thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling

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International Journal of Plasticity 22 (2006) 279–313

Thermomechanics of shape memorypolymers: Uniaxial experiments and

constitutive modeling

Yiping Liu a,*, Ken Gall a, Martin L. Dunn a,Alan R. Greenberg a, Julie Diani b

a Department of Mechanical Engineering, CAMPMODE, University of Colorado, Boulder,

CO 80309-427, USAb Laboratoire d�Ingenierie des Materiaux, UMR 8006 CNRS, 75013 Paris, France

Received 1 September 2004

Available online 2 June 2005

Abstract

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an

elevated temperature and subsequent cooling to a lower temperature. When reheated, the ori-

ginal shape can be recovered. Relatively little work in the literature has addressed the consti-

tutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are

critical for predicting the deformation and recovery of SMPs under a range of different con-

straints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is

systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial

compression. After initial pre-deformation at a high temperature, the strain is held constant

for shape storage while the stress evolution is monitored. Three cases of heated recovery

are selected: unconstrained free strain recovery, stress recovery under full constraint at the

pre-deformation strain level (no low temperature unloading), and stress recovery under full

constraint at a strain level fixed at a low temperature (low temperature unloading). The free

strain recovery results indicate that the polymer can fully recover the original shape when

reheated above its glass transition temperature (Tg). Due to the high stiffness in the glassy state

(T < Tg), the evolution of the stress under strain constraint is strongly influenced by thermal

0749-6419/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2005.03.004

* Corresponding author. Tel.: +1 303 735 2651; fax: +1 303 492 3498.

E-mail address: [email protected] (Y. Liu).

mailto:[email protected]

280 Y. Liu et al. / International Journal of Plasticity 22 (2006) 279–313

expansion of the polymer. The relationship between the final recoverable stress and strain is

governed by the stress–strain response of the polymer above Tg. Based on the experimental

results and the molecular mechanism of shape memory, a three-dimensional small-strain inter-

nal state variable constitutive model is developed. The model quantifies the storage and release

of the entropic deformation during thermomechanical processes. The fraction of the material

freezing a temporary entropy state is a function of temperature, which can be determined by

fitting the free strain recovery response. A free energy function for the model is formulated and

thermodynamic consistency is ensured. The model can predict the stress evolution of the uni-

axial experimental results. The model captures differences in the tensile and compressive recov-

ery responses caused by thermal expansion. The model is used to explore strain and stress

recovery responses under various flexible external constraints that would be encountered in

applications of SMPs.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Polymer network; Shape memory effect; Stress and strain recovery; Thermomechanics; Con-

stitutive model

1. Introduction

Shape memory materials are defined by their capacity to store a deformed (tem-

porary) shape and recover an original (parent) shape. The shape memory behavior

is typically induced by a change in temperature and has been observed in metals,

ceramics, and polymers (Feninat et al., 2002; Gandhi and Thompson, 1992; Otsukaand Wayman, 1998). Owing to their unique shape memory mechanism, the shape

memory effect in polymers differs significantly from that observed in ceramics

and metals. Shape memory polymers (SMPs) experience lower stresses during

deformation and demonstrate much larger recoverable strains. A shape memory

polymer can be a thermoset or thermoplastic with a chemically or physically

cross-linked network structure, which permits a rubbery plateau at a temperature

above the glass transition temperature, Tg. One benefit of SMPs is that Tg can

be easily tailored by the control of chemistry and structure (Davis and Burdick,2003; Jeon et al., 2000; Jeong et al., 2000; Lendlein et al., 2001; Li et al., 1999;

Liang et al., 1997; Liu et al., 2002; Metzger et al., 2002; Takahashi et al., 1996;

Zhu et al., 2003).

Early applications of the shape memory effect in polymers included heat shrink-

able tubes, wraps, foams and self-adjustable utensils. Recently, substantial research

efforts on SMPs are emerging in the areas of biomedical devices, deployable space

structures, and microsystems (Abrahamson et al., 2003; Gall et al., 2000, 2004;

Lendlein and Langer, 2002; Metcalfe et al., 2003). The design of SMP-based devicesdemands thorough characterization and constitutive modeling of the thermome-

chanical shape memory cycle. Constitutive models facilitate the prediction of recov-

erable stress and strain levels under varying degrees of constraint, as will be

invariably experienced in SMP applications. In addition, advanced SMP applica-

tions require optimized storage and recovery properties achieved through a funda-

Y. Liu et al. / International Journal of Plasticity 22 (2006) 279–313 281

mental knowledge of the constitutive relationship for various thermomechanical

conditions.

The mechanical properties of polymeric materials are strongly temperature depen-

dent. An amorphous polymer that is stiff at room temperature can become quite

compliant and behave in a ductile, rubbery manner at higher temperatures. The tran-sition from high stiffness to low stiffness occurs at the glass transition temperature,

Tg. In Fig. 1, the storage modulus, loss modulus and tan delta of an epoxy resin

are plotted versus absolute temperature. The general behavior observed in Fig. 1

is characteristic of amorphous thermoset polymers with appropriate cross-linking

density. All such polymers show a similar response as a function of temperature, ref-

erenced to their respective Tg.

The properties of such polymers near Tg account for the shape memory behavior

during a thermomechanical cycle. At temperatures well above Tg, the polymer is inthe rubbery state. The stiffness of the polymer is low, and the deformation energy is

converted into a free conformational entropy change. With appropriate cross-linking

density, rubbery elastic strains can be on the order of several hundred percent. In the

rubbery state, the elastic deformation of an ideal network polymer produces a

change in the conformational entropic state of the polymer chains, and the interac-

tion between chain molecules is relatively neglectable (Bower, 2002; Matsuoka, 1992;

Ward and Hadley, 1993). When T < Tg, the large-scale conformational changes are

not possible but localized conformational motions are allowed (Strobl, 1997). Attemperatures well below Tg, the polymer is in the glassy state and behaves as an

elastic solid at small strains. The low temperature elasticity is due to primary

bond stretching, which causes a change in internal energy. In both the glassy and

rubbery states, the elastic strain can be stored ‘‘instantaneously’’ and released

100

101

102

103

104

Mod

ulus

(M

Pa)

400380360340320300280

Temperature (K)

1.0

0.8

0.6

0.4

0.2

0.0

Tan delta

Storage modulus

Loss modulus

Tan delta

Cooling Heating

Tg=343K

Fig. 1. Storage modulus, loss modulus and tan delta of the shape memory polymer. The DMA test is

conducted in three-point flexure using a dynamic scan analyzer at a frequency of 1 Hz.


‘‘instantaneously’’ upon removal of the applied stress. The damping factor (Tan

delta) in these two states is relatively small compared to that in the glass transition

region (Fig. 1), in which viscoelasticity governs the deformation and recovery in the

time scale considered.

By invoking a shape memory thermomechanical cycle, a connection between thesetwo deformation states occurs. The entropic state induced by the high temperature

deformation can be stored ‘‘temporarily’’ at low temperatures, i.e. ‘‘frozen,’’ by

emerging atomic interactions, which is defined as the internal stress field. The frozen

strain can only be released when strong molecular interaction has disappeared after

subsequent heating to a high temperature. The driving force for the strain release is

micro-Brownian thermal motion, which becomes increasingly important at higher

temperature (Bower, 2002; Matsuoka, 1992; Strobl, 1997; Ward and Hadley,

1993). The theory of rubber elasticity is reasonable up to relatively large elongationsas long as the cross-link density is moderate. In the rubbery state, the Young�s mod-

ulus predicted by rubber elasticity theory is proportional to the absolute temperature

and the cross-linking density.

Although earlier constitutive modeling efforts using rheological approaches are

able to describe the thermomechanical behavior of SMPs, they do not account for

the different strain storage and release mechanisms at the molecular level and thus

have limited predictive power (Monkman, 2000; Tobushi et al., 2001). In addition,

previous experiments and models have not examined different flexible constraint lev-els under both tension and compression (Gall et al., 2002; Liu et al., 2003, 2004).

SMP applications involve tensile, compressive and bending loads, so it is critical

to understand the effects of stress state on shape storage and release. In particular,

thermal expansion effects can give rise to different storage and recovery mechanics

in tension and compression as will be shown in the present work. With this back-

ground, the primary goal of the present paper is to develop a small-strain constitu-

tive model for SMPs and to explore the uniaxial thermomechanical response of

SMPs under various constraint conditions. When the mechanism-based model isfit to a free strain recovery curve, it is capable of predicting the stress recovery of

the polymer under various constraint levels.

Based on the thermodynamic concepts of entropy and internal energy, it is pos-

sible to interpret the thermomechanical behavior of SMPs from a macroscopic

viewpoint without explicitly incorporating details of the molecular interactions.

In this study, we do not intend to address the chemical aspects of various shape

memory mechanisms. Instead, we adopt the concept of ‘‘frozen’’ strain into a sim-

plified thermomechanical model and illustrate the efficiency of the model by com-paring the experimental and constitutive modeling results for an epoxy resin, a

thermoset polymer. The utility of the model is also demonstrated by simulating

the strain and stress recovery under various flexible constraint levels. Flexible con-

straint levels are always encountered in real applications where the polymers can-

not recover freely, but must perform work on another material or object. For

example, in biomedical applications (Lendlein and Langer, 2002; Metcalfe et al.,

2003; Metzger et al., 2002), the polymers must expand against the constraint of

the surrounding tissue.


