A thermodynamically-consistent 3 D constitutive model for shape memory polymers

International Journal of Plasticity 35 (2012) 13–30

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

A thermodynamically-consistent 3D constitutive model for shape memory polymers M. Baghani a, R. Naghdabadi a,b,⇑, J. Arghavani a,c, S. Sohrabpour a,d a

Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran c Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran d The Academy of Sciences of IR Iran, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 4 June 2011 Received in final revised form 4 December 2011 Available online 10 February 2012 Keywords: Shape memory polymers Continuum thermodynamics Numerical solution Finite element

a b s t r a c t The ever increasing applications of shape memory polymers have motivated the development of appropriate constitutive models for these materials. In this work, we present a 3D constitutive model for shape memory polymers under time-dependent multiaxial thermomechanical loadings in the small strain regime. The derivation is based on an additive decomposition of the strain into six parts and satisfying the second law of thermodynamics in Clausius–Duhem inequality form. In the constitutive model, the evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings. The viscous effects are also fully accounted for in the proposed model. Further, we present the time-discrete form of the evolution equations in the implicit form. The model is validated by comparing the predicted results with different experimental data reported in the literature. Finally, using the finite element method, we solve two boundary value problems e.g., a 3D beam and a medical stent made of shape memory polymers. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Shape memory materials are a class of multi-phase smart materials that have the ability to return from a deformed state (temporary shape) to their original (permanent) shape called as shape memory recovery. The shape memory recovery is typically induced by an external stimulus such as heat, electricity or magnetism. This behavior has been observed in metals, ceramics and polymers (Lendlein and Langer, 2002; Gall et al., 2009; Gao et al., 2011; Diani et al., 2006). Shape memory materials have been researched, developed, and utilized in a wide range of applications such as advanced technologies in the aerospace, medical and oil exploration industries (Lendlein and Behl, 2009; Small IV et al., 2010; Leng et al., 2009). Among all smart materials, we focus on shape memory polymers (SMPs). In contrast to other smart materials such as shape memory alloys, SMPs possess the advantages of large elastic deformation, low energy consumption for shape programming, low cost, low density, potential biocompatibility, biodegradability and excellent manufacturability (Leng et al., 2009; El Feninat et al., 2002). Because of these characteristics, SMPs have attracted a great deal of interest for their potential applications. Moreover, SMPs have a promising future for application in sensors, actuators and smart devices (Monkman, 2000; Poilane et al., 2000; Tey et al., 2001; Saı¯, 2010). Although constitutive modeling of polymers has been carried out in several works (see e.g., Khan and Lopez-Pamies (2002), Reese (2003), Khan et al. (2006), Hasanpour et al. (2009) and Long et al. (2010) among others), but due to complicated ⇑ Corresponding author at: Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. Tel.: +98 21 6616 5546; fax: +98 21 6600 0021. E-mail address: [email protected] (R. Naghdabadi). 0749-6419/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2012.01.007

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

behavior of different polymers, this field of research is still under progress, especially in the study of SMPs. In addition to experimental efforts attempted to characterize the behavior of SMPs (Tobushi et al., 1997, 1998; Lendlein and Langer, 2002; Abrahamson et al., 2003; Gall et al., 2004; Liu et al., 2004, 2006; Baer et al., 2007; Atli et al., 2009; Kolesov et al., 2009; Kim et al., 2010; Volk et al., 2010a,b), considerable part of the research has focused on the prediction and description of the SMP behavior through the development of constitutive models. There have been several attempts to properly reproduce SMP features such as shape memory effect (SME) and timedependent behavior in a predictive modeling frame (see e.g., Liu et al. (2006), Diani et al. (2006), Kim et al. (2010), Srivastava et al. (2010) among others). In order to develop new constitutive models for SMPs, two general approaches have mainly been adopted: micro modeling and macro modeling. Description of micro-scale features, such as cross-linking, chain mobility, interface motion and entanglement of polymer molecules, is the main focus of micro models. They are useful for understanding the fundamental phenomena, although they are not easily applicable to the structural scale (Xu and Li, 2010; Nguyen et al., 2010). Macro approaches, on the other hand, phenomenologically describe the material behavior and, in general, they are appropriate for being utilized within numerical methods, such as the finite element method (FEM), in an efficient manner (Voyiadjis et al., 2010; Arghavani et al., 2010). In the following, we present a brief review on some constitutive models of SMPs, available in the literature. After introducing the SMPs in early 90s, researchers tried to model the characteristic behavior of these materials. Most of the earlier modeling efforts (Tobushi et al., 1997, 2001; Lin and Chen, 1999; Bhattacharyya and Tobushi, 2000; Abrahamson et al., 2003; Morshedian et al., 2005) have adopted rheological models consisting of spring, dashpot, and frictional elements in one-dimensional constitutive models. While such models seem to be simple, they have limitations in certain classes of constitutive relations and usually lead to predictions that agree only qualitatively with experiments. To describe the behavior of the SMPs, several macro or phenomenological constitutive models (Liu et al., 2006; Diani et al., 2006; Chen and Lagoudas, 2008a; Qi et al., 2008; Kim et al., 2010; Reese et al., 2010; Xu and Li, 2010; Baghani et al., 2012), as well as micro or physical based ones (Barot et al., 2008; Nguyen et al., 2008, 2010; Kafka, 2008; Srivastava et al., 2010) have been proposed in recent years. One of the first phenomenological constitutive models for thermoset SMPs within the small deformation regime has been proposed by Liu et al. (2006). They have adopted a first-order phase transition concept and modeled the SMP as a continuum mixture of a glassy and a rubbery phase. This model additively decomposes the strain into thermal, elastic, and a stored term. The stored strain has been introduced to identify the strain storage and release mechanisms. However, their model simplifies the SMP as a special elastic material hence does not consider the time-dependency of the material behavior. Moreover, the model does not present the mathematical evolution law upon heating (Chen and Lagoudas, 2008a,b). Based on this model, Chen and Lagoudas (2008a,b) developed a 1D constitutive model to capture the characteristics of SMP behavior in the large deformation regime. Constitutive equations of crystalline SMPs have been formulated by Barot et al. (2008) where the interaction between an original amorphous phase and a semi-crystalline phase is analyzed. Nguyen et al. (2008) have investigated the SMP behavior through structural and stress relaxation mechanisms. They proposed that the shape memory effect in the SMPs is primarily initiated by the drastic change of molecular chain mobility induced by the glass transition. The chain mobility underpins the ability of the chain segments to rearrange locally to bring the macromolecular structure to equilibrium, suggesting that the structure returns instantaneously to the equilibrium state at a high temperature but responds considerably more slowly in cooling process. Such modeling approach macroscopically freezes the material at a low temperature in a non-equilibrium configuration and allows the material to store a temporary shape. Increasing the temperature, the mobility is restored and the shape recovery happens. This approach can be used to physically interpret the mechanisms responsible for the SME. However, the utilization of this approach requires taking advantage of some elaborate physically-based hypotheses e.g., the free energy activation parameter in the Adams–Gibbs model is assumed to be independent of the temperature and pressure which is still questionable (Andreozzi et al., 2004; Xu and Li, 2010). From a macroscopic point of view, SME can be characterized in a stress–strain-temperature diagram as illustrated in Fig. 1. The thermomechanical cycle starts at a strain- and stress-free state while the temperature is Th (high temperature) (point , permanent shape). At this point, a purely mechanical loading is applied to SMP and the material demonstrates a rubbery behavior up to point . At point , strain is held fixed and the temperature is decreased until the rubber-like polymer drastically turns into a glassy polymer at the low temperature Tl (point , fixed shape). In fact, in the neighbourhood of the transition temperature Tg, SMP exhibits a combination of rubbery and glassy behaviors. Subsequently, the material is unloaded. Regarding much higher stiffness of the glassy phase in comparison to the rubbery phase, after unloading, strains change slightly (point ). Finally, the temperature is increased up to Th. It is seen that the strain will relax and the original permanent shape can be recovered (point ). It is noted that based on the quality of the SMP, in some practical cases, some residual strain may remain in the SMP (point ) (Volk et al., 2010a,). This cycle is called stress-free strain recovery in SMP applications. In practice, other types of recoveries may happen. If at point , the strain is fixed and the temperature is increased, the fixed-strain stress recovery (point ) happens. The dotted line in Fig. 1, illustrates the mentioned behavior schematically. In this study, we present a 3D constitutive model for thermoset SMPs, based on the continuum thermodynamic considerations. We focus on a phenomenological or macro-modeling approach, which is able only to give an average representation of the phenomena occurring at the material micro-mechanical level (Xiao et al., 2006; Voyiadjis et al., 2010). Recently, with the aim of improving the SMP mechanical properties, e.g., strength and stiffness, hard particles (e.g., glass microballoons) are

