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Journal of the Mechanics and Physics of Solids 56 (2008) 1766–1778

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A constitutive theory for shape memory polymers. Part IIA linearized model for small deformations

Yi-Chao Chena,�, Dimitris C. Lagoudasb

aDepartment of Mechanical Engineering, University of Houston, Houston, Texas 77204, USAbAerospace Engineering Department, Texas A&M University, College Station, Texas 77843, USA

Received 2 April 2007; received in revised form 3 December 2007; accepted 13 December 2007

Abstract

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such

materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006.

Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modelling. Int. J. Plasticity 22,

279–313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the

underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for

nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which

describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is

captured by using the concept of frozen reference configuration. The relation between the overall deformation and the

stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear

constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a

shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory

by considering small deformations. The predictions of this model are compared with experimental measurements.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Shape memory polymers; Nonlinear constitutive theory; Large deformations; Neo-Hookean model; Linearization

1. Introduction

In Part I of this work, a three-dimensional constitutive theory for large deformations of shape memorypolymers (SMPs) is developed. The theory is built on the framework of nonlinear thermoelasticity, withsufficient structures to describe thermomechanical properties of the active and frozen phases of the material,as well as the corresponding phase transitions. Such transitions are mathematically formulated based on twobasic assumptions concerning the storage and release of deformations when the material passes through thetransition temperature: first, the deformation must be continuous during cooling, and second, the deformationstored in the material at freezing is instantaneously released when the material is heated to the glass transitiontemperature. These assumptions lead to the derivation of a mathematical model for the constitutive behavior

e front matter r 2008 Elsevier Ltd. All rights reserved.

ps.2007.12.004

ing author.

esses: chen@uh.edu (Y.-C. Chen), lagoudas@aeromail.tamu.edu (D.C. Lagoudas).

ARTICLE IN PRESSY.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–1778 1767

of SMPs under arbitrary temperature/loading paths. The main constitutive equation (Eq. (21) in Part I) soobtained reads

FðtÞ ¼ ½1� fðyðtÞÞ�FaðSðtÞ; yðtÞÞ þZ t

0

Ff ðSðtÞ; yðtÞÞF�1

f ðSðtÞ; yðtÞÞFaðSðtÞ; yðtÞÞf0ðyðtÞÞ~y

0ðtÞdt, (1)

where FðtÞ is the average deformation gradient, SðtÞ the Piola–Kirchhoff stress, yðtÞ the temperature, fðyÞ thevolume fraction of the frozen phase, FaðS; yÞ and Ff ðS; yÞ are, respectively, the constitutive functions of the

active and frozen phases, and ~yðtÞ is the net cooling history obtained by replacing each cooling/heating portionof the original temperature history yðtÞ with a constant temperature.

In this second part of the paper, we derive a linear constitutive model from Eq. (1) by considering smalldeformations. This derivation, presented in Section 2, is carried out by following the standard procedure oflinearizing the constitutive functions about the zero stress state while leaving temperature as the variable of thelinearized constitutive functions. The resulting constitutive equation bears some similarities to that in Liu et al.(2006), and at the same time possesses some differences. The sources and implications of such differences arediscussed.

In Section 3, the linear constitutive model is further reduced for isotropic materials. The resulting modelinvolves seven scalar constitutive functions: the frozen volume fraction, as well as the thermal strains, Young’smoduli and Poisson’s ratios of the active and frozen phases, respectively. They are all functions oftemperature. The uniaxial tension of such a material is considered, for which the axial and lateral strains canbe computed from the histories of the axial stress and temperature.

In Section 4, a detailed analysis of using the experimental measurements to calibrate the model is presented.It is found that not all seven constitutive functions can be determined from the experimental data acquired byLiu et al. (2006). However, certain combinations of the constitutive functions can be determined from theexperimental data, and such combinations are sufficient to make predictions of the material behavior of theSMP for the same type of experiments.

Comparisons of the theoretical predictions and the experimental measurements are presented in theconcluding Section 5. Reasonably good agreements are found, which serves as validation of the model.

