Composites: Part A 64 (2014) 132–138

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Composites: Part A

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Thermal mechanical constitutive model of fiber reinforced shapememory polymer composite: Based on bridging model

http://dx.doi.org/10.1016/j.compositesa.2014.05.0031359-835X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel./fax: +86 451 86402328.E-mail address: lengjs@hit.edu.cn (J. Leng).

Qiao Tan a, Liwu Liu b, Yanju Liu b, Jinsong Leng a,⇑a Centre for Composite Materials and Structures, Harbin Institute of Technology (HIT), No. 2 YiKuang Street, PO Box 3011, Harbin 150080, People’s Republic of Chinab Department of Astronautical Science and Mechanics, Harbin Institute of Technology (HIT), P.O. Box 301, No. 92 West Dazhi Street, Harbin 150001, People’s Republic of China

a r t i c l e i n f o

Article history:Received 7 January 2014Received in revised form 3 May 2014Accepted 6 May 2014Available online 16 May 2014

Keywords:A. Polymer-matrix composites (PMCs)B. ThermomechanicalC. Computational modeling

a b s t r a c t

The ever increasing applications of Shape Memory Polymers (SMPs) and its Composites (SMPCs) havemotivated the development of appropriate constitutive models. In this work, based on composite bridg-ing model, a constitutive model for unidirectional SMPCs under thermal mechanical loadings in the smallstrain range has been developed. The composite bridging model has been adopted to describe the distri-bution of stress–strain between fiber reinforcement and SMPs matrix. Besides, considering the influenceof fiber content and temperature, the storage and release of ‘‘frozen strain’’, the recovery of stress hasbeen quantified as well. The stress–strain curves of SMPCs laminate under axial tensile indicate thatthe theoretical data derived from the developed model are basically accordant with the experimentaldata, and that the proposed model is suitable for machining practice. Furthermore, the model has beenapplied to predict stress recovery, strain storage and releasing with changing of temperature.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Shape Memory Polymer and its Composites (SMPCs) are a classof smart materials that have the ability to return from a deformedstate to their original shape at an external stimulus, such as heat,light, electricity and so forth [1–12]. SMPCs consist of a ‘stiff’ rein-forcement and a ‘softer’ matrix arranged together with one phasedistributed as an array inside of the other phase. The best knownexample of this type of composite is fibrous composite, wherethe ‘stiff phase’ is usually long thin fibers, graphite, glass or boron[13,14]. This type of composite owns many outstanding mechani-cal properties, such as high stiffness, strength and thermo-elasticstability, good corrosion resistance and thermal insulation[15–20]. Besides, for the stiffness and the strength of the carbonfiber are much higher than SMP matrix, the obtained compositegenerates a higher value of stress during process of the deforma-tion recovery, and this is a desirable feature in engineeringapplications.

Because of these advantages, SMPCs are widespread in manybranches of engineering within areas of applications ranging frombiomedical devices, aircrafts, ships, and space vehicles to bridges,buildings, and storage vessels [21,22]. Designing of SMPCs baseddevices involves the need for strain–stress prediction to achieve

a satisfactory performance under the required conditions [23].Thus it is necessary to develop appropriate constitutive modelsthat can be used as simulation tools to assist the designs of SMPCsbased structures.

A number of researchers have described the thermo-mechanicalbehavior of the SMPs and its composites [24,25]. The modelingefforts have split into several directions. In one direction, the shapememory effect is attributed to the formation of phases during cool-ing which serves to lock the temporary shape and a loss of thephases due to heating when shape recovery occurs. Models basedon this concept include the 1D model by Liu et al., 3D models byBarot et al. for semicrystalline SMPs, Qi et al. and Chen and Lagou-das for general SMPs [26–35]. Besides, some earlier modelingefforts have adopted rheological models consisting of spring, dash-pot, and frictional elements in constitutive models [36]. Descrip-tion of micro-scale features, such as cross-linking, chain mobility,interface motion and entanglement of polymer molecules, is thethird direction of SMPs modeling approaches, which includingrecent models by Nguyen et al., Castro et al. and so forth [37,38].

