Finite Strain 3D Thermoviscoelastic Constitutive Modelfor Shape Memory Polymers

Julie DianiLaboratoire d’Ingenierie des Materiaux, UMR 8006 CNRS, ENSAM Paris, 151 bd de l’hopital,75013 Paris, France

Yiping LiuDepartment of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309–427

Ken GallSchool of Materials Science and Engineering and George Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, Georgia 30332

A 3D thermoviscoelastic model is proposed to representthe thermomechanical behavior of shape memory poly-mers. The model is based on a physical understanding ofthe material behavior and a mechanical interpretation ofthe stress–strain–temperature changes observed duringthermomechanical loading. The model is thermodynam-ically motivated and is formulated in a finite strain frame-work in order to account for large strain deformations.Model predictions capture critical features of shapememory polymer deformation and, in some cases, pro-vide very favorable comparisons with experimental re-sults. POLYM. ENG. SCI., 46:486–492, 2006. © 2006 Society ofPlastics Engineers

INTRODUCTION

Shape memory polymers (SMPs) have the capability toretain a temporary shape at low temperature that was cre-ated during high temperature deformation. When the mate-rial in the temporary shape is heated above a critical tem-perature, the material recovers its original shape whenunconstrained. The “shape storage” and “shape recovery”properties of the polymer are due to a reversible change ofstate from rubbery to glassy and from glassy to rubbery,respectively. Consequently, the change of state in mostSMPs is linked to the glass transition temperature of thematerial. The mechanisms controlling the thermally inducedshape memory effect in polymers are very different from

those operating in shape memory metals or ceramics, re-sulting in differences in observed behaviors. For example,in most SMPs, it is not possible to cycle between thetemporary and permanent shape under pure thermal loading,an effect called two-way shape memory in alloy systems.Once the polymer has recovered its original shape underheating, it is necessary to reapply thermomechanical con-straint to rememorize the temporary shape. SMPs have apredominantly amorphous structure with either physical orchemical cross-links, resulting in a network structure. In therubbery state, above the glass transition temperature, Tg, thecross-linked SMP networks can undergo very large elasticdeformations, similar to elastomers, which involve changesin entropy. Below Tg, SMP networks deform to moderatestrains and the stress–strain response is driven by changes ininternal energy.

The majority of research on SMPs has focused on ex-perimental observations, physical understanding, andemerging applications [1–10]. Few studies have addressedthe constitutive modeling of the thermomechanical shapememory cycle in polymers. Essentially, two approacheshave been used to describe the thermomechanical behaviorof SMPs. The first approach is based on micromechanicsmodeling of the material during the change of state. Thematerial is assumed to be “soft” at high temperature (aboveTg) and “frozen” at low temperature (below Tg). During theglass transition, a fraction of the material is in glassy statewhile the remainder of the material is in rubbery state and arule of mixtures is applied as a micromechanical constraint[11]. The second approach is based on well-known standardlinear viscoelastic approaches commonly used to predict thethermomechanical properties of polymers [12, 13].

Despite the fact that one of the main advantages of SMPs

Correspondence to: J. Diani; e-mail: julie.diani@paris.ensam.frContract grant sponsor: DGA; contract grant number: ERE –046000011.DOI 10.1002/pen.20497Published online in Wiley InterScience (www.interscience.wiley.com).© 2006 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—2006

is their large strain deformability, current constitutive mod-els have been developed only in the context of infinitesimalstrains [11–14]. To account for large strain deformations,we develop a 3D thermoviscoelastic constitutive modelformulated in finite strains. This model is based on theviscoelastic properties of crosslinked SMP networks and isthermodynamically motivated.

In the next section, we will summarize the thermome-chanical response of SMPs using the data of Liu et al. [11].In presenting the data, emphasis is placed on physical in-terpretation of the energy transfer and stress evolution dur-ing a shape memory thermomechanical cycle. In the fol-lowing section, 3D, finite-strain, and thermodynamicallymotivated constitutive model is proposed. This section alsoprovides model predictions of the stress and strain changeduring a classic thermomechanical cycle along with com-parison to experimental data from Liu et al. [11]. The articleends with brief conclusions.