2. Materials and experimental methods

Test specimens were prepared from a commercial thermoset epoxy system, DP5.1

supplied by Composite Technology Development (CTD), Inc. This SMP is intended

for deployable composite space structure applications (Abrahamson et al., 2003;Gall et al., 2000). The polymer was synthesized from a mixture of resin and hardener.

These two parts were hand mixed and degassed at a vacuum level of 85 kPa

(25 inHg). The mixture was then cast into a Teflon mold and cured at room temper-

ature, 298 K (25 �C), for 24 h and at 323 K (50 �C) for another 5 h. An amorphous

thermoset polymer material was obtained, with Tg = 343 K (70 �C).A Perkin Elmer Dynamic Mechanical Analyzer (DMA-7) was used to perform a

dynamic thermal scan at a frequency of 1 Hz. The span of the three-point bending

test was 10 mm. The size of the test specimen was approximately 1 · 3 · 15 mm. Un-der a constant dynamic force of 80 mN, the strain amplitude was about 0.03–3% in

the temperature range of 273–400 K. The purpose of the dynamic thermal scan was

to determine the basic mechanical response of the polymer. Heating/cooling rates of

±1 and ±5 K/min, respectively, were used, and the results were comparable.

The mechanical properties of the epoxy polymer vary markedly in the tempera-

ture range of 273–358 K (0–85 �C), while they vary only slightly outside this temper-

ature range. For simplicity, we define a low temperature Tl = 273 K (0 �C) and a high

temperature, Th = 358 K (85 �C). Since the temperature range from Tl to Th spansthe glass transition region between the glassy and rubbery states, it was selected as

the temperature range for all of the experimental tests. An Instron 5869 Testing Sys-

tem and an Instron SFL 3199-408 Temperature Controlled Environmental Chamber

were used to conduct the isothermal and thermomechanical tests of the SMP. An In-

stron 2620-825 Dynamic Strain Gage Extensometer with strain range from �10% to

+50% was used to measure strain. The acquired engineering strain and stress were

converted into true strain and true stress in reporting the experimental data. To cal-

culate the true stress at Th, we assumed the cross-linked polymer is incompressible inthe rubbery state (Ward and Hadley, 1993). During the thermomechanical cycle, the

strain was held constant and thermal expansion was taken into account for calculat-

ing the true stress. The absolute temperature was used to report the thermal history.

Dog-bone shaped specimens with a cross-section of 8 · 7 mm and a gage length of

35 mm were machined from cured polymer sheets. The edges of the samples were

polished using 600 grit silicon carbide sandpaper to remove any edge effects. The rel-

atively large cross section of the specimen made it possible to perform both tension

and compression tests on the same sample. Under a full strain constraint, the stressstate can exchange between tension and compression due to thermal expansion or

contraction during a shape memory thermomechanical cycle. Thus, it is important

to use a sample that can sustain both compressive and tensile stresses during thermo-

mechanical cycling. Isothermal stress–strain curves were generated to quantify the

elastic modulus. The isothermal properties were determined at two temperatures cor-

responding to the selected low, Tl and high, Th, temperatures.

Thermomechanical tests for strain storage and strain/stress recovery were per-

formed using identical sample geometry and load frame setup. In the temperature


range between Tl and Th, the temperature of the environmental chamber was pro-

grammed to have a constant cooling/heating rate. For all of the thermomechanical

tests the heating/cooling rate was selected to be ±1 K/min, which allowed tempera-

ture to reach equilibrium at the center of the sample. The data were collected every

minute during a thermomechanical cycle.

3. Experimental results

3.1. Dynamic thermal scan

The experimental results are presented in Figs. 1–8. Fig. 1 shows the storage mod-

ulus, loss modulus and tan delta as functions of temperature. Tan delta representsthe ratio of the loss to the storage modulus. The tan delta curve shows a glass tran-

sition peak at approximately Tg = 343 K, with a drop in storage modulus of approx-

imately two orders of magnitude from Tl = 273 K to Th = 358 K. In addition, the

storage modulus shows a rubbery plateau in the region beyond T @ 350 K. For this

high strength epoxy, according to the large-strain tests at Th, which are not shown in

this study, the elastic strain before failure was approximately 40% in compression.

These results indicate that the cross-link density is not sufficiently high to inhibit

the large-scale conformational motion of the polymer chain. Therefore, it is reason-able to state that the entropic motions are present in the polymer at T > Tg, and the

strain–stress behavior up to moderate elongations is governed by rubbery elasticity.

The dynamic mechanical results are characteristic of a cross-linked polymer that will

show a useful shape memory response. The glass transition behavior of the polymer

is basically identical during heating and cooling under the specified rate conditions.

Fig. 2. Stress–strain–temperature diagram illustrating the thermomechanical behavior of a pre-tensioned

shape memory polymer under different strain/stress constraint conditions.

8

6

4

2

0

-2

Stre

ss (

MPa

)

-10 -5 0 5 10

Strain (%)

Compression

T = Tl (273 K)E = 750 MPa

T = Th (358 K) Tension

Tension

T = Th (358 K)E = 8.8 MPa

Fig. 3. Isothermal stress–strain tests at temperatures above and below Tg. Th is the pre-deformation

temperature and Tl is the lowest strain storage temperature.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

The

rmal

str

ain

(%)

1.051.000.950.900.850.80

T/Tg (K/K)

= 1.8×10-4

(1/K)

= 0.9×10-4

(1/K)

CoolingHeating

Fig. 4. Thermal expansion strain as a function of temperature in the temperature range from Tl to Th.

Average thermal expansion coefficient is calculated for the polymer in the glassy and rubbery states.


3.2. Schematic diagram of thermomechanical behavior

Prior to presenting the experimental results, we first describe the rather sophisti-

cated thermomechanical shape-memory cycle for polymers. A three-dimensional

4

3

2

1

0

-1

Stre

ss (

MPa

)

1.051.000.950.900.850.80

T/Tg (K/K)

Tensioned, = 9.1%

Compressed, = - 9.1%

Undeformed, = 0

Fig. 5. Stress responses of the polymer during cooling under different pre-strain constraint conditions.

-10

-5

0

5

10

Stra

in (

%)

1.051.000.950.900.850.80

T/Tg (K/K)

Tensioned

Compressed

Undeformed

Fig. 6. Free strain recovery responses during heating for polymers that have experienced different pre-

deformations and strain-storage processes. Responses are at r = 0.


strain–stress–temperature diagram of the shape memory effect under tensile loadingis shown in Fig. 2. A typical tensile pre-deformation and strain-recovery cycle for a

polymer can be described in four steps: process 1 ! 2 ! 3 ! 4 or state

4

3

2

1

0

-1

Stre

ss (

MPa

)

1.051.000.950.900.850.80T/Tg (K/K)

Tensioned, = 9.1%


Undeformed, = 0

Stress evolution during cooling

Fig. 7. Stress recovery responses of the polymer during heating under different strain constraints that are

the same as the pre-deformation strain levels. Dashed lines represent stress evolution during cooling.

3

2

1

0

-1

-2

Stre

ss (

MPa

)

1.051.000.950.900.850.80

T/Tg (K/K)

Tensioned, = 8.6%


Undeformed, = 0

Fig. 8. Stress recovery responses of the polymer under different strain constraints that equal the fixed

strains obtained after low temperature unloading.


A ! B ! C! D ! A. To start the thermomechanical cycle, the material is broughtto state ‘‘A’’ at an elevated temperature, Th, at zero strain and zero stress. The first

step of the cycle (process 1 ) involves a high-strain deformation in the rubbery state,


which is called ‘‘pre-deformation’’ or ‘‘pre-strain.’’ In state ‘‘B,’’ the corresponding

pre-deformation strain and stress are denoted by epre and rpre.The second step (process 2 ) is a ‘‘strain storage’’ process whereby the material is

cooled under ‘‘pre-strain constraint,’’ such that epre is maintained. Because the strain

does not change, this process is denoted as ‘‘fully constrained.’’ Due to thermal con-traction, the tensile stress needed to maintain the pre-deformed shape increases as

the temperature decreases. In the glassy state (T < Tg), the polymer molecular chain

segments are ‘‘frozen’’ by thermally reversible molecular chain interactions. In state

‘‘C,’’ the temperature is at Tl and the stress reaches rl. The deformation history is

stored during cooling and can be released as recovered strain or stress during reheat-

ing, depending on the constraint imposed during heating.

The third step (process 3 ) is a ‘‘low temperature unloading’’ process, which is de-

fined as the removal of the strain constraint in the glassy state. The low temperatureunloading process is sometimes referred to as spontaneous ‘‘springback.’’ Due to the

higher elastic modulus at Tl, the springback strain, el, is very low compared to the

pre-deformation strain, epre, which is mostly stored in the glassy state. In state

‘‘D,’’ the temporary shape of the polymer is fixed. The retained strain, (epre�el), cor-responding to the fixed shape at Tl is called ‘‘fixed strain’’ and denoted by efix, whichis close to epre. The material at this state is a ‘‘pre-deformed’’ shape memory polymer.