Although these models have improved the understanding ofSMPs and its composites, more complicated models are neededto predict the thermo-mechanical characteristic of fiber reinforcedSMPCs. However, a little research about the constitutive relation-ship of unidirectional fiber reinforced SMPCs have been done. Inthis study, based on composite bridging model theory and the con-tinuum thermodynamic considerations, a 3D constitutive model

Fig. 2. Stress–strain diagram illustrating the thermo-mechanical behavior of shape

Q. Tan et al. / Composites: Part A 64 (2014) 132–138 133

has been developed to simulate the stress–strain–temperaturerelationship of unidirectional elastic carbon fiber reinforced SMPCs[39,40]. To capture the thermo-mechanical response of this familyof composites, we assume the material as a mixture of elastic rein-forcement and SMP matrix, which is further divided into a contin-uum mixture of a glassy and a rubbery phase, and the fraction ofeach phase is complementary variation depending on the temper-ature. Using the theory of composite bridging model, we thenincorporate the influence of elastic fiber reinforcement in the con-stitutive model. The mechanical characters of fiber, glassy phaseand rubbery phase are embodied by the Generalized Hooke’s laws.Comparisons between model prediction and experimental resultsare presented too. Additionally, the elongation at break of carbonfiber which we used in this research as reinforcement is 1.8%, thusonly small deformation was investigated in this study.

2. Materials and experiment

The matrix of the sample is a kind of epoxy based shape mem-ory polymer which was developed by one of our research groupmember [41]. The reinforcement material is T700 Toray carbonfiber. A Perkin Elmer Dynamic Mechanical Analyzer (DMA-7) isused to perform a dynamic thermal scan at a frequency of 1 Hzin the temperature range of 25–250 �C. The size of the epoxybased shape memory polymer test specimen is approximately5 � 3 � 1 mm. The purpose of the dynamic thermal scan is todetermine the basic mechanical response, glass transition temper-ature and the elastic modulus of epoxy. According to DMA test, thetrend of variation in storage modulus and tangent delta with thechanging temperature is illustrated in Fig. 1. It is clearly observedthat the glass transition temperature Tg of the SMP samples is100 �C.

To verify the efficiency of the developed model, conventionaluniaxial tension tests are carried out according to ASTM-D638stands, using a Zwick/roell testing machine equipped with a ther-mal chamber. The SMPCs samples used for uniaxial tension tests isa kind of SMPs matrix composites, consist of a ‘stiff’ fiber reinforce-ment and a ‘softer’ SMPs matrix arranged together with one phasedistributed as an array inside of the other phase. To eliminate theeffect of thermal expansion, experiment data will only start toadopt once the temperature reaches the design temperature forabout 10 min. The experiment data can be collected by using soft-ware linked to devices. Tests are running at a constant stretchingspeed of 1 mm/min. Fig. 2(A–C) presents the stress–strain responseof samples with different fiber volume fraction at a lowtemperature (30 �C) corresponding to the glassy state, at a high

0

500

1000

1500

2000

2500

Mod

ulus

(MPa

)

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Tan

delta

Temperature (°C)

Storage modulus

Tan delta

Tg=100°C

Fig. 1. The DMA test: storage modulus and tan delta of the shape memory polymer.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

memory polymer composite at different temperatures: (A) fiber content 7.5%, (B)fiber content 8.2%, (C) fiber content 11.5%. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

temperature (100 �C) corresponding to the rubbery state, and atan intermediate temperature (60 �C, 80 �C).

3. Constitutive model

3.1. Overall model description

In this study, SMPCs has been divided into shape memory poly-mer matrix and fiber reinforcement two segments. Based on thetwo phase theory of Liu, shape memory polymer matrix can bedivided into glassy phase and rubbery phase two parts, whichcan be transformed to each other gradually with changed temper-ature [24]. In this 3-D model, the ‘‘frozen fraction’’ (glassy phase),the ‘‘active fraction’’ (rubbery phase) and the fiber fraction isrespectively defined as:

134 Q. Tan et al. / Composites: Part A 64 (2014) 132–138

/g ¼Vgla

Vmatrix¼ Vgla

Vgla þ Vrub; /r ¼ 1� /g ð1Þ

wg ¼Vgla

V; wr ¼

Vrub

V; wf ¼

Vfiber

V; wm ¼ wg þ wr ð2Þ

V ¼ Vmatrix þ Vfiber ¼ Vgla þ Vrub þ Vfiber; wg þ wr þ wf ¼ 1 ð3Þ

Here V, Vgla, Vrub, Vfiber denote the total volume of the SMPCs, volumeof glassy phase, rubbery phase and, fiber reinforcement separately.And wg denote the volume fraction of the glassy phase and wr

denote the volume fraction of rubbery phase, wf denote the volumefraction of carbon fiber. Vfiber is invariable. The shape memory effectis caused by the transition of a crosslinking polymer from a statedominated by entropic energy (rubbery state) to a state dominatedby internal energy (glassy state) as the temperature changes. Bywhich, the glass transition during a thermo-mechanical cycle isreflected and the shape memory behavior can be captured, as sche-matically illustrated in Fig. 3.

