development of a finite strain two-network model for shape memory polymers using qr decomposition

International Journal of Engineering Science 81 (2014) 177–191

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International Journal of Engineering Science

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Development of a finite strain two-network model for shapememory polymers using QR decomposition

http://dx.doi.org/10.1016/j.ijengsci.2014.02.0050020-7225/� 2014 Published by Elsevier Ltd.

⇑ Corresponding author. Tel.: +1 979 862 3999.E-mail address: [email protected] (A.R. Srinivasa).

Pritha Ghosh, A.R. Srinivasa ⇑Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, United States

a r t i c l e i n f o

Article history:Received 20 April 2013Received in revised form 4 February 2014Accepted 8 February 2014

Keywords:Shape memory polymersActivation stress hysteresisFinite deformationQR decompositionUpper triangular decompositionThermal Bauschinger effect

a b s t r a c t

The aim of this paper is to develop a thermodynamically consistent finite deformation con-tinuum model to simulate the thermomechanical response of shape memory polymers(SMPs). The SMP is modeled as a thermoviscoelastic material whose response in a thermo-mechanical cycle is modeled as a combination of a rubbery (viscoelastic) and a glassy (elas-tic) network in series. The activation criterion for the breakage of temporary networkjunctions is governed by a temperature dependent rate equation (akin to a thermal Bausch-inger effect). We further show how the decomposition of the deformation gradient into anupper triangular matrix and a rotation (the QR) decomposition can be used instead of themore traditional polar (RU) decomposition in the development of models for such materi-als with persistent configurational changes. Such a decomposition has both physical mean-ing as well as computational advantages due to their convenient structure. Using theseassumptions, we propose a specific form for the Helmholtz potential and the rate of dissi-pation. The model is simple (deliberately ignoring some aspects such as anisotropy thataffect the quantitative but not the qualitative response features) and is able to capturethe major phenomena of interest in SMPs. The efficacy of the resulting model, which isin a state evolution form, is demonstrated by comparing with published experimental datafor simple shear. We study the response of the SMP model for monotonic shear deforma-tion, different deformation rates as well as cyclic shear deformation at different tempera-tures. Comparisons with experiments show good agreement. Finally, we implement thethermomechanical cycle under shear deformations and study the behavior of the model.

� 2014 Published by Elsevier Ltd.

1. Introduction

Shape memory polymers (SMPs) belong to a class of stimuli-responsive polymers that can recover a large deformed shapein response to a relatively small environmental stimulus, such as temperature, light and humidity (Lendlein & Kelch, 2002).Thermally responsive SMPs can be easily deformed (with response similar to an elastomeric rubber, with modulus ofmagnitude 10–50 MPa) to give a temporary shape at low-working temperatures below their transition temperature. Herethe SMP is glassy (with modulus of 500–1000 MPa) and holds onto this shape at low temperatures. Upon heating above theirtransition temperature, they return to their original structure, where it is rubbery again. On the morphological level, Kelchand Lendlein (2002) observe that SMPs consist of at least two components: switching domains and permanent networks. Theswitching domains act as a molecular switch with a well-defined glass transition temperature Tg or melting temperature Tm,

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178 P. Ghosh, A.R. Srinivasa / International Journal of Engineering Science 81 (2014) 177–191

and enable the fixation of the temporary shape. These domains form a temporary-network, which can be made to persistover long periods of time by suitably lowering the temperature, below transition. The permanent network that determinesthe permanent shape of the SMP consists of physical netpoints associated with a high thermal transition temperature orcovalent netpoints (covalently crosslinked polymer network). Classification of thermally induced SMPs could be definedby identifying unique characteristics, such as morphology (amorphous or semicrystalline), nature of crosslinks (chemicalversus physical), or the underlying mechanism responsible for the shape memory effect.

SMPs have gained tremendous popularity in the academic research field over the last two decades. In a recent review,Wagermaier, Kratz, Heuchel, and Lendlein (2010) state that the majority of research activities of the last decade on SMPswere focused on the experimental characterization of the shape memory effect and its principal physical understanding.However, they also note that only a few studies concentrated on the development of constitutive theories that describethe thermomechanical properties of SMPs at the macroscopic level. The approaches in modeling techniques that have devel-oped over the last decade from 1997 to 2010 can be classified into rheological and phenomenological approaches as shownin Table 1. Initial rheological modeling attempts mainly focused on capturing the shape recovery characteristics qualita-tively. The more recent advances on the morphological studies encouraged thermodynamically motivated phenomenologicalmodels over the last five years. However in the attempt to encompass all the physical aspects of the material, the resultingphenomenological models are complicated and there is no assistance available to an application designer who can use sim-ple models to predict the SMP responses even approximately.

The model developed by Tobushi, Hashimoto, Hayashi, and Yamada (1997) takes irreversible deformation and thermalstrains of shape memory polyurethanes into account. Tobushi and coworkers have added a friction element into the standardlinear viscoelastic model to simulate the behaviors of shape memory polyurethanes. They have carried out a series of creeptests of shape memory polyurethanes at different temperatures. They propose that if the strain exceeds particular thresholdstrain, irreversible deformation occurs. Also, the temperature change that would cause thermal expansion is accounted forby simply adding a coefficient of thermal expansion to terms involving the rate of strain, without any thermodynamic con-siderations as seen in Eq. (4) in Tobushi et al. (1997). As a result, their model can be shown to be theormodynamically incon-sistent even for pure thermoelasticity. The model proposed by Lin and Chen (1999) employs two Maxwell elementsconnected in parallel to describe the shape memory properties of shape memory polyurethane. Their modeling resultsand experimental data show some deviation, which the authors ascribe to the polydispersed glass transition temperatureof the studied samples. The model can qualitatively explain the occurrence of shape memory behaviors. However, sincethe dampers in the model are both viscous, there is no truly irrecoverable strain at the end of the process. Furthermore,the model was not developed using thermodynamical principles. Abrahamson, Lake, Munshi, and Gall (2003) utilize a fric-tion element in their model to account for the irrecoverable strain at the end of the cycle, that progresses from fully stuck tofully free over a finite range of strain. It was found that the stress–strain curve predicted by the model and that obtained byexperiment agreed well. However, Abrahamson’s model does not take the thermal expansion of the material into consider-ation during the change of temperature,which is a crucial parameter to the material response.