A material can be pre-tensioned or pre-compressed and then kept in ‘‘hibernation’’

at Tl for an extended period without recovery of the fixed strain (Metcalfe et al.,2003; Tey et al., 2001).

The fourth step (process 4 ) of the thermomechanical cycle, which involves reheat-

ing, is denoted ‘‘free strain recovery’’ or ‘‘unconstrained recovery,’’ which implies the

absence of external applied stress and the free recovery of the induced strain. During

the heating process, thermal expansion strain occurs in addition to the fixed strain at

T < Tg. The strain is recovered near Tg. At Th, the remaining strain is called ‘‘resid-

ual strain.’’ Typically, the residual strain is negligible, and the final state of the mate-

rial returns to the starting state ‘‘A.’’Besides the typical free strain recovery cycle ( 1 ! 2 ! 3 ! 4 or A ! B ! C!

D ! A), ‘‘fully constrained’’ stress recovery responses are also shown in Fig. 2.

Continuing from state ‘‘C,’’ the heating process 5 characterizes the stress recovery

under the ‘‘pre-strain constraint.’’ In the case of pre-strain constraint, the strain con-

straint imposed during cooling is kept in place during subsequent heating. The stress

evolution as a function of temperature is comparable to that of the strain storage

process 2 . The thermomechanical process 5 represents an idealized situation. In

practical applications, the strain constraint imposed during cooling will invariablybe removed prior to recovery.

It is also possible to achieve ‘‘fully constrained’’ stress recovery after low temper-

ature unloading. Starting from state ‘‘D,’’ during the heating process 6 , the polymer

specimen is subjected to a new strain constraint level equal to the fixed strain, efix.The newly prescribed constraining strain level is close to the pre-strain constraint le-

vel and is called ‘‘fixed-strain constraint.’’ Beginning from the stress-free state ‘‘D,’’ a

compressive stress occurs due to constrained thermal expansion and reaches a max-

imum in the glassy state between Tl and Tg. With further temperature increase, the


material softens as T ! Tg. At Th, the recovery stress reaches state ‘‘E,’’ where both

the stress and strain are very close to those of the pre-deformation state ‘‘B.’’

From the above description of the thermomechanical processes, we can see that

the pre-deformation temperature is Th while the lower limit of the strain storage tem-

perature is Tl. The cooling and heating rates, which can quantitatively affect theshape memory behavior, are kept constant. During heating, three kinds of constraint

are selected for the strain/stress recovery: unconstrained strain recovery ( 4 ), pre-

strain constrained stress recovery ( 5 ) and fixed-strain constrained stress recovery

( 6 ). The later two are fully constrained recovery. In general, pre-deformed SMPs

can also be recovered under an elastic constraint, which is important for applica-

tions, and will be discussed at the end of this paper. In the following experimental

results, the thermomechanical response data will be presented using normalized tem-

perature, T/Tg. With Tg = 343 K, Tl = 273 K and Th = 358 K, we have Tl/Tg = 0.796and Th/Tg = 1.044.

Finally, we note that Fig. 2 shows the typical strain–stress–temperature relation-

ships for thermomechanical tests with tensile pre-deformation. In these tests, the

samples are tensioned and the ‘‘applied’’ strain states are always tension, while the

stress states may be compressive due to thermal expansion effects. As a convention,

tests with tensile pre-deformation are marked ‘‘tensioned’’ in the figures that follow.

Correspondingly, if a compressive pre-deformation is applied, the sample is macro-

scopically compressed and the test result is marked ‘‘compressed’’ without regard tothe tensile stress state that may develop due to thermal contraction. In order to

understand the thermal effect on the thermomechanical responses, tests without

pre-deformation are also performed. These test results are marked ‘‘undeformed,’’

even though tensile or compressive thermal stresses arise during the thermomechan-

ical cycle.

3.3. Isothermal static stress–strain tests and thermal strain

Fig. 3 provides the isothermal stress–strain responses in tension and compression

at Th = 358 K and Tl = 273 K. Since the pre-deformations were all conducted at Th,

the tests at Th represent the high temperature pre-deformation processes at the

beginning of the shape memory thermomechanical cycle, e.g. process 1 in Fig. 2.

On the other hand, the tensile test at Tl coincides with the low temperature unload-

ing, e.g. process 3 in Fig. 2. Fig. 3 shows that in both the glassy and the rubbery

states, the true stress–strain responses are linear, elastic and symmetric within the

small-strain ranges considered. By a linear fit, we can see that the modulus in theglassy state (750 MPa) is approximately two orders of magnitude larger than

the modulus in the rubbery state (8.8 MPa), which is consistent with the dynamic

mechanical results performed under three-point bending (Fig. 1).

3.4. Thermal strain and thermal expansion coefficient

Fig. 4 shows the unconstrained uniaxial thermal strain (thermal contraction and

expansion) in the temperature range of the thermomechanical cycle. Although not


evident from Fig. 2, this test is essential in understanding the thermomechanical

behavior of the polymer in a shape memory cycle, particularly during strain con-

straint. The thermal strain was measured during both cooling and heating, and

the curves overlap. Within the temperature range Tl to Th, the change in thermal

strain, eT, is approximately 1.15%. The thermal expansion coefficient a is definedas the slope of the thermal strain curve. The average thermal expansion coefficient

in the rubbery state (a = 1.8 · 10�4 K�1) is about two times of that in the glassy state

(a = 0.9 · 10�4 K�1). This phenomenon is caused by the decrease of the fractional

free-volume during cooling below Tg (Ward and Hadley, 1993; Matsuoka, 1992).

For an extremely slow cooling rate (�1 K per day), a should be constant at

T > Tg and T < Tg (Ward and Hadley, 1993). After fitting the thermal strain curve,

we have a ¼ deTdT .

3.5. Stress evolution during cooling under pre-strain constraint

Fig. 5 shows the evolution of stress as a function of temperature under pre-strain

constraint. The pre-strain, epre, is selected to be 9.1% for both pre-tension and pre-

compression. In Fig. 5, the evolution of the stress in the undeformed sample reflects

thermal stress generated under zero-strain constraint. The thermal stress during cool-

ing provides a positive contribution to the total stress, since the sample is attempting

to contract but cannot. The ‘‘undeformed’’ curve indicates that the evolution of thethermal stress is described by two stages. In the first stage, which is within the vicinity

of Tg, the thermal stress is very small. In the second stage, which corresponds to the

glassy state, the thermal stress rises rapidly. The transition between these two stages is

relatively smooth and gradual. Considering the decrease of the thermal expansion

coefficient in the glassy state (Fig. 4), the significant increase of the thermal stress

slope reflects the predominant effect of an increased modulus below Tg.

The stress evolution in the tensioned and compressed samples follows the same

trend as the pure thermal stress except that the starting stresses, rpre, are different(Fig. 5). In the tensioned sample, the thermal stress contribution raises the total

stress level starting from the pre-deformation stress. In the compressed sample, at

Th, the pre-deformation stress, rpre, is negative. With decrease of the temperature

and increase of the thermal stress, the total stress reaches zero and then becomes

increasingly positive.

Despite the different pre-deformation conditions (undeformed, tensioned and

compressed), during the cooling process, the constrained thermal shrinkage ulti-

mately produces tensile thermal stresses. However, if the compressive pre-deforma-tion stress (strain) is large enough, the sample may not achieve tensile stress upon

cooling. At Tl, the stress reaches rl. Following this state, the free strain recovery

or fully constrained stress recovery can be examined in the reheating process.

3.6. Free strain recovery after low temperature unloading

If low temperature unloading is performed after cooling, the fixed strain, efix, isclose to the pre-strain, epre (Fig. 2). For clarity, the unloading curves for the tests


(Fig. 5) are not shown since they would appear as vertical lines in stress–temperature

space. The value of efix is 8.6% for the tension test, �0.4% for the undeformed test

and �9.4% for the compression test, with instantaneous elastic unloading strains

of 0.5%, 0.4% and 0.3%, respectively. The elastic unloading strains are smaller than

the accumulated thermal strain eT = 1.15% (Fig. 4). This comparison implies that inaddition to the pre-deformation, some of the thermal stress-induced tensile confor-

mational changes are also stored during the cooling process. The stored strain that is

induced by thermal stress is called ‘‘additional strain.’’ In Fig. 6, starting from the

fixed strain, efix, free strain recovery is accomplished with a boundary condition of

zero stress and a constant heating rate. For the initially undeformed sample, the

recovery curve shows only the thermal effect during a thermomechanical cycle.

For pre-tensioned or pre-compressed sample, a sigmoidal-type free strain recovery

response is evident. It can be seen that despite the different pre-deformation states,the strain recovery paths of the pre-tensioned and pre-compressed samples are

similar. From Tl to near Tg, the gradual change of strain is dominated by thermal

expansion. Beyond Tg, the strain is rapidly recovered due to the release of both

pre-deformation strain and additional strain. At Th the original shape is fully recov-

ered and the strain converges to zero for all samples. Free strain recovery curves ini-

tiated at larger strains follow paths proportional to the curves presented in Fig. 6.

