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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2013; 94:895–919 Published online 26 April 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4474

Finite element implementation of a macromolecular viscoplastic polymer model‡ S. Kweon1 and A. A. Benzerga2, * ,† 1 Mechanical 2 Department

Engineering, Southern Illinois University, Edwardsville, IL 62026, USA of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA

SUMMARY This paper presents a time-integration method for a viscoplastic physics-based polymer model at finite strains. The macromolecular character of the model resides in (i) the viscoplastic law based on a doublekink molecular mechanism, and (ii) a full chain network model inspired by rubber elasticity to describe the large-strain orientation hardening. A back stress enters the constitutive model formulation. Essential aspects of a three-dimensional finite-element implementation are outlined, the main novelty being in the back stress formulation. The computational efficiency and accuracy of the algorithm are examined in a series of parameter studies. In addition, because a co-rotational formulation of the constitutive equations is employed using the Jaumann rate in the hypoelastic equation and the back stress evolution equation a detailed analysis of stress oscillations is carried out up to very large strains in simple shear. Subsequently, three-dimensional FE analyses of compression with friction and instability propagation in tension are used as a means to demonstrate the robustness of the implementation and the potential occurrence of stress oscillations and shear bands in large-strain analyses. Copyright © 2013 John Wiley & Sons, Ltd. Received 2 July 2012; Revised 30 November 2012; Accepted 3 January 2013 KEY WORDS:

viscoplastic; finite elements; finite deformations; polymer

1. INTRODUCTION The mechanical behavior of polymers is known to be path and time dependent. It generally is dependent on thermal, mechanical, and environmental conditions [1]. Viscoelastic [2, 3] as well as viscoplastic [4] constitutive relations have been developed to model the nonlinear mechanical response of polymers. As pointed out by Argon [4], the yielding behavior of polymers in their glassy state cannot be fully understood on the basis of modifying concepts of viscoelasticity. The nonlinear response of polymeric materials subject to complex loading history has been described using viscoplastic models such as Perzyna’s [5] and Bodner’s [6]. For instance, Kim and Muliana [7] have recently used Perzyna’s model in conjunction with the nonlinear viscoelastic Shapery model. Although the concurrent modeling of viscoelastic and viscoplastic effects is quite appealing, the viscoplastic part was based on a phenomenological model that was originally motivated by dislocation-mediated metal plasticity. In addition, their formulation was limited to infinitesimal strains. Polymer-specific viscoplastic models are often based on Eyring’s [8] theory for thermally activated processes with a linearly stress-dependent free-enthalpy. They generally provide a useful framework in narrow ranges of strain rate and temperature and do not invoke specific molecular-scale mechanisms. Argon [4] proposed a theory that is based on a double-kink molecular mechanism and arguably extends the range of application in terms of temperature and *Correspondence to: A. A. Benzerga, Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA. † E-mail: [email protected] ‡ Supporting information may be found in the online version of this article. Copyright © 2013 John Wiley & Sons, Ltd.

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strain rate. More recent heuristic improvements have been proposed on the basis of cooperative molecular interactions [9]. The aforementioned models adequately describe the plastic straining that is related to chain conformation or configuration changes in the early stages of deformation. They do not account, however, for such changes at large stretches, which usually include molecular alignments. The associated increase in plastic resistance has been described as an entropic effect, as in the theory of rubber elasticity [10]. Boyce and co-workers [11, 12] have developed a three-dimensional model that combines Argon’s viscoplastic law to describe the initial stages of yielding with a non-Gaussian chain network model inspired by rubber elasticity to describe the large strain behavior. The plastic resistance due to molecular alignments is typically introduced as a back stress in this formulation. This back stress represents the storage of internal energy at sufficiently large stretches because the molecules are extended in addition to being uncoiled. Wu and van der Giessen [13] developed this approach further by accounting numerically for the 3D orientation distribution of individual chains and by deriving a composite (or full-network) model that combines the three-chain and eight-chain models in [11] and [12], respectively. This class of viscoplastic models is referred to as macromolecular in this paper. Wu and van der Giessen [14] described a detailed two-dimensional finite-element implementation of their version of the macromolecular model. They used a convected representation of finite strain plasticity and developed a rate formulation of orientation hardening in which the updating of the back stress is based on an evolution law derived in terms of the Jaumann rate. In their actual implementation, however, they followed their earlier works, [11, 15] where the constitutive updating of the back stress is carried out following an explicit scheme bypassing the evolution law they developed. The same strategy was followed in more recent works [16, 17]. Later, Chowdhury et al. [18] have employed an implicit update of the back stress, which is better for enforcing equilibrium; also see [19]. However, their implementation is for dynamic transient problems only. This paper presents a time-integration method for the macromolecular model constitutive relations at finite strains. Its computational efficiency and accuracy are examined in a series of parameter studies. Essential aspects of a three-dimensional finite-element implementation in the commercial code ABAQUS through a user-defined subroutine are outlined. As imposed by the user-defined material subroutine (UMAT) structure, a co-rotational formulation of the constitutive equations is employed using the Jaumann rate in the hypoelastic equation. This precludes the use of a hyperelastic formulation as in [20, 21]. What is of particular importance is that pathological stress oscillations can occur under simple shear loading in hypoelasticity [22] particularly in conjunction with a kinematic hardening model [23, 24]. Because the constitutive model has a strong kinematic hardening component and is formulated using the Jaumann rate both in the hypoelastic equation and in the evolution equation of the back stress tensor, the issue of stress oscillations is investigated. To put this study in perspective, the typical behavior in simple shear is revisited for a class of simple constitutive models (perfect plasticity, isotropic hardening, and kinematic hardening) up to very large stretches. Particular attention is given to strain levels and material parameters at which oscillations take place when using either the simple or macromolecular constitutive models. Subsequently, 3D FE analyses of compression with friction and instability propagation in tension are used as a means to demonstrate the robustness of the implementation and the potential occurrence of stress oscillations in large-strain analyses. 2. CONSTITUTIVE MODEL We employ a viscoplastic model based on the macromolecular chain network theory to describe the thermomechanical behavior of glassy polymers. Two variants of the model are used: in its original form following [12] and [13] and in a modified form aimed at better representing the small-strain nonlinear response [25]. They will be referred to as macromolecular models. Within a finite strain framework, additive decomposition of the rate of deformation tensor is assumed D D D e C D p . The elastic part is obtained from the hypoelastic law r

D e D C 1 W  , Copyright © 2013 John Wiley & Sons, Ltd.

