An anisotropic visco-hyperelastic model for thermally-actuated shape memory polymer-based woven fabric-reinforced composites

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An anisotropic visco-hyperelastic model for thermally-actuated shape memory polymer-based woven fabric-reinforced composites Xiaobin Su , Yingyu Wang , Xiongqi Peng * School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Shape memory polymer composites Constitutive behavior Anisotropic material Viscoelastic material

An anisotropic visco-hyperelastic constitutive model for thermally-actuated woven fabricreinforced shape memory polymer composites (SMPCs) is proposed. The viscoelasticity of the shape memory polymer and anisotropic hyperelasticity of the fabric are separately described. The second law of thermodynamics is used to derive the constitutive relations through decomposing the thermodynamic free energy. A woven fabric-reinforced SMPC is prepared and its thermo­ mechanical and shape memory experiments are conducted to verify the proposed model. Com­ parison between the predicted and experiment results indicates an good consistency, which attests that the developed model can predict the anisotropic mechanical and shape memory properties of thermally induced SMPCs reinforced by woven fabrics effectively. And numerical investigation of the effect of fiber yarn orientation on the shape fixity and recovery is also performed.

1. Introduction Shape memory polymers (SMPs) can deform actively upon external stimuli, among which SMPs with thermal actuation are the most common (Cheng et al., 2019; Leng et al., 2011; Liu et al., 2017; Zhao et al., 2019). Shape memory polymer composites (SMPCs) are developed with SMPs as the matrix to enhance the mechanical property of pure SMPs (Patel and Purohit, 2019; Wang et al., 2019; Zhang et al., 2017). The special shape memory capacity and good mechanical performance make SMPCs have a widespread appli­ cations in engineering areas (Gao et al., 2019; Liu et al., 2014, 2018; Mu et al., 2018). So, constitutive models representing the thermomechanical behaviors of SMPCs are needed. As the matrix of SMPCs, SMPs provide shape memory capacity for SMPCs. SMPs have the essential complex material behaviors of polymers, which are related to time and temperature and have been modeled by some theories (Ames et al., 2009; Anand et al., 2009; Long, 2014; Regrain et al., 2009; Shojaei and Li, 2013; Srivastava et al., 2010a; Uchida et al., 2019). However, modeling SMPs in a shape memory cycle usually needs to consider the load-, time-, and temperature-related boundary conditions comprehensively, which is complicated. Tobushi et al. (1997) developed a constitutive model for SMPs with the standard linear viscoelastic (SLV) model and then extended it into a nonlinear one (Tobushi et al., 2001). Thereafter, viscoelastic models considering intricate polymer structure evolution and stress relaxation mechanisms and description of yield behavior in the finite strain regime were gradually proposed (Nguyen et al., 2008; Srivastava et al., 2010b; Xiao et al., 2013). The above models are based on the essential viscoelasticity of polymers and there are also models representing the shape memory effect through introducing the concept of phase transition. Liu

* Corresponding author. E-mail address: [email protected] (X. Peng). https://doi.org/10.1016/j.ijplas.2020.102697 Received 11 August 2019; Received in revised form 21 January 2020; Accepted 2 February 2020 Available online 7 February 2020 0749-6419/© 2020 Elsevier Ltd. All rights reserved.

Please cite this article as: Xiaobin Su, International Journal of Plasticity, https://doi.org/10.1016/j.ijplas.2020.102697

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et al. (2006) proposed a model considering phase transition in the small strain regime. Afterwards, more complicated phase transition-based models were gradually developed (Baghani et al., 2012; Boatti et al., 2016; Park et al., 2016; Qi et al., 2008; Reese et al., 2010). Besides, there are also modeling works incorporating the viscoelasticity theory and concept of phase transition simul­ taneously (Li et al., 2017; Su and Peng, 2018). The addition of reinforcements usually introduces the complicated anisotropic material behaviors and interaction between the two constitutes of SMPCs, which leads to the complexity of modeling SMPCs. There are already some modeling works for SMPCs, such as Shi et al. (2012)’s numerical analysis for a carbon nanotube-reinforced SMPC shell, Nishikawa et al. (2012)’s simulation analysis for a discontinuous fiber-reinforced SMPC, Tan et al. (2014)’s composite bridging theory model for a SMPC reinforced by unidirectional fibers, Bergman and Yang (2015)’s model for a SMPC cantilever beam reinforced by unidirectional fibers, Gu et al. (2019)’s thermo-viscoelastic finite deformation constitutive with internal state variables for unidirectional fiber-reinforced SMPCs, Hong et al. (2019)’s homogenized constitutive model for woven fabric-reinforced SMPCs with the consideration of thermal residual stress, etc. Among the above-mentioned models, there are fewer modeling studies for SMPCs reinforced by woven fabrics. However, woven fabric-reinforced SMPCs have good balanced enhancement effect and large deformation limit in multiple directions, so they have greater engineering application potential compared with SMPCs reinforced by other reinforcements. To understand their more complex mechanical behaviors in applications, constitutive models for woven fabric-reinforced SMPCs based on continuum mechanics theory are needed. Pure SMPs have isotropic thermomechanical property, however, SMPCs reinforced by woven fabrics exhibit anisotropic material behavior, which is very common and of great application value to many engineering fields. In the past, several works about finite strain anisotropic constitutive models of woven fabric composites have been carried out (Boisse et al., 2011; Gong et al., 2016; Guzman-­ Maldonado et al., 2015; Peng et al., 2013). Though these models are very successful in capturing anisotropic mechanical behaviors of wove fabrics, they cannot represent the shape memory phenomena of SMPCs without the consideration of time- and temperature-related viscous property of SMPs. In this work, a novel anisotropic visco-hyperelastic constitutive model for thermally-actuated SMPCs reinforced by woven fabrics is proposed, which can represent the anisotropic thermomechanical properties and shape memory performance at the same time. The model is developed through decomposing the thermodynamic free energy and considering the anisotropic hyperelastic behaviors induced by woven fabrics and isotropic viscoelastic properties coming from SMP matrix comprehensively. The paper’s outline is listed as follows. In Section 2, the preparation of a SMPC material and related experiments applied to characterize its thermomechanical properties are described. In Section 3, an anisotropic visco-hyperelastic constitutive model for the woven fabric-reinforced thermally induced SMPCs is proposed on account of the thermodynamic free energy decomposition and comprehensive consideration of isotropic viscoelasticity and anisotropic hyperelasticity. In Section 4, the thermomechanical experi­ ment data are used to complete parameter determination and model verification. And the influence of the initial fiber yarn orientation of the woven fabric on the shape memory behavior in tension are numerically investigated with the proposed model in Section 5. Finally, Section 6 give some conclusions. 2. Thermomechanical experiments 2.1. Material and specimen preparations Photo-cross-linking is a facile and versatile polymerization strategy, which has contributed a lot to the development of SMPs (Xie et al., 2019). Photo-cross-linking can improve basic molecular structure, facilitate advanced SMPs design and work together with advanced manufacturing techniques, such as 3D printing. So, a photo-cross-linking SMP based on tert-Butyl acrylate was prepared in this paper. Tert-Butyl acrylate, poly(ethylene glycol) dimethacrylate and di(ethylene glycol) dimethacrylate in liquid form were used in the asreceived condition. 30 wt% di(ethylene glycol) dimethacrylate with 70 wt% poly(ethylene glycol) dimethacrylate were mixed as the crosslinker. Then, the SMP solution was prepared by mixing 60 wt% tert-Butyl acrylate with 40 wt% crosslinker. The 2, 2-dimethoxy-2phenylacetophenone in powder form was added to the SMP solution as photo polymerization initiator with a concentration of 0.1 wt%. Then, the mixture was shaken for 10 min with magnetic stirring apparatus to obtain fully dissolved solution. Balanced plain woven T300 carbon fabric was selected as the reinforcement for the SMPC. The fabric properties are listed in Table 1. To obtain samples with uniform thickness and fiber distribution, a specialized curing mold was set up, as shown in Fig. 1a. The SMPC specimens were prepared according to the following procedure: (i) place the carbon fiber fabric preform on the base plate; (ii) assemble the mold with clips and seal the cavity with silicone grease, which prevents the leakage of the SMP resin; (iii) inject the SMP resin into the mold from the sprue on the glass cover plate; (iv) place the mold into the ultraviolet curing box and polymerize the solution for 15 min, as shown in Fig. 1b; (v) disassemble the mold, overturn the sample, add a frame and assemble the mold again; (vi) repeat procedures (iii) and (iv) to complete the two-step ultraviolet curing to guarantee the symmetrical distribution of the SMP matrix