3.7. Stress response during heating under pre-strain constraint

After strain storage without unloading (‘‘C’’ in Fig. 2), the previous strain con-

straint level, epre, is maintained as the boundary condition. In Fig. 7, the stress recov-

ery process is carried out at constant rate heating. For the undeformed sample,

starting from the tensile stress, rl, the thermal stress decreases rapidly and the recov-

ery process is almost completed as T ! Tg. For the pre-tensioned and pre-com-

pressed samples, the results follow the same trend, except that the recovery

stresses gradually saturate to their initial levels at Th. The stress recovery pathsnearly overlay respective stress evolution curves during cooling under the same strain

constraint levels (Fig. 5), which are redrawn with dashed lines in Fig. 7. The overlay

of curves during cooling and heating reveals a near thermo-elastic process with min-

imal hysteresis effects. We note that previous work (Liu et al., 2003, 2004) has shown

significant hysteresis during cooling and subsequent heating under three-point flex-

ure. We believe that this hysteresis was mainly an artifact of the loading condition

(separation of the probe from the sample) rather than a real material effect.

3.8. Stress response during heating under fixed-strain constraint

After a low temperature unloading, the fixed strain, efix, slightly differs from the

previous pre-strain as indicated in Section 3.6. In Fig. 8, the fully constrained stress

recovery process is carried out with boundary conditions of fixed-strain constraint

(zero initial stress) and a constant heating rate. The results in Fig. 8 are important

since the starting points represent the initial states for the polymer deployment in

SMP applications. Starting from an unloaded state at zero stress (‘‘D’’ in Fig. 2),


the stress recovery path no longer monotonically decreases with increasing temper-

ature but shows a more complex evolution as a function of temperature. At temper-

atures far below Tg, due to constrained thermal expansion, compressive thermal

stress is generated with an increase in temperature. At certain temperatures below

Tg, the material becomes more compliant and the compressive stress ceases toincrease. It can be seen that both the maximum compressive stresses and the corre-

sponding temperatures are different for various samples (undeformed, pre-tensioned

and pre-compressed). With further increase in temperature, the stresses are rapidly

recovered. At Tg, the stresses are almost fully recovered to reflect the initial pre-

deformation stresses.

The experimental results in Section 3 create a foundation for understanding the

shape memory thermomechanics and provide insight into the relative importance

of the operant mechanisms. Based on the experimental results, a simplified small-strain three-dimensional model is developed in Section 4. When fit to a subset of

the experimental results, the constitutive model can capture the basic stress/strain

evolutions described above. We note that, as a first-order approximation, the model

is limited to small strain, linear elastic and rate-independent behavior. Future work is

needed to expand this model to a rate-dependent model. In addition, for low-

strength, high-strain capacity polymers, a large-strain framework is necessary.

4. A constitutive model for shape memory polymers

4.1. Model approach and development

Generally, there are two elastic deformation mechanisms in a polymer; entropic

conformational motion and the non-conformational motion corresponding to an

internal energy change (Bower, 2002; Matsuoka, 1992; Strobl, 1997; Ward and Had-

ley, 1993). The cross-linked epoxy does not readily flow, so permanent plastic defor-mation is precluded. The polymer chains between the cross-linking nodes can be

considered as entropy springs, which implies that the polymer chains accommodate

conformational rotation around C–C bonds. In the rubbery state, when subjected to

deformation, the entropy of the polymer will decrease. With a decrease of tempera-

ture and free-volume, the free conformational rotation of an individual C–C bond

will gradually transform into cooperative conformational rotation with its neigh-

bors, which requires much longer times (Matsuoka, 1992; Strobl, 1997). The inter-

molecular barrier to the entropic motions results in the viscoelastic/viscoplasticbehavior (creep or stress relaxation) and the rate effect of polymers (Arruda and

Boyce, 1993; Colak, 2005; Khan and Lopez-Pamies, 2002). In the viscoelastic state,

the stress response of a polymer will greatly depend on the strain rate or the fre-

quency of the dynamic strain. Furthermore, in a thermomechanical cycle, a shape

memory polymer will experience both a temperature effect and a time effect, which

can result in complex behavior. In this study, we focus on the shape memory

responses instead of the relaxation behavior of the polymer. We will use a simplified

phenomenological approach to model the temperature-dependent shape memory


behavior of a viscoelastic polymer. The temperature effect will be separated from the

time effect by fixing the cooling/heating rate, which is a typical processing approach

in shape memory polymer applications. In the future, rate (time) dependence can be

included in the constitutive model when necessary to predict experiments containing

time as a variable.Since the number of polymer chain segments involved in the cooperative confor-

mational rotation will increase with the decrease of the temperature for T < Tg, the

large-scale entropic changes will be prevented, and only the localized entropic mo-

tions occur in the polymer when a load is applied (Strobl, 1997). The terms ‘‘glassy’’

state and ‘‘rubbery’’ state are usually used referring to the material states in temper-

ature ranges of T < Tg and T > Tg, respectively. In order to avoid ambiguity and spe-

cifically quantify the material state, we define two kinds of idealized C–C bonds, the

‘‘frozen bond’’ and the ‘‘active bond,’’ which coexist in the polymer. The frozenbond represents the fraction of the C–C bonds that is fully disabled with regard to

the conformational motion, while the active bond represents the rest of the C–C

bonds that can undergo localized free conformational motions. Therefore, once

cooled to the glassy state, frozen bonds are predominant.

Fig. 9 shows a schematic of a simplified 3-D shape memory polymer model. At an

arbitrary temperature during the thermomechanical cycle, we assume the polymer

model is a mixture of two kinds of extreme phases: the ‘‘frozen phase’’ (dark shaded

region) and the ‘‘active phase’’ (light shaded region). The frozen phase (hard phase),which is composed of the frozen bonds, implies that the conformational rotation cor-

responding to the high temperature entropic deformation is completely locked

(stored), while the internal energic change such as the stretching or small rotation

of the polymer bonds can occur. In addition, any further conformational motion of

the material is impossible in the frozen phase. The frozen phase is the major phase

of a polymer in the glassy state. In contrast, the active phase (soft phase) of the

model consists of the active bonds, so the free conformational motion can potentially

Fig. 9. Schematic diagram of the micromechanics foundation of the 3-D shape memory polymer

constitutive model. Existence of two extreme phases in the polymer is assumed. The diagram represents a

polymer in the glass transition state with a predominant active phase.


occur and the polymer exists in the full rubbery state. According the model, with the

decrease of the temperature, large-scale conformational motions in the material are

gradually localized in the active phase, which is consistent with the microscopic

mechanism underlying the glass transition (Matsuoka, 1992; Strobl, 1997). By

changing the ratio of these two phases, the glass transition during a thermomechan-ical cycle is embodied and the shape memory behavior can be captured.

During cooling, the material is gradually frozen. In the 3-D model, the ‘‘frozen

fraction’’ (frozen volume fraction) and the ‘‘active fraction’’ (active volume fraction)

are defined as:

/f ¼V frz

V; /a ¼

V act

V; /f þ /a ¼ 1; ð1Þ

where V is the total volume of the polymer, Vfrz is the volume of the frozen phase

and Vact is the volume of the active phase. Note that the frozen and active volumes

account for the overall volume of the material, including the free-volume and the

volume occupied by the polymer chains. The change of the free-volume fraction

of the polymer with temperature is represented by the thermal expansion coefficient,which has been measured experimentally (Fig. 4). In this study, we take the macro-

scopic strain tensor e and temperature T as state variables. In addition, we define the

frozen fraction /f as a ‘‘physical’’ internal state variable that is related to the extent

of the glass transition and the state of the polymer. From Eq. (1), we have

/a = 1 � /f. Furthermore, we assume that under the boundary condition of a suffi-

ciently slow strain rate and the thermal condition of a slow constant heating/cooling

rate, /f and Vfrz are dependent only on temperature T:

/f ¼ /f Tð Þ; V frz ¼ V frz Tð Þ. ð2ÞAt certain temperatures, entropic changes can be frozen and stored ‘‘temporarily’’

after unloading; therefore, if the material has been strained at a high temperature,

/f(T) captures the fraction of strain storage as a function of temperature.