2. Derivation of a linear constitutive model from the nonlinear theory

In the classical theory of mechanics, the infinitesimal strain tensor E is used, which is related to thedeformation gradient through

E ¼ 12ðFþ FTÞ � I. (2)

The constitutive equation for a thermoelastic material is given by

EðX; tÞ ¼ EðSðX; tÞ; yðX; tÞÞ,

where the constitutive function EðS; yÞ gives the strain in terms of the stress and temperature. For smallstrains, we have the first order approximation of the constitutive function:

EðS; yÞ ¼ EðO; yÞ þqEqSðO; yÞ½S�. (3)

The first term on the right-hand side of Eq. (3) is the so-called thermal strain, and the fourth order tensor inthe second term is the elastic compliance tensor. We shall introduce the following notations for these twotensors:

EtðyÞ ¼ EðO; yÞ; MðyÞ ¼qEqSðO; yÞ. (4)

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For an SMP material, we have two thermal strains EtaðyÞ;E

tf ðyÞ and two compliance tensorsMaðyÞ;Mf ðyÞ for

the active phase and the frozen phase, respectively. We then have, keeping Eq. (2) in mind,

12½FaðS; yÞ þ F

T

a ðS; yÞ� � I ¼ EaðS; yÞ ¼ EtaðyÞ þMaðyÞ½S�,

12½Ff ðS; yÞ þ F

T

f ðS; yÞ� � I ¼ Ef ðS; yÞ ¼ Etf ðyÞ þMf ðyÞ½S�: (5)

Since the deformation gradient tensor is close to the identity tensor for small deformations, the first orderapproximation of Eq. (1) reads

FðtÞ ¼ ½1� fðyðtÞÞ�FaðSðtÞ; yðtÞÞ

þ

Z t

0

Ff ðSðtÞ; yðtÞÞF�1

f ðSðtÞ; yðtÞÞFaðSðtÞ; yðtÞÞf0ðyðtÞÞ~y

0ðtÞdt

¼ ½1� fðyðtÞÞ�FaðSðtÞ; yðtÞÞ

þ

Z t

0

½Ff ðSðtÞ; yðtÞÞ � Ff ðSðtÞ; yðtÞÞ þ FaðSðtÞ; yðtÞÞ�f0ðyðtÞÞ~y

0ðtÞdt

¼ ½1� fðyðtÞÞ�FaðSðtÞ; yðtÞÞ þ fðyðtÞÞFf ðSðtÞ; yðtÞÞ

þ

Z t

0

½FaðSðtÞ; yðtÞÞ � Ff ðSðtÞ; yðtÞÞ�f0ðyðtÞÞ~y

0ðtÞdt. (6)

Here we have used the assumption that the material is initially in the active phase, fðyð0ÞÞ ¼ 0. Nowcombining Eqs. (2), (5) and (6), we arrive at

EðtÞ ¼ ½1� fðyðtÞÞ�EaðSðtÞ; yðtÞÞ þ fðyðtÞÞEf ðSðtÞ; yðtÞÞ

þ

Z t

0

½EaðSðtÞ; yðtÞÞ � Ef ðSðtÞ; yðtÞÞ�f0ðyðtÞÞ~y

0ðtÞdt

¼ ½1� fðyðtÞÞ�fEtaðyðtÞÞ þMaðyðtÞÞ½SðtÞ�g þ fðyðtÞÞfEt

f ðyðtÞÞ þMf ðyðtÞÞ½SðtÞ�g

þ

Z t

0

fEtaðyðtÞÞ þMaðyðtÞÞ½SðtÞ� � Et

f ðyðtÞÞ �Mf ðyðtÞÞ½SðtÞ�gf0ðyðtÞÞ~y

0ðtÞdt. (7)

Using the terminology in the paper by Liu et al. (2006), we rewrite Eq. (7) as

EðtÞ ¼ EeðtÞ þ EtðtÞ þ EsðtÞ, (8)

where

EeðtÞ ¼ ½1� fðyðtÞÞ�MaðyðtÞÞ½SðtÞ� þ fðyðtÞÞMf ðyðtÞÞ½SðtÞ� (9)

is the elastic strain,

EtðtÞ ¼ ½1� fðyðtÞÞ�EtaðyðtÞÞ þ fðyðtÞÞEt

f ðyðtÞÞ (10)

corresponds to the thermal strain, and

EsðtÞ ¼

Z t

0

fEtaðyðtÞÞ þMaðyðtÞÞ½SðtÞ� � Et

f ðyðtÞÞ �Mf ðyðtÞÞ½SðtÞ�gf0ðyðtÞÞ~y

0ðtÞdt (11)

corresponds to the stored strain.Eqs. (9) and (10) for the elastic strain and the thermal strain are identical to those in the work of Liu et al.