Table 1 Parameters of the balanced plain woven carbon fabric. Number of filament of a single yarn

Yarn tensile modulus (GPa)

Yarn width (mm)

Yarn space (mm)

Thickness (mm)

Area density (g/m2)

3k

230

1.5

2.0

0.26

200

2

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Fig. 1. Forming system for SMP and SMPC.

on both faces of the fabric; (vii) transfer the mold to an oven and stay for 1 h at 90 � C for the complete polymerization through further thermal curing. The composite sheets were then demolded. The volume fractions of SMP and carbon fabric are 87% and 13%, respectively. SMP samples were also prepared with similar procedure. Rectangular specimens with dimensions of 30 mm � 10.5 mm � 2 mm were used for DMA test. Dog-bone shaped specimens which have the gauge dimensions of 30 mm � 4 mm � 2 mm were used for isothermal uniaxial tensile tests. And rectangular specimens with dimensions of 130 mm � 25 mm � 2 mm were used for shape memory experiments. For SMPC, rectangular specimens with dimensions of 130 mm � 25 mm � 2 mm were used for both isothermal tension and shape memory tests. For each case, three specimens were tested. SMP and SMPC rectangular samples are shown in Fig. 2. 2.2. DMA test DMA test of the SMP was first conducted using a PerkinElmer dynamic mechanical analyzer (DMA 8000) to determine its glass transition temperature range, in which the tensile and shape memory tests were subsequently performed. SMP specimens were placed into the device with a single cantilever bending fixture and the heating rate was set as 2 � C/min. At the same time, the specimen was dynamically bended at a frequency of 1 Hz. The storage moduli, loss moduli and tanδ varying with temperature are shown in Fig. 3. It is determined that the temperature range where shape memory effect takes place is about 30–80 � C. 2.3. Tensile tests of the fabric, SMP and SMPC The material property of the carbon fabric is nearly independent of temperature, so its tensile tests were directly performed at room temperature. Based on the anisotropic material behavior of the woven fabric, tensile tests were conducted using samples with di­ mensions of 115 mm � 230 mm in two different loading modes: uniaxial tension and bias extension. In uniaxial tension, the load is applied along one fiber yarn direction because of the same property between the warp and weft yarns in the plain woven fabric, as shown in Fig. 4 (a); in bias extension, the loading direction is initially oriented at �45� to the warp and weft yarns, as shown in Fig. 4 (b). The thermomechanical behaviors of the SMP and SMPC are dependent on temperature, so their isothermal tensile tests were implemented at multiple temperatures including 30 � C, 40 � C, 50 � C, 60 � C, 70 � C and 80 � C. Before loading, each specimen was experienced a heat preservation process for 10 min to achieve thermal equilibrium in the heating chamber. The tension was conducted at a constant strain rate of 0.01/s and unloading initiated when the specimen failure emerged. Similar to the tensile tests of the fabric, uniaxial tension and bias extension tests were both carried out for the SMPC specimens to investigate the anisotropic material property. 2.4. Shape memory cycle tests of the SMP and SMPC In the shape memory experiments of the SMP and SMPC, the following shape memory cycle procedure was used: (i) heat the samples to 70 � C and make them reach thermal equilibrium in the following 10-min heat insulation process; (ii) stretch the samples to a displacement of u0 at 70 � C; (iii) cool the samples to 30 � C with the stretch fixed; (iv) unload the samples by reducing the tensile force to

Fig. 2. Prepared SMP and SMPC samples. 3

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Fig. 3. Storage moduli, loss moduli and tanδ versus temperature of the SMP.

Fig. 4. Schematic illustration of the loading condition in the tensile tests of the fabric.

zero tardily at 30 � C; (v) reheat the samples to 70 � C for shape recovery. The displacement change of the specimens during the free shape recovery process was measured and the engineering strain ε is calculated as ε ¼ u/L0 � 100%. L0 is the initial length between the clamps and u is the displacement during the reheating step. For all the specimens, L0 ¼ 70 mm was chosen. Both the uniaxial tension and bias extension loading modes were used in the shape memory tests of the SMPC. The deformation limit of the carbon fiber is small, so the loaded engineering strain in the uniaxial tension shape memory cycle of the SMPC is much smaller than that in the bias extension shape memory cycle to avoid fiber breakage. u0 ¼ 3.5 mm (ε ¼ 5%) was used for the SMP and bias tensile SMPC specimens and u0 ¼ 0.35 mm (ε ¼ 0.5%) was employed for the SMPC specimens in uniaxial tension.