Diani, Liu, and Gall (2006) developed a three dimensional thermoviscoelastic constitutive model using a thermodynam-ically based finite strain model. This model was based on the viscoelastic properties of crosslinked SMP networks. The defor-mation gradient is split into elastic and viscous parts, and the stress is split into entropic (following the Neo–Hookean model)and internal energy (defined in terms of Hencky strain measure) parts. The conditions on the evolution of the viscous part ofthe deformation gradient when the material is completely rubber or completely glassy is specified. The mechanical dissipa-tion equation is obtained from the Clausius Duhem inequality, and the evolution of the deformation gradients for temper-atures ranging between above glass transition temperature to below glass transition, are defined to satisfy the mechanicaldissipation equation. However, the model does not predict the details of the response for uniaxial compression (see Fig. 5 oftheir paper). Another recent work is the three-dimensional small-strain internal state variable constitutive model for uniax-ial experiments by Liu, Gall, Dunn, Greenberg, and Diani (2006). The model uses two internal variables to account for themicromechanical structures: frozen fraction describing the volume fraction of the frozen phase, and stored strain describingthe strain that is stored (memorized) in the material during freezing. The entropic strain energy is gradually stored duringcooling and released during reheating as free recovery strain or constrained recovery stress. The model was validated foruniaxial experiments carried out inhouse by the authors, for various loading conditions and strain levels. Following a similarapproach to Liu’s work, Chen and Lagoudas (2008) developed a three-dimensional constitutive model which can account forthe non-linear material response to large deformations. This model, proposed on the framework of thermoelasticity, incor-porates the concepts of stored strain and a frozen volume fraction. The constitutive equation proposed by this model, in itsgeneral form, can be interpreted as a rule of mixtures with contributions due to deformations in the frozen and active phaseas well as a contribution due to deformations which are frozen upon cooling. The linearized version of this model for smalldeformation as a subsequent work was validated using data generated by Liu et al., however the large deformation modelwas not validated due to lack of sufficient data in the literature. Barot and Rao (2006)and Barot et al. (2008) have developedmodels for SMPs undergoing shape setting through partial crystallization using a thermodynamical approach developed byRajagopal and Srinivasa (2000) using the maximum rate of dissipation criterion.

Model of glass transition, based on the notion of a ‘‘frozen or glassy phase’’ and a ‘‘rubbery phase’’, while being able tomodel the response phenomenologically, do not reflect the current understanding the glass transition phenomenon. Suchphase separation phenomena are observed only if there is a structural difference between the glassy and the rubbery phases,i.e., a first order transition according to the Ehrenfest classification. However, as noted by Jackle (1986) (p. 182), ’’Any

Table 1Classification of few models in the current literature that deal with the macroscopic responses of pure SMPs.

Ad-hoc Thermodynamic Concepts

Rheological Phenomenological

Rate type Tobushi et al. (1996, 1997, 1998, 2008), Lin andChen (1999), Bhattacharyya and Tobushi (2000),and Abrahamson et al. (2003)

Ghosh and Srinivasa (2011)and Diani et al. (2006)

Liu et al. (2006) and Qi et al. (2008)

History dependent Hong et al. (2007) – Barot et al. (2008), Chen andLagoudas (2008), and Kim et al. (2010)

P. Ghosh, A.R. Srinivasa / International Journal of Engineering Science 81 (2014) 177–191 179

attempt to classify a GT as a thermodynamic phase transition (PT) is, of course, in conflict with the known fact that the GTtemperature depends on the experimental time scale set by the cooling rate. Further, Schumacher, Herr, Oelgeschlaeger,Traverse, and Samwer (1997) have shown by X-ray diffraction that ‘‘within the uncertainty the total number of nearestneighbors determined by X-ray diffraction remained unchanged’’ during the glass transition, indicating that there is nostructural changes.

Our approach, while similar in the use of thermodynamical ideas to that of Barot et al., differs from these models. Wefocus on shape setting phenomena through glass transition and not through crystallization phenomena. More importantly,while their approach is based on the use of history integrals, our approach is based on a much simpler rate type constitutiveequations using a plasticity like approach, leading to computationally tractable solutions. In this work, a Helmholtz potentialbased approach will be adopted (Rajagopal & Srinivasa, 2000) for the development of the constitutive equations for contin-uum model, using the maximum rate of dissipation criterion for the evolution of viscoelastic strain (Srinivasa & Srinivasan,2009). The primary hypothesis of this model is that the hysteresis of temperature dependent activation criterion plays a leadrole in controlling its main response features (Ghosh & Srinivasa, 2011). This hypothesis has been validated and analyzed forseveral thermomechanical homogenous (uniaxial tension and compression) (Ghosh & Srinivasa, 2012) and boundary valueproblems (Ghosh, Reddy, & Srinivasa, 2012) for small deformations.

In this paper, we will be studying the finite strain model developed here for shear deformations, and evaluating thebehavior of the model under the activation stress hypothesis. The material properties of the material like modulus and vis-cosity depend on heating and cooling processes which affects the material behavior. The approach makes a number of sim-plifying assumptions that nevertheless capture the essential features of the response. These assumptions are not fundamentalto the model but have been adopted in order to illustrate the simplest possible model that will show SMP behavior. If nec-essary, the model can be generalized by systematically relaxing the various assumptions made here.