(1) Int. J. Numer. Meth. Engng 2013; 94:895–919 DOI: 10.1002/nme

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with C the isotropic tensor of elastic moduli (expressed in terms of Young’s modulus E and r

Poisson’s ratio ) and  the Jaumann stress rate defined by 1 r  D P C   W  W   , W D .r v  r vT /, 2

(2)

where W is the skew symmetric part of the velocity gradient. A flow rule is used to provide the plastic part of D as Dp D r PN D

3PN 0  I d D   b 2e d

2 p D W D p I e D 3

r

3 0  W 0 , 2 d d

(3)

(4)

where PN is the effective plastic strain rate, e is the effective stress,  d is the driving stress and b is the back stress with X 0 referring to the deviatoric part of second-rank tensor X . Temperature and strain-rate sensitivity is accounted for through the viscoplastic law [4, 11]   m   A.s  3˛m / e PN D P0 exp  1 , (5) T s  3˛m where P0 is a reference strain rate, s is the athermal strength, A is a material parameter, T is the absolute temperature, ˛ is a pressure sensitivity parameter, m D kk , and m is the strain rate 3 sensitivity exponent, which was derived to be 5=6 in [4] but is here treated as a material parameter. In the original macromolecular model, the evolution of the athermal strength state variable s is given by [11] ! s   PN , sP D h 1  (6) sss T , PN where h is the slope of the yield drop with respect to plastic strain and sss is the saturation strength.   The writing sss T , PN simply means that sss may depend on temperature and strain rate but only through the dependence of elastic moduli upon the latter. The drawback of Equation (6) is that it causes a sharp ‘yield point’, which is rarely seen in polymeric materials. To overcome this drawback, a more realistic smooth transition between the elastic and plastic regimes is modeled, based on [25], as ! ! s s  PN C H2 .N / 1    PN sP D H1 .N / 1   (7) s1 T , PN s2 T , PN         N  Np N  Np  1 , H2 .N / D h2 tanh C1 , H1 .N / D h1 tanh f Np f Np

(8)

where s1 and s2 are adjustable parameters related to the peak and dip stresses, h1 is the slope of the initial rise to the peak stress, h2 is the decreasing slope past the peak stress, f and Np are adjustable parameters related to the width of the peak yield point. For simplicity, one could use h1 D h2 as in [25]. Note that s1 and s2 may depend on temperature T but only through elastic modulus E. The evolution of the back stress was rederived by Wu and van der Giessen [14] as r

bD R W D p ,

R D .1  /R3ch C R8ch ,

(9)

where R3ch and R8ch are the three-chain and eight-chain back stress moduli tensors, respectively, N

and  is defined by  D 0.85 p , N being the average number of links between entanglements and N

Copyright © 2013 John Wiley & Sons, Ltd.

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N the maximum principal stretch. N is calculated from the left Cauchy–Green tensor B D F  F T . It is worth noting that it is D p that is used in (9) and not D as in [25]. The three-chain [11] and eight-chain [12] models were derived based upon different sampling geometries of molecular chains to represent the statistical response of the chain network. In terms of the components of B, those of R8ch are given by    1 Rp ˇc ˇc Bij Bkl c 8ch N C .ıi k Bj l C Bi k ıj l / , (10) Rij kl D C p  3 c Bmm c N where C R is the rubbery modulus and N as mentioned earlier. Also, ıij is the Kronecker delta, and 1 2c D trB, 3

ˇc D L1



c p N

 c D

,

ˇc2 , 1  ˇc2 csch2 ˇc

(11)

where L1 is the inverse Langevin function defined as L.x/ D coth x  x1 . A Pade approximation was used for L1 . The components of R3ch are given in terms of the principal total stretches I , with respect to axes pointing onto the principal stretch directions (no sum on I or J ): 8   ˇI I 1 Rp ˆ 2 ˆ .ıIK ıJL C ıJK ıIL / N  if I D J C C p ˆ I ˆ < 6 I N 3ch RIJKL D (12) ˆ ˆ 1 R p 2I C 2J ˆ ˆ N 2 .I ˇI  J ˇJ /.ıIK ıJL C ıJK ıIL / if I ¤ J : C 6 I  2J If I D J with I ¤ J then .12/1 is used instead. Also, ˇI and I are shorthand notations for ˇI D L

1



I p N

 ,

I D

ˇI2 1  ˇI2 csch2 ˇI

.

(13)

3. TIME INTEGRATION METHOD A semi-implicit integration algorithm is employed to integrate the earlier constitutive relations within a co-rotational formulation [26]. Our implementation differs from previous ones [14, 27], mainly in the way the back stress evolution is tackled. For this reason, we summarize in this section the main elements of the proposed implementation. In essence, the general formulation follows that of [28] for crystal plasticity and [29] for porous metal plasticity. We express all constitutive relations in an intermediate, rotated configuration obtained from the current one based on R  . The latter is the rotation tensor that results from the polar decomposition of the incremental deformation gradient F following: F D R  U  .

(14)

This formulation preserves incremental objectivity and is equivalent to using a Jaumann rate in hypoelastic Equation (1). However, the spin tensor W entering (2) is never actually computed. All kinematic and stress-like quantities defined in the rotated configuration are indicated by the .O/ symbol. Expressions for the constitutive relations in the rotated configuration are listed in Appendix A. Because the rotation part of the incremental deformation is taken care of in this intermediate configuration, constitutive descriptions need to be written only for the stretch part of deformations. In the rotated configuration, the state variables are collected in the vector set h i 0 ŒX T D O 0 , bO , bOm , O m , s , (15) Copyright © 2013 John Wiley & Sons, Ltd.

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where the deviatoric–volumetric decomposition is applied to the Cauchy stress  and the back stress b to facilitate convergence in the Newton–Raphson procedure employed in the following text. The set of differential equations for the aforementioned variables is made of Equations (A.2), (A.4), and (A.5) (or (A.6)). This coupled system is solved using an implicit time integration method following a backward Euler scheme. For example, a scalar variable such as the athermal shear strength s is integrated using s t Ct D s t C t sP t Ct .X / with s t the value taken by s at the beginning of the increment and t the time step. The residual on s is thus obtained as Rs D

s t Ct  s t  sP t Ct .X /, t

where the evolution Equation (7) is used to provide an estimate of sP t Ct . Integrated residuals are collected in the vector set h i ŒR T D R O 0 , R bO 0 , RbOm , RO m , Rs .