Fig. 5. Rheological representation of the anisotropic visco-hyperelastic model. 4

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3. Model development 3.1. Kinematics SMPCs reinforced by woven fabrics have isotropic viscoelastic and anisotropic hyperelastic material behaviors from SMP matrix and woven fabrics, respectively. So, a rheological representation shown in Fig. 5 is used in this paper. A spring element and a Maxwell model arranged in parallel are employed to describe the viscoelasticity of the SMP matrix. Meanwhile, the anisotropic hyperelastic material property from woven fabrics is represented using a single spring element, which is arranged in parallel with the Maxwell model. The thermal deformation of the SMPC generated in the shape memory cycle is anisotropic and its mathematical description is complicated. To keep matters simple, the thermal deformation caused by temperature variation is not considered in this paper. It is assumed that the bonding between the woven fabric and matrix is perfect, so they experience the same deformation in the thermodynamic loading process, that is: (1)

Fmatrix ¼ Ffabric where Fmatrix and Ffabric are the deformation gradients of the SMP matrix and woven fabric, respectively. For annotation convenience, we define:

(2)

Fmatrix ¼ Ffabric ¼ F And the deformation gradient can be further multiplicatively decomposed as (Lubliner, 1985):

(3)

F ¼ Fe � Fv

where Fe and Fv are the elastic deformation gradient coming from the spring and viscous deformation gradient coming from the dashpot, respectively. Then, define the Cauchy-Green tensors as: C ¼ FT � F; ​ ​ ​ ​ Ce ¼ FTe � Fe ¼ Fv T � C � Fv 1

(4)

B ¼ F � FT ; ​ ​ ​ ​ Be ¼ Fe � FTe And define the Green-Lagrange tensors as: 1 E ¼ ðC 2

1 IÞ; ​ ​ ​ Ee ¼ ðCe 2

(5)

IÞ

For the later use, the invariants of C and Ce are defined as: �o 1n I1 ¼ trðCÞ; I2 ¼ ½trðCÞ �2 tr C2 ; I3 ¼ detðCÞ 2 �o 1n ½trðCe Þ �2 tr Ce 2 ; I3e ¼ detðCe Þ I1e ¼ trðCe Þ; I2e ¼ 2

(6)

And the volume-preserving deformation invariants are defined as: I 1 ¼ I 3 1=3 I1 ; ​ ​ ​ ​ ​ I 2 ¼ I 3 2=3 I2 e

I 1 ¼ I e3

1=3

e

I1 ; ​ ​ ​ ​ I 2 ¼ I e3

2=3

(7)

I2

The main deformation during the loading process of fabrics includes tension along the fiber yarn direction and shear between the weft and warp fiber yarns because of their special material structure, so the invariants representing the fiber stretch and fiber-fiber shear are introduced (Peng et al., 2013): I a4 ¼ a0 � C � a0 ; ​ I b4 ¼ b0 � C � b0 � 1=2 I7 ¼ I a4 I b4 a0 � C � b0 a0 � b0

(8)

where a0 and b0 are unit vectors along the original directions of weft and warp yarns, respectively. Ia4 and Ib4 represent the stretches of yarns. And I7 represents the cross-over shearing between the two families of yarns. For the convince of later expression, unit structure tensors are defined for woven fabrics in the initial undeformed configuration: Ma ¼ a0 � a0 ; ​ Mb ¼ b0 � b0 ; ​ Mab ¼ a0 � b0 ; ​ Mba ¼ b0 � a0 And unit structure tensors in the deformed configuration are defined as:

5

(9)

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Ma ¼ a � a ¼

F � Ma � FT F � M b � FT ; ​ ​ ​ Mb ¼ b � b ¼ I a4 I b4

Mab ¼ a � b ¼

(10)

F � Mab � FT F � Mba � FT qffiffiffiffiffiffiffi ; ​ Mba ¼ b � a ¼ qffiffiffiffiffiffiffi a b I4I4 I a4 I b4

where a and b are unit vectors along the current directions of weft and warp yarns, respectively. 3.2. Constitutive equations Constitutive relations can be derived on account of the second law of thermodynamics, so the Helmholtz free energy of the SMPC should be expressed first. As shown in Fig. 5, it is the sum of three parts: ΨðC; Ce ; a0 ; b0 ; TÞ ¼ Ψmatrix ðC; Ce ; TÞ þ Cf ðTÞΨfabric ðC; a0 ; b0 Þ þ ΨT ðTÞ

(11)

where T is temperature. Ψmatrix and Ψfabric are the Helmholtz free energies of the SMP matrix and woven fabric reinforcement, respectively. The bonding strength between matrix and fibers changes with temperature, which is high at low temperature and low at high temperature (Li et al., 2019). So, the energy contribution of the fabric to the whole SMPC also varies in the shape memory cycle. Then, a temperature correction coefficient Cf related to temperature is introduced here to revise the energy contribution of the fabric to the total Helmholtz free energy of SMPCs. ΨT are the only temperature-dependent Helmholtz free energy (Reese et al., 2010): � � T ΨT ðTÞ ¼ Uref T ηref þ C T Tref T ln (12) Tref where Tref is the reference temperature. Uref and ηref are the reference internal energy and reference entropy, respectively. And C is the heat capacity. And Ψmatrix can be further decomposed into two parts: (13)

Ψmatrix ¼ Ψm ðC; TÞ þ Ψe ðCe ; TÞ where Ψm and Ψe are the Helmholtz free energies coming from the single spring and Maxwell model, respectively. _ can be expressed as: Based on Eqs. (11) and (13), Ψ � � � � _ ¼ ∂Ψm : C_ þ ∂Ψe : C_ e þ Cf ∂Ψfabric : C_ þ Ψ ∂C ∂Ce ∂C � � ∂Ψm ∂Ψe ∂Cf ∂ΨT _ ​ ​ ​ ​ ​ ​ ​ ​ ​ þ þ Ψfabric þ T ∂T ∂T ∂T ∂T

(14)

Clausius-Duhem inequality form of the second law of thermodynamics can be expressed as: S : E_

_ þ T_ ηÞ ðΨ

(15)

q � rT = T � 0

where S represents the second Piola-Kirchhoff stress, η represents entropy, q represents heat flux vector and r is gradient operator. Based on Eqs. (5) and (14), the expression of Clausius-Duhem inequality can be further revised as: � � 1 ∂ Ψm ∂Ψe ∂Ψfabric S 2 � Fv T 2Cf 2Fv 1 � : C_ 2 ∂C ∂Ce ∂C � � ∂Ψ ∂Ψ ∂Ψ ∂C η þ T þ m þ e þ Ψfabric f T_ (16) ∂T ∂T ∂T ∂T � ∂Ψe 2Ce � : Lv q � rT T � 0 ∂Ce The above inequality should be always fulfilled, so it is concluded: S¼2

