Computational Materials Science 64 (2012) 96–100

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Two models for gradient inelasticity based on non-convex energy

Benjamin Klusemann a,⇑, Swantje Bargmann b, Bob Svendsen a

a Material Mechanics, RWTH Aachen University, Aachen, Germanyb Institute of Mechanics, TU Dortmund University, Dortmund, Germany

a r t i c l e i n f o

Article history:Received 5 October 2011Received in revised form 18 January 2012Accepted 27 January 2012Available online 19 February 2012

Keywords:Gradient plasticityNon-convexityAlgorithmic variationalDual mixedImplicitExplicit

0927-0256/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.commatsci.2012.01.037

⇑ Corresponding author. Tel.: +49 241 80 25012.E-mail address: benjamin.klusemann@rwth-aache

a b s t r a c t

The formulation of gradient inelasticity models has generally been focused on the effects of additionalsize-dependent hardening on the material behavior. Recently, the formulation of such models has takena step in the direction of phase-field-like modeling by considering non-convex contributions to energystorage in the material (e.g., [1]). In the current work, two such models with non-convexity are comparedwith each other, depending in particular on how the inelastic deformation and (excess) dislocation den-sity are modeled. For simplicity, these comparisons are carried out for infinitesimal deformation in a one-dimensional setting. In the first model, the corresponding displacement u and inelastic local deformationc are modeled as global via weak field relations, and the dislocation density q is modeled as local via astrong field relation. In the second model, u and q are modeled as global, and c as local, in this sense.As it turns out, both models generally predict the same inhomogeneous deformation and material behav-ior in the bulk. Near the boundaries, however, differences arise which are due to the model-dependentrepresentation of the boundary conditions.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Although now a basic tenet of modern material modeling andmaterial science, the idea that material properties and mechanicalbehavior are determined by the underlying microstructure is stillan issue of research in detail (e.g., [2,3]). In particular, with regardsto model development, the focus in recent years has been on theformulation of models encompassing multiple length- and/or time-scales. For the case of the inelastic behavior of single- and polycrys-talline metals, for example, two recent developments in thisdirection are the application of microscopic phase field methods[4] at the single dislocation level and the development of gradientcrystal plasticity (e.g., [5–12]) at the glide-system level in such sys-tems. A prominent aspect of phase field models is the modeling ofenergetic microstructure interaction via non-convex contributionsto the free energy of the system. Recently, gradient plasticity hasbeen extended in this direction in [1], resulting in a type ofcoarse-grained phase field form for gradient plasticity (e.g.,[13,14,4]). In particular, [1] models the inelastic local deformationc as global via a weak field relation, and the excess dislocation den-sity q as local via a strong field relation. Examples of analogous mod-els can be found in the literature [15–19]. A second class of models(e.g., [20,8,21,10,7,22,23]) treat q as global, and c as local, in thissense (the latter representing the usual flow rule). The purpose ofthe current work is to compare both of these models in the context

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n.de (B. Klusemann).

of their predictions for stress relaxation and the development ofdeformation inhomogeneity. As will be discussed below, for theboundary conditions considered, complete agreement is obtainedin the bulk; on the other hand, small differences arise near theboundaries. In Section 2 basic model aspects are reviewed and theformulations of the two models are given. This is followed by a com-parison of results in Section 3 and a brief summary in Section 4.

2. Model formulation

The model formulation is carried out in the framework of con-tinuum thermodynamics (e.g., [24]) and rate variational methods(e.g., [13,12]) which are most natural for history-dependent behav-ior. For simplicity, attention is restricted to isothermal, supply-free,quasi-static, infinitesimal deformation processes as well to one-dimension. This can be associated for example with the classicalcase of a one-dimensional bar or alternatively with a prototypemodel for glide or deformation on a single glide system. Let B rep-resent such a bar of length l0 with boundary @B. Again, besides thedisplacement field u, the principle global unknown is either theinelastic local deformation c or the dislocation density q, depend-ing on the model. To be precise, a global quantity is modeled via aweak field relation, whereas a local quantity is determined by astrong field relation or an ordinary differential equation in thesense of an internal variable.