THERMOMECHANICAL BEHAVIOR OF SMPS

A heavily crosslinked epoxy network is considered in theexperimental studies of Liu et al. [11]. The epoxy materialdemonstrates a rubbery plateau above Tg on a plot of mod-ulus versus temperature, a requirement of a polymer todemonstrate a useful shape memory effect. The elastic prop-erties of the epoxy have been characterized above andbelow the glass transition temperature, and the materialbehaves as an isotropic material. The glassy elastic modulusis �750 MPa, while the rubbery modulus is on the order of8.8 MPa at Th � 358 K. The thermal expansion coefficientof the polymer was measured to be 9 � 10�5 K�1 below Tg

and 1.8 � 10�4 K�1 above Tg.The shape memory epoxy has been submitted to a 5 step

thermomechanical loading cycle outlined below:

● Step 1: Temperature is raised above Tg.● Step 2: The material is subjected to a set “pre-strain”.● Step 3: Temperature is decreased below Tg (at a cooling

rate of 1 K � min�1) while the prestrain is held constant.● Step 4: Stresses are released by removing the prestrain

constraint.● Step 5: Temperature is raised above Tg (at a heating rate

of 1 K � min�1) while strain is held constant (the samplesize is not allowed to change except for the release ofstresses that occurred in Step 4).

Three different prestrain conditions were considered in Step2: 9.1% uniaxial extension, 9.1% uniaxial compression, orno strain. As above Tg, the material is in the rubbery state,it can undergo very large elastic deformations with resultinglow stresses. Although the applied experimental strains aresmall, they are sufficient for formulation of the constitutivemodel, which will then be capable of predicting geometri-cally nonlinear large strain deformations. Nonlinear defor-mations resulting from material nonlinearity (e.g. chainalignment) would need to be added to the present frame-

work if they were observed within the imposed deformationlimits. Often, SMPs are only deformed within the limits ofgeometric nonlinearity and not material nonlinearity, sincethe latter often leads to material damage.

During Step 3, the stress changes were monitored as afunction of temperature, as shown in Fig. 1. One observesthat the thermal expansion does not generate appreciablestress in the direction of stretching as long as the material isin the rubbery state. The volume change of the polymer isaccommodated by expansion in the transverse directionsand variation of the stress due to the thermal contraction inthe direction of tension is small due to the low value of theYoung’s modulus above Tg. When the material is in theglassy state, the thermal contraction during cooling inducesa positive stress, which increases linearly with the temper-ature. The comparison between the three applied loadingstates in Fig. 1 is critical to the formulation of a constitutivemodel, and has not been previously discussed by Liu et al.[11]. In particular, the stress difference imparted above Tg,at the beginning of the cooling in Step 3, is identical to thestress offset below Tg at the end of the cooling. The unifor-mity of the stress offset during strain storage (Step 3) iscrucial for defining stresses and stress evolution in themodel. During cooling and material “freezing”, the stressesthat come from the prestrain at high temperature, or equiv-alently from an entropy change, do not vanish because ofthe material change of state but remain intact. This indicatesthat contributions to the stress due to entropy change in therubbery state remain while the material is transforming to aglassy state. This result is consistent with the physics of thematerial and must be incorporated into modeling efforts. Atthe molecular level, changes in entropy are caused by al-terations of chain conformation. At high temperature, thefree volume is large enough to promote conformationalchain motions. While temperature decreases, cooperativechain motion become impractical and the chain conforma-tions do not change unless the polymer is deformed pastyield at a temperature below Tg. The latter situation isbeyond the range of application of SMPs, since they are

FIG. 1. Stress evolution vs. temperature during cooling test at fixedstrain.

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typically not submitted to large deformation below Tg.Therefore, while a prestrain is applied above Tg and tem-perature is subsequently decreased below Tg, the chainconformation remains the one imposed by the prestrain andso the entropy contribution is converted to internal energystored in the polymer.

