006). In other words, the strain ratetensor, d, is the corotational rate of the Hencky strain tensorassociated with the logarithmic spin tensor. This result hasbeen presented by Reinhardt and Dubey (1995) as D-rateand by Xiao et al. (1997) as log-rate. The work-conjugatepair of Eulerian Hencky strain and Cauchy stress are ofmostinterest in constitutivemodeling in large strain regime (Xiaoet al., 2006). Due to these remarkable properties, Henckystrain has been used in constitutive modeling of solids bymany researchers (see, e.g., Naghdabadi et al., 2005; Linand Schomburg, 2003). There are still more motivations todevelop a Hencky-strain based constitutive model. Forexample, for a Hookean-type constitutive relation, onlythe Hencky-based one is useful at moderately large elasticstretches (Anand, 1979, 1986).Large strain constitutive models are normally devel-oped based on appropriate small strain ones (Arghavaniet al., 2010, 2011). The recently proposed 3D model byBaghani et al. (2012c), based on the classical frameworkof continuum thermodynamics, is a good candidate to beextended to large strains. This model is interesting for itscapability to capture SME, beside its thermodynamicalconsistency. Starting from this basis, the development ofa Hencky based finite deformation model, which is alsoable to take into account large deformation effects, is themain goal of the present work.

    The current paper is presented in the following sections.Section 2 describes the behavior of an SMP in a full cyclefor different (strain or stress) recovery processes. Section3 presents the kinematic formulations. In Section 4, wedevelop a thermodynamically-consistent constitutivemodel for SMPs in the large deformation regime. Time-dis-crete form of the introduced model is presented within thefinite element framework in Section 7. We then, in Section6, apply the proposed constitutive model to some exam-ples and investigate the ability of the model to reproducethe features of SMPs. Using the proposed finite elementmodel, two boundary value problems, i.e., a 3D beam anda medical stent are simulated. Finally, we present asummary and draw conclusions in Section 7.

    2. Shape memory effect in a thermomechanical SMPcycle

    In this section, we describe SME in a stressstrain-tem-perature diagram as depicted in Fig. 1. The thermome-chanical cycle starts at a strain- and stress-free state ata high temperature Th (point a , permanent shape). Atthis state, SMP undergoes a mechanical loading andexhibits a rubbery behavior up to point b . At point b ,strain is held fixed and the temperature is decreased untilthe rubbery polymer gradually converts to a glassy poly-mer at a low temperature Tl (point c , temporary shape).In fact, around the glassy temperature Tg , SMP possesses acombination of rubbery and glassy behaviors. In this step,the material is unloaded. According to the very high stiff-ness of the glassy phase compared to the rubbery phase,after unloading, strains vary slightly (point d). Finally,the temperature is increased up to Th. It is observed that

  • M. Baghani et al. /Mechanics of Materials 73 (2014) 110 3the strain relaxes and the permanent shape is recovered(point a ). This cycle is called a stress-free strain recoveryin SMP applications. Commonly, other kinds of recoveriesmay occur. If at point d, the strain is kept fixed and thetemperature is increased, the fixed-strain stress recovery(point e ) happens (shown in Fig. 1 with a dotted-line).3. Kinematics

    In accordance with the polar decomposition theorem,the deformation gradient F with J detF > 0 may bedecomposed as:

    F RU VR 1

    where U and V are symmetric positive definite right andleft stretch tensors, respectively. Also, R is a proper orthog-onal tensor. Moreover, the right and left Cauchy-Greendeformation tensors are, respectively, defined as:

    C FTF U2; b FFT V2 2

    The material (Lagrangian) and spatial (Eulerian) Henckystrain tensors H and h are defined, respectively, as Lubarda(2001):

    H logU 12logC; h logV 1

    2logb 3

    The velocity gradient tensor l is also defined in the follow-ing form:

    l _FF1 4

    The strain rate tensor d and the vorticity tensorw are sym-metric and skew symmetric parts of l, respectively, asfollow1:

    d syml; w skewl 5

    Taking the time derivative of relation C FTF yields:

    _C 2FTdF 6

    Corotational rate of an Eulerian tensor A associated withthe rotating frame having spin X is defined as Naghdabadiet al. (2012):

    A _A XA AX 7

    where _A is the material time derivative of A in the fixedframe and A

    is the corotational rate of A. Double contract-

    ing (7) with an arbitrary second-order symmetric tensor B(we also assume A to be symmetric) and after some math-ematical manipulations, we obtain2:

    A: B _A : B 2X : AB 8

    If A and B are coaxial3 (and symmetric), AB is also sym-metric. Thus, we arrive at the following identity for any arbi-trary spin tensor X:1 symA 1=2A AT and skewA 1=2A AT compute thesymmetric and skew symmetric parts of an arbitrary tensor A, respectively.