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

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Fig. 1. Stress–strain-temperature diagram illustrating the thermomechanical behavior of a pre-tensioned shape memory polymer under different strain or stress recovery conditions.

dispersed in SMP matrices (Li and Nettles, 2010). To capture the behavior of this family of composites, we assume the material as a mixture of a hard segment and an SMP matrix. Using the rule of mixture, we then incorporate the effect of hard segment in the constitutive model. As the stiffness of the hard segment is much higher than the stiffness of the rubbery or glassy phases (Li and Nettles, 2010), the obtained composite generates a higher value of stress in the fixed-strain stress recovery which is a desirable feature in engineering applications. Moreover, taking into account the importance of timedependent behavior in modeling of polymer materials, three tensorial internal variables (corresponding to the rubbery, glassy and hard phases) are incorporated into the presented constitutive model. The article is organized as follows. In Section 2, a three-dimensional thermodynamically-consistent constitutive model of a thermoset SMP-based material is developed in the small strain regime. In Section 3, time-discrete form of the introduced model is presented within the finite element framework. In Section 4, we apply the proposed constitutive model to several examples and investigate the ability of the model to reproduce the features of the SMPs. In Section 5, using the proposed finite element model, two boundary value problems, i.e., a 3D beam and a medical stent are simulated. Finally, in Section 6, we present a summary and draw conclusions. 2. Constitutive model development 2.1. Kinematic description In this study, we separate the material domain into shape memory polymer and hard segments. Further, we divide the shape memory polymer segment into glassy and rubbery phases. In this regard, we introduce the equivalent representative volume element (RVE) of the material as schematically illustrated in Fig. 2. Assuming small strains, we decompose the total strain additively into four parts; shape memory polymer, hard, irreversible and thermal parts. This decomposition has already been applied in one-dimensional modeling of SMP based foams by Xu and Li (2010).

Fig. 2. Equivalent representative volume element for shape memory polymer and hard segments: dots represent the hard segment in all cases. Left: at T = Tl, dominant glassy phase. Middle: at T = Tg, combination of all phases. Right: at T = Th, dominant rubbery phase.

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

e ¼ /p ep þ /h eh þ ei þ eT

ð1Þ i

T

where superscripts p and h stand for the SMP and hard segments, respectively. Also, e and e are the irreversible and thermal strains, respectively, while ep describes the strain in SMP segment of the RVE. /p and /h denote the volume fractions of the SMP and hard segments of the RVE, respectively with the constraint /p + /h = 1. Further, the strain in the SMP segment is decomposed into two components. Such a decomposition has already been applied in other works (Liu et al., 2006; Chen and Lagoudas, 2008a; Kim et al., 2010).

ep ¼ ur er þ ug eG

ð2Þ

where subscripts r and g stand for the rubbery and glassy phases, respectively. eG represents the total strain in the glassy phase. Also, ur and ug are volume fractions of the rubbery and glassy phases, respectively with ur + ug = 1. It is assumed that /p and /h are constants, while ur and ug are functions of temperature. 2.1.1. Strain storage and recovery We now consider the phase transformation (rubbery to glassy and vice versa) in the representative volume element (RVE). Assuming temperature decreasing, the strain in the newly generated glassy phase, already been in the rubbery phase, had experienced the er previously. Then ugeG is defined as:

ug eG ¼ ug ðeg þ eg Þ ¼ ug eg þ

1 Vg

!