∂Ψm ∂Ψe ∂Ψfabric � F T þ 2Cf þ 2Fv 1 � ∂C ∂Ce v ∂C

1 ∂Ψe Lv ¼ Ce � ζ ∂Ce

η ¼ ηref þ C ln q¼

(17) T Tref

∂ Ψm ∂T

∂Ψe ∂T

Ψfabric

∂Cf ∂T

krT; ​ k > 0 6

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where Lv ¼ F_ v � Fv 1 . ζ is viscosity parameter. k is called as heat conduction coefficient guaranteeing the non-negative value of the thermal dissipation. Based on Eq. (17), the second Piola-Kirchhoff stress S of the SMPC includes two parts: (18)

S ¼ Smatrix þ Cf Sfabric

where Smatrix and Sfabric represent the second Piola-Kirchhoff stresses coming from the SMP matrix and woven fabric reinforcement, respectively. And Smatrix and Sfabric are expressed as: � � ∂Ψm ∂Ψe Smatrix ¼ 2 � Fv T þ Fv 1 � ∂C ∂Ce (19) ∂Ψfabric Sfabric ¼ 2 ∂C 3.3. Specific Helmholtz free energy function for the SMP matrix Helmholtz free energy of the SMP matrix can be expressed by a variety of hyperelastic laws and the proper energy function form can be chosen based on the specific material. Mooney-Rivlin model is used here. Then, Ψm and Ψe can be given as: �2 1 � 1=2 Ψm ¼ Cm10 ðI 1 3Þ þ Cm01 ðI 2 3Þ þ 1 I3 Dm (20) �2 � � 1 � e1=2 e e e e I Ψe ¼ C10 I 1 3 þ C01 I 2 3 þ 1 De 3 m e e where Cm 10 , C01 , C10 , C01 , Dm and De are material parameters depending on temperature. According to the isochoric flow assumption (Nguyen et al., 2008), that is jFv j ¼ 1, I3 ¼ Ie3 can be obtained with Eq. (3). Then, we can define Dm ¼ De ¼ D for annotation convenience. Based on Eqs. (17) and (19), σmatrix and Lv can be derived as:

σmatrix ¼ I 3 1=2 F � Smatrix � FT � � � 1 2 I1 I þ 2Cm01 I 3 7=6 ðI1 I CÞ � B I2 I 3 3 � � � � 1e ​ ​ ​ ​ ​ ​ ​ ​ ​ þ2Ce10 I 3 1=2 I e3 1=3 Be I I þ 2Ce01 I 3 1=2 I e3 2=3 I e1 I Ce � Be 31

� ​ ​ ​ ​ ​ ​ ​ ​ ​ ¼ 2Cm10 I 3 5=6 B

​ ​ ​ ​ ​ ​ ​ ​ ​ þ 1 Lv ¼ Ce10 I e3 ζ ​ ​ ​ ​ ​ þ

1=3

2 � 1=2 I D 3

1 1 e 1=2 e 1=2 I I3 ζD3

�

(21)

� 1 I

� 1e 1 I 1 I þ Ce01 I e3 3 ζ

� Ce

2e I I 32

� 2=3

I e1 I

� Ce � Ce

2e I I 32

�

� 1 I

(22)

m e e Equations (21) and (22) contain six parameters including Cm 10 , C01 , C10 , C01 , D and ζ. And the same parameter weight function as in Su and Peng (2018) is introduced:

φ¼

1 1 þ exp½gðT

(23)

Tr Þ�

where φ is the weight of material parameter values at low temperature and the weight of material parameter values at high tem­ perature is 1 φ. Tr and g are constant parameters. Then, the six temperature-dependent material parameters are expressed as: Cm10 ¼ φCm10 l þ ð1

φÞCm10 h ; ​ ​ ​ ​ Cm01 ¼ φCm01 l þ ð1

φÞCm01

Ce10

φÞCe10 h ;

φÞCe01

h

φÞζ

h

¼

φCe10 l

þ ð1

D ¼ φD l þ ð1

​ ​ ​ ​

Ce01

¼

φCe01 l

þ ð1

φÞD h ; ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ζ ¼ φζ l þ ð1

h

(24)

where subscript h denotes the corresponding parameter value at high temperature and subscript l at low temperature. 3.4. Specific Helmholtz free energy function for the woven fabric Based on the special deformation characteristic of the woven fabric, the Helmholtz free energy can be decomposed into two parts (Peng et al., 2013): 7

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� Ψfabric ¼ Ψten I a4 ; I b4 þ Ψshe ðI7 Þ

(25)

where Ψten and Ψshe represent the Helmholtz free energies contributed by the fiber stretch and fiber-fiber shear, respectively. The Helmholtz free energy from the fiber stretch can be represented by a polynomial function of Ia4 and Ib4 (Peng et al., 2013): h h � �4 �4 i �3 �3 i þ k2 I a4 1 þ I b4 1 Ψten I a4 ; I b4 ¼ k1 I a4 1 þ I b4 1 h (26) �2 �2 i ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ þ k3 I a4 1 þ I b4 1 where k1 , k2 and k3 are constant material parameters with unit of stress. And the Helmholtz free energy coming from the fiber-fiber shear can be also expressed by a polynomial function of I7 (Peng et al., 2013): (27)

Ψshe ðI7 Þ ¼ k4 ðI7 Þ4 þ k5 ðI7 Þ3 þ k6 ðI7 Þ2 where k4 , k5 and k6 are constant material parameters with unit of stress. Then, the Cauchy stress σfabric of the woven fabric reinforcement can be calculated as follows with Eq. (19):

σfabric ¼ I 3 1=2 F � Sfabric � FT � ∂Ψten ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ¼ I 3 1=2 2I a4 a ∂I 4 � ∂Ψten 1=2 2I b4 b ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ þ I3 ∂I 4 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ þ I 3 1=2

�

ðI7 þ a0 � b0 Þ

∂Ψshe Ma ∂I7

ðI7 þ a0 � b0 Þ

∂Ψshe Mb ∂I7

(28)