(16)

(17)

Their expressions are as follows: R O 0

1 D 2



O 0  O 00 t



C PN

3  0 O 0 O 0 O  b  D 2e

  0 1 O0 O0 O C .O m / I  O 00  .O m /0 I 1 O O O R bO 0 D b  b0  J W R W D C J W R W C W t t RbOm D

(18)

(19)

  0 0  1 1 O O WD O C 1I W R O W C 1 W O C .O m / I  O 0  .O m /0 I (20)  I WR bm  bOm 0 t 3 3 t 1 ŒO m  .O m /0  K DO kk t     s O s O s  s0 PN  H2 1  PN ,  H1 1  Rs D t s1 s2 RO m D

(21)

(22)

where the subscript 0 refers to values at the beginning of a time increment t , J is the fourthrank projector into the deviatoric hyperplane, that is, Jij kl D 12 .ıi k ıj l C ıi l ıj k /  13 ıij ıkl , I is the second-rank identity tensor and and K are the shear and bulk moduli, respectively. In the aforementioned example, Equation (16), the time-integrated equation obtained by discretizing Equation (7), or equivalently Equation (A.6), is given above as Equation (22). In Equations (18)–(22), the state variables at the beginning of the increment are collectively denoted by ŒX 0 , and the variables at the end of the increment by X , that is, dropping the superscripts t and so forth in the earlier example. The constitutive updating is then based on   @ŒR 1 ŒX D ŒX 0  ŒR . (23) @ŒX The solution of ŒR T D 0 using a Newton–Raphson method requires that the Jacobian matrix @ŒR =@ŒX be evaluated. To ensure fast convergence, a consistent tangent matrix (denoted by L) is computed to be used in subsequent equilibrium steps. To compute L, we first write       @X @ŒR 1 @R D , (24) O O @ŒX @D @D Copyright © 2013 John Wiley & Sons, Ltd.

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where 

@X O @D

and 

@R O @D

T

T

"

0 @O 0 @bO @bOm @O m @s , , , , D O @D O @D O @D O @D O @D

# (25)

"

# @R O 0 @R bO 0 @RbOm @RO m @Rs D , , , , . O O O O O @D @D @D @D @D

(26)

Note that the same Jacobian matrix @ŒR =@ŒX in Equation (23) is used again in Equation (24). Extracting the terms of interest from (24), L is obtained as   1 @O 0 @O m LD CI ˝ , (27) O O t @D @D where I is the second-rank identity tensor. The algorithm used in the implementation is summarized in Figure 1. A key difference with previous implementations resides in the treatment of the back stress evolution. Wu and van der Giessen [14] developed a rate formulation of orientational hardening in which the updating of the back stress is based on the current moduli tensor R in (9). In their actual implementation, however, they followed their earlier work [13] (also see [11, 17]) where they compute the current value of b using the eigenvalues of the plastic left stretch tensor V p , that is based on the updated state of deformation. Although this may be computationally efficient,

Figure 1. Flowchart of the integration algorithm in the user-defined material subroutine (UMAT). Copyright © 2013 John Wiley & Sons, Ltd.

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because it does not require recomputing R at every time step, it sacrifices accuracy in that small errors build up in enforcing equilibrium. Therefore, in this work, we have used the format (9) in the time-integration procedure so that the components of the Jacobian in (24) are directly affected by this choice. For brevity, only those components of the Jacobian that are related to the back stress are provided (Appendix B). Besides accuracy, some merits of using Equation (9) include simplicity in computation and ease of implementation within the co-rotational framework. Indeed, if only the eight-chain model is used (i.e.,  D 1 in (9)), then one does not need to decompose the deformation gradient into stretch and rotation parts to solve the eigenvalue problem on V p then rotate the b components accordingly. On the other hand, when the full network model is used (i.e.,  ¤ 1) one cannot avoid solving an eigenvalue problem on B to compute the components of R3ch with respect to the principal stretch directions. The reason for this is that unlike R8ch , the three-chain moduli tensor does not admit a nice tensorial representation. Once that is carried out, R3ch is rotated to the current base and summed up with R8ch following .9/2 . Finally, the resulting tensor R is rotated to the intermediate O Note that R O 6D R is due to induced anisotropy. configuration using Equation (A.1) to obtain R. We also emphasize that the use of Equation (9) is only valid within the co-rotational framework based upon the Jaumann rate. If another objective rate were to be used, the back stress evolution equation would need to be rederived accordingly. It is worth noting that Chowdhury et al. [18] have used the format (9) in their dynamic implementation using a convective representation of finite strain plasticity. 4. NUMERICAL IMPLEMENTATION AND VERIFICATION The constitutive model was implemented in the finite-element code ABAQUS as a user-defined subroutine§ . The flowchart of the UMAT is shown in Figure 1. The consistent tangent moduli tensor L is calculated inside the UMAT and supplied to ABAQUS. The proposed time-integration method is not fully implicit, which warrants numerical tests to verify the (time) convergence of the method. This is carried out in Section 4.1 in the succeeding text. In addition, the structure of the UMAT imposes a co-rotational formulation of the constitutive equations using the Jaumann rate. This imparts potential shortcomings to the formulation. Among these, we investigate in Section 4.2 the onset of stress oscillations in simple shear because of the strong kinematic hardening component. Also, it has long been known that the Jaumann rate of Cauchy stress is not work-conjugate with any finite strain measure and this leads to errors. Such errors can be remedied, as for example proposed by Bazant et al. [30]. However, the error for a plastically incompressible polymer is less than a fraction of a percent. Note that in our previous dynamic implementations of the macromolecular model [18], we have used the Jaumann rate of Kirchoff stress, which is work-conjugate with the Hencky strain. Finally, any hypoelastic formulation potentially leads to spurious dissipation in a closed stress cycle. This drawback can be remedied in hyperelastic formulations, for example, [20, 21]. 4.1. Stability, accuracy, and convergence The use of a backward Euler scheme in the constitutive updating is good for stability as well as accuracy. Also, convergence of solutions is much improved using a tangent stiffness matrix that is consistent with the earlier implicit scheme. In theory, convergence would be guaranteed for very large time steps if the time-integration method had been fully implicit. However, the algorithm used here includes inevitable explicit updates for tensor R entering the back stress evolution equation, because it is calculated based on the left Cauchy–Green tensor B. The latter is itself calculated based on the deformation gradient F , which is explicitly updated at the beginning of a time increment and supplied to the UMAT by ABAQUS. This sets restrictions on the time step. An iterative procedure is used whereby the time step is adaptively reduced when the residual equations do not converge §

The developed UMAT is made available upon request.

Copyright © 2013 John Wiley & Sons, Ltd.

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200 25˚C 50˚C

True Stress (MPa)

150

75˚C Model

100

50

0

0

0.2

0.4

0.6

0.8

1

1.2

True Strain

Figure 2. Best fit of PMMA data in axisymmetric compression from [27] using the macromolecular model with E D 3.20 GPa,  D 0.34, s0 D 81.3 MPa, s1 =s0 D 1=0.75, s2 =s1 D 0.75, C R =s1 D 0.138, N D 7.39 D exp .2/, P0 D 2  1013 /s, A D 350ı K/MPa, m D 1, ˛ D 0.2, h1 =s1 D 36.9, h2 =s1 D 1.85, Np D 0.05, and f D 0.1.