Z Vg

er dv ¼ ug eg þ

1 Vp

Z

er dv

ð3Þ

Vg

where Vg and Vp are volumes of the glassy phase and the SMP segment, respectively. In (3), strain in the glassy phase is divided into two parts: strain in the old glassy phase, eg, and strain in the newly generated glassy phase, eg . We now recast (3) as:

ug eG ¼ ug eg þ

Z

er dug ¼ ug eg þ eis

ð4Þ

Consequently in a cooling process eis is defined as:

eis ¼

Z

er dug

ð5Þ

Such a strain storage in the cooling process has previously been introduced by Liu et al. (2006). In contrast to the cooling process, in a heating process, the stored strain in the glassy phase should be relaxed. This can be mathematically shown as: G

g

g

ug e ¼ ug ðe þ e Þ ¼ ug e

g

1 þ Vg

!

Z Vg

eis 1 dv ¼ ug eg þ Vp ug

Z Vg

eis dv ug

ð6Þ

We may write (6) in a more compact form as:

ug eG ¼ ug eg þ

Z

eis du ¼ ug eg þ eis ug g

ð7Þ

As a result, in a heating process eis is defined as:

eis ¼

Z

eis du ug g

ð8Þ

We remark that the strain storage/release occurs only in the glassy phase. However, by definition, eis is assigned to the whole RVE. As a result, a division by ug appears in the integrals of Eqs. (6)–(8). From (5) and (8), it is concluded that eis is a fully thermal-driven variable. Moreover, we combine (5) and (8) to obtain:

is

e ¼ ks1

Z

r

e dug þ ks2

Z

is

e du ; ug g

8 > > < ks1 ¼ 1; ks1 ¼ 0; > > : ks1 ¼ 0;

ks2 ¼ 0; T_ < 0 ks2 ¼ 1; T_ > 0 ks ¼ 0; T_ ¼ 0

ð9Þ

2

where ð_Þ ¼ @=@t represents the derivative with respect to time. Also, ks1 and ks2 are constants to be utilized to identify the heating and cooling processes. Based on the experimental observations, time-dependent effects are of importance in constitutive modeling of polymeric materials (see e.g., Khan and Zhang (2001), Huber and Tsakmakis (2000) and Reese (2003) among others). Experimental observations also reveal that the storage/release process is a highly temperature-driven phenomenon (see e.g., Diani et al. (2011) and Volk et al. (2010a, among others). It is noted that although some SMPs may show time-dependent storage during the cooling process (Li and Xu, 2011) but this is beyond the scope of this paper.

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

17

In fact, to capture the viscoelastic behavior of SMPs in each phase, another decomposition should be employed. In this work, we assume that the material in all three phases behaves viscoelastically. Utilizing the small strain assumption, we may additively decompose strains in the glassy, rubbery and hard phases as:

eb ¼ eeb þ eib ; b ¼ r; g; h

ð10Þ

where the superscripts eb and ib denote the elastic and inelastic (viscous) parts of the strain in all phases, respectively. For instance, eer and eir denote elastic and inelastic (viscous) strains in the rubbery phase. Such a decomposition in the rubbery and glassy phases has been previously utilized in one-dimensional constitutive modeling of SMP behavior by Kim et al. (2010) and will be used in the following. We emphasize that the viscoelastic internal variables eib, (b = r, g, h), are employed to describe just the material time-dependent behavior and are not able to describe the storage and release mechanisms in SMPs. A schematic rheological illustration as shown in Fig. 3 could be helpful to follow the derivation of the equations in this Section. We remark that the total strain is the weighted summation of the strain in each phase (weights have been shown on the left hand side of each element in Fig. 3). 2.2. Free-energy density function for the RVE We now express the convex free-energy density function W for an amorphous SMP material. Based on the mixture rule, we express the following form for the energy function:

Wðe; T; /p ; /h ; ug ; er ; eer ; eg ; eeg ; eh ; eeh Þ ¼ /h Wh ðeh ; eeh Þ þ WT ðTÞ þ /p ½ur Wr ðer ; eer Þ þ ug Wg ðeg ; eeg Þ

ð11Þ

where Wr, Wg and Wh are Helmholtz free-energy density functions of the rubbery, glassy and hard phases, respectively. Also, WT denotes the thermal energy and temperature is represented by T. In order to enforce the kinematic constraints (1) and (2) in the formulation, we use the method of Lagrange multipliers and add the following term Wk to the free energy (11) (see e.g., Arghavani et al. (2010) and Zaki (2007) among others):

Wk ðe; T; /p ; /h ; ur ; ug ; er ; eg ; eis ; eh ; ei Þ ¼ k : ½e  /p ður er þ ug eg þ eis Þ  /h eh  ei  eT 

ð12Þ

where k is the (tensorial) Lagrange multiplier. Therefore, the free energy function is re-expressed as follows to consider the kinematic constraints (1) and (2) in the formulation:

W ¼ W þ Wk ¼ /h Wh þ WT þ /p ður Wr þ ug Wg Þ þ Wk

ð13Þ

We also define: neq eb b Wb ðeb ; eeb Þ ¼ Weq b ðe Þ þ Web ðe Þ;

b ¼ r; g; h

ð14Þ

where the superscripts eq and neq stand for equilibrium and non-equilibrium parts of Wb(eb, eeb). In the present model, it is emphasized that the internal variables are ug, eir, eig, eih, ei and eis. Thus, we should define evolution equations for the internal variables in the context of continuum thermodynamics. It is noted that we use a prescribed evolution equation for ug. This equation is derived using the unconstrained strain recovery of the material as a function of temperature (see the Appendix A). 2.3. Thermodynamic considerations The intrinsic mechanical dissipation in the Clausius–Duhem form is defined as (Haupt, 2002; Lubarda, 2001):

_ P0 _ þ gTÞ Dmech ¼ r : e_  ðW where g is the entropy. Substituting (13) and (14) into (15) leads to the following inequality:

Fig. 3. Schematic rheological illustration of the proposed constitutive model.