1. The material is assumed to consist on ONLY two networks (the glassy (low temperature) network and a rubbery (hightemperature network)) rather than as a superposition of a large number of networks.1 However unlike the history integralapproaches which assume that the viscous effects are linear (leading to exponential kernels) we assume that the dissipativeresponse is nonlinear. This minimalist model drastically simplifies the computations but retains the essential features of theSMP response.

2. Rather than introduce a ’’volume fraction of frozen or glassy regions’’, we develop a model where the activation criterioncontrols the effect of the glassy and rubbery states. This is in line with the overwhelming majority of models for glasstransition in the literature.

3. While it is well known that the Rubbery state has a temperature dependent modulus, the modulus change is much smal-ler when compared to that between the glassy and rubbery states and so we ignore it in the implementation (although itis possible to account for it in the formulation). We do this to show that these features are NOT essential to the SMPbehavior.

4. Although in recent experimental findings (Behl, Razzaq, & Lendlein, 2010a; Behl, Zotzmann, & Lendlein, 2010b) someresearchers have observed large strain induced anisotropy, these ideas are new and complex in the SMP field (Beblo &Weiland, 2008). Few hypotheses regarding the material behavior that cause the observed anisotropic effects have beenoffered, but these incipient concepts will be dealt with in future extensions of the general modeling method describedin this work. Thus we deliberately ignore these anisotropy effects in the model, going for simpler isotropic response func-tions. There is currently no comprehensive data on the anisotropic moduli. However, as we will indicate, it is possible toextend the model to account for the strain induced anisotropy.

5. We propose to use a Hencky Elasticity model where the strain energy is a quadratic function of the logarithmic strain.This is suitable for the range of deformations that are considered.

For the development of the finite strain model, we follow the general technique of upper triangular or QR decomposition(Srinivasa, 2012) for deriving the constitutive equations of the SMP model. This technique involves the decomposition of a

1 Even if the history integral models do not explicitly invoke a finite number of different networks, at the implementation stage, it is converted into a Pronyseries, resulting in a large but finite number of different superimposed networks.


matrix into an orthogonal matrix Q and a upper triangular matrix R. We use this technique for its simpler and faster imple-mentation benefits. The response of the model is studied for shear deformation subject to glass transition and low temper-atures. We compare these results with shear experimental observations from the literature. We then explore the behavior ofthe model under shear deformation at different initial temperatures, and gain a deeper insight of how the model works. Thisexperiment is extended to cyclic shear deformation, and the results are in general agreement with those found in the liter-ature. We also study the effect of deformation rates on the model response. Finally, we study the response of the model for ashear deformation thermomechanical cycle and compare it with the experimental findings in the literature.

2. Kinematics

Consider a body B which at time t occupies a configuration jtðBÞ. The position of any particle X in jt is given by x. Themotion of the body, measure from some fixed configuration jr , wherein the position vector is given by X, and the deforma-tion gradient are given by the following two equations, respectively.

Fig. 1.motionconvecaffects

x ¼ Xjr ðX; tÞ ð1Þ

F :¼ @Xjr

@Xð2Þ

We consider an evolving natural configuration jpðtÞ that reflects the evolving microstructure due to network breaking andreformation as shown in Fig. 1.

For homogeneous deformations, the gradient of the mapping from jr to jpðtÞ is denoted by G. We shall define the mappingof the line elements from jpðtÞ to jt as Fe, so that the deformation gradient is multiplicatively decomposed (Lee, 1969) asbelow:

Fe ¼ FG�1 ð3Þ

A few clarifying remarks are in order with regard to this decomposition. This notion is generally attributed to Lee (1969)although forms similar to this have been used by others (see Naghdi, 1990), and the tensor G is referred to as Fp, and it refersto the permanent or stress free state. AS noted by Truesdell and Noll (2004), it appears that this decomposition was originallyproposed in full generality for elastoplastic materails by Eckart (1948) in a series of papers on the thermodynamics of inelas-tic materials. As noted in the review, the meaning associated to this tensor in different contexts is different.

Specifically, in the context of elasto-plasiticity, Fp is not considered a state variable and there is a long discussion aboutthe fact that the strain energy function for elastoplastic materials cannot depend upon Fp. Several authors have developedtheories where this variable is completely absent (see Srinivasa & Srinivasan, 2009, for a detailed discussion in the context ofcrystallographic slip). However, in current work, G refers to the ‘‘temporarily frozen’’ deformation of the permanent network(which have been locked in place due to secondary bonds) and hence IS a state variable and does not refer to permanentdeformations due to crystallographic slip, as we shall see later. We hence use the symbol G to honor Eckart and to differen-tiate the notion introduced here (where it is NOT permanent slip but temporary frozen stretch) from that of elastoplasticitywhere it is permanent.

Returning to the development of the model, when we differentiate the above deformation, we get,

_Fe ¼ LFe � FeLve; ð4Þ

Schematic diagram to illustrate the current natural configuration of the material. Figure (a) depicts the reference configuration jr of the body. TheXjr ðX; tÞ takes the material points to their respective positions in the current configurations jt shown in (b) Simultaneously the material fibers are

ted by the deformation gradient F. The instantaneous relaxation process takes the material fibers to their natural state jpðtÞ as indicated in (c) F�1e

only the line elements and the resulting configuration is shown in dotted lines. Finally the tensor G maps the reference line elements to those in jpðtÞ.