Further, the total strain in SMPCs has been divided into fiveparts: glassy phase strain, rubbery phase strain, fiber strain, storedstrain and thermal strain. Thus the strain can be expressed in Eq.(3) as:

ei ¼ efi wf þ eg

i wf þ eri wf þ eT þ en

gs ð4Þ

where superscript g, r and f stands for glassy phase, rubbery phaseand fiber separately, and eT represent thermal strain, en

gs denote thestrain stored and released during the cooling and heating process.Correspondently, the stress can be divided into five parts, whichcan be expressed in Eqs. (5)–(7) as:

ri ¼ rTi þ rrec

i þ rCi ð5Þ

rCi ¼ wf r

fi þ wgr

gi þ wrrr

i ð6Þ

rmi ¼ /gr

gi þ /rrr

i ð7Þ

where superscript C and m denote composite and matrix separately.And rrec

i represent recovery stress.

3.2. Composite bridging model

A unidirectional fiber reinforced composite micro-mechanicsmodel must follow the relationship below [39,40].

rCi ¼ wf r

fi þ wmrm

i ð8Þ

eCi ¼ wf e

fi þ wmem

i ð9Þ

Here wf is volume fraction of fiber, wm is volume fraction of matrixand rj denote the external stress exert on SMPCs. And em

i ; efi ;rm

i ;rfi

denote the strain and stress of shape memory polymer matrix andfiber reinforcement separately. Additionally, the fiber has to satisfyits own constitutive relation when the strain within the extent ofelasticity

Fig. 3. Representative body element for SMPCs: transition between glassy state and rureader is referred to the web version of this article.)

efi ¼ ½S

fij�r

fj ð10Þ

Here ½Sfij� is the flexibility matrix of fiber reinforcement.

It is assumed that the surface of the matrix is in direct contactwith the fiber and bond together. Until the composite has beendamaged the interface will not be de-bonding, or relatively dis-placement. Suppose the composite undamaged, these two internalstresses, rm

i and rfj , can be related by a non-singular matrix.

rmi ¼ ½Aij�rf

j ð11Þ

Here [Aij] is bridging matrix. Insert Eq. (11) into Eq. (8), internalstress in fiber and matrix can be derived as:

rfi ¼ ðwf ½I� þ wm½Aij�Þ�1rj ¼ ½Bij�rC

j ð12Þ

rmi ¼ ½Aij�ðwf ½I� þ wm½Aij�Þ�1rj ¼ ½Aij�½Bij�rC

j ð13Þ

Here [I] is identity matrix. [Bij] is an inverse matrix of (wf[I] +wm[Aij]), and the explicit expression of [Bij] will be given in the latersection. Substitute Eq. (12) into Eq. (10), the internal strain in fibercan be derived as following:

efi ¼ ½S

fij�r

fj ¼ ½S

fij�½Bij�rC

j ð14Þ

Here ½Sfij� denotes the flexibility matrix of fiber, the explicit expres-

sion will be given later. Although a material with two phases isinhomogeneous, we assume that the corresponding stresses inglassy phase and rubbery phase are equal to the stress in matrix[24]:

rri ¼ rg

i ¼ rmi ð15Þ

3.3. Strains in the matrix

When at a temperature well below SMPs’ glass transition tem-perature, hardness of SMPs is very high. At this point, the deforma-tion is correlated to the external force, once the force has beenremoved, the deformation will recover immediately. At this tem-perature, large-scale conformational changes are not possible butlocalized conformational motions are allowed. Thus it is appropri-ate to adopt the Generalized Hooke’s law to describe the glassyphase mechanical characteristics.

egi ¼ Sg : rg

i ð16Þ

Here Sg denote the flexibility matrix of glassy phase. When the tem-perature is well above the SMPs’ glass transition temperature, thepolymer is in rubbery state. The stiffness of rubbery phase can beas low as several MPa, while the rubbery elastic strain can be onthe order of several hundred percent with appropriate cross-linkingdensity. Thus it is necessary to adopt a nonlinear hyper-elastic con-stitution equation to describe the rubbery phase large deformation.Yet the non-linearity of rubbery elasticity can be ignored for smallstrains, we can also assume that the material behaves in a linear

bbery state. (For interpretation of the references to color in this figure legend, the

Q. Tan et al. / Composites: Part A 64 (2014) 132–138 135

elastic manner in the rubbery phases. Hence the strain of rubberyphase can also be related to the stress tensor through the General-ized Hooke’s laws:

eri ¼ Sr : rr

i ð17Þ

where Sr denote the flexibility matrix of rubbery phase.