    2 Double contraction is represented by :, defined as A : B AijBij .3 Tensors A and B are coaxial if AB BA._A : B A: B 9

    The corotational rate of the Eulerian Hencky strain h asso-ciated with the so-called logarithmic spin Xlog is the sameas the strain rate tensor d, and h is the only strain with thisproperty. In view of (7), we obtain (Naghdabadi et al.,2012; Teeriaho, 2013):

    hlog _h Xlogh hXlog d 10

    Finally, the second PiolaKirchhoff stress tensor S isobtained from the Cauchy stress as:

    S JF1rFT 114. Constitutive model development

    Employing a multiplicative decomposition, we first sep-arate the thermal expansion effects. The total thermome-chanical deformation gradient, F, is defined as F FmFthwhere Fm is the mechanical part and Fth represents thethermal deformation gradient and is defined as:

    Fth Fth1 1Z TTh

    aeff dT

    !1 12

    where aeff denotes the effective thermal expansion coeffi-cient. The second-order identity tensor is also representedby 1.

    Considering Eq. (12), total velocity gradient tensor isobtained by d dm dth.

    In this study, we separate the material domain intoglassy and rubbery phases as previously performed in sim-ilar works (e.g., Liu et al., 2006; Baghani et al., 2012c,b).Following a well-established approach adopted in plastic-ity (Lubarda, 2001; Haupt, 2002), we assume a local multi-plicative decomposition of the mechanical deformationgradient in the rubbery phase, Fr as well as the glassyphase, Fg , into elastic parts Fer and Feg , defined with respectto intermediate configurations, and inelastic stored part Fs,defined with respect to the reference configuration.Accordingly:

    Fm FeFs; Fr FerFs; Fg FegFs 13

    Assuming finite deformations, we employ the averagingscheme (on the current volume) and decompose the elasticHencky strain tensor, he, into two parts: elastic Henckystrain tensor in the rubbery phase, her , and elastic Henckystrain tensor in the glassy phase, heg .

    he urher ugheg ;ur ug 10 6 ur ;ug 6 1

    14

    where superscripts r and g stand for the rubbery and glassyphases, respectively. Volume fractions of the rubbery andglassy phases are represented by ur and ug , respectively.Such an averaging scheme has previously been adoptedfor constitutive modeling of multi-phase materials for dif-ferent quantities (for example, for F by Chen and Lagoudas,2008a,b; Sengupta and Papadopoulos, 2011, for d by Ahziet al., 2003; Makradi et al., 2005; Wu et al., 2005, for E

  • 4 M. Baghani et al. /Mechanics of Materials 73 (2014) 110(Green strain) by Baghani et al., 2013 or for stretch, k, byKim et al., 2010).

    In view of (13)1, the strain rate tensor could be rewrit-ten as:dm sym _FmFm1

    de sym FelsFe1

    15

    where ls _FsFs1. Taking derivative of (14), we obtain:_he ur _her ug _heg _ug heg her

    16

    To satisfy the principle of material objectivity, the Helm-holtz free energies, in the both phases, should depend onFer and Feg only via the elastic Hencky strain tensors her

    and heg , respectively:

    W Fm; T W her;heg ; T

    17

    Using the averaging scheme, the Helmholtz free energyfunction is introduced as:

    W her;heg ;ug ; T

    urWr her

    ugWg heg

    WT T 18

    where Wr and Wg are hyperelastic strain energy functionsof the rubbery and glassy phases, respectively, and WT isthe thermal energy. It is also assumed that Wr and Wg areisotropic functions of her and heg , respectively.

    s : d _W g _T

    0 19

    in which g represents the entropy. In this section it is as-sumed that the Kirchhoff stress, s, and elastic Henckystrain, he, are coaxial. Thus, we obtain:

    s : de s : _he 20

    Substituting (15), (20) and (16) in (19) gives:

    s : dth urdother ug _heg _ug heg her

    symFelsFe1 h i

    @W@her :