For a 3-D small-strain model, when the material is subjected to a stress tensor r,

the stress-induced strain tensor is e. In Eq. (3), we define the total stress r. Although

a material with two phases is inhomogeneous, we make the basic assumption that thecorresponding stresses in the two phases equal r:

r ¼ /frf þ ð1� /fÞra; rf ¼ ra ¼ r. ð3ÞWe define the total strain e to be:

e ¼ /fef þ ð1� /fÞea; ð4Þwhere ef is the strain in the frozen phase and ea is the strain in the active phase. In the

frozen phase, the entropic portion of the pre-deformation is assumed to be completely

locked and stored during cooling. Due to the localized freezing process, the entropic

frozen strain, eef , can be anisotropic inside the ever-growing frozen phase and should

be considered as a function of the position vector, x. An integral over the frozen frac-

tion volume should be used to sum the local contribution of eef . From the model, the

deformation of the frozen phase arises from three parts: the average of the frozen(stored) entropic strain, the internal energetic strain eif and the thermal strain eTf :


ef ¼1

V frz

Z V frz

0

eef ðxÞdvþ eif þ eTf . ð5Þ

In the active phase, the deformation is composed of two parts: the external stress-

induced entropic strains eea and the thermal strain eTa :

ea ¼ eea þ eTa . ð6ÞIncorporating Eqs. (5) and (6) into (4) we have:

e ¼ 1

V

Z V frz

0

eef ðxÞdv� �

þ /feif þ ð1� /fÞeea

� �þ /fe

Tf þ ð1� /fÞeTa

� �. ð7Þ

In Eq. (7), since the strain variables, eef , eif and eea, describe the local states in small

regions of a polymer, they are internal state variables. For simplicity we define the

first bracketed term on the right-hand side of Eq. (7) to be the ‘‘stored strain,’’ es,

which reflects the strain storage as a function of temperature and is often called a

‘‘history’’ variable. Letting d/ ¼ dvV represent a small volume fraction around the po-

sition x, the internal variables eef and /f can be incorporated into the evolution of thisnew internal variable es in a cooling or heating process:

es ¼1

V

Z V frz

0

eef ðxÞdv ¼Z V frz

0

eef ðxÞdvV

¼Z /f

0

eef ðxÞd/. ð8Þ

In Eq. (7), eif and eea are elastic strains. Since the glassy state yield behavior is

not considered and the non-linearity of rubbery elasticity (Boyce and Arruda,2000) can be ignored for small strains (Fig. 3), we assume that the material behaves

in a linear elastic manner in both the frozen and active phases. Thus, the internal

variables, eif and eea, can be related to the stress tensor through the Generalized

Hooke�s laws:

eif ¼ Si : r; eea ¼ Se : r; ð9Þwhere Si is the elastic compliance fourth-order tensor corresponding to the internal

energetic deformation, while Se is the elastic compliance fourth-order tensor corre-

sponding to the entropic deformation. Here, we define the second additive term

on the right-hand side of Eq. (7) to be the mechanical (elastic) strain em:

em ¼ /feif þ ð1� /fÞeea ¼ /fSi þ ð1� /fÞSeð Þ : r. ð10Þ

The total thermal strain eT defined by the third part of Eq. (7) can be expressed interms of thermal expansion coefficient a of the equivalent homogeneous material,

while a can be defined by the ‘‘rule of mixtures:’’

eT ¼/feTf þ ð1� /fÞeTa ¼

Z T

T 0

/faf hð ÞdhþZ T

T 0

ð1� /fÞaa hð Þdh� �

I ¼Z T

T 0

a hð Þdh� �

I;

a ¼/faf þ 1� /fð Þaa; ð11Þ

where af and aa are the thermal expansion coefficients of the frozen phase and the

active phase, respectively, and I is the identity tensor. Because the constitutive rela-

tion is derived for the thermomechanical cycle starting from Th, for the temperature


integral above we have T0 = Th. Using the definitions of es (Eq. (8)), em (Eq. (10)) and

eT (Eq. (11)), we simplify Eq. (7) as:

e ¼ es þ em þ eT. ð12ÞSubstituting this into Eq. (10) and rearranging, we obtain the overall constitutive

equation for the polymer in a thermomechanical cycle:

r ¼ /fSi þ ð1� /fÞSeð Þ�1: e� es � eTð Þ. ð13Þ

In the constitutive equation (13), only two internal variables es and /f remain. We

can see that the stress is dependent on a set of variables j: r = r(j) = r(e, es,T,/f).

From Eq. (13), the elastic stiffness fourth-order tensor of the equivalent homoge-

neous material is defined as:

C ¼ /fSi þ ð1� /fÞSeð Þ�1. ð14Þ

4.2. Thermodynamic considerations

In this study, our attention is focused on the stress–strain–temperature relations

of the shape memory polymer thermomechanical responses. Because the conceptsof internal energy and entropy are used to identify the deformation mechanisms,

the thermodynamic framework of the constitutive modeling is given in this section.

However the internal energy, entropy and the thermomechanical dissipation will not

be further discussed in detail. Since the frozen fraction is analogous to the martensite

fraction in shape memory alloys, we will follow the development of a shape memory

alloy constitutive model (Helm and Haupt, 2003) in formulating the free energy

function of the proposed shape memory polymer model.

From Eq. (12), the total strain e can be decomposed additively into three parts:an elastic strain em, a storable inelastic strain es and a thermal strain eT. Let us

now introduce the Helmholtz free energy (Helm and Haupt, 2003; Holzapfel,

2000): w(em, es,T,/f) = U � Tg where U is the internal energy, and g is the entropy.

Because em = e � es � eT, in fact, the free energy also depends on the set of vari-

ables j = (e, es,T,/f). The free energy function is used to describe the energy stor-

age in a polymer due to thermomechanical loading. Corresponding to the

deformation mechanisms of the shape memory polymer model described in

Section 4.1, the Helmholtz free energy is decomposed into two parts (Helm andHaupt, 2003):

wðe; es; T ;/fÞ ¼ wmTðem; T ;/fÞ þ wsðes; T Þ. ð15ÞThe first part, wmT, consists of the mechanical (elastic) energy, the thermal energy

and the initial (without strain storage) free energy of the material (Helm and Haupt,

2003; Holzapfel, 2000):

wmTðem; T ;/fÞ ¼1

q1

2C : em : em

� �þ cd0 ðT � T 0Þ � T log

TT 0

� �

þ wf0 þ 1� /fð ÞDw0ð Þ; ð16Þ


where q is the mass density, cd0 the specific heat at constant deformation and T0 the

reference temperature. In Eq. (16), the mechanical strain em includes the entropic

strain in the active phase and the internal energetic strain in the frozen phase. The

second bracketed term of Eq. (16) is the pure thermal contribution. The third brack-

eted term of Eq. (16) represents the initial free energy of the material. wf0 is the initialfree energy of the frozen phase, while Dw0 is the difference of the initial free energy

between the active phase and the frozen phase. Dw0 is temperature dependent and

can have a positive or negative value (Helm and Haupt, 2003). In Eq. (15), the sec-

ond part of the free energy, ws, corresponds to the entropic strain storage in the fro-

zen phase due to an internal stress field. We assume ws is a quadratic function of es(Helm and Haupt, 2003; Holzapfel, 2000):

wsðes; T Þ ¼1

2qCs : es : es; ð17Þ

where Cs is a fourth-order temperature-dependent elasticity tensor governing the

storage strain and the internal stress tensor.

The second law of thermodynamics applied in the form of the Clausius–Duhem

inequality is given as (Holzapfel, 2000):

r : _e� q _wþ g _T� �

� q � rTT

P 0; ð18Þ

where q is the heat flux vector. Introducing the strain decomposition (Eq. (12)) and

the free energy function (Eq. (15)) into the inequality, we obtain:

r� qowmT

oem

� �: _em þ r : _eT � q

owoT

þ g

� �_T

þ r� qows

oes

� �: _es � q

owmT

o/f

_/f � q � rTT

P 0. ð19Þ

Since the inequality must be satisfied for any arbitrary thermomechanical process,

one gets the potential relations for the stress tensor and the entropy (Lion, 1997):

r ¼ qowmT

oemand g ¼ � ow

oTþ 1

qr :

deT

dT. ð20Þ

Note that the general form of the constitutive equation for the stress in Eq. (20) is

equivalent to Eq. (13). Furthermore, the change of entropy of a thermomechanical

process can be determined if the free energy w as a function of temperature is known.

The internal stress field rs, which stores the entropic strain in the frozen phase (Helm

and Haupt, 2003), can be defined as:

rs ¼ qows

oes. ð21Þ

Then the Clausius–Duhem inequality simplifies into:

r� rsð Þ : _es � qowmT

o/f

_/f � q � rTT

P 0. ð22Þ


The internal dissipation is obtained as dm ¼ ðr� rsÞ : _es � q owmT

o/f

_/f . In shape

memory polymers, the inelastic storage strain es can be decomposed into two parts:

the stress induced rate dependent storage strain and the temperature induced rate

independent storage strain. It can be understood that the driving force for the rate

evolution of the storage strain, _es, is the difference between the external and the inter-nal stress tensors, (r � rs). In our present model, we do not consider the rate depen-

dence and assume that the storage strain es depends only on temperature. Therefore,

we have _es ¼ 0, which is consistent with the experimental observation from Fig. 7,

i.e., the shape memory response at a sufficiently slow constant cooling/heating rate

is a near thermo-elastic process with a minimal hysteresis effect. From Eq. (22),

the driving force for the evolution of the frozen fraction, _/f , is � owmT

o/f, which is the

difference of the initial free energy between the active phase and the frozen phase,

Dw0 (Eq. (16)). The dissipation inequality, dm P 0, leads to the restriction thatDw0 has the same sign as _/f .

In Eq. (22), the thermal dissipation is given by dT ¼ �q � rTT . By the Fourier law,

one defines q = �k$T with k > 0, so that the heat conduction inequality is valid:

dT P 0. As a result, the Clausius–Duhem inequality is satisfied and the proposed

constitutive model in Section 4.1 is thermomechanically admissible for an arbitrary

thermomechanical process.