At the end of unloading in Step 4, all three samples havesustained a slight compressive strain relative to the valuethey were deformed to during Step 2. The sample previouslystretched to 9.1% strain was unloaded to a strain of 8.6%.Final strains of �0.4% and of �9.4% are observed afterunloading in Step 4 for the samples that have been origi-nally submitted to no applied strain, and a �9.1% compres-sive strain, respectively. If these contractions were exclu-sively due to the thermal strains in all three samples, theywould be identical. On the contrary, we note that the con-tractions are 0.3% for the sample initially in deformed incompression, 0.4% for the sample that was initially unde-formed, and 0.5% for the sample initially deformed intension. Another source that contributes to the compressivestrains during unloading is the material accommodation tothe stresses resulting from the entropy change at high tem-perature. For the undeformed sample, no stress comes fromthe entropy change; therefore, the value of 0.4% is due tothe thermal contraction only. For the sample that was ini-tially deformed in tension, the positive stress due to theentropy change has to be accommodated by a contraction ofthe material in the glassy state, which explains a highercontraction for the sample initially in tension compared tothe undeformed sample. On the other hand, for the sampleinitially deformed in compression, the material has to ac-commodate a compressive stress and thus expands. Thisexplains the contraction of 0.3% only in the compressedsamples.

In Step 5, stresses have been measured at a constantstrain (sample size is fixed after unloading in Step 4) as afunction of heating and are plotted in Fig. 2. At the begin-ning of the individual thermal cycles, the materials are in theglassy state and the restricted thermal expansion creates

compressive stresses. Then, with increasing temperature,the glass transition occurs. Thermal contributions to thestresses vanish while the values of stress tend toward theexpected responses of a material in the rubbery state. Thetemperatures at which the stresses start to increase dependon the state of stored strain in the material (tensile versuscompressive versus no stored strain). This indicates that theglass transition temperature is modified with the storedprestrain of the specimen. The specimen initially deformedin tension has a Tg lower than the initially undeformedspecimen, which in turn has a lower Tg than the sampleinitially deformed in compression. This phenomenon issimilar to the phenomenon observed in physical aging ex-periments [15]. Actually, the glass transition temperature isknown to increase with compaction of chains. Also, itappears in Fig. 2 that the thermal stress at the beginning ofthe cooling process depends on the specimens’ state ofstrains. As regards physical aging, an increase of the expan-sion coefficient, caused by the compaction of the chains, hasbeen observed (see discussion in van der Linde et al. [16].The higher thermal stress measured in the specimen initiallydeformed in compression is likely due to a higher expansioncoefficient caused by chain compaction.

As a first approximation, the changes in the glass tran-sition temperature Tg and in the thermal expansion coeffi-cient will be neglected in the course of the modeling. Thisimplies that the model will generate the same response forthe three specimens during the heating process from the lowtemperature to Tg, at which temperature, different stresswould be obtained according to the specimen state of strain.For this reason, only one complete cycle will be provided.More accuracy can be readily added to the proposed frame-work as more quantitative data is gathered at larger strainlevels and the effects of strain state on glass transitiontemperature are even more significant. Finally, one can notethat above Tg, the stress values resulting from the recoveryof the sample are consistent with the initial stresses im-parted above Tg caused exclusively by changes in entropy.This implies that the stored strain and energy can cyclebetween entropic and enthalpic states without being quan-titatively disturbed by the glass transition process.

MODELING

Viscoelastic Model

Above Tg, polymers exist in a rubbery state, and changesin energy are due almost exclusively to changes in entropy(here we are restricting ourselves to the case of isovolumeloading above Tg, since at constant temperature above Tg,volume change would result from internal energy change).Below Tg, in the glassy state, conformational chain motionsare restricted and the material primarily undergoes changesin internal energy. We will assume that in both rubbery andglassy states, strains are elastic and no plasticity occursthrough mechanisms such as chain slippage and crazing.

FIG. 2. Stress evolution vs. temperature during heating test at fixedstrain.

488 POLYMER ENGINEERING AND SCIENCE—2006 DOI 10.1002/pen

The experimental results demonstrate that during the cool-ing process, stresses caused by a prior change of entropyremain. This indicates that a change in entropy must betaken into account in the overall temperature range. It islikely that the entropy change below Tg will be very small(almost negligible) but nothing in the material behaviorindicates that contributions should theoretically withdrawbelow Tg. Therefore, the entropy change may be defined asa function of the total deformation for all temperatures.

On the other hand, changes in internal energy seem tooccur only when changes of entropy are impractical. Hence,changes in internal energy are assumed to be involved instress evolution only when the overall material is not in therubbery state. We can thus define a reference state for theinternal energy, as a state causing no contribution to thestress from the internal energy. This state is described by thestate of the material while passing the glass transition tem-perature.