ð15Þ

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

" ! #  eq   @ Weq @ Wneq @ Wneq @ Wr @ Wneq @ Wneq g g g r r _g  _r  _ ir  /p ug _ ig : : þ e : e þ e : e @ er @ eer @ eer @ eg @ eeg @ eeg  eq   h   i @ Wh @ Wneq @ Wneq @W 0 _ h h  /h : e_ h  þ : e_ ih  ug T  W0T T_  e  /p ur er þ ug eg þ eis  /h eh  ei  eT : k_ h eh eh @ ug @e @e @e h   i 0 r g 0 g r _ is _ _ _ _ ð16Þ  e  /p ur e þ ug e þ ug ðe  e ÞT þ e  /h e_ h  eT T_  e_ i : k  gT_ > 0

r : e_  /p ur

where ()0 = @/@T denotes the derivative with respect to temperature. Regarding (9), the evolution equation of eis in the rate form, is obtained as:

_ is

e ¼u

0 _ gT

eis k s1 e þ k s2 ug

!

r

ð17Þ

It is noted that stored strain eis inherently consists of thermal-driven time-dependent parts. For example, we may decompose eis in the form of eis ¼ eise þ eisv where eise and eisv denote the stored strains related to the elastic and viscus effects during the storage process, respectively, with the following evolution equation laws:

8   is > < e_ ise ¼ u0g T_ ks1 eer þ ks2 uee g   is > : e_ isv ¼ u0g T_ ks1 eir þ ks2 uev

ð18Þ

g

However, Eq. (18) are consistent with Eq. (17) which does not consist of the elastic or viscus terms. Since we did not require the components of the stored strain (eise and eisv ), we may alternatively combine these equations and arrive at Eq. (17). _ Inequality (16) must be fulfilled for arbitrary thermodynamic processes, i.e. for arbitrary e_ ; e_ r ; e_ ir ; e_ g ; e_ ig ; e_ h ; e_ ih ; e_ i ; k_ and T. _ we may conclude (Coleman and Gurtin, 1967): For arbitrary choices of the variables e_ ; e_ r ; e_ g ; e_ h ; k_ and T,

8 eq neq eq neq eq neq @W @W @W @W > r ¼ k ¼ @@Werr þ @@Werer ¼ @egg þ @egeg ¼ @ehh þ @eheh > > < e ¼ /p ður er þ ug eg þ eis Þ þ /h eh þ ei þ eT > h   i > 0 > : g ¼ W0T  @@uW u0g þ /p u0g eg þ ðks1  1Þer þ ks2 ueis þ eT : r g

ð19Þ

g

Furthermore, the Clausius–Duhem inequality (16) is reduced to:

(

neq

/p ur

@ Wg @ Wneq r : e_ ir þ ug : e_ ig @ eer @ eeg

) þ /h

@ Wneq h : e_ ih þ r : e_ i P 0 @ eeh

ð20Þ

In accordance with the viscoelastic behavior of polymers, the following evolution equations are sufficient conditions for satisfaction of (20):

e_ ir ¼

1 @ Wneq r ; gr @ eer

e_ ig ¼

neq 1 @ Wg ; gg @ eeg

e_ ih ¼

neq 1 @ Wh ; gh @ eeh

e_ i ¼

1

gi

r

ð21Þ

where gr, gg, gh, and gi are positive viscous coefficients of the rubbery, glassy, hard and irreversible parts, respectively. We now summarize the constitutive equations in the time-continuous frame in Table 1. 3. Time-discrete form of the constitutive model In this section, we investigate the numerical solution of the constitutive model derived in Section 2 and summarized in Table 1, with the final goal of using it within a finite element program. The main task is to apply an appropriate numerical time-integration scheme to the evolution equations of the internal variables. It is noted that, in general, implicit schemes are preferred because of their stability at larger time step sizes. Moreover, the present section provides some details about the stress update and the computation of the consistent tangent matrix. These are two main points where the material model is directly connected to the finite element solution procedure. We treat the non-linear problem described in Section 2 as an implicit time-discrete deformation-driven problem. Accordingly, we subdivide the time interval of interest [0, t] in sub-increments and solve the evolution problem over the generic interval [tn, tn+1] with tn+1 > tn. To simplify the notation, we indicate with the subscript n a quantity evaluated at time tn, and with no subscript a quantity evaluated at time tn+1 (Arghavani et al., 2011). Further, we show the increment of time by Dt. Assuming to know the solution and the strain en at time tn as well as the strain e at time tn+1, the stress and the internal variables should be updated from the deformation history. 3.1. Time integration scheme In this section, we introduce quadratic forms for the free-energy density functions as follow:

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

19

Table 1 Time-continuous form of the proposed constitutive model. External variables: e, T Internal variables: ug, eis, eir, eig, eih, ei Kinematics: e = /pep + /heh + ei + eT ep = urer + ugeg + eis er = eer + eir, eg = eeg + eig, eh = eeh + eih Stress: eq

neq

r ¼ @@Werr þ @@Werer ¼

@ Weq g @ eg

@ Wneq

@ Weq

@ Wneq

h þ @ egeg ¼ @ ehh þ @ eeh Evolution equations: viscous evolution equations: neq

neq

e_ ir ¼ g1r @@Werer ; e_ ig ¼ g1g @@Wegeg ; e_ ih ¼ g1h

@ Wneq h @ eeh

; e_ i ¼ g1 r i

shape memory evolution equation:   is e_ is ¼ u_ g ks1 er þ ks2 ue g

with

8 > > < ks1 ¼ 1; ks1 ¼ 0; > > : ks1 ¼ 0;

ks2 ¼ 0; T_ < 0 ks2 ¼ 1; T_ > 0 ks2 ¼ 0; T_ ¼ 0

Entropy:





0

is

g ¼ W0T þ r : eT þ /p u0g eg þ ðks1  1Þer þ ks2 ue g

1 2

b b Weq eb : Keq b ðe Þ ¼ b : e ;