F*11

F*12

F*13 F*23

F*22

F*33

a b

c

A

B

C

Fig. 2. Schematic diagram of the deformation of a unit cube by an upper triangular deformation F� , resulting from the QR decomposition of the inelasticdeformation G. Notice that the deformation is such that (a) the X axis is not rotated, (b) the X–Y plane is the same before and after the deformation, and (c)the deformation consists for three stretches and three shears. It can be shown that F� is the finite deformation equivalent of the engineering strain tensor.


where, L :¼ _FF�1; Lve :¼ _GG�1, and _G�1 ¼ �G�1 _GG�1.The first term represents the rate of change of Fe when the microstructure is fixed i.e., due to non-dissipative processes,

and the second term is due to the changes in the underlying microstructure (breaking and reforming of the secondary bonds)when the current configuration of the material is frozen i.e., a inelastic change. We can now introduce the elastic squaredstretch tensor and the viscoelastic squared stretch tensor

Be :¼ FeFte ð5Þ

Bve :¼ GGt ð6Þ

In this development, we will use the following strain measures:

ee :¼ 12 lnBe, the logarithmic elastic squared right stretch tensor,

Bve :¼ GGt , the viscoelastic squared stretch tensor,Dve :¼ 1

2 ðLve þ LTveÞ, the viscoelastic flow rate.

3. Constitutive theory for finite deformation

As has been stated by Rajagopal and Srinivasa (2004), viscoelastic and viscoplastic materials are not only character-ized by the way in which they store energy but also the way in which work is converted into heat (i.e., their mode ofdissipation). Following their work, the two key ingredients that are required for developing a finite deformation versionof the theory are the form for the Helmholtz potential which involves both finite elastic and inelastic strains, and therate of dissipation function which depends on the generalization of the viscoelastic strain rate. The response of theSMP in a thermomechanical cycle is represented as a two network model. This consists of a permanent network thatis responsible for shape recovery and a temporary network that is responsible for the shape fixing phenomena. The per-manent network represents the uncoiling of the polymer chains between the crosslinks when the SMP is deformed.When two chains come close enough they stick together momentarily and form temporary nodes because of the elec-trostatic attraction between the individual chains. The connections between these temporary nodes form the temporarynetwork. The temporary network can ‘‘lock in’’ the shape of the polymer at low temperatures, as the mobility of thepolymer chains decreases and the temporary nodes that are formed hold parts of the chains immobile. At high temper-atures mobility of the polymer chains increase, and the temporary network breaks, thus ‘‘unlocking’’ the shape of thepolymer. The permanent network takes over at higher temperatures and recoils the polymer back to its original state.We begin the thermodynamical modeling of the response of the material by assuming that the Helmholtz potential iscomposed of two parts: the permanent network w1ðFeÞ and the temporary network w2ðGÞ. In view of the fact that theglassy network deforms much less than the rubbery network, we assume that the Helmholtz potential for the glassy net-work is given by the Hencky elastic model, which is quadratic in the logarithmic elastic strain of the glassy network. Itshould be emphasized that (a) this is not a linearized or small strain model and (b) it can be completely generalized asneeded.


We thus propose the total Helmholtz potential for the permanent and temporary response as

w ¼ w1ðFeÞ þ w2ðGÞ ð7Þ

w1ðFeÞ ¼12

CGðee � aðh� hhÞIÞ � ðee � aðh� hhÞIÞ ð8Þ

w2ðGÞ ¼12lRðI � Bve � 3Þ ð9Þ

where the isotropic glassy stiffness matrices are given as follows,

CG ¼ kGI� Iþ 2lG I� 13

I� I� �

ð10Þ

where kG is the glassy bulk modulus, lG and lR are the glassy and rubbery shear modulus, a is the thermal expansion.We hasten to add that there is absolutely no difficulty in extending the above definition to anisotropic media by requiring

that CG be a function of G and requiring to be orthotropic with axis coinciding with that of the eigenvalues of G . This willneedlessly complicate the model without changing its qualitative behavior (and we lack experimental data with which tocompare) and so we choose to not do this in this paper.

In order to account for the viscoelastic response as well as for the ‘‘locking phenomenon’’ that is key to the shape memoryeffect, we assume that the rate of dissipation is composed of two distinct terms. the dissipative viscous behavior and acti-vation for temporary network breakage and formation behavior in the second term:

n ¼ nðDveÞ ¼ gGDve � Dve þ jkDvek; Dve :¼ ð _GG�1Þsymm ð11Þ

where gG is the viscosity, and j is the network activation threshold which depends upon the temperature, stress and also theprior history of the process. It is important to note that the second term is a positively homogenous first power term in Dve

and it is this that provides the shape locking feature (see Srinivasa & Srinivasan, 2009). It will be shown presently that untilthe magnitude of the deviatoric stress exceeds this threshold the shape change will be purely thermoelastic and recoverableupon unloading.

Here again, there are indications that the yield pehnomenon is history dependent and anisotropic. It is again a simplematter to deal with this by assuming the n is an anisotropic function of Dve, with anisotropy depending upon G Again, sincethis does not add any qualitatively new feature to the response while compicating the equations substantially, we willchoose to suppress this in this paper.

We begin the development of constitutive equations by stating the mechanical energy dissipation equation, where therate of dissipation is given by the difference of the mechanical power supplied and the rate of increase of the isothermalwork function:

T � L � q _w ¼ nðDveÞ

T � L � q@wFe� _Fe � q

@w@G� _G ¼ nðDveÞ ð12Þ

Now L can be written in the following form, where Le ¼ _FeF�1e and Lve ¼ _GG�1.