4.3. Evolution equations for internal variables and material properties

From the model in Fig. 9, at a certain temperature T, the frozen volume is Vfrz(T).

The frozen entropic strain eef ðxÞ describes the internal entropic strain state of the

polymer as a function of position x inside Vfrz. On the other hand, the entropic strain

eeaðT Þ in the active phase is governed by the Hooke�s law (Eq. (9)). During cooling

with a change of temperature dT, a fraction of the active material d/ at x(T) will

be frozen and transform into a new fraction of the frozen material. As a result,

the current active entropic strain eeaðxðT ÞÞ is fixed as the stored strain eefðxÞ. Thuswe have drawn a relation between the position vector and the temperature history

in the sense that the frozen strain at position x is converted from the active strain

at temperature T. Consequently we deduce that:

eef ðxÞ ¼ eeaðxðT ÞÞ ¼ eeaðT Þ ¼ Se Tð Þ : r Tð Þ. ð23ÞWith the definition of es in Eq. (8) and the constitutive relationship in Eq. (13) we

have the temperature derivative of the stored strain as a function of temperature:

des

dT¼eef ðxÞ

d/f

dT¼ Se Tð Þ : r Tð Þ d/f

dT

¼Se Tð Þ : /fSi þ ð1� /fÞSeð Þ�1 : e� es � eTð Þ d/f

dT. ð24Þ

It can be seen that during cooling, the strain-storage also depends on the set of

variables j: desdT ¼ gðjÞ ¼ gðe; es; T ;/fÞ. Given /f, Si and Se, if a strain boundary con-

dition e is prescribed for a cooling process, the stored strain as a function of temper-

ature, es(T), can be solved from the first-order differential equation (Eq. (24)). Thus,


es(T) reflects the strain storage history during cooling. During the subsequent heat-

ing, with the decrease of /f, the stored strain will be released in a process, which is

the reverse of the strain storage. The change of es during heating inherits the resul-

tant es(T) determined previously during the cooling process. During heating, the

internal variable es is assumed to depend only on the current temperature, whilethe current strain and stress is assumed not to affect the strain releasing process:

es ¼ es Tð Þ. ð25ÞIn a thermomechanical cycle, the strain storage and release is controlled by the

change of /f, which represents the temperature/time-dependent glass transition of

the polymer model. In this study, we hold the cooling/heating rate to a constant va-

lue and assume that /f depends only on temperature. In order to characterize /f, thefree strain recovery result of the uniaxial compression test (Fig. 6) is used. If the

internal energetic strain and the thermal strain are omitted, /f is in fact the ratio

of scalar es to epre. In the free strain recovery test, the measured uniaxial strain e isthe strain retained during the recovery process, which qualitatively reflects the

change of the stored strain es. If we assume that thermal expansion is not affected

by the pre-deformation, it is possible to subtract the thermal strain from the total

strain e, and obtain a ‘‘modified recovery strain’’ e* which reflects the real strain stor-

age level. To this end, e*/epre gives the path of /f as a function of temperature. Basedon the experimental result, for the frozen fraction /f, we propose a phenomenolog-

ical function of temperature with two variables, cf and n, which can be determined by

fitting the strain ratio data, e*/epre:

/f ¼ 1� 1

1þ cf T h � Tð Þn ¼e�

epre. ð26Þ

To apply the model in predicting the uniaxial test results described in Section 3,

mechanical properties of the material should be specified. Based on the uniaxial

strain–stress test (Fig. 3), we can give 1-D forms for the elastic moduli of the polymermodel. From the definition of the 3-D elastic stiffness in Eq. (14), the Young�s mod-

ulus E of the polymer is given as:

E ¼ 1

/f

Ei

þ 1� /f

Ee

; ð27Þ

where Ei is the modulus corresponding to the internal energetic deformation and Ee

the modulus corresponding to the entropic deformation. We assume Ei is a constant

in the temperature range considered. According to the theory of rubber elasticity, thestress is a nonlinear function of deformation at large extensions (Boyce and Arruda,

2000). However in this study, the maximum strain is limited to 10% and it is in the

linear range (Fig. 3). From the rubbery elasticity of a network polymer (Ward and

Hadley, 1993), Ee is a linear function of absolute temperature and becomes zero

at 0 K. So we have:

Ei ¼ constant; Ee ¼ 3NkT ; ð28Þ


where N is the cross-link density and k is Boltzmann�s constant

(k = 1.38 · 10�23 N m/K). Eq. (26) specifies the frozen fraction /f as a function of

temperature. If we also know the values of Young�s modulus E at Tl and Th, then

Ei, Ee and E as functions of temperature can be obtained. The cross-link density

of the polymer can be subsequently calculated from Eq. (28).In summary, the 1-D small-strain constitutive equations and material property

functions of the shape memory polymer model are listed below.

Constitutive equation : r ¼e� es �

R TT ha dT

/f

Ei

þ 1� /f

Ee

¼ E e� es �Z T

T h

a dT� �

; ð29Þ

Temperature derivative of stored strain :desdT

¼e� es �

R TT ha dT

Ee

/f

Ei

þ 1� /f

Ee

� � d/f

dT

� �.

ð30ÞFrozen fraction: /f ¼ 1� 1

1þcf ðT h�T Þn

Young�s modulus: E ¼ 1/fEiþ1�/f

Ee

Modulus of the internal energetic deformation: Ei = constant

Modulus of the entropic deformation: Ee = 3NkT

Coefficient of thermal expansion: a ¼ deTdT

Based on a subset of the uniaxial test results including one free strain recovery test

and two isothermal strain–stress tests, in the next section we will derive the material

parameters /f, Ei and Ee as functions of temperature. Subsequently, the uniaxial

stress evolution of the shape memory polymer under various thermomechanical con-

ditions will be predicted.

5. One-dimensional modeling results and discussion

5.1. Fitting and determination of material parameters

First, we determine the frozen fraction function /f(T). Since we assume the phys-

ical internal variable /f depends only on temperature, it can be treated as a macro-scopic material property unaffected by the strain and stress state of the material. The

uniaxial free strain recovery curve of the pre-compressed sample and the thermal

strain curve were used to derive /f (Fig. 10). Comparison of these two curves sug-

gests that the low temperature part of the strain recovery curve mainly reflects ther-

mal expansion. Therefore, we assume that in a free strain recovery test, the

macroscopic recovery strain e arises from two sources: the thermal expansion strain,

eT, and the ‘‘actual’’ release of the stored strain. In order to capture the magnitude of

the actual strain recovery during heating, we define the modified recovery strain, e*in Eq. (26), to be:

-10

-8

-6

-4

-2

0

Stra

in (

%)

1.051.000.950.900.850.80

T/Tg (K/K)

Free recovery strain(Compressed)

Thermal strain

Modified recovery strain, e∗

Fig. 10. Free recovery strain, thermal strain and modified recovery strain for fitting of the frozen fraction

that is a physical internal state variable.


e� ¼ c e� eTð Þ; c ¼ epree T lð Þ � eT T lð Þ . ð31Þ

In this study, c is calculated to be c = 1.1. Furthermore, we note that e*(Tl) = epre,which indicates that the pre-deformation strain is completely stored at Tl. The result-

ing modified recovery strain curve is also shown in Fig. 10.

Using Eq. (26), we can fit /f(T) to the modified strain recovery curve divided by

the pre-strain, e*/epre (Fig. 11). Although the e*/epre curve has a sigmoidal shape, the

sigmoidal function does not fit the curve especially well. Instead, if we assign n = 4,

the nonlinear power law function (Eq. (26)) fits the curve very well. In Fig. 11, the

key feature of the /f(T) curve is that the storage/recovery occurs in the vicinity ofTg. When a material is cooled from Th, the frozen fraction /f rapidly rises from 0

to 0.8 at Tg. At temperatures far below Tg, /f flattens and has a limiting value at

1. Thus, the two limiting cases are /f(Th) = 0 and /f(Tl) = 1. The frozen fraction

curve mimics the phase fraction in other transformations such as a martensitic phase

change in polycrystalline shape memory alloys (Gall and Sehitoglu, 1999; Helm and

Haupt, 2003).

Substituting Eq. (28) into Eq. (27), the Young�s modulus of the polymer at an

arbitrary temperature can be rewritten as:

E Tð Þ ¼ 1

/f

Ei

þ 1� /f

3NkT

. ð32Þ

From the uniaxial strain–stress test results (Fig. 3), we know two extreme values

of the modulus: E(Th) = 8.8 MPa and E(Tl) = 750 MPa. Using /f(Th) = 0 and

/f(Tl) = 1, two equations based on Eq. (32) are solved together, so Ei and the

1.0

0.8

0.6

0.4

0.2

0.0

Froz

en f

ract

ion,

f

1.051.000.950.900.850.80

T/Tg (K/K)

Fitting of the frozen fraction, f

pre

Fig. 11. Frozen fraction, /f, as a function of temperature, derived from curve fitting of the modified

recovery strain curve divided by the pre-deformation strain.


cross-link density N can be obtained. The three moduli, E, Ei and Ee are plotted in

Fig. 12. Since only the modulus endpoints were used in fitting the model parameters,

the temperature dependence of the modulus in Fig. 12 is a prediction of the model.