�

∂Ψshe ðMab þ Mba Þ ∂I7

4. Model verification The mechanical and shape memory experiments in Section 2 provide a series of experimental data representing the anisotropic thermomechanical material behaviors of the prepared SMPC, which can be used to verify the proposed model. 4.1. Parameters determination Based on the engineering stress-strain curves of the SMP from the uniaxial tension tests, as shown in Fig. 6, the initial elastic moduli at different temperatures can be calculated with the stresses and strains in the initial tension curves (less than 1%). Then, the SMP matrix parameters listed in Table 2 can be determined based on the parameter determination guideline described in Appendix. The SMP matrix is isotropic, so all simulations of the SMP matrix in tension and shape memory tests were conducted with MATLAB in 1D analysis model for computational efficiency. The comparison between the simulated initial elastic moduli and experiment data is shown in Fig. 7 and good consistency is achieved. The parameters in the woven fabric part are determined using the uniaxial tension and bias extension tests of the fabric based on the

Fig. 6. Experimental engineering stress-strain curves of the SMP. 8

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Table 2 Material parameters of the proposed constitutive model. Parameters

Values

Units

m m m Cm 10 l ,C01 l ,C10 h ,C01

1.0, 0.2, 1.0, 0.2

MPa

Ce10 l ,Ce01 l ,Ce10 h ,Ce01 D l ,D

ζ l ,ζ g

Tr

h

h

k1 ,k2 ,k3 ,k4 ,k5 ,k6

Cf

h h

106, 24, 0.12, 0.024

2.54 � 10 3, 1.67 � 10

3

MPa MPa

1

5.0 � 104, 150

MPa⋅s 1

0.45

�

C

43.5

�

C

2090, 2280, 2.15, 0.15, 0.001, 0.06

MPa

7.6 � 10 3Tþ0.628

Fig. 7. Initial elastic moduli of the SMP at different temperatures.

parameter determination guideline described in Appendix, as listed in Table 2. The simulation of the woven fabric in uniaxial tension was conducted with MATLAB in 1D analysis model. The woven fabric in bias tension was simulated using ABAQUS with the anisotropic hyperelastic part of the model compiled into UANISOHYPER_INV. In the finite element simulations, a rectangular plate with the same dimensions as the fabric specimens was meshed with shell elements S4R. In the loading step, one end of the rectangle was fixed with the other end loaded. Comparison between the simulations and experiments is shown in Fig. 8 (a) and (b). And the consistency is good. Based on the engineering stress-strain curves of the SMPC from the uniaxial tension and bias extension tests, as shown in Fig. 9 (a) and (b), the initial elastic moduli at different temperatures can be calculated. With the parameters of the SMP matrix and woven fabric determined, the temperature correction coefficient is obtained using the initial elastic moduli of the SMPC in the isothermal uniaxial

Fig. 8. Engineering stress-strain curves of the woven fabric from tension experiment. 9

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Fig. 9. Experimental engineering stress-strain curves of the SMPC.

tension at different temperatures based on the parameter determination guideline described in Appendix, as listed in Table 2. Fig. 10 shows the comparison between the simulated and experimental results. Overall, the predicted initial elastic moduli have good con­ sistency with the experiment data. 4.2. Verification With all the parameters determined, the initial elastic moduli of the SMPC in isothermal bias extension at different temperatures are simulated, as shown in Fig. 11. The finite element simulations of the SMPC in bias tension and shape memory tests were implemented using ABAQUS with the anisotropic visco-hyperelastic constitutive model compiled into UMAT. Moreover, a rectangular plate with the same dimensions as the SMPC specimens was meshed with solid elements C3D8. In the loading step, one end of the rectangular plate is fixed with the other end loaded. The good consistency between the simulation results and experiment data demonstrates the effec­ tiveness of the proposed model in predicting the anisotropic mechanical behavior. For the shape memory materials, shape recovery property is the most important. So, shape memory cycle tests of the SMP and SMPC are simulated to verify the proposed model further. Being restricted by the temperature-controlled chamber used in the experiments, the cooling and heating rates are not constant. So, the profiles of temperature changing with time in the cooling and reheating steps of the shape memory cycle tests were recoded and shown in Fig. 12. The profiles were then used as boundary conditions in the numerical simulations. Fig. 13 shows the numerical engineering strain versus temperature and time during the unconstrained shape recovery step (the

Fig. 10. Initial elastic moduli of the SMPC in uniaxial tension at different temperatures. 10

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Fig. 11. Initial elastic moduli of the SMPC in bias extension at different temperatures.

Fig. 12. Temperature histories in the cooling and reheating steps.

deformation is free) of the SMP matrix. And the numerical results reproduce the shape recovery tendency of the experiments, which demonstrates that the proposed model can describe the fundamental shape memory character of the SMPC coming from the matrix. The simulated engineering strains changing with temperature and time during the unconstrained shape recovery processes of the SMPC with two different loading modes are also shown in Figs. 14 and 15. And the comparison with the experiment data shows that the proposed model has the ability to predict the anisotropic shape recovery tendencies of the SMPC with different initial loading direction accurately. So, the validity of the proposed model is demonstrated. In order to verify the proposed model further, the shape fixity and recovery ratios of the above-mentioned shape memory cycles were also calculated: Rf ¼

εi εi εf ; ​ Rr ¼ ε0 εi

(29)

where Rf and Rr are the shape fixity and recovery ratios, respectively. ε0 is the initial loading strain. εi and εf are the strains at the beginning and end of the reheating process, respectively. Fig. 16 shows the fixity and recovery ratios of the SMP and SMPC with two different loading modes and good consistency between the simulation results and experiment data is achieved, which verifies the effectiveness of the proposed model further.

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Fig. 13. Shape recovery curves during the unconstrained shape recovery process of the SMP matrix.

Fig. 14. Shape recovery curves during the unconstrained shape recovery process of the SMPC in uniaxial tension.