In the current rate-dependent thermodynamic context, dissipa-tive/kinetic processes are represented by a simple rate-dependentpower-law form

γ [%]

0 0.5 1 1.5 2-0.5

0

0.5

1

Fig. 1. Characterization of the non-convexity of wc. Solid line: wc; dot-dashed line:@cwc; dashed line: @c@cwc; dotted line: c@cwc � wc. All quantities are scaled by theirvalues for c = 2%.

B. Klusemann et al. / Computational Materials Science 64 (2012) 96–100 97

v ¼ 1m0 þ 1

rD0 _c0j _c= _c0jm0þ1; ð1Þ

for the dissipation potential. Here, _c0 denotes the material referencerate, rD0 the drag stress, and m0 is the rate sensitivity. Since thisform of v is non-negative and convex in _c, it satisfies the dissipationprinciple (e.g., [24], Chapter 9) sufficiently. This form tacitly as-sumes zero activation energy or stress for initiation of inelasticdeformation. Since the current work is concerned with purely qual-itative effects, m0 = 1 is chosen for simplicity, analogous for exam-ple to discrete dislocation modeling as based on linear drag. Aswill be seen below, in the case that c is modeled as global, this re-sults in a Ginzburg–Landau-/phase-field-like relation for c. Otherchoices for m0, including those relevant to crystal plasticity, forexample, would result in a non-Ginzburg–Landau form. Other val-ues of m0 influence the strength, but not the qualitative effect, ofrate-dependence on the material behavior and microstructuredevelopment (e.g., [1]).

Turning next to energetic processes, these are represented inthe current context by the free energy density w. In the case ofnon-convex gradient inelasticity, this consists of elastic, non-con-vex wc, and gradient parts, i.e.

w ¼ 12

E0jru� cj2 þ wc þ12

aE0 ‘2E0jrcj2: ð2Þ

Here, E0 is the Young’s modulus, aE0 is the energetic gradient hard-ening modulus, and ‘E0 represents the corresponding materiallengthscale. Taking our cue from the phase-field world, and for easeof comparison with the work of [1], attention is focused here on theLandau-Devonshire form

wc ¼ C1c4 þ C2c3 þ C3c2: ð3Þ

This is a classical form often used in the context of phase fieldsimulations (often with even exponents 2, 4, 6) (e.g., [25]), whereit describes the configurational energy of a two-phase material.In the current context, the phases involved are regions in the mate-rial of low or high inelasticity, i.e., deformation. As usual, non-con-vexity is realized by choosing corresponding values for C1, C2, C3;these along with the other material parameter values from [1] usedin this study are given in Table 1. Given these values for C1, C2, C3,one can characterize the non-convexity of wc graphically as shownin Fig. 1. For example, the convexity criterion cocwc � wc (Fig. 1,dashed curve) changes from positive to negative when wc changesfrom convex to concave. At the spinodal points, @c@cwc changesfrom positive to negative. Points of negative curvature are unstablewith respect to small fluctuations leading to instability of homoge-neous deformation and the onset of deformation inhomogeneity.

Given the above constitutive relations, application of contin-uum thermodynamic methods (e.g., [24,13]) yields in the system

0 ¼ d _uf; 0 ¼ d _cfþ @ _cv; ð4Þ

of evolution-field relations for the continuum fields u and c in termsof the variational derivative dzf = @zf � div @rzf and the energy stor-age rate density

f ¼ @ruw � r _uþ @cw _cþ @rcw � r _c; ð5Þ

induced by w in the form (2). Note that (4)1 represents the quasi-static momentum balance, and (4)2 is the flow rule. To these weadd the purely kinematic evolution relation

Table 1Material and process parameter values used in the simulations [1].