3.4. Stored strain

The following specify stored strain egs in the representative vol-ume element dv. With a drop in temperature, a part of rubberyphase will translates into glassy phase gradually, at the same timethe deformation previously in rubbery phase will be ‘frozen’ andstored. Here Vfroz denotes the glassy phase volume which translatesfrom rubbery phase. It is assumed that the elastic fiber reinforce-ment is evenly distributed in the SMPs matrix and the fiber inglassy phase and rubbery phase is evenly interlaced. During theprocess of strain storage, the fiber, which evenly distributed in rub-bery phase, its deformation will fixed too as this part of rubberyphase translate into glassy phase. In SMPCs application, firstlythe composite is raised to above its glass transition temperatureby external heater, it is means that at this point the SMPs iscompletely in rubbery phase. Secondly a load has been appliedon SMPCs based structure, and then it will display elastic deforma-tion. Maintain this deformation and cooled down the structure, thedeformation will be fixed. So the total volume which has beenfrozen can be expressed as:

Vfroz ¼ Vg þVg

Vg þ VrVf ð18Þ

With the decreasing of temperature, a part of rubber phase (DV)translate into frozen phase. The growth of stored strain is DVer, andit can be expressed as following:

engs ¼

Vfrozegs þ DVer

Vð19Þ

Here er denote the strain in rubber phase. When rubber phase trans-lates into frozen phase continuously, the stored strain changed con-tinuously, and it can be expressed as:

engs ¼

Vfrozegs þR

VfrozerdV

Vð20Þ

Because Vfroz ¼ Vg þ Vg

VgþVrVf ¼ V : wg

1�uf; thus the expression of en

gs

can be simplified as:

engs ¼

11� wf

½wg1egs þ ðwg2 � wg1Þer� ð21Þ

In the above equation, wg1 and egs denote the volume fractionand strain of frozen phase at initial temperature, wg2 and en

gs denotethe volume fraction and strain of frozen phase at current temper-ature. It can be seen that the carbon fiber content has no impactson stored strain.

With the increasing temperature, the stored strain will releasegradually, include the part of frozen fiber. It will cause the reduc-tion of stored strain in the whole body element. From macrocosmicpoint of view, the deformation recovered. Therefore, the storedstrain can be expressed as follow during the temperature increas-ing process.

engs ¼

egsVfroz

V¼

wgegs

1� wfð22Þ

When the temperature is increased from T1 to T2, the storedstrain which has been released is ðegs � en

gsÞ. The recovery stressduring the temperature increasing process can be extrapolated as:

rrec ¼ egsðwr1ðT1Þ � wr2ðT2ÞÞðEf wf þ Erð1� wf ÞÞ ð23Þ

Here wr1(T1) and wr2(T2) denote the volume fraction of rubber phaseat the initially temperature and current temperature separately.

3.5. Thermal strain

The total thermal strain eT of SMPCs is constitutively defined as:

eT ¼ wrerT þ wge

gT þ wf e

fT ð24Þ

Here we make an assumption that rubbery phase, glassy phaseand carbon fiber are homogenous materials, and their thermalstrains only related to thermal expansion coefficient separately,which means that the sole internal straining mechanism is thermalstrain, thus the thermal strain can be simply defined as:

eT ¼Z T

T0

adT ð25Þ

The thermal strain in glassy phase can be expressed as:

egT ¼

Z T

T0

agdT ð26Þ

where ag denote the thermal expansion coefficient of glassy phase.The thermal strain in rubbery phase can be expressed as the integralof thermal expansion coefficient of rubbery phase:

erT ¼

Z T

T0

ardT ð27Þ

And the thermal strain in the fiber can be expressed asfollowing:

efT ¼

Z T

T0

af dT ð28Þ

Here ar and af are the thermal expansion coefficient of rubberyphase and fiber separately. Thus the whole thermal strain can beexpressed as following:

eT ¼Z T

T0

½wrðTÞar þ wgðTÞag þ wf ðTÞaf �dT ð29Þ

4. Results and discussion

4.1. Parameter identification

Based on the investigation of Liu YP, the young’s modulus ofshape memory polymer matrix can be related to /g as follows [24]:

EðTÞ ¼ 1/g

ELþ 1�/g

3NkT

ð30Þ

Here /g is the volume fraction of glassy phase, and N denote cross-link density. By analyzing the empirical data obtained in part 2, therelation between the elastic modulus and temperature is found tofollow a power function. Thus the elastic modulus is obtained by fit-ting experimental data, and expressed as follows:

EðTÞ ¼ leb1�T

c1

� �2

þmeb2�T

c2

� �2

þ neb3�T

c3

� �2

ð31Þ

The volume fraction of glassy phase can be derived from Eq.(30) as following:

/gðTÞ ¼ELð3NkT � EðTÞÞEðTÞð3NkT � ELÞ

ð32Þ

The volume fraction of rubbery phase and glassy phase in SMPcomposite are expressed as following separately:

136 Q. Tan et al. / Composites: Part A 64 (2014) 132–138

wrðTÞ ¼ ð1� wf Þð1� /gÞ ð33Þ

wgðTÞ ¼ ð1� wf Þ/g ð34Þ

The volume fraction of glassy phase, rubbery phase and fiberreinforcement is demonstrated in Fig. 4 separately. The fiber con-tent remains invariable. At a temperature well above the SMPsglass transition temperature, volume fraction of rubbery phase isat its peak, wr(Th) = 1 � wf. It is can be seen from Fig. 4 that, the vol-ume fraction of rubbery phase is reduced with the falling of thetemperature, until reaching a low point at wr(Tl) = 0. The changingtrend of glassy phase volume fraction is opposite to that of rubberyphase.

Based on composite bridging model, the unidirectional compos-ite flexibility matrix can be derived as:

½Sij� ¼ ðwf ½Sfij� þ wm½S

gij�½Aij�Þðwf ½I� þ wm½Aij�Þ�1 ð35Þ

When the strain in the fiber and matrix is within the extent ofelasticity, the flexibility matrix of both can be expressed asfollowing:

½Sfij� ¼

1Ef

11

; � v f12

Ef11

; 0

� v f12

Ef11

; 1Ef

11

; 0

0; 0; 1Gf

12

266664

377775; ½Sm

ij � ¼

1Em ; � vm

Em ; 0

� vm12

Em ; 1Em ; 0

0; 0; 1Gf

12

2664

3775 ð36Þ

The above matrixes have been partitioned and substituted intoEq. (35), and the bridging matrix can be deduced in the flowingform:

20 40 60 80 100 120-20

0

20

40

60

80

100

Volu

me

fract

ion

(%)

Ag

Af

Ar

Temperature (°C)

Fig. 4. Glassy phase/rubbery phase volume fraction, wg, wr, a function of temper-ature; wf, a constant value. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Table 1Material parameters adopted from experiments and data fitting.

Material parameters Values

Eg 2029.62Er 14.11Ef 230l, m, n 0, 33.36, 0.0073b1, b2, b3 1949, 7.738, 40.36c1, c2, c3 856.6, 50.61, 28.6ar 2.35 � 10�4

ag 1.17 � 10�4

af �0.3 � 10�6

Tl, Tg, Th 27.8, 100, 120a 0.4b 0.32

½Aij� ¼a11 a12 0a21 a22 00 0 a33

24

35 ð37Þ

The explicit expression of these parameters in Eq. (37) can beexpressed as:

a11 ¼ Em=Ef11 ð38Þ

a12 ¼ðSf

12 � Sm12Þða22 � a11Þ

Sm11 � Sf

11

ð39Þ

a22 ¼ bþ ð1� bÞ Em

Ef22

; 0 < b < 1 ð40Þ

a33 ¼ aþ ð1� aÞ Gm

Gf12

; 0 < a < 0 ð41Þ

[Bij], the converse matrix of bridging matrix (wf[I] + wm[Aij]), can beexpressed as:

½Bij� ¼b11 b12 b13

0 b22 b23

0 0 b33

24

35 ð42Þ

The flowing are the specifically expressions of each parameter:

b11 ¼ ðwf þ wma11Þ�1 ð43Þ

b12 ¼ �ðwma12Þ=½ðwf þ wma11Þðwf þ wma22Þ� ð44Þ

b13 ¼ ½ðwma12Þðwma23Þ � ðwf þ wma22Þðwma13Þ�=c ð45Þ

b22 ¼ ðwf þ wma22Þ�1 ð46Þ

b23 ¼ �ðwma23Þðwf þ wma11Þ=c ð47Þ

b33 ¼ ðwf þ wma33Þ�1 ð48Þ

c ¼ ðwf þ wma11Þðwf þ wma22Þðwf þ wma33Þ ð49Þ

Here, a part of material parameters in Table 1 come from DMAexperimental result as illustrated in part 2, other parameters comefrom the coefficient of thermal expansion (CTE) experiments anddata fit.