(Figure 1). In this section, accuracy and convergence properties of the proposed algorithm are addressed using two sets of simple parametric studies. In the first parametric study, we examine the effect of the initial time increment t on the material response. Material parameters calibrated on the PMMA data of Arruda et al. [27] were used (Figure 2). As shown in the figure, the typical behavior exhibits an up-down-up response where the flow stress is temperature and strain-rate dependent. Figure 3 shows the peak flow stress for various time-increment sizes and several values of temperature (Figure 3(a)) or strain rate (Figure 3(b)). Clearly, the choice of t can affect the accuracy of the results and a large time increment may lead to a diverged solution. At a nominal strain rate of about 1/s, the solution becomes inaccurate when the time step size gets larger than 103 s. At higher rates of loading, even smaller values of the time step must be used to ensure stability (Figure 3(b)). When converged states were not reached (open symbols), the reported stresses are not peak stresses but the maximum stresses that were attained. In all cases, a tolerance of 103 times the initial residual was used. From this parametric study, it is concluded that an initial time increment within the range 105 /s–103 s allows to represent accurately the response of this material for wide ranges of temperature and strain rate. In the second parametric study, we examine the convergence rate by plotting the error in the computed stress versus the number of iterations needed to reach convergence (Figure 4). Here, the Euclidean norm of the stress residual is used as a measure of the error, normalized by the norm of the rate of deformation, the latter remaining constant over iterations. Assessing the quality of the results on the basis of residuals as a measure of error is not uncommon in viscoplasticity problems [31]. The results in Figure 4 correspond to that stage of straining when it takes the largest number of iterations before convergence, that is, near the ‘elastic–plastic transition’. Results are shown for two values of the time increment. When a relatively large time step is used (t D 2  103 ) the amount of error is initially large, and it takes some iterations to reach convergence. The number of iterations needed is typically 6–7 but quickly decreases to 2 for subsequent time steps. For a sufficiently small time step (t D 104 ), fewer iterations are needed during the ‘elastic–plastic transition’. In both cases, the error rapidly decreases following near quadratic convergence because the Newton–Raphson scheme is used in the internal solving procedure of the UMAT. 4.2. Response to simple shear The case of simple shear is an interesting benchmark for large-strain elasto-plasticity problems. Nagtegaal and De Jong [23] have obtained an oscillatory response when using a Prager–Ziegler kinematic hardening rule together with the Jaumann rate for the stress and back stress. Johnson and Copyright © 2013 John Wiley & Sons, Ltd.

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80

(a)

Peak Stress (MPa)

70 60 50 25˚C

40

50˚C 75˚C

30 20 10-5

10-4

10-3

10-2

10-1

Δt (second) 80

(b)

Peak Stress (MPa)

70 60 50 40

25˚C, ΔU/(L*Δt)=1.44/sec 25˚C, ΔU/(L*Δt)=14.4/sec

30 20 10-7

25˚C, ΔU/(L*Δt)=144/sec

10-6

10-5

10-4

10-3

10-2

10-1

Δt (second)

Figure 3. Effect of time-increment on the peak stress at various values of (a) temperature (at UP =L0 D 1.44=s); and (b) strain rate (at T D 25ı C). Open symbols correspond to diverged solutions using a tolerance of 103 times the initial residual. 100 90

Stress error (%)

80 70

Δt=2*10-3 sec

60

Δt=10-4 sec

50 40 30 20 10 0

1

2

3

4

5

6

7

Iteration number

Figure 4. Error in the computed stress,

kR O 0 k , at the elastic–plastic transition versus the number of iterations 0 O k kD

before convergence for two values of the time increment.

Bammann [24] presented analytical solutions of the previous problem using either the Jaumann or Green–Naghdi rates. Their results illustrate the removal of oscillations when the Green–Naghdi rate is used and suggest that an oscillatory response is characteristic of kinematic hardening models. Later, Cescotto and Habraken [32] solved analytically the simple shear problem for an Copyright © 2013 John Wiley & Sons, Ltd.

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isotropically hardening material and demonstrated the possibility of oscillations depending on the ratio of yield stress to elastic modulus. For a more complex constitutive model, such as the ratedependent macromolecular model, analytical solutions are not available and one must recourse to numerical solutions. To set the stage, we revisit the onset of stress oscillations in the context of simple rate-independent elasto-plastic constitutive models that are readily taken from the ABAQUS material library. Three plasticity models are used (i) perfect plasticity (no hardening); (ii) linear isotropic hardening; and (iii) linear kinematic hardening. An associated flow rule is used in all. A value of 37.5 MPa is used for the hardening rate, where relevant. In every case, two values of the key material parameter y =E are used with y being the initial yield stress and E Young’s modulus. A relatively low value of 0.0218 is typical for glassy polymers (much lower values would be typical for metals) and a very high value of 0.7 is used to explore limit cases. In all simulations, a 2D plane strain element (CPE4) is used with the nonlinear geometric analysis option for the large strain analysis. The boundary conditions consisted of fixing the bottom two nodes and applying a displacement  in the x1 direction to the top two nodes. This preliminary study allows to establish a hierarchy of responses to simple shear when the Jaumann rate is used for the stress. Figure 5 shows the three model responses to simple shear. The shear stress as well as in-plane normal stresses are plotted against the engineering shear strain "12 D =2. To illustrate the role of elastic strain, all stress components are normalized by the yield strength. Plots on the left of

1

σ12/σy σ11/σy

0.5

Normalized Stress

Normalized Stress

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σ22/σy 0 -0.5 -1

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Shear strain Γ/2

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σ11/σy

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σ22/σy

0.5 0 -0.5 0

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-0.5

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Shear strain Γ/2

σ22/σy

-0.5

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(e)

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Shear strain Γ/2

Normalized Stress

0

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(b)

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σ22/σy

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σ11/σy

(c)

σ12/σy

0.5

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σ12/σy

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σ11/σy

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Shear strain Γ/2

Normalized Stress

Normalized Stress

σ12/σy

Shear strain Γ/2

Normalized Stress

-0.5

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-1

σ22/σy 0

(a)

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σ11/σy

0.5

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σ12/σy

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(d)

σ12/σy σ11/σy

0.5

σ22/σy 0 -0.5 -1

0

1

2

3

4

5

6

Shear strain Γ/2

7

8

9

(f)

Figure 5. Response to simple shear using three ABAQUS built-in finite-strain constitutive models and two values of y =E: 0.0218 (left plots) and 0.7 (right plots). (a) and (b) Elastic-perfectly plastic, (c) and (d) elasto-plastic with isotropic hardening, (e) and (f) elasto-plastic with kinematic hardening. Copyright © 2013 John Wiley & Sons, Ltd.