rD0 (MPa) aE0 (GPa) E0 (GPa) lE0/l0

35 14.7 210 10�1

C1 C2 C3 _�0= _c0

1.525 � 108 �5.2 � 106 5.0 � 104 4.0 � 10�3

0 ¼ _qþr _c; ð6Þ

for q due to Ashby [26].On the basis of these relations, one can formulate different

models for the initial boundary-value problem (IBVP) to solve foru, c and q. To this end, for simplicity, attention is restricted tothe kinematic boundary conditions

_uj@B given; _cj@B ¼ 0; ð7Þ

here. In particular, the latter are generally referred to in the litera-ture as ‘‘micro-hard’’ boundary conditions. Given these, the firstmodel for the IBVP to be considered here is based on weak modelrelations for u and c derived from the weak stationarity of the ratefunctional

R ¼Z

Br dv ¼

ZBðfþ vÞdv; ð8Þ

(see [13]) with respect to admissible variations d _u and d _c, i.e., suchthat d _uj@B ¼ 0 and d _cj@B ¼ 0 for u and c, respectively. These togetherwith the local relation (6) for q compose Model 1 in Table 2. A sec-ond model for the IBVP may be based instead on the strong sta-tionarity condition

rD0 _c= _c0 ¼ E0 ðru� cÞ � @cwc � aE0 ‘2E0 rq; ð9Þ

of R with respect to _c, i.e., (4)2, and the weak form of (6) for _q con-sistent with the boundary condition (7)2; this is summarized asModel 2 in Table 2. Differences between the two models includefor example the fact that q is modeled as C0 in Model 1 and C1 inModel 2. In addition, the boundary condition (7)2 is realized differ-ently in the two models. In particular, in Model 1, _c is directly set tozero at the boundary, whereas in Model 2, _cj@B ¼ 0 is only weaklyenforced via omission of the corresponding boundary term in theweak field relation for _q. These difference results in differences inthe corresponding solutions to be discussed below. Both modelshave been implemented numerically via forward and backwardEuler time integration in the context of the finite element method.

Table 2Relations for two models for the IBVP.

Model 1 Model 2

0 ¼R

B @r _uf � rd _udv 0 ¼R

B @r _uf � rd _udv0 ¼

RBf@ _crd _cþ @r _cf � rd _cgdv 0 ¼ @ _cvþ d _cf

0 ¼ _qþr _c 0 ¼R

Bf _q d _q� _c rd _qgdv

300

98 B. Klusemann et al. / Computational Materials Science 64 (2012) 96–100

For more details about the numerical implementation the interest-ing reader is referred to [1] for Model 1 and to [21] for Model 2. Allsimulation have been carried out with a constant time size for max-imum possible numerical compatibility.

global strain [%]

0 0.5 1 1.5 2

glob

alst

ress

[MPa

]

0

100

200

Fig. 3. Comparison of the global stress–strain behavior from Model 1 and forward-Euler time-integration for different time step sizes. Solid line: implicit,_�0 Dt ¼ 4� 10�5 :¼ rref ; dashed line: explicit, _�0 Dt=rref ¼ 10; dotted line: explicit,_�0 Dt=rref ¼ 1; dot-dashed line: _�0 Dt=rref ¼ 10�1.

0 0.5 1 1.5 20

50

100

150

200

250

Fig. 4. Local stress–strain behavior from Model 1 at three points along the bar. Solidline: x/l0 = 0.1; dashed line: x/l0 = 0.25; dotted line: x/l0 = 0.5.