4.2. Model verification

Tension fatigue tests are carried out at different temperature toobtain the stress–strain curve of SMPCs samples with differentfiber content, as mentioned before. To show the validity and accu-racy of the model under different thermo-mechanical loadingsconditions, the comparisons between the results of simulationand experiments have been made as show in Fig. 5(A–C). It isobserved from Fig. 5 that there are deflections in varying degreebetween the experimental curve and the theoretical calculatedcurves. That mainly because the elastic modulus calculated by

Units Description

(MPa) Young’s modulus in the glass phase(MPa) Young’s modulus in the rubber phase(GPa) Young’s modulus of carbon fiber(–) Parameter for storage modulus(–) Parameter for storage modulus(–) Parameter for storage modulus(�C) Rubber phase coefficient of thermal expansion(�C) Glass phase coefficient of thermal expansion(�C) Carbon fiber coefficient of thermal expansion(�C) Temperature(–) Bridging parameter(–) Bridging parameter

Fig. 5. Comparison between experiments and simulations: stress–strain diagramillustrating the thermo-mechanical behavior of shape memory polymer compositeat different temperatures: (A) fiber content 7.5%, (B) fiber content 8.2%, (C) fibercontent 11.5%. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Fig. 6. Numerical simulations of store strain changing with decreasing of temper-ature. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Fig. 7. Numerical simulations of store strain release with increasing of tempera-ture. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

20 40 60 80 100 1200

10

20

30

40

50

60

Rec

over

y st

ress

ψ f = 11.5%

= 8.2%

= 7.5%

= 0%

Temperature (°C)

ψ f

ψ f

ψ f

Fig. 8. Numerical simulations of stress–temperature behavior of the SMP compos-ite from free shape recovery process: with different fiber content (0%, 7.5%, 8.2%,11.5%). (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Q. Tan et al. / Composites: Part A 64 (2014) 132–138 137

bridging model deviate from experiment results. And becauseerrors are in an acceptable range, the correctness of the modelhas been verified generally.

It is can be seen from Fig. 5 that the mathematical model cap-tures the isothermal stress–strain behaviors of SMPCs at differenttemperatures and with different fiber contents. The results indi-cated that the stress r which we need to sustain the same valueof strain e is increased with increase of fiber content wf. This isbecause, based on the composite bridging model stress distributionprinciple rf

i ¼ ½Bij�rj, most part of the stresses are supported by thecarbon fiber. Because the elastic modulus of carbon fiber is as largeas 230 GPa, even it undertaking most of the load, the incensementof strain is very little. This result indicates that the fiber contentbrings great effect to mechanical properties of SMPCs, especiallythe strength of the material. Fig. 5 also illustrated that, comparedwith at temperature T = 60 �C and T = 80 �C, the stress r whichwe need to sustain the same value of strain e is increased

significantly at temperature T = 30 �C. That because with theincreasing of temperature, the storage modulus of SMP decrease.Thus the modulus at T = 30 �C is much higher that at T = 60�C,and for the time being the polymer translates from glass state intorubber state gradually. When at same load, the deformation ofrubbery phase is much larger than glassy phase.

4.3. Recovery behaviors prediction

Using parameters in Table 1, the model is able to predict freerecovery behaviors and the strain storage process. Fig. 6 showsthe non-isothermal free recovery behaviors of SMP composite withdifferent storage strain. Besides, the change trend of stored strain

138 Q. Tan et al. / Composites: Part A 64 (2014) 132–138

with temperature is also predicted in Fig. 7. By contrasting thestrain releasing in Fig. 6 and strain storage process in Fig. 7, it iscan be see that, the strain storage process coincides with the non-isothermal free recovery behaviors, and these two curves almostoverlap. In the application of SMPCs, such as SMPCs based spacedeployable structures, recovery force is an important index thatevaluates mechanical performances of structures. The higher ofthe recovery force, the less of the hinge in need. Hence, it is vitalto investigate how to improve the recovery force by changing fibercontent. Fig. 8 shows the change trend of recovery stress under con-dition of constraint deformation of 2% and increasing temperature.It is observed that, with the rise of temperature, the recovery stressincrease continuously as well. When the temperature is above SMPsglass transition temperature Tg, the recovery stress reaches theequilibrium point. Besides, the recovery stress is closely related tofiber content, and the recovery stress increase with fiber content.