(2006). However, the expression for the stored strain given in Eq. (11) has some differences from theirs (seeEq. (8) of their paper), as discussed below.

�

The integral variable of the stored strain in Liu et al. (2006) is the frozen fraction f, and their expression for thestored strain is valid for cooling processes only. In the present work, the release of the stored strain duringheating is also accounted for by using the net cooling history ~y in Eq. (11). The present mathematicaldevelopment is in agreement with the following statement (with the present notation) in Liu et al. (2006, p. 299)‘‘The change of Es during heating inherits the resultant Es determined previously during the cooling process.’’

ARTICLE IN PRESSY.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–1778 1769

�

The integrand of Eq. (11) includes the thermal strains at freezing, which are absent in Liu et al. (2006). Thepresent form reflects the idea that the strain, whether it is mechanical or thermal, is being stored during theglass transition. Conceptually, this appears to be consistent with the freezing mechanism of the material, asalluded by the following statement in Liu et al. (2006, p. 291): ‘‘in addition to the pre-deformation, some ofthe thermal stress-induced tensile conformational changes are also stored during the cooling process.’’ � Of significant importance is the term �Mf ðyðtÞÞ½SðtÞ� in Eq. (11). This term, absent in Liu et al. (2006),

stems from the concept, which has been discussed in detail in Part I Section 3, that a further deformationafter freezing should have the frozen state, not the initial state, as the reference configuration. The use ofthis frozen reference configuration is essential for a consistent theory. As an illustrative example, consideran SMP with constant elastic compliances Ma and Mf , respectively, in the active and frozen phases, as wellas zero thermal strains. Suppose that the material is loaded to a prescribed stress S at the initial temperature(in active phase), followed by cooling to the frozen phase while holding the stress constant at S. We thenhave, during cooling,

EðtÞ ¼ ½1� fðyðtÞÞ�Ma½S� þ fðyðtÞÞMf ½S� þ

Z t

0

fMa½S� �Mf ½S�gf0ðyðtÞÞ~y

0ðtÞdt

¼ Ca½S�. (12)

That is, the strain remains constant during cooling. This is the expected result when the material is beingfrozen under constant stress. Without the term �Mf ½S� in the integrand of Eq. (12), the strain duringcooling would be Ma½S� þ fðyðtÞÞMf ½S�, implying that the mechanical strain would increase during coolingunder constant stress.

Liu et al. (2006) conducted uniaxial tension tests of an epoxy SMP and collected a wealth of experimentaldata. We can determine the constitutive functions of the present model using part of their experimental data,and compare the model predictions with the remaining data. To this end, we shall consider isotropic materialsunder uniaxial tension in the next section. It is noted that the linear model for isotropic SMPs can be obtainedin an alternative way by first deriving from Eq. (1) a nonlinear model for isotropic SMPs, and then linearizingit for small deformations.

3. Isotropic SMPs, uniaxial tension

For an isotropic linear elastic SMP, the elastic compliance tensors MaðyÞ and Mf ðyÞ are given by

MaðyÞ½S� ¼1

EaðyÞf½1þ naðyÞ�S� naðyÞðtrSÞIg,

Mf ðyÞ½S� ¼1

Ef ðyÞf½1þ nf ðyÞ�S� nf ðyÞðtrSÞIg; (13)

where Ea and Ef are Young’s moduli of the active and frozen phases, respectively, and na and nf Poisson’sratios. They are all assumed to depend on temperature. Moreover, the thermal strains of an isotropic SMP aregiven by

EtaðyÞ ¼ �

taðyÞI; Et

f ðyÞ ¼ �tf ðyÞI, (14)

�taðyÞ and �tf ðyÞ are the scalar thermal strain functions for the active and frozen phases, respectively.