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Figure 5 are for y =E D 0.0218, whereas those on the right correspond to y =E D 0.7. For y =E D 0.0218, no oscillations are found in the case of perfect plasticity (Figure 5(a)) or isotropic hardening (Figure 5(c)). On the other hand, an oscillatory response is observed when using the kinematic hardening rule (Figure 5(e)), in keeping with previous findings in the literature. By way of contrast, for y =E D 0.7, a non-monotonic response is obtained for all material models. Comparison between the y =E D 0.0218 and y =E D 0.7 cases clearly shows that the propensity for stress oscillations is very much dependent on the amount of elastic strains, consistent with the analytical solution in [32]. The results suggest that the critical amount of elastic strain is lower when using a kinematic hardening rule. It is also remarkable that a non-monotonic response can occur for a perfectly plastic material, a behavior which, to our knowledge, had not been noticed before. Another noticeable feature of the finite-strain simple shear response is the development of normal stresses 11 and 22 (Figure 5). Recall that in nonlinear elasticity, a simple shear deformation cannot be produced by pure shear forces alone. Very recently, Destrade et al. [33] have derived the homogeneous deformation due to a pure shear stress for a homogeneous, isotropic nonlinearly elastic material, thus extending an earlier work of Moon and Truesdell [34]. It follows from their solution that normal as well as shear tractions are necessary on the inclined faces of an initially square material element in order to maintain a state of homogeneous shear deformation. In the particular case of an incompressible (neo-Hookean) material, the applied shear stress also leads to an uniaxial compression in the direction of shear (x1 ) and thus an expansion in the transverse plane (x2 –x3 ). Such findings provide a means to rationalize the FE results in Figure 5 despite some differences. Besides the use of an elasto-plastic constitutive relation, the solutions here are for an applied shear strain (not stress) such that no width reduction along the vertical x2 direction was possible. This explains the emergence of a compressive stress transverse to the shear direction (22 < 0 always in Figure 5) and thus to a tensile stress in the direction of shear (11 > 0 always in Figure 5). For the type of isochoric plane deformation that is simulated, the constraint kk D 0 is obeyed through 11 D 22 so that the out-of-plane stress 33 vanishes. Thus, on the basis of pure kinematics, it seems impossible to avoid normal stresses under simple shear deformation. As shown in Figure 5, the magnitude of these stresses depends on the type of constitutive model used and their non-monotonic character is determined by the behavior of the shear stress. Along the same lines, it would be of interest to investigate the hierarchy of simple shear responses corresponding to different stress rates for a given constitutive model. As noted earlier, the UMAT structure imposes that the Jaumann rate be used. Another type of subroutine (user-defined element, UEL) allows multiple stress rates. However, the convergence properties of UEL subroutines and concurrent support with other ABAQUS functions (such as friction) are still limited. With the above findings in mind, the same plane simple shear problem is now analyzed using the macromolecular model. Figure 6 shows all relevant Cauchy stress components against the applied shear strain. Normal stresses develop, as with the previous simpler models but the state of triaxial stress is more complex here. More importantly, an oscillatory response is found to settle, but at a shear strain  no smaller than 4.0 (compare with a value of 1.0 for the linear kinematic hardening model (Figure 5e) for the same y =E ratio). Parametric studies show that varying non-dimensional model parameters such as s2 =s1 (softening parameter) or C R =s1 (orientation hardening parameter) does not decrease significantly the critical strain for oscillations. What controls the latter is the amount of elastic strain through y =E, consistent with the general trends in Figure 5 and previous analytical treatments [32]. Evidently, the constitutive relations inherent to the macromolecular model include an isotropic hardening/softening part in addition to a strongly nonlinear kinematic hardening. This alone may explain the much delayed onset of oscillations compared with the linear kinematic hardening model. Wu and van der Giessen [15] have reported some rather detailed analyses of the simple shear problem employing a variant of the macromolecular model. In particular, they derived 2D boundaryvalue problem solutions accounting for heterogeneous deformation modes. Their calculations were however limited to shear strains of about 2.0 (compare with 10 in Figure 6), that is much smaller than the levels at which stress oscillations could be seen (refer to Figure 6(b) versus Figure 6(a)). Also, the formulation in [15] included a locking stretch beyond which all plastic flow is deactivated and the response is purely (hypo-)elastic. As noted in Section 3, an important difference between our Copyright © 2013 John Wiley & Sons, Ltd.

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Shear strain Γ/2 120

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(b)

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80

σ33 σkk/3

60 40 20 0 -20 -40

0

0.2

0.4

0.6

Shear strain Γ/2

Figure 6. Response to simple shear using the macromolecular model and the Jaumann rate. (a) Up to a shear strain  of 10, (b) up to a shear strain  of 2. Note that kk D 0 throughout the deformation.

implementation and that of Wu and Van der Giessen [13,15] is in the way of updating the back stress b. In the present paper, b is updated according to (9), whereas in [13, 15], b is directly computed based on the updated deformation gradient tensor. While the present method is more accurate (better for enforcing equilibrium), it is possible that the use of the Jaumann rate of b enhances the oscillations. Investigating this is beyond the scope of the paper. However, because oscillations also occur in the absence of a back stress (Figure 5(a–d)), their onset cannot be traced back to the use of a Jaumann rate of b in the present model. In addition, Wu and van der Giessen [15] only reported the development of the normal stress transverse to the shear direction (22 ), which they attributed to the development of anisotropy. In fact, normal stresses are unavoidable in simple shear at finite strains, irrespective of anisotropy; see Figure 5 and [33] for a rationale. The anisotropy is induced by the orientational hardening at large strains. It is responsible for the more complex stress state in comparison with the isotropic models in Figure 5. In particular, an axial stress (11 ) gradually develops and becomes dominant at very large shear strains. The induced anisotropy breaks the symmetry between 11 and 22 and thus manifests itself in the development of an out-of-plane normal stress (33 ), a phenomenon that was absent in Figure 5. Such behavior illustrates the fact that simple shear is not so simple [33]. The critical shear strain levels (at which stress oscillations occur) are much greater than achievable in practice. For instance, in polymer networks with a constant density of entanglements (hence constant N in Equations (10)–(12)), each molecular chain can sustain a maximum stretch equal p to N (that is 2.7 in the treated example) [14]. For larger stretches, the network locks thereby suppressing all further plastic flow. Eventually, fracture would take place at strains lower than the critical shears for oscillations. Yet, from a purely computational standpoint, the onset of oscillations Copyright © 2013 John Wiley & Sons, Ltd.