1 2

eb Wneq eeb : Kneq : eeb ; b ðe Þ ¼ b



þ @@uW u0g g

b ¼ r; g; h

ð22Þ

neq where Keq are fourth-order positive definite tensors of equilibrium and non-equilibrium parts of each phase. In the b and Kb case of isotropic materials, we only need to know elastic modulus of equilibrium part, Eeq b , elastic modulus of non-equilibrium part, Eneq b , and Poisson’s ratio, mb, of each phase. We emphasize that the same Poisson’s ratio is used for equilibrium and non-equilibrium tensors. We now apply a backward-Euler integration algorithm to the model presented in Table 1 and obtain:

eib ¼ Sb : eibn þ Jb : eb

ð23Þ

with b

S ¼

Iþ

Dt

gb

!1 Kneq b

Jb ¼

;

Dt

gb

Sb : Kneq b ;

b ¼ r; g; h

ð24Þ

where I is the symmetric fourth-order identity tensor. Moreover:

ei ¼ ein þ

Dt

gi

r

ð25Þ

Regarding (17), the discrete form of the evolution equation for the stored strain is obtained as:

eis e ¼ e þ Dug ks1 e þ ks2 ug is

is n

!

r

ð26Þ

where Dug = ug  ugn. Consequently, we obtain:

is

e ¼ 1  Dug

ks2

ug

!1

n

eisn þ Dug ks1 er

o

ð27Þ

Finally, we conclude:

8 is e ¼ ks3 eisn þ ks4 er > > <  1 ks ks3 ¼ 1  Dug u2 g > > : ks4 ¼ ks1 ks3 Dug

ð28Þ

20

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

3.2. Consistent tangent matrix In this section, we report the construction of the tangent matrix. Using the discrete form of (19)1, we may write:





neq b r ¼ Keq : eb  eib ; b ¼ r; g; h b : e þ Kb

ð29Þ

Moreover, we have: b r ¼ Keq b : e þ

gb  Dt





eib  eibn ¼ Keq b þ

gb Dt

Jb



: eb þ

gb  Dt

Sb  I : eib n

ð30Þ

We now recast (30) in a more compact form as:

r ¼ Hb : eb þ Q bn

ð31Þ

where

Hb ¼ Keq b þ

gb Dt

Q bn ¼

Jb ;

gb  Dt

Sb  I : eib n;

b ¼ r; g; h

ð32Þ

We now write strains eg, eh and ei in terms of er as:



1



1



eg ¼ Hg : Hr : er þ Q rn  Q gn ; eh ¼ Hh : Hr : er þ Q rn  Q hn ei ¼ ein þ

Dt 

gi

Hr : er þ Q rn

1

ð34Þ





e ¼ /p ug Hg : Hr : er þ Q rn  Q gn þ ur er þ ks3 eisn þ ks4 er þ /h Hh

1

ð33Þ



Substituting (33) and (34) into (19)2, we obtain:







  Dt  r r : Hr : er þ Q rn  Q hn þ eT þ ein þ H : e þ Q rn

ð35Þ

gi

Solving (35) for er, we obtain the following relation:

A : er ¼ b

ð36Þ

where

8  1 1 < A ¼ /p ug Hg : Hr þ /h Hh : Hr þ Dgt Hr þ ur þ ks4 I i     : b ¼ e  / u Hg1 : Q r  Q g þ k eis  / Hh1 : Q r  Q h  eT  ei  Dt Q r s3 n p h g n n n n n n g

ð37Þ

i

Now, we can find an expression for stress using (30). We may write:



1

1

r ¼ Hr : /p ug Hg : Hr þ /h Hh : Hr þ  :



e  /p ug H

g 1

Dt

gi

1 Hr þ ður þ ks4 ÞI

     1 Dt : Q rn  Q gn þ ks3 eisn  /h Hh : Q rn  Q hn  eT  ein  Q rn þ Q rn

gi

ð38Þ

Also, the consistent tangent matrix is calculated as follows:

CTan ¼

   1 1 dr Dt 1 ¼ Hr : /p ug Hg : Hr þ /h Hh : Hr þ Hr þ ur þ ks4 I de gi

ð39Þ

For more details and discussions on the numerical solution, we refer to Baghani (2012). Remark. In most engineering applications, SMP components are used to provide a force over some large displacement via the shape memory effect e.g., in artery stents. Structural components such as beams and torque tubes usually exhibit large global displacements with small strains. We should remark that small-strain constitutive models can be successfully applied to the solution of such large displacement problems, where strains are small though rotations can be arbitrarily large. From theoretical point of view, it is done by replacing the Cauchy stress, r, and infinitesimal strain, e, tensors by the second Piola– Kirchhoff stress and Green–Lagrange strain tensors, respectively, in the formulation (Crisfield, 1997). However, the numerical manipulation of this procedure follows a standard approach in the literature (Hartl and Lagoudas, 2009), thanks to the well-known Hughes–Winget algorithm (Hughes and Winget, 1980). In fact, for the large displacement problems (where the updated Lagrangian formulation is employed) the time-discretization of the rate form equations should satisfy the objectivity requirements (such algorithms are called incrementally-objective). For a typical Jaumann objective  rate form equation A ¼ Fðt; T; A; . . .Þ, the incrementally-objective time-discrete form via Hughes–Winget algorithm is obtained as (Hughes and Winget, 1980):

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

Anþ1 ¼ QAn Q T þ DtFðt nþ1 ; T nþ1 ; Anþ1 ; . . .Þ

21

ð40Þ

where A and F are second order tensors in the current configuration. Also, Q is the incremental rotation tensor defined as (Hughes and Winget, 1980):