Substituting Eqs. (13) and (14) into Eq. (7), we find that

EðtÞ ¼ ½1� fðyðtÞÞ�f�taðyðtÞÞIþ1

EaðyðtÞÞ½ð1þ naðyðtÞÞÞSðtÞ � naðyðtÞÞðtrSðtÞÞI�g

þ fðyðtÞÞf�tf ðyðtÞÞIþ1

Ef ðyðtÞÞ½ð1þ nf ðyðtÞÞÞSðtÞ � nf ðyðtÞÞðtrSðtÞÞI�g

ARTICLE IN PRESSY.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–17781770

þ

Z t

0

f½�taðyðtÞÞ � �tf ðyðtÞÞ�Iþ

1

EaðyðtÞÞ½ð1þ naðyðtÞÞÞSðtÞ � naðyðtÞÞðtrSðtÞÞI�

�1

Ef ðyðtÞÞ½ð1þ nf ðyðtÞÞÞSðtÞ � nf ðyðtÞÞðtrSðtÞÞI�gf

0ðyðtÞÞ~y

0ðtÞdt. (15)

In a uniaxial tension experiment, the stress tensor has the following component form:

SðtÞ ¼

sðtÞ 0 0

0 0 0

0 0 0

0B@

1CA, (16)

where sðtÞ is the axial stress at time t. The corresponding strain tensor has the component form

EðtÞ ¼

�ðtÞ 0 0

0 �lðtÞ 0

0 0 �lðtÞ

0B@

1CA, (17)

where �ðtÞ is the axial strain at time t, and �lðtÞ the lateral strain at t.Substitution of Eqs. (16) and (17) into Eq. (15) yields

�ðtÞ ¼ ½1� fðyðtÞÞ� �taðyðtÞÞ þsðtÞ

EaðyðtÞÞ

� �þ fðyðtÞÞ �tf ðyðtÞÞ þ

sðtÞEf ðyðtÞÞ

� �

þ

Z t

0

�taðyðtÞÞ � �tf ðyðtÞÞ þ

sðtÞEaðyðtÞÞ

�sðtÞ

Ef ðyðtÞÞ

� �f0ðyðtÞÞ~y

0ðtÞdt, (18)

�lðtÞ ¼ ½1� fðyðtÞÞ� �taðyðtÞÞ �naðyðtÞÞsðtÞ

EaðyðtÞÞ

� �þ fðyðtÞÞ �tf ðyðtÞÞ �

nf ðyðtÞÞsðtÞEf ðyðtÞÞ

� �

þ

Z t

0

�taðyðtÞÞ � �tf ðyðtÞÞ �

naðyðtÞÞsðtÞEaðyðtÞÞ

þnf ðyðtÞÞsðtÞ

Ef ðyðtÞÞ

� �f0ðyðtÞÞ~y

0ðtÞdt. (19)

Eqs. (18) and (19) can be used, among other things, to determine the constitutive functions from uniaxialtension experiments, or to predict the material behavior in a uniaxial tension test once the constitutivefunctions have been determined. We now turn our attention to this matter.

4. Experimental determination of the constitutive functions. Model prediction

Liu et al. (2006) carried out a careful experimental program to calibrate the constitutive model that theydeveloped. In the process, they have made a number of assumptions on the constitutive functions: theyassumed that Young’s modulus of the frozen phase is constant, and that Young’s modulus of the active phaseis a linear function of absolute temperature. Moreover, they chose an empirical formula for the frozen volumefraction. These assumptions require a prior knowledge of the constitutive behavior of the material and greatlyreduce the need of the experimental data, as all the constitutive functions are characterized by a few materialconstants. In particular, they were able to determine these material constants through curve fitting by usingthe strain recovery data. The stress measurements during cooling were not used at all.

Here we attempt to calibrate the present model by using the experimental data obtained by Liu et al. (2006),with a somewhat different methodology. Specifically, we do not use the above-mentioned assumptions, andattempt to determine the constitutive functions with full use of the available experimental data. In particular,we used both the stress measurements during cooling and the strain measurements during heating. Also, thesemeasurements in the whole temperature range are fully utilized, since we make no assumption on the form ofthe constitutive functions. In fact, it is possible to use the current approach to assess the reasonableness of theassumptions made in Liu et al. (2006). For example, by assuming that Young’s modulus of the frozen phase isconstant, we can examine to what extent Young’s modulus of the active phase can be approximated as a linearfunction of absolute temperature, and examine the accuracy of the empirical formula proposed by Liu et al.

ARTICLE IN PRESSY.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–1778 1771

Since no measurements of lateral deformation were made in the experiments of Liu et al., no information onPoisson’s ratios can be obtained. We thus focus on Eq. (18) which relates the axial stress to the axial strain.