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is a limitation of the hypoelastic/co-rotational formulation of the macromolecular model that should be kept in mind. It will manifest more severely in a soft polymer or near the glass transition temperature. Typically, attempts to remove such pathological oscillations from numerical solutions have consisted of employing hyperelastic formulations or using alternative stress rates, for example, [24, 35, 36]. If one were to pursue either option¶ , the evolution Equation (9) of the back stress would have to be modified accordingly. It remains that the literature on this subject appears to be still incomplete. While the occurrence of oscillations with the Jaumann rate is not ubiquitous (in the sense it does not occur with all material models at realistic shear strain levels), other stress rates do not seem to be immune from pathological responses, including in some cases oscillations; the reader is referred to a mathematical analysis by Liu et al. [37]. In practice, therefore, an alternative solution is to monitor on-the-fly the occurrence of oscillations. This will be carried out in all subsequent finite element simulations. 5. SIMULATION OF INHOMOGENEOUS DEFORMATION IN POLYMERS In this section, fully 3D finite-element simulations are presented using the material parameters in Figure 2, unless otherwise noted. Compression of a cylindrical pin is addressed first to illustrate the concurrent use of the friction function in ABAQUS. Next, the tension of a prismatic bar is analyzed laying emphasis on the concurrent propagation of geometric and material instabilities. Both illustrations aim at demonstrating the performance of the proposed implementation. 5.1. Compression The nominally uniaxial quasi-static compression of a cylindrical pin is simulated using a standard geometry. Initially, the pin has a diameter to height ratio of D0 =H0 D 0.5. It is reported to a Cartesian coordinate system with the pin’s axis being along x2 . The finite element mesh used is shown in Figure 7(a). It consists of 7740 eight-noded fully integrated linear hexahedral (C3D8) elements with eight integration points. Application of the displacement boundary conditions on the top and bottom surfaces is carried out in such a way to simulate dry frictional contact with compression dies. The applied nominal strain rate is UP =H0 D 0.0007=s, and the temperature is kept at 25ı C . A Coulomb friction model is used so that the tangential traction Ts on the contact surface is related to the compressive normal traction Tn through jTs j D Tn using the usual sign conventions with the (constant) coefficient of friction. Because of friction, an inhomogeneous state of deformation eventually ensues in the form of barreling (Figure 7(b)). To account for that while comparing compressive responses for various values of the friction coefficient, the following measures of stress and strain are used: " D ln

A0 I A

D

F , A

(28)

where F D Tn > 0 is the net axial reaction force on the top surface, and A is the current area of the transverse cross-section at the barrel apex. Figure 7(b) shows contours of the effective plastic strain N in the meridian plane x1 D 0 at a deformation level given by U=H0 D 0.698 using D 0.2. Friction causes the deformation to become highly inhomogeneous with bulging at the center of the pin and sliding at the die–specimen contact surfaces. The effective plastic strain peaks at the center of the specimen. The stress versus strain responses obtained with various values of the friction coefficient are compared in Figure 7(c). Without friction ( D 0), the deformation remains homogeneous throughout, as expected for an axisymmetric specimen. By way of contrast, the compressive response of a prismatic bar, say with a rectangular cross-section, would inevitably lead to shear band initiation, thickening and subsequent propagation, as analyzed dynamically by Chowdhury et al. [18], ¶

This would have to be performed outside of the UMAT structure of a commercial code such as ABAQUS/Implicit. For example, Benzerga et al. [29] have used and compared the Jaumann and Green–Naghdi rates for void rotation in porous metals.

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(a)

(b)

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μ=0

180

μ=0.05

160

μ=0.1

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μ=0.2

120 100 80 60 40 20 0

0

0.2

0.4

0.6

0.8

1

1.2

Strain ε

(c) Figure 7. (a) Cylindrical specimen. (b) Effective plastic strain N after deformation D 0.2. (c) Stress  versus logarithmic strain " as defined in (28) for various friction coefficients.

because of the intrinsic strain-softening. Such phenomena are precluded under axisymmetric loading. With friction, barreling always occurs and the amount of apparent softening in the  –" curves is greater than the true softening of the material. This behavior hints to the kind of difficulty one faces in extracting the intrinsic behavior of a polymer from a standard compression test [38]. Interestingly, the amount of friction does not seem to affect the post-yield softening behavior as much as it affects the large strain response. Higher amounts of friction lead to larger stresses beyond a strain of 0.8 or so. As the outer edges of the top and bottom surfaces start sliding on the die surfaces, more deviation from the homogeneous response ( D 0) arises. With respect to the onset of stress oscillations treated in the previous section, we note that the maximum (engineering) shear strain ("12 D 12 =2) attained anywhere in the specimens was about 0.52, which is much smaller than the critical strains reported in Section 4.2. The highest values are reached near the specimen-die interface because of constrained sliding. 5.2. Tension For a polymer whose nominal stress–stretch curve exhibits an up-down-up behavior, the tensile response of a bar is usually accompanied with the propagation of a neck [39]. Necking is a geometric instability. If, in addition, the true stress–strain curve exhibits an intrinsic strain-softening Copyright © 2013 John Wiley & Sons, Ltd.

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Figure 8. Geometry of the tension problem. In 3D analyses, the out-of-plane thickness is t0 . L0 D 57, W0 D 19, t0 D 3.2, w0 D 29, l0 D 136, and R D 76.

(cf. the post-yield drop in Figure 2 for PMMA), then the propagation of the neck may be accompanied with that of material instabilities, that is, shear bands. The simulation of shear-band mediated necking in polymers requires full numerical treatments. In this section, we analyze some fully 3D analyses of neck initiation and propagation and discuss the propensity of formation of shear bands during neck propagation, depending on the amount of intrinsic softening. The geometry of the basic problem is sketched in Figure 8. 5.2.1. Convergence analysis. To assess the quality of the numerical results, a convergence analysis was carried out with respect to the spatial discretization. This is important because an optimal mesh size should be used in the subsequent 3D analyses where intense, propagating shear bands may form. We assume plane strain deformation of one half of the tensile specimen shown in Figure 8. Four structured uniform meshes were used as shown in Figure 9, all with CPE4 linear elements. The element size h is divided by two at each refinement step, so that the number of elements is increased by a factor of about four. In all, the nominal strain rate is UP =L0 D 1.44=s and T D 25ı C. While a rich literature is available on a posteriori error estimators for the finite element approximation of linear elliptic problems, for example [40, 41], that available for nonlinear problems remains limited, even more so in viscoplasticity; see [31] and references therein. Therefore, here, we simply examine the global responses obtained with different values of h=W0 , where W0 denotes the width of the gage section, Figure 8. In addition, in softening materials, the loss of ellipticity of the incremental boundary-value problem leads to ill-posedness. In numerical solutions, the latter manifests through a pathological mesh sensitivity even if rate-dependence is known to decrease it [42]. In this section, therefore, we also investigate the effect of mesh density on the shear band pattern formation. It is expected that the large-strain hardening will have a ‘stabilizing’ effect. To increase the propensity for shear banding, we use s2 =s1 D 0.61 with s1 =s0 D 1. Figure 10 summarizes specimen-level responses obtained with all four meshes over a large strain window. It is found that the first two meshes are too coarse to provide a good representation of the post-yield drop. However, the response obtained using the h=W0 D 0.025 mesh is very close to that of the finest mesh (which defines the reference solution). This mesh size is then considered to be sufficient to obtain a converged solution. To understand the aforementioned trends examine the distributions of effective plastic strain at two levels of overall deformation. Figure 11 shows the shear band patterns that develop after the Copyright © 2013 John Wiley & Sons, Ltd.