L ¼ _FF�1 ¼ Le þ FeLveF�1e ð13Þ

Eq. (13) can be used in Eq. (12) to get:

T� q@w@Fe

FTe

� �� Le þ q FT

e TF�Te �

@w@G

GT� �

� Lve ¼ nðDveÞ ð14Þ

Now based on the notion that the network possesses instantaneous elasticity, we stipulate that the stress in the material as

T ¼ q@w@Fe

FTe ð15Þ

With this definition of stress, the reduced energy equation in Eq. (14) now becomes

Ave � Lve ¼ nðDveÞ ð16Þ

where we set the thermodynamic driving force Ave for viscoelastic response as

Ave ¼ q FTe@w@Fe� @w@G

GT� �

ð17Þ

Following Rajagopal and Srinivasa (1998) who modified and extended the work of Ziegler (1963), we now introduce the‘‘maximum rate of dissipation assumption’’ which states that the system will evolve such that the actual value of Dve is thatwhich maximizes nðDveÞ subject to the constraint in Eq. (16). We first form the function h in Eq. (18).

h ¼ nþ k1ðAve � Lve � nÞ þ k2ðtrðLveÞÞ ð18Þ


Applying the method of constrained maximization with Lagrange multiplier to maximize Eq. (18), we first differentiate hwith Dve using the chosen form for n from Eq. (11) and then equate it to zero. After some tedious but straight forward manip-ulations we arrive at

Dve ¼0; if kAdev;sym

ve k 6 j;

1gGkAdev;sym

ve k � j� �

Adev ;symve

kAdev ;symve k

; if kAdev;symve k > j;

8<: ð19Þ

4. QR decomposition

Until now, we have been following the classical procedures based on the use of the multiplicative decomposition. How-ever, when dealing with materials whose instantaneous elastic response is isotropic, there is an essential degeneracy in therelaxed configuration jp since isotropy demands that the constitutive response is invariant to any rotation of the configu-ration jp.

This is a well known issue that has been dealt with in the plasticity literature (see e.g., Lee, 1969). Two common ways ofdealing with this degeneracy is to require that (1) Fe ¼ Ve or G ¼ Up, thus eliminating the redundancy. While theoreticallysound, both these approaches have numerical complications: When carrying out calculations with either Ve or Up in a finitedeformation setting, it is necessary to constantly symmetrize certain tensors (see Simo & Hughes, 1998, as well as Srinivasa &Srinivasan, 2009, for a detailed discussion).

The reasons for this difficulty lie in the fact that the set of symmetric tensors, while being closed under addition are notclosed under multiplication. Recently, Srinivasa (2012) has shown the efficacy of using a different decomposition of thedeformation gradient. Rather than using the polar decomposition theorem, which decomposes F into a rotation R and a sym-metric tensor U, he proposed the use of the well known QR decomposition, which will decompose F into a rotation Q and anupper triangular matrix R. To be specific, the inelastic deformation G is decomposed as

G ¼ QF�; F� :¼ F�iAei � eA; i > A ð20Þ

Notice that F� is upper triangular and contains six terms. These terms have direct physical meaning in terms of distortion andstretching of a unit cube as shown in Fig. 2.

From the figure it is evident that the deformation F� can be very simply interpreted as being composed of three stretchesF�11; F

�22 and F�33 along each of the axes together with three shearing deformations j12 :¼ F�12=F�22;j13 :¼ F�13=F�33 and

j23 :¼ F�23=F�33 in the X–Y, X–Z and Y–Z planes. The total volume change is simply F�11F�22F�33. A detailed discussion of the nat-ure of this decomposition, its efficacy for modeling anisotropic response and its use in finite elasticity can be found inSrinivasa (2012).

Unlike the polar decomposition, which, in general requires the use of eigenvalues and eigenvectors of C, the QR decom-position can be directly and explicitly computed (see Srinivasa, 2012). It may also be observed that the upper triangularmatrix has exactly the same number of elements as the Cauchy green stretch tensor C and contains the same information.Indeed they are related through a Cholesky Factorization.

The set of upper triangular matrices have a number of advantages over the set of symmetric matrices obtained by polardecomposition. First, upper triangular matrices are closed under both addition and multiplication, so that discretizing differ-ential equations that utilize the QR decomposition is simple. For example, if G is upper triangular so is G�1 and Lve. On theother hand, if G is symmetric, Lve is not.

Further, the determinant and inverse of an Upper triangular matrix are trivial to compute and hence make conditionssuch as incompressibility easy to enforce. Furthermore, thermal expansion (see Baek & Srinivasa, 2003) is easier to enforcewith this approach due to the explicit computation of the determinant.

Thus, the isotropy of the Helmholtz potential implies that it should satisfy the requirement that

w ¼ w1ðFeÞ þ w2ðGÞ ¼ w1ðFeQ Þ þ w2ðQ tGÞ ð21Þ

for every orthogonal tensor Q , then, without loss of generality, we will choose Q t to be a rotation that makes G upper tri-angular. Based on this discussion, and in view of the isotropy of the elastic response, we assume without loss of generalitythat G is upper triangular.
eAsym
ve � eDve ¼ nðeDveÞ ð22Þ

In upper triangular form we can write Eq. (19) as

eDve ¼1gGkeAdev ;sym

ve k � j� � eAdev;sym

ve

keAdev;symve k

ð23Þ

where eAdev ;symve can be derived from the deviatoric part of the relationship in Eq. (17) below

eAsymve ¼ eFT

e CGðee � aðh� hhÞIÞeF�Te � lR

eGeGT� �

ð24Þ


Thus the equations of the finite strain model are enlisted below:

1. State Variables: F;G; h;T2. Elastic Response:

T ¼ CGðee � aðh� hhÞIÞ

3. Flow Rule: eDve ¼ 1gG

/eAdev ;symve

keAdev ;symve k

4. Activation Conditions:

Table 2Dimens

Dim

Non

/ ¼ 0; 8 keAdev;symve k 6 j;

keAdev;symve k � j; 8 keAdev;sym

ve k > j;

(ð25Þ

5. Non-dimensionalization

We select the following non-dimensionalization parameters:

(1) The typical rubbery modulus ER, from experimental results.(2) The maximum strain applied �0 from experimental results.(3) The glass transition temperature hg .(4) The non-dimensionalization of the time, since this is connected with the kinetic response (see Table 2).