There are five constitutive functions in Eq. (18), namely, fðyÞ; �taðyÞ; �tf ðyÞ;EaðyÞ, and Ef ðyÞ. However, it is

found that for the uniaxial tension experiment not all five functions are independent, in the sense that not all ofthem can be determined by the experiment, and that not all of them are needed to describe the materialbehavior in such an experiment. Precisely, three combinations of these five functions constitute all theinformation accessible and amenable to the experiment. A possible choice of these combinations is given bythe following three constitutive functions:

�

Overall thermal strain function

�tðyÞ ¼ ½1� fðyÞ��taðyÞ þ fðyÞ�tf ðyÞ þZ y

y0½�taðzÞ � �

tf ðzÞ�f

0ðzÞdz. (20)

�

Instant elastic compliance function

CðyÞ ¼1� fðyÞ

EaðyÞþ

fðyÞEf ðyÞ

. (21)

�

Distributed elastic compliance function

DðyÞ ¼1

EaðyÞ�

1

Ef ðyÞ

� �f0ðyÞ. (22)

For an arbitrary temperature history yðtÞ with yð0Þ ¼ y0, we have

Z t

0

½�taðyðtÞÞ � �tf ðyðtÞÞ�f

0ðyðtÞÞ~y

0ðtÞdt ¼

Z yðtÞ

y0½�taðzÞ � �

tf ðzÞ�f

0ðzÞdz. (23)

In virtue of Eqs. (20)–(23), we can rewrite Eq. (18) as

�ðtÞ ¼ �tðyðtÞÞ þ CðyðtÞÞsðtÞ þZ t

0

DðyðtÞÞsðtÞ~y0ðtÞdt. (24)

Eq. (24) indicates that the material behavior of an SMP under uniaxial tension is completely described by thethree constitutive functions defined in Eqs. (20)–(22). From a practical point of view, it then suffices todetermine these constitutive functions in an experimental program.

It is obvious from Eq. (24) that the overall thermal strain function �tðyÞ can be readily obtained bymeasuring the strain as a function of temperature with zero stress in a cooling or heating process. Fig. 1 showsthe experimental measurements of the thermal strain in the temperature range of 273–358K, which isreproduced from Liu et al. (2006, Fig. 4). They measured strains for both cooling and heating, and reportedslight difference between the two sets of data. Only the data points for cooling are shown in Fig. 1 with a curvefitting to remove obvious experimental scattering. Here and henceforth, we use the smooth fitted curves in thecalibration of the model.

A typical shape memory cycle has been described in Part I Section 5. Relevant measurements in such a cyclewere made by Liu et al. (2006). We now discuss determination of the constitutive functions by using theirmeasurements.

The first step of the cycle is loading from zero strain to a prescribed strain �pre at the initial temperaturey0 ¼ 358K. The overall thermal strain is zero. Eq. (24) then becomes

�ðtÞ ¼ Cðy0ÞsðtÞ. (25)

Fig. 2 shows the stress and strain measurements by Liu et al. (2006, Fig. 3) and a straight line fitting to the datapoints. The linear relation between the stress and strain as indicated in Eq. (25) is seen to hold well. Since

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Fig. 1. Overall thermal strain function.

Fig. 2. Stress–strain relation during initial loading at high temperature y0 ¼ 358K.

Y.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–17781772

fðy0Þ ¼ 0, Cðy0Þ is the elastic compliance of the active phase at the initial temperature, and is found to beapproximately ð8:8MPaÞ�1.

The second step is cooling while the strain is held at �pre. It follows from Eq. (24) that

�pre ¼ �tðyðtÞÞ þ CðyðtÞÞsðtÞ þ

Z t

0

DðyðtÞÞsðtÞy0ðtÞdt. (26)

Since the temperature is a monotone decreasing function of time in this step, we can rewrite Eq. (26) withtemperature being the independent variable:

�pre ¼ �tðyÞ þ CðyÞ �sðyÞ þ

Z y

y0DðzÞ �sðzÞdz, (27)

where �s is such that

sðtÞ ¼ �sðyðtÞÞ. (28)

The temperature reaches y2 ¼ 273K at the end of the second step. The function �sðyÞ was measured by Liuet al. (2006, Fig. 5), and is shown in Fig. 3.

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Fig. 3. Stress–temperature relation during cooling with constant strain �pre ¼ 9:1%.