5. Numerical investigation of the effect of initial fiber yarn orientation on the shape fixity and recovery For the woven fabric-reinforced SMPCs, the anisotropic mechanical property comes from the special material structure of fabrics. So, shape memory cycles with different initial fiber yarn orientations, which have similar processes with the experiments described in Section 2, are simulated with the proposed model and determined material parameters to investigate the influence of initial fiber yarn orientation on the shape fixity and recovery. Initial fiber yarn orientations of 45� /45� , 30� /60� , 15� /75� , 0� /90� are used. In these shape memory cycles, the strains in the loading step are adjusted based on the different initial fiber yarn orientations considering the elongation limit of carbon fiber (10% for 45� /45� , 7% for 30� /60� , 4% for 15� /75� , 1% for 0� /90� ). And physical experiments of shape memory cycles with initial fiber yarn orientations of 45� /45� and 0� /90� , which are the upper and lower boundaries of the fiber orientation field, were also conducted to verify the correctness of the simulations. All the engineering strain-temperature and engineering strain-time curves of different initial fiber yarn orientations in the reheating step are shown in Fig. 17. And the simulation results have a good consistency with the experimental data. So, the reliability of the numerical results is demonstrated. In order to evaluate the effect of fiber yarn orientation on the shape memory behaviors further, the shape fixity and recovery ratios of the aforementioned shape memory cycles were also calculated. Fig. 18 shows the fixity ratio versus initial fiber yarn orientation and recovery ratio versus initial fiber yarn orientation curves. And we can see that the simulated fixity and recovery ratios of the two shape memory cycles with upper- and lower-boundary fiber orientations have a good consistency with the experiment data. So, the 12

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Fig. 15. Shape recovery curves during the unconstrained shape recovery process of the SMPC in bias extension.

Fig. 16. Fixity and recovery ratios of the SMP and SMPC in the shape memory cycles.

effectiveness of the proposed model and numerical simulation results is further demonstrated. Figs. 17 and 18 show that fiber yarn orientation has great effect on the shape memory performance. When the fiber yarn orientation varies from 45� /45� to 0� /90� , shape recovery initiates ahead of time step by step and the shape fixity and recovery ratios gradually decrease and arise, respectively. Rigid rotation of fiber yarns emerges upon loading in the shape memory cycle, which has destructive influence on the recovery process and can be represented by the variation of shear angle between the weft and warp fiber yarns. Recovery initiates after the rigid rotation is overcome, which needs large recovery force at high temperature. The rotation effect at the beginning of the reheating step decreases with the fiber yarn orientation varying from 45� /45� to 0� /90� , as shown in Fig. 19. So, the recovery processes initiate increasingly early. Carbon fiber has temperature-independent elasticity, which impedes the shape fixity and promotes the shape recovery. The spring-back effect strengthens with fiber yarn orientation varying from 45� /45� to 0� /90� . So, the final fixity ratio decreases while recovery ratio increases. 6. Conclusions An anisotropic visco-hyperelastic constitutive model for thermally induced woven fabric-reinforced SMPCs was developed. The thermomechanical behaviors were decomposed into isotropic viscoelasticity and anisotropic hyperelasticity. A spring element and a Maxwell model arranged in parallel were engaged to represent the viscoelastic material behavior while a single spring element was employed to describe the hyperelastic material property. A reasonable temperature correction coefficient was also introduced to represent the varying contribution of fabric reinforcements to the SMPCs. Through decomposing the thermodynamic free energy, constitutive relations were derived with the Clausius-Duhem inequality. Parameters were determined with the tensile experiment data. And the shape memory cycle tests with different loading modes were used to verify the developed model. At last, the effect of fiber yarn orientation on the shape fixity and recovery was numerically investigated with the proposed model. 13

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Fig. 17. Shape recovery curves in the shape recovery processes of the SMPC with different initial fiber yarn orientations.

Fig. 18. Fixity ratio-initial fiber yarn orientation and recovery ratio-initial fiber yarn orientation curves.

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Fig. 19. Contours of fiber yarn rotation at the beginning of the reheating step.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Xiaobin Su: Conceptualization, Methodology, Writing - original draft. Yingyu Wang: Data curation, Investigation. Xiongqi Peng: Supervision, Writing - review & editing. Acknowledgements The supports from the National Natural Science Foundation of China (11972225) is gratefully acknowledged. Appendix. Guideline for determination of material parameters The developed constitutive model has twenty-two material parameters all together. Though the number of the material parameters 15

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seems to be large, they can be determined separately step by step with dividing the parameters into three parts: the SMP matrix, woven fabric and temperature correction coefficient. A1. The SMP matrix parameters m m m e e e e There are fourteen parameters in the SMP matrix part including Cm 10 l,C01 l,C10 h,C01 h,C10 l,C01 l,C10 h,C01 h,D l ,D h ,ζ l ,ζ h , g and Tr . All the parameters can be determined with the initial uniaxial tension curves of the SMP at high and low temperatures together with the initial elastic moduli at different temperatures. m A1.1 Cm 10 h,C01 h and D h m m m At high temperature, Cm 10 ¼ C10 h, C01 ¼ C01 h and D ¼ D h . And the spring of the Maxwell element has no stress contribution to the SMP matrix because of the small viscosity of the dashpot. Then, we have Be ¼ I. With Eq. (21), the stress expression at high temperature is revised as: � � � � � 1 2 7=6 I1 þ 2Cm01 h I 3 I2 σ matrixi ¼ 2Cm10 h I 3 5=6 λ2i I1 λ2i λ2i 3 3 (A.1) � � 2 1=2 ​ ​ ​ ​ ​ ​ ​ ​ ​ þ I3 1 ; ​ ði ¼ 1; 2; 3Þ D h m where λi are the stretches. Fitting the initial tension curve at high temperature with Eq. (A.1) can determine Cm 10 h,C01 h and D h , as shown in Fig. A1.

Fig. A1. Initial tensile curves of the SMP matrix at high and low temperatures. m e e A1.2 Cm 10 l,C01 l,C10 l, C01 l and D l m m m e e e e At low temperature, Cm 10 ¼ C10 l, C01 ¼ C01 l, C10 ¼ C10 l, C01 ¼ C01 l and D ¼ D l . And the spring of the Maxwell element un­ dertakes the complete deformation because of the large viscosity of the dashpot. Then, we have Be ¼ B. With Eq. (21), the stress expression at low temperature is revised as: � � � � � 1 2 7=6 I1 þ 2Cm01 l I 3 I2 σ matrixi ¼ 2Cm10 l I 3 5=6 λ2i I1 λ2i λ2i 3 3 � � � � � 1 2 5=6 7=6 (A.2) I1 þ 2Ce01 l I 3 I2 ​ ​ ​ ​ ​ ​ ​ ​ þ 2Ce10 l I 3 λ2i I1 λ2i λ2i 3 3

​ ​ ​ ​ ​ ​ ​ ​ þ

2 � 1=2 I D l 3

� 1 ; ði ¼ 1; 2; 3Þ

m e e where λi are the stretches. Fitting the initial tension curve at low temperature with Eq. (A.2) can determine Cm 10 l, C01 l, C10 l, C01 l and D l , as shown in Fig. A1.