5. Conclusion

The thermal mechanical behaviors of thermally induced unidi-rectional SMPCs had been investigated in this paper. Variousexperiments had been performed to fully characterize the complexthermal–mechanical response of unidirectional SMPCs and guidethe development of the constitutive model. Composite bridgingmodel had been employed to describe the relationship betweenfiber reinforcement and SMPs matrix by the constitutive model.Additionally, the concept that glassy phase and rubbery phasecan exist in harmony and transform to each other as temperaturechange had been adopted.

It is assumed further that the stress in SMPs matrix is equalwithin both glassy phase and rubbery phase, and the GeneralizedHooke’s laws had been adopted to describe the elastic deformationof glassy phase and rubbery phase. The storage and releasing ofstrain had been further investigated during the cooling/heatingprocess. Simulations of uniaxial tensile experiments under isother-mal conditions agreed with the experimentally results. The ther-mal mechanically coupled constitutive model developed in thispaper offers the potential for the development of a robust simula-tion-based capability for the design of devices made from unidirec-tional SMPCs for a variety of applications.

Acknowledgement

This work is supported by the National Natural Science Founda-tion of China (Grant Nos. 11225211, 11272106, and 11102052).

References

[1] Lin JR, Chen LW. Study on shape-memory behavior of polyether basedpolyurethanes. I. Influence of the hard-segment content. J Appl Polym Sci1998;69(15):63–74.

[2] Lin JR, Chen LW. Study on shape-memory behavior of polyether basedpolyurethanes. II. Influence of the soft-segment molecular weight. J ApplPolym Sci 1998;69(15):75–86.

[3] Jeong HM, Ahn BK, Cho SM, Kim BK. Water vapor permeability of shapememory polyurethane with amorphous reversible phase. J Polym Sci: Part B:Polym Phys 2000;38(30):09–17.

[4] Lendlein A, Jiang H, Junger O, Langer R. Light induced shape memory polymers.Nature 2005;434(8):79–82.

[5] Leng JS, Lu H, Liu Y, Huang W, Du S. Shape-memory polymers a class of novelsmart materials. MRS Bull 2009;34(11):848–55.

[6] Leng JS, Lan Xin, Liu YJ, Du SY. Shape-memory polymers and their composites:stimulus methods and applications. Prog Mater Sci 2011;56(7):1077–135.

[7] Leng JS, Lv HB, Liu YJ, Du SY. Electroactivate shape memory polymer filled withnanocarbon particles and short carbon fibers. Appl Phys Lett 2007;91:144105.

[8] Liu YJ, Lv HB, Lan X, Leng JS, Du SY. Review of electro-active shape-memorypolymer composite. Compos Sci Technol 2009;69(13):2064–8.

[9] Lv HB, Leng JS, Liu YJ, Du SY. Shape-memory polymer in response to solution.Adv Eng Mater 2008;10(6).

[10] Leng JS, Lan X, Liu YJ, Du SY, Huang WM, Liu N, et al. Electrical conductivity ofthermoresponsive shape-memory polymer with embedded microsized Nipowder chains. Appl Phys Lett 2008;92:014104.

[11] Leng JS, Huang WM, Lan X, Liu YJ, Liu N, Phee SY, et al. Significantly reducingelectrical resistivity by forming conductive Ni chains in a polyurethane shape-memory polymer/carbon-black composite. Appl Phys Lett 2008;92:204101.

[12] Leng JS, Zhang Dawei, Liu Yanju, Yu Kai, Lan Xin. Study on the activation ofstyrene-based shape memory polymer by medium-infrared laser light. ApplPhys Lett 2010;96:111905.

[13] Gall K, Mikulas M, Munshi NA, Beavers F, Tupper M. Carbon fiber reinforcedshape memory polymer composites. J Intell Mater Syst Struct2000;11(11):877–86.

[14] Wei ZG, Sanstrom R. Shape memory materials and hybrid composites forsmart systems: Part I. Shape-memory materials. J Mater Sci1998;33(37):43–62.

[15] Leng JS, Lv HB, Liu YJ, Du SY. Electro-activate shape memory polymer filledwith nanocarbon particles and short carbon fibers. Appl Phys Lett2007;91:144105.

[16] Ta Ohki, Ni Q-Q, Ohsako N, Iwamoto M. Mechanical and shape memorybehavior of composites with shape memory polymer. Composites: Part A2004;35(9):65–73.