Y.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–1778 1773

The third step is unloading while temperature is being held at y2 ¼ 273K. It follows from Eq. (24) that

�ðtÞ ¼ �tðy2Þ þ Cðy2ÞsðtÞ þZ y2

y0DðzÞ �sðzÞdz. (29)

Eq. (29) corresponds to a straight line on the stress–strain diagram, which was measured by Liu et al. (2006).The elastic compliance Cðy2Þ of the frozen phase can be found from the slope of this line, and is reported to beapproximately ð750MPaÞ�1.

The fourth and the last step of the cycle is heating at zero stress. As discussed in Part I Section 5, we againdefine sðtÞ such that sðtÞot and that yðsðtÞÞ ¼ yðtÞ. Eq. (24) then leads to

�ðtÞ ¼ �tðyðtÞÞ þZ sðtÞ

0

DðyðtÞÞsðtÞy0ðtÞdt. (30)

Define �� such that

�ðtÞ ¼ ��ðyðtÞÞ. (31)

By using Eq. (28), we can rewrite Eq. (30) as

��ðyÞ ¼ �tðyÞ þZ y

y0DðzÞ �sðzÞdz. (32)

Function ��ðyÞ, as measured by Liu et al. (2006, Fig. 6) is shown in Fig. 4.Combining Eqs. (27) and (32), we find the instant elastic compliance function and the distributed elastic

compliance function as

CðyÞ ¼�pre � ��ðyÞ

�sðyÞ, (33)

DðyÞ ¼1

�sðyÞd

dy½��ðyÞ � �tðyÞ�. (34)

By using Eqs. (33), (34), and the experimental data presented in Figs. 1, 3, and 4, we find functions CðyÞ andDðyÞ. We leave these function as computed without fitting them into empirical formulas. Upon substitutingthem into Eq. (24), we can then predict the thermomechanical behavior of the material in any temperature-loading path. In the next section, we present a few predictions and compare them with the experimentalmeasurements of Liu et al. (2006). Among other things, this provides validation of the linear model.

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Fig. 4. Strain–temperature relation during heating with zero stress.

Y.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–17781774

5. Theoretical predictions. Validation of the model

The first set of predictions is for the same types of temperature-loading paths as described above, withdifferent pre-strain levels. In addition to the experiment with �pre ¼ 9:1%, which is used to calibrate the model,experiments with �pre ¼ 0 and �9:1% were also carried out by Liu et al. (2006). In particular, for �pre ¼ 0, theinitial loading is absent, and the sample is cooled while being held in its initial length. It is subsequentlyunloaded at the low temperature, and finally heated at zero stress. For �pre ¼ �9:1%, the initial loading iscompressive, although the stress during cooling changes from compressive to tensile.

The stresses �sðyÞ as functions of temperature during cooling for these values of pre-strain are found bysolving the integral Eq. (27) and using the functions CðyÞ and DðyÞ obtained above. They are plotted in Fig. 5as model predictions, together with the experimental data. Since no approximation is used here, the modelprediction for �pre ¼ 9:1% is exactly the same as the smooth curve obtained from the experimental datathrough curve fitting. Other two curves, on the other hand, provide true comparison of the theory and theexperiment. It is observed the theoretical predictions are in reasonable agreement with the experimentalmeasurements.

After cooling, the sample is unloaded at the low temperature, and then heated at zero stress. The strain ��ðyÞduring heating is computed by using Eq. (32), DðyÞ and �sðyÞ as previously obtained. The resulting predictionsof ��ðyÞ are plotted in Fig. 6. Good agreements with the experimental data are observed.

Liu et al. (2006) also carried out experiments of different temperature-loading paths than those describedabove. One such experiment involves the same initial loading and cooling processes, but followed with heatingat the pre-strain without unloading. Since the present model asserts full recovery when the material returns tothe active phase, the final heating process will be simply the reverse of the previous cooling process. Indeed, thepredicted stress–temperature relations are exactly the same as those presented in Fig. 5. We plot these curves inFig. 7, along with the experimental data during heating. While the model predictions in Figs. 5 and 7 are thesame, the experimental data are slightly different since the material exhibited small hysteresis effects, ascommented by Liu et al. (2006).