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(a)

(b)

(c)

(d)

Figure 9. Meshes used for the convergence study of the plane strain tensile specimens. (a) 295, (b) 1180, (c) 4720, and (d) 18880 elements. 140

Stress σ (MPa)

120 100 80 60 295 elements

40

1180 elements 4720 elements

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18880 elements

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Strain (-ε)

Figure 10. Computed curves of stress  versus logarithmic strain ", as in (28), for the plane strain tensile specimens for various mesh densities. Also, s2=s1 D 0.61 and s1=s0 D 1.

yield drop (U=L0 D 0.088). The first two meshes pick up the formation of two crossed shear bands, whereas the other two pick up two sets of crossed shear bands. The band width clearly depends on the mesh size, and strain localization is more intense in the finest mesh. The maximum values of N are about 0.92 and 1.0 in the h=W0 D 0.025 and h=W0 D 0.0125 meshes, respectively. In dynamic problems, viscosity introduces a length scale into the problem; and hence, the shear band width is limited [42, 43]. On the other hand, in quasi-static problems, rate-dependence does not make the solution fully regular in the presence of continuous softening. However, for polymers with the peculiar up-down-up behavior, the shear bands do not remain stationary and propagate instead, somewhat reminiscent of Lüders or Portevin-Le-Châtelier bands. Copyright © 2013 John Wiley & Sons, Ltd.

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(b)

(c)

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(d)

Figure 11. Contours of effective plastic strain N of the plane strain tensile specimens with the grip parts for various mesh densities at U=L0 D 0.088. UP =L0 D 1.44=s, and T D 25ı C: (a) 295, (b) 1180, (c) 4720, and (d) 18880 elements.

In fact, upon continued deformation, the shear bands that accompany the early stages of neck propagation disappear gradually (Figure 12). During neck stabilization, the magnitude of the difference between the maximum plastic strains in the last two meshes decreases significantly (compare 2.13 versus 2.17) although some differences in the deformation pattern and the resulting surface roughness remain due to history effects. 5.2.2. Fully 3D computations. All subsequent calculations were carried out using h=W0 D 0.025, which was found to be sufficient for a converged solution as per the earlier considerations. To reduce further the cost of 3D calculations, we conduct two series of simulations exploiting partial symmetries. In the first set, we use symmetry with respect to the mid-thickness plane x3 D 0 so that one half of the specimen is modeled. The corresponding mesh is shown in Figure 13(a). A displacement rate UP is imposed to the top surface in the x2 direction. The lateral movement in the x1 and x3 directions is not constrained. The remaining boundary conditions are       l0 l0 l0 T1 x1 , , x3 D 0, uP 2 x1 , , x3 D UP , T3 x1 , , x3 D 0, (29) 2 2 2 where Ti denote traction components. No axial displacement is applied to the bottom surface so that       l0 l0 l0 T1 x1 ,  , x3 D 0, uP 2 x1 ,  , x3 D 0, T3 x1 ,  , x3 D 0. (30) 2 2 2 Copyright © 2013 John Wiley & Sons, Ltd.

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(a)

(b)

(c)

(d)

Figure 12. Contours of effective plastic strain N of the plane strain tensile specimens with the grip parts for various mesh densities at U=L0 D 0.5. UP =L0 D 1.44=s and T D 25ı C: (a) 295, (b) 1180, (c) 4720, and (d) 18880 elements.

All lateral surfaces of the specimen remain traction free and the following symmetry conditions were used T1 .x1 , x2 , 0/ D 0, T2 .x1 , x2 , 0/ D 0, uP 3 .x1 , x2 , 0/ D 0.

(31)

Also, one node at the left bottom corner was fixed in the width direction to prevent rigid-body motion. In the second set of 3D calculations, we use symmetry with respect to the mid-width plane x1 D 0 in addition to the earlier symmetry so that one quarter of the specimen is analyzed. Other boundary conditions are similar and shall not be repeated. The analysis of the half-specimen allows in principle to explore modes of localization that are fully unsymmetric in the x1 –x2 plane, as evidenced in experiments, for example, [44]. The mesh shown in Figure 13(a) consists of 28320 eight-noded brick elements (C3D8). To capture localized deformations in the necking area, 40 elements were used in the width direction and three elements in the thickness direction. For the quarter specimen, the number of elements was half the number quoted earlier. Distributions of the effective plastic strain N at incipient necking and at some arbitrary stage of neck propagation are depicted in Figure 13(b,c), respectively. In fact, right at the maximum load, two necks develop across the thickness direction (see Movie S1 online). Subsequently, they form Copyright © 2013 John Wiley & Sons, Ltd.

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(d)

Figure 13. Contours of effective plastic strain N of the half specimen for PMMA 3D tension (Standard Specimen) for UP =L0 D 1.44=s and T D 25ı C: (a) Mesh, (b) U=L0 D 0.140, (c) U=L0 D 1.0, and (d) quarter specimen U=L0 D 1.0.

in the width direction. In the x1 –x2 plane, they are mediated by two pairs of crossed shear bands. Only one of the two necks, the bottom one, propagates, whereas the other remains stationary until it gets absorbed by the propagating neck. The symmetry breaking here is due to small round-off errors in mesh generation. In experiments, such instances are reported due to material inhomogeneities or geometric imperfections. For comparison, the N contours in the quarter specimen are provided in Figure 13(d). Because of the inherent symmetries, the pattern of shear bands in the quarter specimen (omitted for brevity) is almost identical to that in the half specimen. Specimen-level measures of stress and strain may be defined in a manner similar to compression, that is, using Equation (28). Figure 14 shows the computed curves of  versus ". Despite small differences in the patterns of shear bands during neck propagation, the stress–strain curves are essentially the same for the half and quarter specimens. An additional parameter sensitivity analysis is conducted using two-dimensional specimens assuming plane strain conditions. The objective here is to investigate the effect of a stronger amount of intrinsic softening (as affected by the ratio s2 =s1 ) on the onset and propagation of instability. The greater amount of softening may represent PMMA under different physical aging conditions or, alternatively, a different material such as polystyrene. More specifically, we used s2 =s1 D 0.61 with Copyright © 2013 John Wiley & Sons, Ltd.

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Stress σ (MPa)

80

60 Quarter Specimen Half Specimen

40

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Strain (-ε)

Figure 14. Computed curves of stress  versus strain ", as in (28), for the quarter and half specimens in 3D tension of PMMA.

(a)

(b)

(c)

(d)

Figure 15. Contours of effective plastic strain N for the full specimen under plane strain tension for UP =L0 D 1.44=s and T D 25ı C: (a) U=L0 D 0.077, (b) U=L0 D 0.085, (c) U=L0 D 0.497, and (d) U=L0 D 1.0.

s1 =s0 D 1 so as to have a sharp yield point. The full specimen is modeled in order to capture unsymmetric localization modes. The total number of CPE4 plane strain elements was 9440, keeping the same in-plane mesh density as for the 3D problem in Figure 13(a). Copyright © 2013 John Wiley & Sons, Ltd.