    _her @W@heg :

    _heg @W@T

    _T g _T

    P 0

    21

    Using relation (18), we have:

    ur s @Wr@her

    : _her ug s @Wg

    @heg

    : _heg

    s : dth _ug heg her

    symFelsFe1 h i

    @W@T

    _T g _T

    P 0

    22

    Following standard arguments, we finally may write:

    s @Wr

    @her @W

    g

    @heg23

    From (23), it is observed that Kirchhoff stress and elasticstrains her and heg are coaxial. Thus, as the storage/releasemechanism is considered fully thermal-driven, entropy, g,is easily obtained.

    Motivated from the small strain model (Baghani et al.,2012b,c), it is assumed that the Hencky stored strain fol-lows the evolution Eq. (24), in the intermediate configura-tion, during the cooling and heating processes:hsD

    ds _ugher; _T < 0_ugug h

    s; _T > 0

    0; _T 0

    8>>>: 24

    where hs log Vs and her log Uer . The appropriateobjective corotational rate is denoted by D, also meansthat the quantity is defined with respect to the intermedi-ate configuration. In fact, in the cooling process, evolutionEq. (24)1 allows the stored strain to take the value of theelastic strain in the rubbery phase, her . On the other hand,during the heating process, relation (24)2 makes it possiblefor the stored strain to be recovered in a trend similar toug . It is noted that we employ a prescribed evolution equa-tion for ug . This equation is derived using the uncon-strained strain recovery of the material as a function oftemperature. The evolution Eq. (24) can be transferred tothe initial configuration in the following form:

    _Cs 2FsTdsFs _ug log CrCs1

    Cs; _T < 0

    _ugug C

    s log Cs

    ; _T > 0

    8 tn. We denotewith the subscript n a quantity evaluated at time tn, andwith no subscript a quantity evaluated at time tn1. Wealso show the increment of time by Dt. Having the solutionas well as Cn at time tn and C at time tn1, the stress and theinternal variables are updated from the deformationhistory.

    Discretization of Eqs. (25)1 and (30), during the coolingprocess yields:

    Cs Csn Dug log CrCs1

    Cs

    log FmCs1FmT

    B : log FerFerT

    8>: 31where Dug ug ugn. Also, it is assumed that the rotation ineach phase is equal to the total rotation, Rr Rg R. It couldbe shown that log RUmCs1UmRT

    R log UmCs1Um

    RT

    and also log RUrCs1UrRT

    R log UrCs1Ur

    RT . Further-more, for any fourth order isotropic tensor such as Bwe haveBijkl RpiRqjRrkRslBpqrs, in which R is an orthogonal arbitrarytensor (de Souza Neto et al., 2011). As a result, Eq. (31) canbe recast in the following form:

    Cs Csn DugUs log Us1CrUs1

    Us

    log UmCs1Um

    B : log UrCs1Ur

    8>: 32On the other hand, discretization of Eqs. (24)2 and (32)2during the heating process yields:0.8 0.85 0.9 0.95 11

    0

    1

    2

    3

    4

    T* [ K/K ]

    Stre

    ss [

    MPa

    ]

    0 = 9.1%

    0 = 9.1%

    0 = 0%

    Stre

    ss [

    MPa

    ]

    Fig. 2. Left: Stresstemperature response in the cooling process in different pre-denote large strain model, small strain model and experimental data, respectdifferent pre-strains e0s of 9:1%; 0% and 9:1% in fixed-strain stress recovery. ACs Csn Dugug C

    s log Cs

    log UmCs1Um

    B : log UrCs1Ur

    8>>>>>>>>>>>>:

    A:1

    where dij is the Kronecker delta function. Eliminating Ur in

    Eq. (A.1), we arrive at a nonlinear equation in terms of Us

    which can be solved employing the NewtonRaphsonmethod as:

    R R : dUs 0 A:2

    in which:

    Rijpq @Rij@Uspq

    @L1ij@Uspq

    Bijkl@L2kl@Uspq

    L1ijpq BijklL2klpq A:3

  • M. Baghani et al. /Mechanics of Materials 73 (2014) 110 9L1ijpq @L1ij@Uspq

    @L1ij@Yrs

    @Yrs@Uspq

    L3ijrsL4rspq;L3ijrs @L1ij@Yrs

    A:4

    L4rspq @Yrs@Uspq

    UrtUsm@ Us2

    tm

    @Uspq

    0@

    1A

    UrtUsmL6tmpq;L6tmpq @ Us2

    tm

    @UspqA:5

    L2klpq @L2kl@Uspq

    @L2kl

    @Zvw

    @Zvw@Uspq

    L11klvwL12vwpq;L11klvw @L2kl@Zvw

    A:6

    L12rspq L13rxpq Us2

    xmUrms U

    rrxL

    6xmpqU

    rms U

    rrx U

    s2

    xmL13mspq

    h iA:7

    L13txpq @Urtx@Uspq

    @Urtx

    @Xij

    @Xij@Uspq

    L14txij IiwpqMwrUsrj U

    siwMwrIrjpq U

    siwL

    15wrpqU

    srj

    A:8

    L14txij @Urtx@Xij

    ; L15wrpq @Mwr@Uspq

    @Mwr@Nxy

    @Nxy@Uspq

    L16wrxyL17xypq;L16wrxy @Mwr@Nxy

    A:9

    L17xypq @Nxy@Uspq

    Csn;wm

    DuL6xwpqU

    s1

    my Us1

    xw L6mypq

    A:10

    where I is the fourth order symmetric identity tensor.Now, in view of (11), calculating Cs, the second PiolaKir-chhoff stress is identified as:

    S F1sFT F1 Kr : G : 12 log Ve

    FT

    U1R1 Kr : G : 12 log RUCs1URT

    RU1

    12U1

    Kr : G : log UCs1U

    U1A:11

    In order to construct the consistent tangent matrix in thecooling process, we may write:

    D 2 dSdC

    2 @S@C

    2 @S@Us

    :@Us

    @C 2X 2T : Y A:12

    Taking derivative from (A.1), we obtain:

    dR Us;C @R@C

    : dC @R@Us

    : dUs 0 A:13

    Recasting (A.13) into a matrix form, we obtain:

    dUs R 1 Q dC A:14

    We also have Q @R@C. Now, using (A.14), we can compute

    the matrix [Y] (the matrix form of the fourth-order tensorY) such that:

    dUs Y dC dUs Y : dC A:15

    The indicial form of fourth order tensorsX and T are calcu-lated as follows:Xijxy @Sij@Cxy

    L1rsFmkrs Iimpq U

    1

    kj U1

    imIkjpq

    U1

    imFmkrs

    @L1rs@Upq

    U1

    kj

    8>>>:

    9>>=>>;

    @Upq@Cxy

    A:16

    Tijkl @Sij@Uskl

    U1

    imFmprs U

    1

    pj

    @L1rs@Yuv

    @Yuv@Uskl

    U1

    imFmprs U

    1

    pjL3rsuvL

    4uvkl A:17

    in which:

    @L1rs@Upq

    @L1rs

    @Yuv

    @Yuv@Upq

    L3rsuv Cs1

    rt

    IurpqUtv UurItvpq

    A:18

    We now construct the spatial tangent matrix using the fol-lowing relation (Wriggers, 2008):

    Cmnkl 1JFmMFnNFkKFlLDMNKL A:19

    where C is the fourth-order spatial consistent tangenttensor.

    Time integration scheme for heating process: In a similarway, Eq. (33) can be solved to find Cs. Substituting Cs from(33)1, in light of (A.11), the stress is also obtained.

    In order to construct the consistent tangent matrix inthe heating process, we may write:

    D 2 dSdC

    2X A:20

    where X is previously defined in (A.16).

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    A finite deformation constitutive model for shape memory polymers based on Hencky strain1 Introduction2 Shape memory effect in a thermomechanical SMP cycle3 Kinematics4 Constitutive model development5 Time-discrete form of the SMP constitutive model6 Numerical results and model verification6.1 Example 1: Reproduction of stress-free strain recovery6.1.1 Example 2: Simple shear test in stress-free strain recovery process6.1.2 Example 3: double-clamped 3D SMP beam under distributed pressure: SME test6.1.3 Example 4: medical SMP stent

    7 Summary & conclusionsAcknowledgmentAppendix A Time-discrete form of the proposed modelReferences

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