Yet another set of experiments involves the same initial loading, followed by cooling at the fixed pre-strain,and unloading at the low temperature. Then instead of heating at zero stress, the specimen is heated at theconstant strain �fix that has occurred in the end of unloading. Denoting by �sðyÞ the stress as a function oftemperature during heating, and by �sðyÞ again the stress as a function of temperature during initial heating, wefind, using the same arguments leading to Eqs. (27) and (32), that

�fix ¼ �tðyÞ þ CðyÞ �sðyÞ þ

Z y

y0DðzÞ �sðzÞdz. (35)

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Fig. 6. Predictions of the strain–temperature relations during heating at various pre-strain levels.

Fig. 7. Predictions of the stress–temperature relations during heating immediately after cooling at various pre-strain levels.

Fig. 5. Predictions of the stress–temperature relations during cooling at various pre-strain levels.

Y.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–1778 1775

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Fig. 8. Predictions of the stress–temperature relations during heating at constant strain levels after unloading.

Y.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–17781776

Using measured �tðyÞ and previously determined CðyÞ, DðyÞ and �sðyÞ, we can solve Eq. (35) for �sðyÞ. It is expectedthat during heating, a compressive thermal stress will first be developed due to the strain constraint. As thetemperature approaches the transition temperature, the level of this compressive stress will decrease as the materialbecomes more compliant. For the case where the pre-strain is tensile, the stress eventually should turn to tensilefrom compressive. Such behaviors were observed in both experiments and model predictions, as shown in Fig. 8.However, the agreement between the experimental measurements and model predications is not as good as those inthe previous comparisons. Similar discrepancies were also observed by Liu et al. (2006, Fig. 16). This discrepancycould be caused by either the inaccuracy of the model or the inaccuracy of the measurements.

We wish to examine the assumptions that Liu et al. (2006) made on the constitutive functions, which greatlysimplify the task of calibrating the model. Specifically, they assumed that

Ef ðyÞ ¼ const:; (36)

EaðyÞ ¼ 3Nky, (37)

fðyÞ ¼ 1�1

1þ cðy0 � yÞn, (38)

where N is the cross-link density, k Boltzmann’s constant ðk ¼ 1:38� 10�23 Nm=KÞ, and c and n are materialconstants to be determined by experiment. Eq. (38) is an empirical formula, and Eq. (37) follows from a classicaltheory of rubber elasticity for network polymers. Since the material is much stiffer in the frozen phase, theassumption (36) of Young’s modulus in frozen phase being constant is likely to have a negligible effect on the overallaccuracy of the model. As discussed above, these three constitutive functions, when being left in general form, cannotbe individually determined by the uniaxial tension experiment. However, if we assume the function form for one ofthese three constitutive functions, the remaining two can be determined from the experimental data.

Here, we take assumption (36), and examine to what extent the constitutive functions EaðyÞ and fðyÞ asdetermined by the experiment can be approximated by the functions on the right-hand sides of Eqs. (37) and(38). It follows from Eq. (21) that

Ef ðy2Þ ¼1

Cðy2Þ, (39)

where y2 ¼ 273K is the temperature at which the entire material is in the frozen phase. We shall take Ef ðy2Þ asthe constant value for Ef . We can then solve Eqs. (21) and (22), arriving at

fðyÞ ¼ 1� e�R y

yoðDðzÞ=ðCðzÞ�Cðy2ÞÞÞ dz; EaðyÞ ¼

1

Cðy2Þ þ ½CðyÞ � Cðy2Þ�eR y

yoðDðzÞ=ðCðzÞ�Cðy2ÞÞÞ dz

. (40)

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The constitutive functions fðyÞ and EaðyÞ are computed from (40), with use of CðyÞ and DðyÞ previouslydetermined from the experimental data. It is found that the predicted frozen volume fraction fðyÞ can beapproximately expressed by the empirical formula (38) with reasonable accuracy. However, the predictedYoung’s modulus for the active phase is qualitatively inconsistent with the classical result (37). A number offactors could attribute to this inconsistency, including the accuracy of Eq. (37) itself, reasonableness of theassumption of constant Young’s modulus for frozen phase, accuracy of the experimental data, and accuracyof the model.

In concluding this paper, we have developed a nonlinear constitutive theory for SMPs, with a primary goalto describe the large deformations of such materials. A linear model is derived from the nonlinear theory andis examined by using the existing experimental data. The experiments for large deformations of SMPs arebeing currently conducted. Such experiments can be used to calibrate and validate the nonlinear constitutivemodels. Also under investigation are the models that incorporate the rate dependent and time dependentbehaviors of the material.