Non-dimensional form of Stress Equation:

T ¼ CGð��e � �að�h� �hhÞIÞ ð26Þ

Non-dimensional form of Flow Equation:

Dve ¼1�gGkAdev;sym

ve k � �j� � Adev;sym

ve

kAdev;symve k

ð27Þ

6. Implementation

We solve the system of ODEs for the evolution of activation stress, temperature, thermal expansion and eG in mass matrixform P _x ¼ rþ Qx, as described in detail in Algorithm 1, where x ¼ ½j; h;a;G6�1

0 �. We implement the algorithm in MATLAB

using the ode45 solver, where we calculate the current value of eGðtÞ given the input for the deformation eF for all time steps

and the initial condition for eGð0Þ. The ODE system consists of these equations _�j ¼ fj _�h; _�h ¼ gðtÞ; _�a ¼ @a@h

_�h;_eG ¼ eLve

eG. The ODEsystem in mass matrix form P _x ¼ rþ Qx looks as below

1 � @j@h 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 00 � @a

@h 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

266666666666666664

377777777777777775

_�j_�h_�a

_G11

_G22

_G33

_G12

_G13

_G23

26666666666666666664

37777777777777777775

¼

0g

0000000

266666666666666664

377777777777777775þ

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 Lve11 0 0 0 0 00 0 0 0 Lve23 0 Lve22 0 00 0 0 0 0 Lve33 0 0 00 0 0 0 Lve12 0 Lve11 0 00 0 0 0 0 Lve13 0 Lve11 Lve12

0 0 0 0 0 Lve23 0 0 Lve22

266666666666666664

377777777777777775

jh

aG11

G22

G33

G12

G13

G23

266666666666666664

377777777777777775ð28Þ

ional quantities and corresponding non-dimensional quantities.

T �e t h a lR gG k

dim �T ¼ TER�0

��e ¼ �e�0

�t ¼ tt0

�h ¼ hhg

�a ¼ ahg

�0�lR ¼ lR

ER�gG ¼

gGER t0

�kR;G ¼ kR;GER


Note that for the stress control case (Algorithm 1, line 14), knowing the Cauchy stress alone is not enough to specify thedeformation uniquely. We need additional assumptions about orientation. In this algorithm, we will assume without loss ofgenerality that Fe ¼ Ve. If orientation information is known this can be incorporated.

Algorithm 1. SMP model implementation

1: Input by user:� TimeData = [0:MaxTime], Initial Condition: x ¼ ½j; h;a;G6�1

0 �� Material Parameters: EG; ER; mG; mR; �0; hh; hl; hg ;g; t0

� Control Parameters: gðtÞ; f j; @a=@h� For a strain control problem, input: eFðtÞ, or for a stress control problem, input: TðtÞ

2: Calculate Non dimensional Material Parameters: �kR;�kG; �aG; �aR; �gG; �lR; �lG

3: function _x ¼ ODESOLVERðt; xÞ4: Assign G6�1

n ¼ G6�1n�1 as initial approximation . G6�1 rearranged as upper triangular G3�3

5: Call relevant function, depending on strain control (line 6) or stress control (line 14) problem:

6: function STRAIN CONTROL input: eFn;Gn, output: Tn;Gnþ1

7: Fen ¼ FnG�1n

8: Be ¼ Fen FTen

9: lnBe ¼ R3i¼1Vi

Belog kiBe

� �Vi

Be

� �T. ki

Be;ViBe: eigen values, vectors of Be

10: �e ¼ 12 ln Ben . lnB3�3

e rearranged as ln Be6�1n

11: T6�1n ¼ CGð�e � anðhn � hhÞIÞ. . rearranged as upper triangular T3�3

n

12: Continue to line 2113: end function14: function STRESS CONTROL input: Tn;Gn, output: Fn;Gnþ1

15: M6�1 ¼ 2 C�1G T6�1

n þ anðhn � hhÞI� �

. Tn rearranged as vector T6�1n

16: Be ¼ eM . M6�1 rearranged as upper triangular M3�3

17: Fen ¼ Ve ¼ffiffiffiffiffiBep

. Matrix square root18: Fn ¼ Fen Gn

19: Continue to line 2120: end function

21: Adev ;symven

¼ devðFTen

TnF�Ten� �lRGnGT

nÞ . Adev ;symven

rearranged as dev A6�1ven

22: kdev Avenk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdev AT

vendev Aven j

q. Calculating norm of dev Aven

23: if kdev Avenk 6 jn then . Activation threshold not reached

24: D6�1ven¼ 0 . No update in Dve

25: else kdev Avenk > jn . Activation threshold reached

26: D6�1ven¼ 1

gGðkdev Avenk � jnÞ dev Aven

kdev Aven k. Update Dve

27: end if

28: . Calculate eLveðnÞ

eLveðnÞ ¼D11

ven2D41

ven2D51

ven

0 D21ven

2D61ven

0 0 D31ven

26643775

29: P _x ¼ rþ Qx . Solve ODE system in mass matrix form (ref Eq. (28))30: end function

7. Results for shear deformation

7.1. Isothermal shear deformation at different temperatures

In this section, we analyse the shear deformation behavior of the SMP model for different temperature cases, and studythe normal stresses developed.