A1.3 Ce10 h, Ce01 h, ζ h , ζ l , g and Tr Initial elastic moduli at different temperatures can be used to determine the remained parameters. And the fitting procedure contains the following steps: e m 4 ∘ (1) Give the initial values: Ce10 h¼ Cm 10 h , C01 h¼ C01 h , ζ h ¼ 50MPa � s, ζ l ¼ 10 MPa � s, g ¼ 1:0 and Tr ¼ 50 C. All the initial 16

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values are chosen based on the experiment data and representative material parameters in the literature. (2) Calculate the initial elastic moduli at different temperatures using the mechanical equations (3), (21) and (22) and parameter equations (23) and (24). (3) Check the consistency between the predicted results and experimental data: good consistency leads to (4) and poor consistency leads to (2) with adjusted values. (4) The parameters are determined. A2. The woven fabric parameters There are six parameters in the woven fabric part including k1 ,k2 ,k3 ,k4 ,k5 and k6 . k1 , k2 and k3 are for the fiber stretch energy. k4 , k5 and k6 are for the fiber-fiber shear energy. Based on Eqs. (26)–(28), the stress of the woven fabric can be expressed as follows: 2 �3 �2 �3 a a 1 þ 6k2 I a4 I a4 1 þ 4k3 I a4 I a4 1 1=2 4 8k1 I 4 I 4 5Ma σfabric ¼ I 3 � I7 4k4 I 37 þ 3k5 I 27 þ 2k6 I7 2 �3 �2 �3 b b (A.3) 1 þ 6k2 I b4 I b4 1 þ 4k3 I b4 I b4 1 1=2 4 8k1 I 4 I 4 5Mb ​ ​ ​ ​ ​ ​ ​ ​ ​ þ I3 � 3 2 I7 4k4 I 7 þ 3k5 I 7 þ 2k6 I7 � 1=2 4k4 I 37 þ 3k5 I 27 þ 2k6 I7 ðMab þ Mba Þ ​ ​ ​ ​ ​ ​ ​ ​ ​ þ I3 Then, all these parameters can be determined with the following procedure: (1) Through fitting the uniaxial tension experiment data, k1 , k2 and k3 can be determined based on Eq. (A.3) with I7 ¼ 0 and Ib4 ¼ 1(orIa4 ¼ 1). (2) Through fitting the bias extension experiment data, k4 , k5 and k6 can be obtained based on Eq. (A.3) with k1 ,k2 ,k3 determined. A3. Temperature correction coefficient Cf is temperature correction coefficient for the woven fabric. Thermomechanical property of the SMP changes significantly with temperature while the carbon fiber fabric is nearly unchanged, so the contribution of the fabric to the whole SMPC also varies with temperature. Here, we assume that the correction coefficient is a linear function of temperature for simplicity. Then, Cf has the following expression: (A.4)

Cf ¼ kf T þ if

Through fitting the initial elastic moduli of the SMPC in the tensile experiments at different temperatures, Cf can be determined with the following procedure: (1) Give the initial values: kf ¼ 0 and if ¼ 1. (2) Calculate Cf at different temperatures with Eq. (A.4). (3) Predict the initial elastic moduli of the SMPC in the uniaxial tension along one fiber yarn direction at different temperatures with Eqs. (3), (21), (22) and (28). (4) Check the consistency between the predicted results and experimental data: good consistency leads to (5) and poor consistency leads to (2) with adjusted values. (5) The parameters are determined. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijplas.2020.102697.

References Ames, N.M., Srivastava, V., Chester, S.A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications. Int. J. Plast. 25, 1495–1539. Anand, L., Ames, N.M., Srivastava, V., Chester, S.A., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation. Int. J. Plast. 25, 1474–1494. Baghani, M., Naghdabadi, R., Arghavani, J., Sohrabpour, S., 2012. A thermodynamically-consistent 3D constitutive model for shape memory polymers. Int. J. Plast. 35, 13–30. Bergman, D., Yang, B., 2015. A finite element model of shape memory polymer composite beams for space applications. Int. J. Numer. Methods Eng. 103, 671–702. Boatti, E., Scalet, G., Auricchio, F., 2016. A three-dimensional finite-strain phenomenological model for shape-memory polymers: formulation, numerical simulations, and comparison with experimental data. Int. J. Plast. 83, 153–177. Boisse, P., Hamila, N., Vidal-Sall�e, E., Dumont, F., 2011. Simulation of wrinkling during textile composite reinforcement forming. Influence of tensile, in-plane shear and bending stiffnesses. Compos. Sci. Technol. 71, 683–692.