During the change of state, which is recognized in arange of temperatures �Tg � �T,Tg � �T�, only part of thematerial is in the glassy state and contributes to the stress interms of a change of internal energy. The heterogeneity ofthe material can be simulated by a partial contribution of thetwo deformation mechanisms. On a rheological basis, this isperformed by decomposition of the deformation into a vis-cous part and an elastic part. Let us now formalize theseremarks as equations to propose a rheological model corre-sponding to the described behavior.

The Helmholtz free energy � is defined as

� � U � T� (1)

where U is the internal energy, T is the temperature, and �is the entropy. The entropy is defined as a function of thetotal deformation characterized by the deformation gradientF. For the internal energy contribution, the deformationgradient is assumed to split into a viscous part Fv and anelastic part Fe defined by

F � Fe � Fv (2)

and U is only a function of Fv. Therefore, the Helmholtzfree energy is written as

� � U�Fe � T��F. (3)

On a rheological basis, relations in Eqs. 2 and 3, definingthe Helmholtz free energy, are sketched by a Zener model,as shown in Fig. 3.

We will now detail the constitutive equations describingthe model and use the model to capture basic thermome-chanical shape memory cycles.

Constitutive Equations

During a thermomechanical load, the total deformation Fdecomposes into

F � Fm � Fth (4)

where Fm accounts for mechanical deformation and Fth

accounts for thermal deformation. Assuming isotropy, Fth isgiven by

Fth � ��T � T0I (5)

where � is the thermal expansion coefficient and T0 char-acterizes the reference temperature. Considering the rheo-logical scheme in Fig. 3, the elastic deformation in theentropy and the internal energy branches are defined by

Fe � F � Fth�1in the elastic branch (entropy) (6)

Fe � F � Fth�1Fv�1in the viscoelastic branch

(internal energy).

Also from scheme 3, the total Cauchy stress is given by

� � �n � �U (7)

where �� is the portion of the stress due to the entropychange and �U is the portion due to the internal energychange.

The entropy changes are based on the theory of rubberelasticity. Hence the entropy function may be defined by aneo-Hookean law [17]:

� �T �Er

6

T

Th�I1 � 3 (8)

where Il is the first invariant of the right Cauchy-Greentensor of the elastic deformation, C � FeT�Fe, and Er is the

FIG. 3. Model rheological scheme.

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Young’s modulus of the material at Th. Young’s modulus isassumed to depend linearly on temperature; therefore, stressdependence on temperature is taken into account for theentropy contribution. The contribution of the entropic en-ergy to hydrostatic deformation can be assumed as negligi-ble and the Cauchy stress derived from Eq. 8 is given by

�� �Er

3

T

ThB � p1 (9)

where B � Fe�FeT is the left Cauchy-Green tensor and p isa Lagrange multiplier related to the boundary conditions.

In a finite strain framework, the stress contribution fromthe internal energy may be defined by

�U � Le�ln�Ve� (10)

where Le is the fourth-order elastic constant tensor, whichcan be reasonably considered to be temperature indepen-dent. The left stretch tensor, Ve, is obtained from the polardecomposition Fe � VeRe where Re is a rotation tensor. Tocompute �U, it is necessary to know Fe, or equivalently F.We will now define the evolution of F in light of remarksand observations in made earlier and consistency with theClausius-Duhem inequality.

When the material is completely in the rubbery state, ithas been noted that contributions of internal energy changesto stress evolution are null, which implies

For T Tg � �T Fe � 1N Fv � F. (11)

While the material is entirely in the glassy state, appliedstrain is accommodated by a change of internal energy onlyat strains below the material yield strain. This is understoodin terms of F by the relation:

For T � Tg � �T Fv � 1. (12)

To define the evolution equations during the change of state,we consider thermodynamic requirements. The Helmholtzfree energy must satisfy the Clausius-Duhem inequality:

1

2S:C � �0�� � T� �

q0

TgradT 0 (13)

where S is the second Piola-Kirchoff stress tensor. Intro-ducing

� � �

TT �

�

CC �

�

Fv Fv (14)

and, after some straightforward steps (see Govindjee andReese [18], for example) Eq. 13 transforms into