The shear deformation is controlled via the input F ¼ Iþ f e1 � e2.The initial conditions are set as fj; h;a;G0 ¼ Ig.Reference temperature for this problem hmax > hg .Activation stress: j constant, depending on temperature case.Thermal Load: Constant, for the following temperature hinit cases:

1. Greater than reference temperature: hinit > hmax.2. At reference temperature: hinit ¼ hmax.
3. At low temperature: hinit < hg .
As shown in Fig. 3(b), for large deformations, the shear deformation develops normal forces in the material. These normalstresses also contribute to strain-softening in this model (where the yield behavior is composed of a von-Mises type criterionwhich is strongly dependent upon normal stress differences, as seen in Fig. 3(a). Even though the activation stress is keptconstant in these three cases, the different strain softening trends may be attributed to the temperature at which the defor-mation is applied and the form of the flow potential Ave (refer to Eq. (24)) that competes with the activation stress. Thereforeeven though the activation stress is constant, the evolution of the flow potential results in strain softening behavior in thismodel. Of course, if further experimental evidence can be obtained on the actual multiaxial stress–strain behavior of thesematerials (tension–torsion experiments, for example) then improved yield models can be used which will show eithergreater or lesser normal stress difference effects. At this stage we lack published experimental data on this effect.

Now let us compare the isothermal shear deformation with the experimental data available in the literature. We refer tothe work by Khan, Koo, Monk, and Eisbrenner (2008), where the shear strain is applied at glass transition and low temper-atures for maximum applied shear strain of 65% on a Veriflex sample. In order to compare with the trends of the results withthe experimental data, we will apply the shear deformation F ¼ Iþ f e1 � e2 that is equivalent to 65% engineering strain. Forthis case, we define the corresponding activation stress values are kept constant at 0.1 and 4 at glass transition temperatureand low temperature, respectively, to observe the relative behavior. We set the reference temperature hmax ¼ hg . The initialconditions are set as fj; h;a;G0 ¼ Ig.

As can be seen from the shear stress response in Fig. 4(a), there is a dramatic change in the mechanical properties, whenthe material is subjected to the same shear deformation, depending on the temperature of the experiment. At glass transitiontemperature the ‘‘effective’’ shear modulus (42.5 GPa) is lower than that at low temperature (68 GPa) as can be seen inFig. 4(a). The trends of these results are similar to those observed by Khan et al. (2008), where they carry out shear exper-iments at low temperature and glass transition temperature on the Veriflex sample. They also observe that the ‘‘effective’’shear modulus of the sample at low temperature (60 GPa) is almost four times that at glass transition temperature. Duringthe low temperature experiment, Khan et al. observed material failure after yielding around shear strain of about 10%.Although the model does show yielding around the same shear strain, the calculated shear stresses are higher because ofthe assumed material properties, and the material continues the response trend beyond the yield regime.

7.2. Rate dependent strain softening behavior

Isothermal shear deformation is carried out for three different shear rates at low temperature i.e., glassy state of the poly-mer and at high temperature i.e., rubbery state of the polymer in Fig. 5. With increasing strain rates, higher stress responselevels are observed. Strain softening is observed in all three cases of strain rates, however, with increasing strain rates, thestress accumulated is higher before softening occurs. At low temperatures, the stress rise before softening is almost same forall three shear rates (Fig. 5(a)), whereas at high temperatures, the stress rise before softening, increases with increasingstrain rates (Fig. 5(b)).

(a) (b)

Fig. 3. Results of the response for shear deformation at different temperatures. (a) Shear stress at different temperatures. Activation stress at lowtemperatures is high, and thus yielding at different temperatures occurs at different stress limits. (b) The trends of normal stress differences at differenttemperatures are similar,but the values increase with decreasing temperature.

(a) (b)

Fig. 4. Results of the response for shear deformation at low and glass transition temperature, with respective activation stresses. (a) Shear stress for h < hg

and h ¼ hg . The lines represent model results, while the circle represent data. (b) Normal stress difference for h < hg and h ¼ hg . Note the different trends inthe Poynting effect, depending on the temperature of deformation.

(b)(a)

Fig. 5. Results of the response for shear deformation at different shear rates at high temperature. (a) Low temperature glassy case: Shear stress response forthree different shear rates. Shear stress rise before softening is almost same for all three shear rates. (b) High temperature rubbery case: Shear stressresponse for three different shear rates. Shear stress rises with increasing shear rate.


7.3. Cyclic isothermal shear deformation

Isothermal cyclic shear deformation is carried out for rubbery and glassy cases. The shear stress response and the normalstress differences for each cycle is plotted in the Fig. 6(a) and (b). From the characteristics of the shear stress response weobserve that the shear stress versus shear strain response is repeatable after multiple cycles, and this behavior is in agree-ment with the experimental observation made by McKnight and Henry (2008) wherein cyclic shear experiments have beencarried out on a composite thermoplastic shape memory polymer (MHI Diaplex). Also, The ratio of the maximum stress riseon loading in the first cycle, to the stress rise after multiple cycles increases with temperature. In the rubbery phase (hightemperature), the stress rise after multiple cycles is almost equal to that in the first cycle. In the glassy phase (low

(b)(a)

Fig. 6. Results of the response for cyclic shear deformation at low and high temperature. (a) Low temperature glassy case: Shear stress on loading aftermultiple cycles is less than the maximum shear stress attained in the first cycle. (b) High temperature rubbery case: Shear stress on loading after multiplecycles is almost equal to the maximum shear stress attained in the first cycle.


temperature), the stress rise after multiple cycles is lower than that in the first cycle. From the characteristics of the normalstress difference, we observe that the behavior has a different trends after multiple cycles for low temperatures, but con-verges rapidly for high temperature cases.

7.4. Thermomechanical shear cycle

In this section, we simulate the response of the SMP undergoing a shear deformation thermomechanical cycle. The glasstransition of the material is hg ¼ 70 C. The processes involved are as listed below:

1. Initial conditions: the material is considered at a stress free state at a temperature above the glass transition temperaturehmax ¼ hg þ 5.

2. Process A: high temperature deformation: The temperature is held fixed at hmax and the shear deformation is increasedsteadily at a constant prescribed rate to give the temporary shape to the material, where the maximum shear deformationis approximately 60% in about 50 s.

3. Process B: cooling and fixing the temporary shape: the shear strain is fixed for about 100 s while the temperature is low-ered to hmin ¼ hg � 25.