100

101

102

103

104

Mod

ulus

(M

Pa)

1.051.000.950.900.850.80T/Tg (K/K)

Ei

E

Ee

Fig. 12. Prediction of the elastic moduli of the polymer as functions of temperature. Ei is the modulus for

internal energic deformation; Ee is the modulus for free entropic deformation.

Table 1

Values of coefficients

n 4

cf (1/K4) 2.7610�5

Ei (MPa) 813

N (mol/cm3) 9.86 · 10�4

a (K�1) �3.16 · 10�4 + 1.42 · 10�6 T


Although E does not precisely match the response of the storage modulus (Fig. 1),

the characteristics of these two curves are in qualitative agreement. We did not fit

the storage modulus curve since the dynamic mechanical test was performed under

three-point flexure at a relatively high frequency. If the modulus values are deter-

mined by careful uniaxial experiments at various temperatures, it is possible to fit

the frozen fraction to these data rather than the free strain recovery. However, we

found it more appropriate to fit the free strain recovery since this response was ob-

tained via a more quantitative testing method in our experiments, and is a much sim-pler test to conduct. The coefficients, n, cf, Ei and cross-link density N determined for

the model are listed in Table 1. For the sake of completeness, the thermal expansion

coefficient a(T) is also listed in the table.

5.2. Prediction of the strain storage and stress response during cooling

Using the 1-D constitutive equation, the temperature derivative of the stored

strain and the material properties determined for the shape memory polymer, thestrain recovery responses are reproduced and the stress responses are predicted in

following sections. Selected uniaxial experimental results and modeling results of

the stress/strain responses under various thermomechanical conditions are shown

in Figs. 13–16.

The thermomechanical cycle of the shape memory polymer starts at Th, in the rub-

bery state. Subjected to an external stress, the material undergoes an elastic pre-

deformation, resulting in both entropic strain and internal energetic strain changes.

Since /f(Th) = 0, the initial condition for the stored strain is es(Th) = 0. In the subse-quent cooling process, the pre-deformation strain is held as a boundary condition:

e = epre. By solving the first-order differential equation (Eq. (30)) with the above ini-

tial and boundary conditions, the evolution of stored strain, es(T), as a function of

temperature is obtained. According to the constitutive relationship in Eq. (29), the

stress response during cooling, r(T), under pre-strain constraint can be derived as:

r ¼ Eðepre � es �R TT ha dT Þ. The predictions of the stress responses for different sam-

ples (undeformed, pre-tensioned and pre-compressed) are shown in Fig. 13. The pre-

dictions show qualitative overall agreement with the experimental data.We now discuss the change of the material state during the fully constrained

cooling process. In the first stage, with the decrease in temperature from Th and

the increase of /f, the conformational positions corresponding to the pre-strain

are gradually frozen. Because the material is relatively compliant (low E), the

10

8

6

4

2

0

-2

Stra

in (

%)

1.051.000.950.900.850.80

T/Tg (K/K)

Tensioned

Undeformed

Experimental result Model prediction

Fig. 14. Prediction of the free strain recovery responses during heating for polymers pre-deformed at

different levels.

4

3

2

1

0

-1

Stre

ss (

MPa

)

1.051.000.950.900.850.80

T/Tg (K/K)

Experimental resultModel prediction

Tensioned, = 9.1%


Undeformed, = 0

Fig. 13. Predictions of the stress responses of the polymer during cooling under different pre-strain

constraint conditions.


thermal stress is insignificant. As /f converges to 1, at temperatures in the range of

1.00–0.95Tg, further conformational motion is no longer possible and the materialstiffens (high E). The thermal stress then begins to accumulate noticeably as marked

4

3

2

1

0

-1

Stre

ss (

MPa

)

1.051.000.950.900.850.80T/Tg (K/K)

Experimental resultModel prediction

Tensioned, = 9.1%

Undeformed, = 0


Fig. 15. Prediction of the stress recovery responses during heating for polymers under different pre-strain


3

2

1

0

-1

-2

Stre

ss (

MP

a)

1.051.000.950.900.850.80

T/Tg (K/K)

Experimental result Model prediction

Tensioned, = 8.6%


Undeformed, = 0

Fig. 16. Prediction of the stress recovery responses during heating for polymers under different fixed-strain



by an increase in the total stress. The final stresses are tensile for three different

samples (Fig. 13). Note that some of the thermal stress induced conformational mo-

tions are incorporated into the stored strain as additional strain.


After the strain storage process, low temperature unloading produces a small

‘‘springback,’’ whereby Hooke�s law at the low temperature governs the magnitude

of this strain. From the constitutive equation (Eq. (29)), the resulting fixed strain is

calculated as:

efix ¼ epre �r T lð ÞE T lð Þ ¼ es T lð Þ þ eT T lð Þ; eT T lð Þ ¼

Z T l

T h

a dT .

5.3. Reproduction of the free strain recovery for undeformed and pre-tensioned sample

The experimental result for free strain recovery of a pre-compressed sample has

been used to fit /f. In Fig. 14, the experimental and modeling results for undeformed

and pre-tensioned samples are shown. After unloading, e(Tl) = efix. The boundary

condition is r = 0. During the stress-free heating, Tl ! Th, strain is released from

the stored strain, es(T), which is gained during the cooling process. It follows from

the constitutive equation (Eq. (29)) that during heating, the recovery strain, e(T),can be calculated as: e ¼ es þ

R TT ha dT . This equation shows that the total recovery

strain of a pre-tensioned sample is composed of two parts. Because /f does not

change much at low temperatures, only the thermal strain appears before the actual

strain recovery initiates (Fig. 14). The stored strain, es, begins to release at high tem-

peratures corresponding to the significant change of /f. At Th, the stored strain and

the overall thermal effect reach zero.

5.4. Prediction of the stress recovery under pre-strain constraint

The experimental and modeling results of stress recoveries without unloading are

shown in Fig. 15. The boundary condition is e = epre. The stored strain, es(T) is accu-mulated during the cooling process and then released during heating. Eq. (9) is used

to calculate the stress response, r(T), which follows the same trend as that of the

fully constrained cooling: r ¼ Eðepre � es �R TT ha dT Þ. Since the model does not ac-

count for any material hysteresis, the predicted cooling (Fig. 13) and heating (Fig.

15) curves perfectly overlay. In reality the material shows some hysteresis, which is

evident in the experimental data of heating versus cooling (Fig. 7).

5.5. Prediction of the stress recovery under fixed-strain constraint

If unloading is carried out at Tl, the subsequent boundary condition during heat-

ing changes to the fixed-strain constraint: e = efix, while es(T) is inherited from the

cooling process. From Eq. (29), the stress recovery response is: r ¼ Eðefix � es �R TT ha dT Þ.The experimental and modeling results of fully constrained stress recovery after

low temperature unloading are shown in Fig. 16. Again, at low temperatures, the

thermal effect dominates; at high temperatures, the stresses gradually recover to

the pre-deformation levels. The ability of the model to reasonably predict the


relatively complex shape of the thermomechanical experimental curves in Fig. 16

suggests that the overall modeling approach is promising.

5.6. Prediction of the 1-D strain and stress recovery under flexible constraint

Thus far, we have used the constitutive model to predict the basic shape-memory

thermomechanical cycles under full constraint. In reality, a shape memory polymer is

typically pre-deformed, maintained at a low temperature, and then utilized in recov-

ery in an application environment. For example, in a minimally invasive surgical

application (Metzger et al., 2002), the pre-compacted shape-memory polymer coil

is taken out from the low temperature storage environment and then inserted into

an artery without deforming the artery at this point. During the temperature-con-

trolled recovery process, the polymer is subjected to a flexible constraint from thesurrounding tissue, and a reactive compressive stress is produced inside the polymer

device. In contrast, when a pre-stretched suture is used to close a skin laceration

(Lendlein and Langer, 2002), a reactive tensile stress in generated. These situations

suggest the use of 1-D bi-material models, in which two materials, the pre-deformed

shape memory polymer and a constraining material, are assembled in parallel (Fig.

17(a) and (b)) or serially (Fig. 17(c)), as long as the small-strain condition is satisfied.

In Fig. 17(a) and (b), the stresses in the two materials are opposite while the strain

change in the polymer and the strain in the constraining material are equal. In con-trast, for the serial arrangement (Fig. 17(c)), the stresses in the two materials are

equal while the changes in strain are opposite. Despite the difference in the strain/

stress direction, the two models are basically equivalent. Consequently, only the

‘‘parallel’’ cases are discussed below.

We now wish to examine the effect of constraining material modulus on the recov-

ery of the shape memory polymer. For simplicity we assume that the shape memory

polymer and the constraining material maintain the same cross-section area and

length during recovery. For the pre-deformed polymer at Tl, the fixed strain is efixand the length equals the original length of the undeformed constraining material,

Lc; this can be expressed using the original length of the polymer,L, asLc = L(1 + efix).