 1   1 1 Q ¼ I w Iþ w 2 2

ð41Þ

where I represents the second order identity tensor and w is the vorticity tensor (asymmetric part of the velocity gradient tensor). Finally, we should highlight that, to use a small-strain constitutive model for solution of large displacement problems in software ABAQUS/Standard, the user should activate the option NLGEOM. However, in this case, only stress and strains are automatically rotated incrementally by the software. The rotation of user-defined tensorial internal variables (eir, eig, eih, ei and eis) are left to the user, while the tensor Q is passed trough the UMAT as a 3  3 matrix Drot. 4. Model verification The present section deals with several uni-axial and multi-axial loading paths. In particular, Examples 4.1, 4.2.1 and 4.2.2 present the results for uni-axial tests to show the model capability of reproducing basic effects such as shape memory effect and time-dependent behavior in different recovery conditions, comparing them also with experimental data available in the literature. Examples 4.3.1 and 4.3.2 present the results for multi-axial combined tension–torsion tests to show the model ability to predict the material behavior under complicated multi-axial loading paths. Furthermore, the identification of material parameters involved in the constitutive equations is reported in Appendix A. 4.1. Example 1: reproduction of stress-free strain recovery and fixed-strain stress recovery In order to show the ability of the model to show the shape memory effect in different strain and stress recovery paths, we ignore the time-dependent effects as well as the hard segment of the equivalent RVE (/h = 0). We use the experiments done by Liu et al. (2006) and adopt the material parameters reported in Table 2. As shown in Fig. 4, the model is able to reproduce the characteristic shape memory effect in stress-free strain recovery as well as in fixed-strain stress recovery. 4.2. Time-dependent aspects in modeling of SMP behavior 4.2.1. Example 2: according to experiments done by Li and Nettles (2010) To show the validity and accuracy of the model in time-dependent regimes of thermomechanical loadings, we simulate experiments reported by Li and Nettles (2010) already used by Xu and Li (2010). During the experiment, an SMP-based syntactic foam sample is compressed under a constant stress r0 = 263 kPa at Th, held for 30 min. Then the sample is cooled down to Tl while the stress is held constant. The cooling rate is determined by Newtons law of cooling: dT ¼ 4:6  105 ðT  293Þ. Once the temperature reaches Tl (at t = 1256 min), the load is removed dt K and the sample is heated up to Th in a stress-free manner with heating rate= 0:3 min . Corresponding material parameters are presented in Table 3. The strain–time behavior is illustrated in Fig. 5 (left). These results show good correlation between experimental results and numerical simulation. Now, if the recovery happen in a fixed-strain stress recovery, the stress–temperature behavior of the material will be obtained as shown in Fig. 5 (right). As results show, in spite of some discrepancy, the overall trend is in good agreement with the experimental data. 4.2.2. Example 3: according to experiments done by Volk et al. (2010a,b) Recently, Volk et al. (2010a,b) have performed a series of time-dependent experiments on Veriflex™. They have also calibrated the model presented previously by Chen and Lagoudas (2008a, 2008 b). This model has been developed for fully thermo-elastic loadings. Therefore, it does not consider irreversible strains in an SMP stress-free strain recovery cycle. The test conditions (time and temperature dependency of the test) was fully reported in work done by Volk et al. (2010 b) for the K case of heating rate of 2 min . Corresponding material parameters are presented in Table 4. Fig. 6 shows the strain-temperature relationship during a stress-free strain recovery in two different prescribed strains eo. It is observed that the proposed model can successfully predict the time-dependency and the amount of irreversible strains. In order to compare model predictions, results predicted by Chen and Lagoudas (2008a) are also illustrated in Fig. 6. 4.3. Multiaxial loadings 4.3.1. Example 4: multiaxial loading-path 1 In this section, we simulate a multiaxial loading path. We use the material parameters reported in Table 3 except that /h = 0.33. First, at T = Th, we apply a strain, e0 = 0.1 for Dt1 = 50 min. Then, we hold the normal strain constant and apply a shear strain, e12 = e13 = 0.15 for Dt2 = 50 min. Holding the strains fixed, we let the material relax the viscous effects for

22

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30 Table 2 Material parameters adopted from experiments reported by Liu et al. (2006). Material parameters

Values

eq Eeq r ; Eg

8.8, 813

[MPa]

/h mr, mg Tl, Tg, Th

0 0.4, 0.3 273, 343, 358    3:14  104 ðT  T h Þ þ 0:7  106 T 2  T 2h I

[–] [–] [K] [–]

1/(1 + 2.76  105(Th  T)4)

[–]

eT

ug

Units

Fig. 4. Reproduction of the shape memory effect: stress-free strain recovery (left). fixed-strain stress recovery (right).

Table 3 Material parameters adopted from experiments reported by Li and Nettles (2010). Material parameters

Values

eq neq eq neq neq Eeq r ; Eg ; Eh ; Er ; Eg ; Eh

1.3, 15,70000, 0.2, 247,1000

[MPa]

gr, gg, gh

5, 30  106, 30  109 0.4 0.4, 0.3, 0.3 296, 344, 353 (0.5542  103T  0.01083456  7  107T2)I 1 1  1þexpð0:66ðTT g ÞÞ

[MPa min] [–] [–] [K] [–] [–]

/h

mr, mg, mh Tl, Tg, Th

eT ug

Units

Fig. 5. Reproduction of the shape memory effect: stress-free strain recovery (left). Fixed-strain stress recovery (right). Experiments reported by Li and Nettles (2010).