17

International Journal of Plasticity xxx (xxxx) xxx

X. Su et al.

Cheng, X.Y., Chen, Y., Dai, S.C., Bilek, M.M.M., Bao, S.S., Ye, L., 2019. Bending shape memory behaviours of carbon fibre reinforced polyurethane-type shape memory polymer composites under relatively small deformation: characterisation and computational simulation. Journal of the Mechanical Behavior of Biomedical Materials 100, 103372. Gao, J.F., Chen, W.J., Yu, B., Fan, P.X., Zhao, B., Hu, J.H., Zhang, D.X., Fang, G.Q., Peng, F.J., 2019. Effect of temperature on the mechanical behaviours of a single-ply weave-reinforced shape memory polymer composite. Compos. B Eng. 159, 336–345. Gong, Y.K., Peng, X.Q., Yao, Y., Guo, Z.Y., 2016. An anisotropic hyperelastic constitutive model for thermoplastic woven composite prepregs. Compos. Sci. Technol. 128, 17–24. Gu, J.P., Leng, J.S., Sun, H.Y., Zeng, H., Cai, Z.B., 2019. Thermomechanical constitutive modeling of fiber reinforced shape memory polymer composites based on thermodynamics with internal state variables. Mech. Mater. 130, 9–19. Guzman-Maldonado, E., Hamila, N., Boisse, P., Bikard, J., 2015. Thermomechanical analysis, modelling and simulation of the forming of pre-impregnated thermoplastics composites. Compos. Appl. Sci. Manuf. 78, 211–222. Hong, S.B., Jang, J.H., Park, H., Kim, J.-G., Goo, N.S., Yu, W.-R., 2019. Three-dimensional constitutive model of woven fabric-reinforced shape memory polymer composites considering thermal residual stress. Smart Mater. Struct. 28 (3), 035023. Leng, J.S., Lan, X., Liu, Y.J., Du, S.Y., 2011. Shape-memory polymers and their composites: stimulus methods and applications. Prog. Mater. Sci. 56, 1077–1135. Li, F.F., Scarpa, F., Lan, X., Liu, L.W., Liu, Y.J., Leng, J.S., 2019. Bending shape recovery of unidirectional carbon fiber reinforced epoxy-based shape memory polymer composites. Compos. Appl. Sci. Manuf. 116, 169–179. Li, Y.X., He, Y.H., Liu, Z.S., 2017. A viscoelastic constitutive model for shape memory polymers based on multiplicative decompositions of the deformation gradient. Int. J. Plast. 91, 300–317. Liu, T.Z., Liu, L.W., Yu, M., Li, Q.F., Zeng, C.J., Lan, X., Liu, Y.J., Leng, J.S., 2018. Integrative hinge based on shape memory polymer composites: material, design, properties and application. Compos. Struct. 206, 164–176. Liu, T.Z., Zhou, T.Y., Yao, Y.T., Zhang, F.H., Liu, L.W., Liu, Y.J., Leng, J.S., 2017. Stimulus methods of multi-functional shape memory polymer nanocomposites: a review. Compos. Appl. Sci. Manuf. 100, 20–30. Liu, Y.J., Du, H.Y., Liu, L.W., Leng, J.S., 2014. Shape memory polymers and their composites in aerospace applications: a review. Smart Mater. Struct. 23, 023001. Liu, Y.P., Gall, K., Dunn, M.L., Greenberg, A.R., Diani, J., 2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plast. 22, 279–313. Long, K.N., 2014. The mechanics of network polymers with thermally reversible linkages. J. Mech. Phys. Solid. 63, 386–411. Lubliner, J., 1985. A model of rubber viscoelasticity. Mech. Res. Commun. 12, 93–99. Mu, T., Liu, L.W., Lan, X., Liu, Y.J., Leng, J.S., 2018. Shape memory polymers for composites. Compos. Sci. Technol. 160, 169–198. Nguyen, T.D., Qi, H.J., Castro, F., Long, K.N., 2008. A thermoviscoelastic model for amorphous shape memory polymers: incorporating structural and stress relaxation. J. Mech. Phys. Solid. 56, 2792–2814. Nishikawa, M., Wakatsuki, K., Yoshimura, A., Takeda, N., 2012. Effect of fiber arrangement on shape fixity and shape recovery in thermally activated shape memory polymer-based composites. Compos. Appl. Sci. Manuf. 43, 165–173. Park, H., Harrison, P., Guo, Z.Y., Lee, M.-G., Yu, W.-R., 2016. Three-dimensional constitutive model for shape memory polymers using multiplicative decomposition of the deformation gradient and shape memory strains. Mech. Mater. 93, 43–62. Patel, K.K., Purohit, R., 2019. Improved shape memory and mechanical properties of microwave-induced shape memory polymer/MWCNTs composites. Materials Today Communications, 100579. Peng, X.Q., Guo, Z.Y., Du, T.L., Yu, W.-R., 2013. A simple anisotropic hyperelastic constitutive model for textile fabrics with application to forming simulation. Compos. B Eng. 52, 275–281. Qi, H.J., Nguyen, T.D., Castro, F., Yakacki, C.M., Shandas, R., 2008. Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers. J. Mech. Phys. Solid. 56, 1730–1751. Reese, S., B€ ol, M., Christ, D., 2010. Finite element-based multi-phase modelling of shape memory polymer stents. Comput. Methods Appl. Mech. Eng. 199, 1276–1286. Regrain, C., Laiarinandrasana, L., Toillon, S., Saï, K., 2009. Multi-mechanism models for semi-crystalline polymer: constitutive relations and finite element implementation. Int. J. Plast. 25, 1253–1279. Shi, G.H., Yang, Q.S., Zhang, Q., 2012. Investigation of buckling behavior of carbon nanotube/shape memory polymer composite shell. Proc. SPIE-Int. Soc. Opt. Eng. 8409, 542–545. Shojaei, A., Li, G.Q., 2013. Viscoplasticity analysis of semicrystalline polymers: a multiscale approach within micromechanics framework. Int. J. Plast. 42, 31–49. Srivastava, V., Chester, S.A., Ames, N.M., Anand, L., 2010a. A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition. Int. J. Plast. 26, 1138–1182. Srivastava, V., Chester, S.A., Anand, L., 2010b. Thermally actuated shape-memory polymers: experiments, theory, and numerical simulations. J. Mech. Phys. Solid. 58, 1100–1124. Su, X.B., Peng, X.Q., 2018. A 3D finite strain viscoelastic constitutive model for thermally induced shape memory polymers based on energy decomposition. Int. J. Plast. 110, 166–182. Tan, Q., Liu, L.W., Liu, Y.J., Leng, J.S., 2014. Thermal mechanical constitutive model of fiber reinforced shape memory polymer composite: based on bridging model. Compos. Appl. Sci. Manuf. 64, 132–138. Tobushi, H., Hashimoto, T., Hayashi, S., Yamada, E., 1997. Thermomechanical constitutive modeling in shape memory polymer of polyurethane series. J. Intell. Mater. Syst. Struct. 8, 711–718. Tobushi, H., Okumura, K., Hayashi, S., Ito, N., 2001. Thermomechanical constitutive model of shape memory polymer. Mech. Mater. 33, 545–554. Uchida, M., Wakuda, R., Kaneko, Y., 2019. Evaluation and modeling of mechanical behaviors of thermosetting polymer under monotonic and cyclic tensile tests. Polymer 174, 130–142. Wang, H.N., Fang, L., Zhang, Z., Epaarachchi, J., Li, L.Y., Hu, X., Lu, C.H., Xu, Z.Z., 2019. Light-induced rare earth organic complex/shape-memory polymer composites with high strength and luminescence based on hydrogen bonding. Compos. Appl. Sci. Manuf. 125, 105525. Xiao, R., Choi, J., Lakhera, N., Yakacki, C.M., Frick, C.P., Nguyen, T.D., 2013. Modeling the glass transition of amorphous networks for shape-memory behavior. J. Mech. Phys. Solid. 61, 1612–1635. Xie, H., Yang, K.-K., Wang, Y.-Z., 2019. Photo-cross-linking: a powerful and versatile strategy to develop shape-memory polymers. Prog. Polym. Sci. 95, 32–64. Zhang, X., Geven, M.A., Grijpma, D.W., Peijs, T., Gautrot, J.E., 2017. Tunable and processable shape memory composites based on degradable polymers. Polymer 122, 323–331. Zhao, W., Liu, L.W., Zhang, F.H., Leng, J.S., Liu, Y.J., 2019. Shape memory polymers and their composites in biomedical applications. Mater. Sci. Eng. C 97, 864–883.

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