4. Process C: relaxing the stress: the temperature is fixed at hmin and the shear stress is gradually relaxed to zero in 3 s.5. Process D: heating under different strain or stress conditions:

(1) Stress Recovery (refer to Section 7.5): The body is heated back to hmax ¼ hg þ 5 in 500 s under constrained shear defor-mation to observe the stress recovery characteristics.

(2) Shape Recovery (refer to Section 7.6): The body is heated back to hmax ¼ hg þ 5 in 500 s under no load conditions toobserve the shape recovery characteristics.

In order to implement this cycle, the model is evaluated in a strain controlled regime (Process A and B) followed by astress controlled regime (Process C), followed by a stress or strain controlled regime (Process D (1) or D (2)). Process Band D have different thermal loads which affect the evolution of the activation threshold. Depending on whether the SMPis being heated under load or cooled under constant deformation, the activation stress evolves in a hysteretic manner whichis key in controlling the response of the SMP model. Since the thermomechanical cycle involves heating and cooling cycles,the rate of activation stress will evolve depending on specifics of the thermal processes, details of which can be found in thepaper by Ghosh and Srinivasa (2011, 2012), and is summarized here for convenience.

7.4.1. Rate form for the activation stressThe activation stress of the material is sensitive to temperature, and the material yields differently depending the current

value of the temperature, the amount of strain the material is subjected to, and on whether the temperature of the materialdropped or increased from the previous time-step. Thus there is a hysteresis of the activation stress from the cooling to theheating cycle, which gives the different trends of the stress-rise during cooling and the strain-recovery during heating. Theseconsiderations suggest that the rate of activation stress has the following functional form

_j ¼ f ð�h; signð _�hÞ; ��Þ _�hfcool ¼ y1ðev þ sinhð�y2ðh� h1ÞÞÞd1

fheat ¼ ð�y3ev � y4ð1� ðy5 tanhðmhþ nÞÞ2ÞÞd2

where, ev is the von mises strain corresponding to the strain ðee � aðh� hhÞIÞ. ev is the scalar strain value selected in 3D con-text, such that it is affected by pure thermal strains or pure mechanical strains. Also, m ¼ 2=ðhmax � h2Þ;n ¼ 1�mhmax. Hereh1 and h2 are limiting values of h until which there is no rise/fall in the stress/strain during the cooling/heating cycle.

d1 ¼1; 8 h 6 h1;

0; 8 h > h1;

�d2 ¼

0; 8 h < h2;

1; 8 h P h2;

�

7.5. Thermomechanical response of the SMP model – Stress recovery

In Fig. 7, the model is subjected to a shear thermomechanical cycle for the stress recovery case.

� The behavior of the model during process A, shear deformation at high temperature and process B, cooling at constantdeformation are as follows:– The stress rises non-linearly during process A.– The stress reduces during process B.– During process C, i.e., unloading at low temperature, the material springs back slightly.

Fig. 7. Results of the response for shear deformation thermomechanical cycle for stress recovery: note that the stress rise during loading and stress fallduring cooling is non-linear. During unloading at low temperatures, material shows slight spring back. The stress recovery during heating remains near zerovalues for a while, after which it rises speedily.

Fig. 8. Response of the model for shear deformation thermomechanical cycle for shape recovery.


� In process D, heating at constant deformation, we notice that the stress of the material falls below zero and remains atzero for a while before finally rising again, to show a stress recovery response. The flow potential of the material is grad-ually rising with temperature until it reaches a value that can compete with the activation stress, after which the stress ofthe material starts rising.– Around 56 �C, the activation stress in the SMP model reaches a value that can now start competing with the stress

developed, and results in the stress recovery response.– The SMP model shows near steady values of stress until the end of the cycle.


7.6. Thermomechanical response of the SMP model – Shape recovery

In Fig. 8, the model is subjected to a shear thermomechanical cycle for the shape recovery case. The experimental andmaterial parameters are kept the same as Section 7.5, except during the heating cycle, the stress is maintained at zero tostudy the shape recovery characteristics.

8. Concluding remarks

In this paper, the details of the development of a Helmholtz potential based 3D finite deformation constitutive model forthe SMP were presented, using the QR decomposition technique. The model was set up in the form of ordinary differentialequations and implemented in a suitable algorithm using an ode solver in MATLAB. The response of the model was studiedfor shear deformation and the characteristics of the isothermal shear response were compared with experimental studies ofKhan et al. (2008) for high temperature and room temperature conditions. The model response was reasonable compared tothese experimental findings. The isothermal shear deformation studies were then studied in detail for different initial tem-perature conditions as compared to the reference temperature of the model. The shear behavior of the model showed similartrends at different initial temperatures. However the Poynting effect exhibited by the normal stresses at these initial condi-tions were radically different, and gave an insight into how the model works. We extrapolated this behavior for cyclic sheardeformation, and although the experimental data in the literature is minimum for these cases, we have made comments onrelated cyclic shear data for SMP composites by McKnight and Henry (2008). We also study the model for different defor-mation rates in rubbery and glassy phase, and notice that the amount of stress rise before strain softening is different forthese two cases, but the strain softening behavior occurs at higher stress values with increasing shear rates. Finally we haveimplemented a shear deformation thermodynamic cycle for the SMP model and studied its response for the various combi-nations of mechanical and thermal loadings within this cycle. For the stress recovery case, the model is able to show typicaldata behavior for loading, cooling and unloading processes. The stress recovery response for the model shows steadyincrease in stress till the end of the cycle. The model is also implemented for a shape recovery thermomechanical cycle undershear loading, where the material shows complete shear recovery at the end of the cycle, the recovery characteristics beingtypical of the shape memory behavior.

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development of a finite strain two-network model for shape memory polymers using qr decomposition

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