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

23

Table 4 Material parameters adopted from experiments reported by Volk et al. (2010b). Material parameters

Values

Units

eq neq neq Eeq r ; Eg ; Er ; Eg

0.39, 1100, 0.02, 150

[MPa]

gr, gg, gi

1, 4000, 10000 0.0 0.4, 0.3 298, 345, 363 (ugag + urar) (T  Th)I 0.7  104, 5.9  104

[MPa min] [–] [–] [K] [–] [K1] [–]

/h mr, mg Tl, Tg, Th

eT ag , ar ug

tanhððTh Tg Þ=bÞtanhððTTg Þ=bÞ ; tanhððTh Tg Þ=bÞtanhððTl Tg Þ=bÞ

b ¼ 7:33

Dt3 = 150 min. We now decrease the temperature down to Tl for Dt4 = 100 min. After that, we unload the stresses and allow the system to relax for Dt5 = 100 min. In this step, the material is in temporary shape. Finally, the temperature is increased to T = Th for Dt6 = 50 min. Increase of the temperature leads to recovery of the strain, thus the material recovers its initial shape. Fig. 7a–d present the time and temperature histories of stress and strain components, respectively. As one may see, due to time-dependent behavior of SMPs, applying a strain produces an over-stress in the material as depicted in Fig. 7a. Besides, holding strains fixed during cooling leads to an increase in normal stresses which happens due to negative thermal strains and glassy-rubbery phase transformation. Moreover, even after heating and recovery, some irreversible strains remain in the material as shown in Fig. 7b and d. 4.3.2. Example 5: multiaxial loading-path 2 In this section, we simulate another multiaxial loading path. The structure is initially under an external load r33 = 500 kPa. First, at T = Th, we apply a strain, e11 = 0.1 in Dt1 = 50 min. Then, we hold the normal strain e11 constant and apply strain e22 = 0 in Dt2 = 50 min. Holding the strains (e11 = 0.1, e22 = 0) and stress (r33 = 500 kPa) fixed, we allow the structure to relax the viscous effects for Dt3 = 150 min. We now decrease the temperature to Tl in Dt4 = 100 min. Subsequently, we unload the stresses (by removing the external constraints and loads) and let the system relax for Dt5 = 100 min. In this stage, the system is in a temporary shape. Finally, the structure is heated up to T = Th in Dt6 = 50 min. Fig. 7(a, b, c, d) present the time and temperature history of stress and strain components, respectively. We use the material properties reported in Table 2 except that /h = 0.33. Similar to previous example, because of time-dependent behavior of the SMP, applying a strain produces an over-stress in the material as shown in Fig. 8a. Also, holding strains fixed during cooling leads to an increase in normal stresses which happens due to negative thermal strains and glassy-rubbery phase transformation. Furthermore, after shape recovery, some irreversible strains remain in the material as illustrated in Fig. 8b and d. 5. Boundary value problems In this section, we solve two boundary value problems to validate the proposed model as well as the numerical solution procedure. A 3D beam and a medical stent are simulated to show the model capability of capturing the shape memory effect. In the boundary value problems, we use the material properties reported in Table 2 except that /h = 0.33. For all simulations, we use the commercial nonlinear finite element software ABAQUS/Standard, implementing the described algorithm within a user-defined subroutine UMAT.

Fig. 6. Reproduction of the shape memory effect: stress-free strain recovery. Experiments reported by Volk et al. (2010b).

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

(a)

(b)

(c)

(d)

Fig. 7. Reproduction of the shape memory effect: stress-free strain recovery.

5.1. 3D SMP beam with distributed force on upper surface: shape memory effect test In this example, a 3D SMP beam is simulated. To this end, a beam with 100 mm length, 12.5 mm width and 10 mm height has been modeled under fully clamped boundary conditions along one end. The history of the applied distributed force on the upper surface and temperature are illustrated in Fig. 9 (left). Owing to the low temperature, while the external load is removed, the beam remains in a deformed state. The structure is in the temporary shape at t = 600 min. However, the initial shape is recovered after heating. Fig. 10 illustrates the steps of the shape memory effect in a stress-free strain recovery cycle. Moreover, the displacement-temperature behavior of the upper middle node at the free end of the beam is shown in Fig. 9 (right). Further, to have a better comparison among different steps of an SMP cycle, the two dimensional illustration of the beam is plotted in Fig. 11. It can be obviously seen that the beam recovers its original shape. 5.2. Medical SMP stent: shape memory effect test In this example, an SMP stent is simulated. Initial configuration of the stent has been shown in Fig. 12-left. Due to the symmetry of the problem in our simulation, we considered only a quarter of the stent geometry. The stent has a length of 20 mm, an inner radius of Ri = 4.8 mm and an outer radius of Ro = 5 mm. The diameter of the holes is d = 0.5 mm. At T = Th, during t1 = 200 min, a displacement uy = 5 mm along line AB is applied to the stent while the displacement uz = 2 mm is applied on the face C simultaneously. We now let the stent relax up to t2 = 300 min. Between t2 and t3 = 500 min, the stent is being cooled down to T = Tl. Consequently, during t3 and t4 = 500 min the whole structure is abruptly being unloaded. The temporary shape of the stent is shown in Fig. 12-right. As shown in Fig. 13, in spite of removing the external loads, the stent memorizes its temporary shape. In this step we let the system relax prior to t5 = 700 min. This step is followed by heating the stent up to T = Th until t6 = 1000 min. The contours in Fig. 13 refer to the von-Mises stress at different steps of a stress-free strain recovery cycle. 6. Summary and conclusions In this paper, we presented a 3D constitutive model for shape memory polymers which reasonably captures the essential features of the shape memory behavior under time-dependent multiaxial loadings. In many applications shape memory

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

(a)

(b)

(c)

(d)

25

Fig. 8. Reproduction of the shape memory effect: stress-free strain recovery.

polymers are used in accompanying with another material such as glass particles as a reinforcing phase. In order to capture the behavior of this family of composites, a hard segment was incorporated into the proposed constitutive model. For this purpose, assuming small strains, we used an additive decomposition of the strain into six parts using a first order mixture rule. In fact, the material was considered as a mixture of rubbery, glassy and hard phases. It was also assumed that the volume fraction of the hard phase was constant while the rubbery and glassy phases were able to be transformed to each other through external stimuli of heat. The evolution laws for internal variables were derived in an arbitrary thermomechanical loading. Moreover, the polymer in each phase (the rubbery, glassy or hard phase) was considered as a viscoelastic material.

Fig. 9. History of the distributed force and temperature applied to the beam (left). Displacement-temperature behavior of the upper middle node at free end of the beam (right). juj, jumaxj and jPmaxj denote the magnitude of displacement, maximum magnitude of displacement and maximum distributed force, respectively.