1

2�S � 2 T�

C� 2Fv

U

CFv�T�:C � 2

U

Ce :�CeLv

�q0

Tgrad T 0. (15)

From which one can extract three relations:

(a) Stress–strain relation: S � 2 T�

C� 2Fv

U

CFv�T

(b) Mechanical dissipation: 2 U

Ce:�Ce � Lv 0

(c) Thermal dissipation: �q0

Tgrad T 0 (16)

where Lv � FvFv�1 is the velocity gradient corresponding toFv. For an isotropic material, Eq. 16b transforms into

�Ce � U

Ce�:Dv 0 (17)

where D is the symmetric part of L. For conditions in Eqs.11 and 12, Eq. 17 is trivially satisfied. For T�(Tg��T,Tg��T], we define changes in Fv to satisfyEq. 17, and

Dv �1

��Ce � U

Ce� (18)

is one simple solution. Parameter � is a parameter of vis-cosity.

The constitutive equations required to solve a thermo-mechanical cycle are given by Eqs. 6, 7, 9–12, and 18.Considering that the material elastic constants and thermalexpansion coefficients have been estimated above and be-low Tg. The only remaining free parameters are �T and �. Tg

and �T can be estimated in Figs. 1 and 2 and � will becalculated while fitting the material response.

Prediction of Experimental Data

The experimentally measured material properties arepresented in Table 1. The theoretical responses correspond-ing to the thermomechanical cycle described earlier arecompared to the experimental data.

First, Steps 2 and 3 have been estimated using parame-

TABLE 1. Material parameters.

Above Tg Below Tg

Young’s modulus (MPa) 8.8 750.0Expansion coefficient (K�1) 1.8 � 10�4 9.0 � 10�5

490 POLYMER ENGINEERING AND SCIENCE—2006 DOI 10.1002/pen

ters in Table 1, and noting that in Figs. 1 and 2, the glasstransition seems to take place between 313 and 339 K.Therefore, Tg and �T are set equal to 326 and 13 K respec-tively. Comparisons between the model and the experimentsare plotted in Fig. 4. A fairly good correlation is shown inFig. 4. The experimental results demonstrate a smoothertransition, caused by spatial variation in the glass transitionbehavior due to local heterogeneity in the epoxy networkstructure. The shape of the gradual transition could be bettercaptured with the present model by using an evolutionfunction for the transition region of another form than theone proposed in Eq. 18. Parameter � has been fitted fortemperature within the glass transition T��Tg � �T,Tg

� �T�. The cooling rate is of � � 1 K � min�1. It is not thestudy’s objective to evaluate the ability of the model tocapture the cooling rate effect, since the experimental data wasavailable only at one rate. Therefore, we have made the as-sumption that � was cooling rate independent by setting �� ���, � being constant and equal to � � 100 MPa �K�1.

Good agreement with experiments has been obtained interms of the strain contraction after stresses are released inStep 3. Comparisons between the theoretical results and theexperimental values are given in Table 2.

An in-depth comparison is now made for the sample thathas been submitted to a compressive prestrain. The stress isevaluated as a function of temperature change for the 5

loading steps described in the second section using theparameters mentioned here. The thermoviscoelastic modelprovides a reasonable prediction of the overall materialbehavior as shown in Fig. 5. Compaction effects on thermalexpansion and glass transition temperature are not presentlyaccounted for. For this reason, the same curve is predictedby the model for the three samples below Tg, which explainsthe single result presented for the heating process. Also, thepredicted shape of the stress evolution during heating andchange of state is different than experimental observations.This discrepancy may be corrected by proposing anotherform of evolution of F from Eq. 18. This last equation is ofa very simple form, which may be improved by betterknowledge of the evolution of the material between glassyand rubbery state.