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Figure 15 shows the distributions of effective plastic strain N at different stages of necking. Unlike the previous case, deformation is not symmetric from the outset. There is no artificial defect in the simulation. Necking takes place because the end grip sections are modeled to trigger instability. The initially coarse, somewhat symmetric shear bands become finer with subsequent straining and mediate the early stages of neck propagation (Figure 15(b)). When the simulation was terminated at U=L0 D 1.0, the drawn polymer has an irregular surface. Such surface roughness is a typical problem encountered in thin film drawing. Complementary information can be visualized in terms of effective strain rate in Figure 16. Bands of peak values of PN represent the transition fronts of the propagating bands. Throughout the deformation, not only there is asymmetry of the neck with respect to the x2 axis (left to right), there also is asymmetry with respect to the x1 axis (top vs bottom). Note that the fact the strain rate localizes does not necessarily entail that the plastic strain localizes. Figure 17 shows the computed curves of stress versus strain of the full and half specimens. It is remarkable that the two curves almost fall on top of each other in spite of the asymmetry in the deformation patterns of the full specimen. None of the aforementioned shear band simulations have exhibited an oscillating local response when shear strain components became large. The maximum shear strain "12 attained in the most severe shear bands was about 0.4, and a typical value was about 0.03. These values are smaller than those encountered in the compression problem near the die-specimen interface because the shear

(a)

(b)

(c)

(d)

Figure 16. Contours of effective plastic strain rate NP for the full specimen under plane strain tension for UP =L0 D 1.44=s and T D 25ı C: (a) U=L0 D 0.077, (b) U=L0 D 0.085, (c) U=L0 D 0.497, and (d) U=L0 D 1.0. Copyright © 2013 John Wiley & Sons, Ltd.

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140 120

Stress σ (MPa)

100 80 60 Half Specimen

40

Full Specimen

20 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Strain (-ε)

Figure 17. Computed curves of stress  versus logarithmic strain ", as in (28), for the full and half specimens in plane strain tension of the material with strong softening, s2=s1 D 0.61 and s1=s0 D 1.

bands were not persistent. Chowdhury et al. [18] reported much more intense shear bands under plane strain compression where values of "12 larger than 2.0 were not uncommon. Such values however are believed to be still lower than strain levels at which stress oscillations can occur on the basis of Figure 6.

6. CONCLUSIONS  A three-dimensional finite-element implementation of a macromolecular viscoplastic polymer

model was developed within a co-rotational formulation of the constitutive equations. To improve accuracy, the explicit update of the back stress used in previous implementations was avoided. Convergence of the time integration algorithm and the global solution was improved by accounting for the back stress dependence of the consistent tangent matrix.  Because the Jaumann rate was used in the hypoelastic law as well as the back stress evolution equation, an analysis of sensitivity to pathological stress oscillations was carried out under simple shear loading. It was shown that oscillations are unlikely to occur for common values of material parameters and achievable strains. However, care should be taken in applying the method to very soft polymers or near the glass transition for very large shears.  Fully three-dimensional illustrative boundary-value problem solutions were obtained for the inhomogeneous deformation of polymers under compressive and tensile loadings. The analyses included a convergence analysis and demonstrate the robustness of the implementation.

APPENDIX A: CO-ROTATIONAL FORMULATION OF MACROMOLECULAR MODEL This formulation of the macromolecular model is derived specially for a user material subroutine for ABAQUS [45]. Rotated tensors are indicated by a hat and follow standard transformation rules T AO D R   A  R  I

BO ijpq D R ki R lj R mp R nq Bklmn .

(A.1)

The constitutive equations are then recast in the rotated configuration in the usual way. For instance, the hypoelastic constitutive law written in the current configuration in Equation (1) is transformed to O e D C 1 W PO D Copyright © 2013 John Wiley & Sons, Ltd.

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O D C in the case of isotropic elasticity, which is assumed. in the rotated configuration. Note that C Also, the objective rate of  is replaced with the time rate of change of the rotated stress. This is the main advantage of utilizing the co-rotational formulation in that the constitutive updating in the rotated configuration is carried out just like for a small-strain implementation without resorting to any objective rate in the calculations. Obviously, the definition of R  depends on the type of objective rate used in the formulation. The flow rule (3) is rewritten as O p D OPN 3 O 0d , D 2O e

(A.3)

where hats are kept on O e and ONP to emphasize that they are calculated on the basis of the rotated driving stress and plastic part of the rate of deformation, respectively. Of particular importance to the formulation is the back stress evolution Equation (9) that transforms as P O WD O p. bO D R

(A.4)

Formally, (A.4) is obtained in a similar way as the hypoelastic law mentioned earlier, except that O ¤ R because of induced anisotropy. here R It is worth noting that evolution Equation (9) was derived in [14] by utilizing specifically the Jaumann rate. By way of contrast, various stress rates could be used in (1). Therefore, consistency with the constraint set by (9) requires that the Jaumann rate be used throughout. This is consistent with the choice of R  as that resulting from the polar decomposition of the incremental deformation gradient F . Finally, we also state for completeness the differential equation governing the evolution of state variable s as 0 1 s  A OPN sP D h @1  (A.5) s T , OPN ss

corresponding to (6) for the original macromolecular model or as 0 0 1 1   s s A OPN C H2 ON @1   A OPN sP D H1 ON @1   O O s1 T , PN s2 T , PN

(A.6)

corresponding to (7) for the modified macromolecular model. APPENDIX B: JACOBIAN MATRIX The components of the Jacobian matrix related to the back stress bO are given as follows, with A RA,B D @R @B R bO 0 ,O 0 D

1 O W C 1 J WR t

(B.1)

1 J t

(B.2)

R bO 0 ,bO 0 D

R bO 0 ,bO D 0

(B.3)

m

R bO 0 ,O D m

Copyright © 2013 John Wiley & Sons, Ltd.

1 O W C 1 W I J WR t

(B.4)

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R bO 0 ,s D 0 RbOm ,O 0 D

1 1 O W C 1 I WR 3 t

RbO

O0 m ,b

D0

(B.6) (B.7)

1 1 3 t

(B.8)

1 1 O W C 1 W I I WR 3 t

(B.9)

RbOm ,bOm D RbOm ,O m D

(B.5)

RbOm ,s D 0

(B.10)

where 0 is the second-rank zero tensor. APPENDIX C: SUPPLEMENTARY MATERIAL Movie S1: Neck formation and propagation in 3D tension of material with mild small-strain softening (PMMA) pulled at UP =L0 D 1.44=s and T D 25ı C. Shown are contours of effective plastic strain N . ACKNOWLEDGEMENTS

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Int. J. Numer. Meth. Engng 2013; 94:895–919 DOI: 10.1002/nme