Fig. 17. Simplified models for prediction of the strain and stress recovery response of pre-deformed shape

memory polymer under flexible constraint. Dark shaded parts represent shape memory polymers; light

shaded parts represent flexible constraint materials.


During recovery, the change of displacement is Dd. Then the strain change in the poly-

mer is De ¼ DdL . The total strain in the polymer equals the pre-deformed strain plus the

strain change: e = efix + De. If the thermal expansion of the constraining material is

omitted, the total strain in the constraining material can be derived as

ec ¼ DdLc¼ Dd

LLLc¼ De

1þefix¼ e�efix

1þefix. We assume that the modulus of the constraining mate-

rial, Ec, is a constant within the temperature range of interest. From equilibrium,

the constitutive equation governing the recovery process becomes:

r ¼ E e� es �Z T

T h

a dT� �

¼ �Ec

e� efix1þ efix

.

In Figs. 18–21, both the strain and stress recovery are simulated for various con-

straining moduli, Ec, normalized with regard to the high temperature modulus of the

shape memory polymer, Eh = E(Th). Figs. 18 and 19 predict the recovery responsesof pre-compressed materials, while Figs. 20 and 21 predict the recovery responses of

the pre-tensioned materials under various flexible constraints. The high-temperature

pre-strains are selected to be �9.1% and +9.1%, respectively. For generalization, the

recovery strain and stress are normalized with respect to the absolute values of the

pre-deformation strain and stress. The dashed lines in the figures represent the ex-

treme cases of the recovery: unconstrained free strain recovery or fully constrained

stress recovery under fixed-strain constraint, which have been described in detail

in Sections 3.6 and 3.8. We observe that with an increase in magnitude of the con-straining modulus, the response of the polymer transforms from free strain recovery

to fully constrained stress recovery. The thermal expansion effect at low tempera-

tures increases with an increase of the constraining stiffness.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

ε/|ε

|pr

e

1.051.000.950.900.850.80

T/Tg (K/K)

Ec/Eh= 0.1

Ec/Eh= 1

Ec/Eh= 10

Ec/Eh= 100

Free strain recovery

Pre-compressed SMP

Ec = 0

Ec/Eh= 1000

Fig. 18. Prediction of the strain response of pre-compressed shape memory polymers under flexible

constraint with various modulus levels.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

σ/|σ

pre|

1.051.000.950.900.850.80

T/Tg (K/K)

Ec/Eh= 0.1Ec/Eh= 1

Ec/Eh= 10

Ec/Eh= 100

Ec/Eh= 1000

Fully constrained stress recoveryEc >> Eh

Pre-compressed SMP

Fig. 19. Prediction of the stress response of pre-compressed shape memory polymers under flexible


1.0

0.8

0.6

0.4

0.2

0.0

pre|

|

1.051.000.950.900.850.80

T/Tg (K/K)

Ec/Eh= 0.1

Ec/Eh= 1

Ec/Eh= 10

Ec/Eh= 100

Ec/Eh= 1000

Free strain recovery

Pre-tensioned SMP

Ec = 0

Fig. 20. Prediction of the strain response of pre-tensioned shape memory polymers under flexible



The results in Figs. 18–21 constitute a tool for the design of uniaxially pre-

deformed deployable shape memory polymer devices. Given the modulus of the con-

straining material and the rubbery modulus of the shape memory polymer, it is

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

σ/|σ

pre|

1.051.000.950.900.850.80

T/Tg (K/K)

Ec/Eh= 100

Ec/Eh= 1000

Ec/Eh= 1

Ec/Eh= 0.1

Ec/Eh= 10

Fully constrained stress recoveryEc >> Eh

Pre-tensioned SMP

Ec/Eh= 10

Fig. 21. Prediction of the stress response of pre-tensioned shape memory polymers under flexible



possible to predict how much relative stress (force) and strain (displacement) will re-

sult. Depending on the level (relative modulus) of constraint, the shape recovery re-

sponse transforms from perfect free strain recovery to perfect constrained stress

recovery. Of course, the results in Figs. 18–21 only hold for samples with equivalent

sizes (cross-sectional area and length), and would need to be scaled appropriately if

the dimensions of the shape memory polymer were different than those of the con-

straining material. Given the assumed linear nature of rubbery elasticity in the model,the results show that the ultimate force or displacement, which is reached after

deployment, scales linearly with the elastic modulus ratio of the two materials.

For example, if the stiffness of the constraining material is equal to the stiffness of

the shape memory polymer in the rubbery state, the polymer will only recover 1/2

its total strain (Figs. 18 and 20) and the stress generated will be 1/2 of the total ap-

plied stress during pre-deformation (Figs. 19 and 21). Furthermore, the bi-material

models can simulate the effect of the constraint modulus on the evolution of the con-

strained stress as a rather complicated function of temperature (Figs. 19 and 21).

5.7. Additional insights

From the model development it can be seen that the two internal variables, /f and

es, are vital components of the shape memory constitutive relationship. Based on the

internal variables and the physics of the glass transition, the shape storage and the

thermal effect underlying the thermomechanical responses are reasonably repro-

duced by the model. In this study, under a slow constant cooling/heating rate, /f

is treated only as a function of temperature. In fact /f is also time/rate dependent,


which should be investigated in the future to develop predictive capacity at different

cooling/heating rates.

The experimental and modeling results have important implications for future re-

search efforts and emerging applications. In reality, the shape memory (recovery) al-

ways occurs under complicated environmental constraints, e.g. surrounding tissuesin biomedical applications. The prediction of the stored strain and the releasing pro-

cess as a function of temperature/time under various strain/stress constraint condi-

tions is crucial for successful design and control of SMP active devices. The

simulation results (Figs. 18–21) indicate that the constitutive model can be expanded

to predict the recovery of pre-deformed shape memory polymers under various flex-

ible external constraints. Polymer materials generally have properties that lie be-

tween those of hard and soft biological tissues (Feninat et al., 2002). Subjected to

a strain less than 10%, the modulus of an artery is �0.3 MPa, which is equivalentto Ec/Eh = 0.03 for the shape memory polymer used in this study. If a pre-compacted

polymer device and the surrounding artery wall have an equal cross-section area, an

almost free strain recovery will occur accompanied with a very small recovery stress

in the polymer and the tissue (Figs. 18 and 19). The example shows that the deploy-

ment process and functionality of SMP devices can be assessed and designed by con-

stitutive modeling.

6. Conclusions

A 3-D constitutive model to predict the thermomechanical behavior of shape

memory polymers is proposed. The model reasonably captures the essential elements

of the shape memory responses. The model simplifies the classical viscoelastic or

pseudo-plastic problem and transforms it into a special elastic problem under the

condition of a constant cooling/heating rate. For completeness, the free energy func-

tion of the model is developed, which is compatible with the Second Law of thermo-dynamics in the sense of the Clausius–Duhem inequality. The thermodynamics

(changes in internal energy, entropy and dissipation) of the thermomechanical cycle

is discussed.

The development of the model is motivated by the shape memory mechanism of

the polymer network. The foundation of the modeling approach is that the entropic

strain energy is gradually stored during cooling and released during reheating as free

recovery strain or constrained recovery stress. With the ‘‘freezing’’ of the conforma-

tional motions, the polymer undergoes the glass transition and becomes significantlystiffer. As a function of temperature, the frozen fraction (an internal state variable)

captures the micromechanics and physics underlying the glass transition and the

shape memory behavior of the polymer. Based on the 1-D experimental results,

the frozen fraction function is developed using a phenomenological approach. When

fit to the free strain recovery curve, the proposed constitutive model can predict the

uniaxial stress responses under various thermomechanical conditions. Insights devel-

oped from the model include:


(i) During a thermomechanical cycle, both the initial high temperature pre-defor-

mation and the final recovery state are governed by the rubbery elasticity of the

shape memory polymers.

(ii) The stored strain, a phenomenological internal state variable, reflects the defor-

mation history during cooling and serves as an initial condition for the laterpart of the thermomechanical cycle (unloading and reheating).

(iii) The energy releasing during heating is exactly the reverse process of energy

storage during cooling, with minimal mechanical dissipation.

(iv) The recovery of strain occurs in the vicinity of the glass transition when pre-

deformed at T > Tg.

(v) For a uniaxial test, due to the high stiffness in the glassy state, the evolution of

the stress under strain constraint is strongly influenced by thermal expansion of

the polymers.(vi) For a uniaxial test, the ‘‘springback’’ at a low temperature is mainly caused by

the thermal effect. Thus, elastic springback should not be used to assess the

shape fixity ratio (stored strain versus pre-deformation strain) of a shape mem-

ory polymer.

(vii) The constitutive model can be used to predict the strain and stress recovery

responses of pre-deformed shape memory polymers under a flexible external

constraint with various compliance levels.

Acknowledgements

The work is funded by the National Science Foundation, DMII, Nanomanufac-

turing, Grant number DMI-0200495. The authors gratefully thank Mark Lake and

Naseem Munshi of Composite Technology Development (CTD) in Lafayette, Colo-

rado for their supply of the CTD-DP5.1 Epoxy System. Dr. Diani thanks the DGAfor financial support, Grant number ERE-046000011.

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thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling

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