The present constitutive model provides a useful frame-work for predicting the thermomechanical response ofSMPs at large strains. The constitutive model captures over-all trends in the complicated thermomechanical response ofSMPs (Fig. 5). The model is fit to basic material parametersderived from uniaxial stress–strain tests, free thermal ex-pansion tests, and glass transition data. Such data is readilyavailable for various polymers that show shape memory,and can be easily determined for emerging polymer sys-tems. The shape of the stress and strain evolution curvesduring the thermomechanical cycle is strongly dependent onthe evolution equations used in the model. As long asproposed evolution equations satisfy thermodynamic con-straints, more sophisticated mathematical equations willimprove point-to-point agreement between experimental re-sults and modeling predictions (Fig. 5). The choice ofspecific evolution equations within the modeling frameworkdepends strongly on the desired accuracy and intendedapplication of the constitutive model. Future work shouldemphasize comparison between modeling predictions andexperimental results for various applied temperature andstrain rates and for large strain deformations, variables thatare all relevant to emerging applications of SMPs.

FIG. 4. Theoretical model prediction of the experimental data during thecooling process at fixed strain.

TABLE 2. A comparison between the model predictions and theexperimental observations of contractions due to the stress release at lowtemperature.

Permanent strain after unloading

Experimentalvalue

Modelprediction

9.1% Prestrain (tension) 8.6% 8.55%0% Prestrain 0.4% 0.43%�9.1% Prestrain (compression) �9.4% �9.42%

FIG. 5. Comparison of the experimental data and model estimate of athermomechanical cycle in terms of stress vs. temperature.

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CONCLUSION

A 3D thermoviscoelastic model has been proposed in thecontext of finite strains to represent the thermomechanicalbehavior of SMPs. This model is based on thermodynamicconsiderations and has been motivated by a mechanicalunderstanding of stress–strain–temperature behaviors inSMPs. The model was formulated using the standard ther-moviscoelastic theory in finite strains and shows favorablepredictions of experimental data. In particular, the modelaccurately estimates the remaining strain during stress re-lease and provides a reasonable prediction of the stress–temperature curves during constrained thermomechanicalrecovery of SMPs.

REFERENCES

1. H. Tobushi, H. Hara, E. Yamada, and S. Hayashi, SmartMater. Struct., 5, 483 (1996).

2. C. Liang, C.A. Rogers, and E. Malafeew, J. Intell. Mater. Syst.Struct., 8, 380 (1997).

3. H.M. Jeong, S.Y. Lee, and B.K. Kim, J. Mater. Sci., 35, 1579(2000).

4. A. Lendlein, A.M. Schmidt, and R. Langer, Proc. Natl. Acad.Sci. USA, 98, 842 (2001).

5. A. Lendlein and S. Kelch, Angew. Chem. Int. Ed., 41, 2034(2002).

6. F. El Feninat, G. Laroche, M. Fiset, and D. Mantovani, Adv.Eng. Mater., 4, 91 (2002).

7. T. Ohki, Q.Q. Ni, N. Ohsako, and M. Iwamoto, Compos. PartA: Appl. Sci. Manuf., 35(9), 1065 (2004).

8. Y. Liu, K. Gall, M.L. Dunn, and P. McCluskey, Mech. Mater.,36, 929 (2004).

9. H. Tobushi, R. Matsui, S. Hayashi, and D. Shimada, SmartMater. Struct., 13, 881 (2004).

10. M. Bertmer, A. Buda, I. Blomenkamp-Hofges, S. Kelch, andA. Lendlein, Macromolecules, 38(9), 3793 (2005).

11. Y. Liu, K. Gall, M.L. Dunn, A.R. Greenberg, and J. Diani, Int.J. Plasticity, 22(2), 279 (2006).

12. H. Tobushi, T. Hashimoto, S. Hayashi, and E. Yamada, J. In-tell. Mater. Syst. Struct., 8, 711 (1997).

13. E.R. Abrahamson, M.S. Lake, N.A. Munshi, and K. Gall,J. Intell. Mater. Syst. Struct., 14, 623 (2003).

14. H. Tobushi, K. Okumura, S. Hayashi, and I. Norimitsu, Mech.Mater., 33, 545 (2001).

15. B. Fayolle and J. Verdu, Techniques de l’Ingenieur AM,3–150, 1 (2004).

16. R. Van der Linde, E.G. Belder, and D.Y. Perera, Prog. Org.Coat., 40, 215 (2000).

17. L.R.G. Treloar, The Physics of Rubber Elasticity, OxfordUniversity Press, England, 3rd ed. (1975).

18. S. Reese and S. Govindjee, Int. J. Solids Struct., 35, 3455(1998).

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