[17] Gall K, Dunn ML, Liu Y, Finch D, Lake M, Munshi NA. Shape memory polymernanocomposites. Acta Mater 2002;50(20):15–26.

[18] Lu HB, Yu K, Sun SH, et al. Mechanical and shape-memory behavior of shape-memory polymer composites with hybrid fillers. Polym Int2010;59(6):766–71.

[19] Li G, Nettles D. Thermomechanical characterization of a shape memorypolymer based self-repairing syntactic foam. Polymer 2010;51(3):755–62.

[20] Li G, Xu W. Thermomechanical behavior of thermoset shape memory polymerprogrammed by cold-compression: testing and constitutive modeling. J MechPhys Solids 2011;59(6):1231–50.

[21] Tobushi H, Hayahi S. Properties and application of shape memory polymerpolyurethane series. Res Mach 1994;46(6):646–52.

[22] Lan X, Liu YJ, Lv HB, Wang XH, Leng JS, Du SY. Fiber reinforced shape-memorypolymer composite and its application in a deployable hinge. Smart MaterStruct 2007;18(2):024002.

[23] Zhang CS, Ni QQ. Bending behavior of shape memory polymer basedlaminates. Compos Struct 2007;78(2):153–61.

[24] Liu Y, Gall K, Dunn M, Greenberg A, Diani J. Thermomechanics of shapememory polymers: uniaxial experiments and constitutive modeling. Int JPlasticity 2006;22(2):279–313.

[25] Diani J, Liu Y, Gall K. Finite strain 3D thermoviscoelastic constitutive model forshape memory polymers. Polym Eng Sci 2006;46(4):486–92.

[26] Chen YC, Lagoudas DC. A constitutive theory for shape memory polymers. PartI. Large deformations. J Mech Phys Solids 2008;56(5):1752–65.

[27] Chen YC, Lagoudas DC. A constitutive theory for shape memory polymers. PartII. A linearized model for small deformations. J Mech Phys Solids2008;56(5):1766–78.

[28] Qi H, Nguyen T, Castro F, Yakacki C, Shandas R. Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers. J MechPhys Solids 2008;56(5):1730–51.

[29] Kim J, Kang T, Yu W. Thermo-mechanical constitutive modeling of shapememory polyurethanes using a phenomenological approach. Int J Plasticity2010;26(2):204–18.

[30] Reese S. A micromechanically motivated material model for the thermo-viscoelastic material behaviour of rubber-like polymers. Int J Plasticity2003;19(7):909–40.

[31] Xu W, Li G. Constitutive modeling of shape memory polymer based self-healing syntactic foam. Int J Solids Struct 2010;47(9):1306–16.

[32] Baghani M, Naghdabadi R, Arghavani J, Sohrabpour S. A constitutive model forshape memory polymers with application to torsion of prismatic bars. J IntellMater Syst Struct 2012;23(7):107–16.

[33] Liu YP, Gall K, Dunn ML, Greenberg AR, Diani J. Thermomechanics of shapememory polymers: uniaxial experiments and constitutive modeling. Int JPlasticity 2006;22(2):279–313.

[34] Barot G, Rao IJ. Constitutive modeling of the mechanics associated withcrystallizable shape memory polymers. Z Angew Math Phys2006;57(4):652–81.

[35] Barot G, Rao IJ, Rajagopal KR. A thermodynamic framework for the modeling ofcrystallizable shape memory polymers. Int J Eng Sci 2008;46(4):325–51.

[36] Tobushi H, Hara H, Yamada E, Hayashi S. Thermomechanical properties in athin film of shape memory polymer of polyurethane series. Smart Mater Struct1996;2716(5):483–91.

[37] Nguyen TD, Qi HJ, Castro F, Long KN. A thermoviscoelastic model foramorphous shape memory polymers: incorporating structural and stressrelaxation. J Mech Phys Solids 2008;56(9):2792–814.

[38] Castro F, Westbrook K, Long K, Shandas R, Qi H. Effects of thermal rates on thethermomechanical behaviors of amorphous shape memory polymers. MechTime-Depend Mater 2010;14(3):219–41.

[39] Huang ZM. Simulation of the mechanical properties of fibrous composites bythe bridging micromechanics model. Composites Part A 2001;32(2):143–72.

[40] Huang ZM. Strength formulae of unidirectional composites including thermalresidual stresses. Mater Lett 2000;43(1–2):36–42.

[41] Leng Jinsong, Xuelian Wu, Liu Yanju. Effect of linear monomer onthermomechanical properties of epoxy shape memory polymers. SmartMater Struct 2009;18(9):095031.