Acknowledgments

We would like to thank Dr. Ken Gall for helpful discussions on an earlier version of this work. Thenumerical and graphic assistances provided by Liang Li and Joao Paulo Poupard are acknowledged. We alsoacknowledge the supports of the Texas Institute for Intelligent Bio-Nano Materials and Structures forAerospace Vehicles, funded by NASA Cooperative Agreement No. NCC-1-02038. Chen wishes toacknowledge the support of ARO Grant 46828-MS-ISP.

Reference

Liu, Y., Gall, K., Dunn, M.L., Greenberg, A.R., Diani, J., 2006. Thermomechanics of shape memory polymers: uniaxial experiments and

constitutive modeling. Int. J. Plasticity 22, 279–313.

Further reading

Behl, M., Lendlein, A., 2007. Shape-memory polymers. Mater. Today 10, 20–28.

Bhattacharyya, A., Tobushi, H., 2000. Analysis of the isothermal mechanical response of a shape memory polymer rheological model.

Polymer Eng. Sci. 2000, 2498–2510.

Bueche, F., 1954. The viscoelastic properties of plastics. J. Chem. Phys. 22, 603–609.

Chen, Y.C., Hoger, A., 2000. Constitutive functions of elastic materials in finite growth and deformation. J. Elasticity 59, 175–193.

Dupaix, R.B., Boyce, M.C., 2007. Constitutive modeling of the finite strain behavior of amorphous polymers in and above the glass

transition. Mech. Mater. 39, 39–52.

Gall, K., Dunn, M.L., Liu, Y., 2004. Internal stress storage in shape memory polymer nanocomposites. Appl. Phys. Lett. 85,

290–292.

Gurtin, M.E., 1970. An Introduction to Continuum Mechanics. Academic Press, New York.

Lendlein, A., Kelch, S., 2002. Shape-memory polymers. Angew. Chem. Int. Ed. 41, 2034–2057.

Liu, Y., Gall, K., Dunn, M.L., McCluskey, P., 2003. Thermomechanical recovery couplings of shape memory polymers in flexure. Smart

Mater. Struct. 12, 947–954.

Liu, Y., Gall, K., Dunn, M.L., McCluskey, P., 2004. Thermomechanics of shape memory polymer nanocomposites. Mech. Mater. 36,

929–940.

Pokrovskii, V.N., 2000. The Mesoscopic Theory of Polymer Dynamics. Kluwer Academic Publishers, Dordrecht.

Rao, I.J., 2002. Constitutive modeling of crystallizable shape memory polymers. ANTEC Proceedings 2002, 1936–1940.

Rouse, P.E., 1953. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1272–1280.

Shaw, M.T., MacKnight, W.J., 2005. Introduction to Polymer Viscoelasticity. Wiley-Interscience, New York.

Tobushi, H., Hara, H., Yamada, E., Hayashi, S., 1996. Thermomechanical properties in a thin film of shape memory polymer of

polyurethane series. Smart Mater. Struct. 5, 483–491.

Tobushi, H., Hashimoto, T., Hayashi, S., Yamada, E., 1997. Thermomechanical constitutive modeling in shape memory polymer of

polyurethane series. J. Intell. Mater. Syst. Struct. 8, 711–718.

Tobushi, H., Hashimoto, T., Ito, N., Hayashi, S., Yamada, E., 1998. Shape fixity and shape recovery in a film of shape memory polymer of

polyurethane series. J. Intell. Mater. Syst. Struct. 9, 127–136.

ARTICLE IN PRESSY.-C. Chen, D.C. Lagoudas / J. Mech. Phys. Solids 56 (2008) 1766–17781778

Tobushi, H., Matsui, R., Hayashi, S., Shimada, D., 2004. The influence of shape-holding conditions on shape recovery of polyurethane—

shape memory polymer foams. Smart Mater. Struct. 13, 881–887.

Truesdell, C., Noll, W., 1992. The Non-Linear Field Theories of Mechanics, second ed.. Springer, Berlin.

Zimm, B.H., 1956. Dynamics of polymer molecules in dilute solution: viscoelasticity, flow birefringence and dielectric loss. J. Chem. Phys.

24, 269–278.