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

Fig. 10. Reproduction of the shape memory effect: stress-free strain recovery at different steps of an SMP cycle. (a) t = 0 min, structure at rest at Th; (b) t = 10 min, load has been fully applied at Th; (c) t = 200 min, beam has been relaxed at Th; (d) t = 500 min, system has been cooled down to Tl and relaxed; (e) t = 510 min, external load has been removed at Tl; (f) t = 600 min, system has been relaxed at Tl; (g) t = 770 min, glass transition at Tg; (h) t = 800 min, temperature has been increased up to Th; (i) t = 1000 min, the structure is relaxed until the permanent shape has been fully recovered at Th.

Fig. 11. Reproduction of the shape memory effect: stress-free strain recovery at different steps of an SMP cycle. steps are in correspondence with Fig. 10. (a) t = 0 min, structure at rest at Th; (b) t = 10 min, load has been fully applied at Th; (c) t = 200 min, structure has been allowed to relax at Th; (f) t = 510 min, external load has been removed at Tl; (g) t = 770 min, glass transition at Tg; (i) t = 1000 min, permanent shape has been fully recovered at Th.

M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

27

Fig. 12. Initial configuration of the simulated medical stent (left). Temporary shape of the stent (right). Due to symmetry, only a quarter of the stent has been modeled.

For the sake of consistency and completeness, free energy function of the model is introduced which is compatible with the second law of thermodynamics, in the sense of the Clausius–Duhem inequality. We also presented the time-discrete form of the evolution equations in an implicit form and presented the time integration scheme as well as the construction of the tangent matrix which are the two main issues in the finite element modeling of the material behavior. The model was validated by comparing the predicted results with different experimental data available in the literature. It was shown that the model is capable of capturing the main features reported in experimental observations. Implementing the proposed model within a user-defined subroutine (UMAT) in the commercial non-linear finite element software ABAQUS, we solved two boundary value problems e.g., a 3D beam and a medical stent made of shape memory polymers and showed the capability of the proposed constitutive model. In fact, the model is a useful and appropriate computational tool for design, analysis and optimization of structures made of shape memory polymers.

Appendix A The material parameters used in Sections 4 and 5, as listed in Tables 2–4, were mainly identified using various experimental results reported in the literature (Liu et al., 2006; Li and Nettles, 2010; Volk et al., 2010b). In the following, we present some guidelines on material parameter identification.  Characteristic temperatures Tl, Tg and Th are measured using DMA (Dynamic Mechanical Analysis) tests (Liu et al., 2006).  It is observed that the thermal strain (under no external load) exhibits a nonlinear behavior as the temperature traverses through the glass transition region (Liu et al., 2006; Li and Nettles, 2010; Volk et al., 2010a,). Different nonlinear relations have been utilized in the literature to capture this effect. Commonly, a second order polynomial expression for the thermal strain gives a good approximation. As an example, such a curve fitting has been performed in the following form for experimental data reported by Liu et al. (2006) as follows:



eT ¼ a1 ðT  T h Þ þ a2 T 2  T 2h



ð42Þ

where a1 and a2 are material parameters and have been calculated using a curve fitting method as a1 = 3.14  104 K1 and a2 = 0.7  106 K2. Fig. 14 (left) shows the fitted curve and the experimental data reported in Liu et al. (2006) for the thermal strain.  Volume fractions /h and /p are constant parameters. In order to specify the evolution equation for the volume fraction of the glassy phase ug as a function of temperature, we use the fact that Eq. (17) during heating in a 1D stress-free strain recovery (while the stored strain after unloading is e0, corresponding to point in Fig. 1) reduces to:

e_ is 

u_ g is e ¼ 0; eis ug ¼1 ¼ e0 ) eis ¼ e0 ug ug

ð43Þ

Eq. (43) shows that the expression for ug should follow the same trend as stored strain eis. Thus using a curve fitting method, we define a function which fits the experimental data in the best way. A combination of exponential or power terms normally leads to a successful curve fitting. For instance, such a curve fitting has been utilized for the experimental data reported by Liu et al. (2006) in Fig. 14 (right). In this case, we used a relation for ug in the following form:

ug ¼ 1=ð1 þ c1 ðT h  TÞc2 Þ

ð44Þ

where c1 and c2 are calculated applying a curve fitting method. The obtained values are c1 = 2.76  105 K1 and c2 = 4.

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

Fig. 13. Stress-free strain recovery at different steps of an SMP cycle. (a) t = 0 min, structure at rest at Th; (b) t = 200 min, load has been fully applied at Th; (c) t = 300 min, system has been relaxed at Th; (d) t = 500 min, system has been cooled down to Tl; (e) t = 500 min, external load has been removed at Tl; (f) t = 770 min, system has been relaxed at Tl; (g) t = 800 min, during glass transition; (h) t = 980 min, during glass transition; (i) t = 1000 min, temperature has been increased up to Th; (j) t = 1200 min, we let the system relax until the permanent shape be fully recovered at Th.

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M. Baghani et al. / International Journal of Plasticity 35 (2012) 13–30

10

0 Curve fitting result Experment by Liu et. al (2006)

−0.2

8

ε is [ % ]

εT[%]

−0.4 −0.6

6

ε

is

=ε−ε

T

4

−0.8 2

Curve fitting result Experment by Liu et. al (2006)

−1 −1.2 270

0 280

290

300 310 320 330 Temperature [ K ]

343 350

360

280

300 320 Temperature [ K ]

343

360

Fig. 14. Thermal expansion strain as a function of temperature (left). Stored strain vs. temperature in a free-stress strain recovery process (required for calibration of the volume fraction of the glassy phase, ug) (right).

 Elastic moduli and viscosity coefficients of each phase is calculated using one-dimensional stress–strain curves. The stress–strain responses of a pure SMP (/h = 0) at Th as well as Tl give the elastic moduli and viscosity coefficients of the rubbery and glassy phases, respectively. These parameters for the hard segment are calculated using stress–strain response of the pure hard segment (before producing the composite and mixing the hard segment in the SMP matrix). Due to the lack of experimental data, we assumed values of 0.4, 0.3 and 0.3 for mr, mg and mh, respectively.

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