3. Results

We now turn to a comparison of results obtained with the mod-els just discussed. All simulation results to follow are obtained for aone-dimensional bar of initial length l0 subject to monotonicextension via displacement boundary conditions. Convergednumerical solutions in the context of implicit time integrationand the finite-element method are obtained for a discretizationwith 100 elements. Unless otherwise stated, all results are basedon backward Euler time integration. In order to focus on non-con-vex effects like transition to inhomogeneous deformation andstress relaxation (e.g., [1]), quasi-static loading is assumedthroughout (see Table 1). Consider first the global stress–strainbehavior of the bar subject to monotonic tension shown in Fig. 2.For both models, the stress shows first an increase and than a sharpfall with a following stress plateau. This is qualitatively similar tophenomena observed experimentally such as upper and loweryield points, or Lüders band development (e.g., [27,28]). The pointof sharp stress reduction, as well as the end of the stress plateau, inFig. 2, are correlated with the spinodal points of wc (e.g., [1]).

Turning next to the issue of time-integration, global stress–strain results for Model 1 obtained via forward-Euler time integra-tion are compared in Fig. 3 with those from Fig. 2 obtained viabackward-Euler integration. As can be seen here, converged resultsshowing stress relaxation and inhomogeneous deformation devel-opment basically in quantitative agreement with those of the im-plicit case are obtained in the explicit case for a time step size_�0 Dt=rref ¼ 10�1. This is one order of magnitude less than the timestep rref :¼ _�0 Dt ¼ 4� 10�5 used in the implicit case.

Consider next the local stress–strain behavior at three represen-tative points in the bar, corresponding to the three curves shown inFig. 4. Since the results are the same for both models, only the re-sults for Model 1 are shown. Whereas the local stress correspondsto the global stress at each point, the results in Fig. 4 demonstratethat the transition to inhomogeneous deformation results in localdeformation states which are different from the homogenized oraveraged state of the bar as a whole. For example, near the bound-ary (x/l0 = 0.1), the boundary condition _cj@B ¼ 0 dominates, result-ing in nearly elastic stress relaxation. As one moves toward themiddle (x/l0 = 0.5), the increase in c upon transition becomes so

global strain [%]

glob

alst

ress

[MPa

]

0 0.5 1 1.5 20

50

100

150

200

250

Fig. 2. Global stress–strain behavior fir Model 1 (dashed line) and Model 2 (solidline).

significant that it even exceeds the externally applied deformation.Note that the behavior is symmetric with respect to the middle ofthe bar (dotted curve).

Consider next the distribution of c over the bar as shown inFig. 5 for Model 1 and in Fig. 6 for Model 2. As expected, the ‘‘mi-cro-hard’’ boundary conditions _cj@B ¼ 0 are satisfied in both casesby the development of boundary layers at the ends of the bar.The initially homogeneous c becomes unstable as the stress maxi-mum is reached (Fig. 4), transforms to inhomogeneous (0.49%) andrelaxes the stress (Fig. 4). Further inhomogeneous deformationinhibits stress increase until sufficient hardening has occurred torender inhomogeneous deformation energetically unfavorableover further boundary layer development.

Lastly, consider the profiles for q from Model 1 in Fig. 7 andfrom Model 2 in Fig. 8. Typical of this distribution is its ‘‘dipole’’-like character. Further, the transition to inhomogeneous deforma-tion results in the development of a density profile suggestingwall-like dislocation concentrations separated by regions of littleor no dislocation concentration reminiscent of dislocation cells.As loading proceeds, these dislocation ’’walls’’ disappear or are ‘‘ab-sorbed’’ into the corresponding boundary layers.

A slight difference in boundary layer development predicted bythe two models with respect to c (Figs. 5 and 6) is even more

γ[%

]

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Fig. 5. Spatial variation of c along the bar from Model 1 for different global strainlevels. Solid line: 0.24%; dot-dashed line: 0.49%; dotted line: 0.59%; dashed line:1.2%; thick-dotted line: 2%.

γ[%

]

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Fig. 6. Spatial variation of c along the bar from Model 2 for different global strainlevels. Solid line: 0.24%; dot-dashed line: 0.49%; dotted line: 0.59%; dashed line:1.2%; thick-dotted line: 2%.

-0.2

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1

Fig. 7. Spatial variation of q along the bar from Model 1 for different global strainlevels. Solid line: 0.24%; dot-dashed line: 0.49%; dotted line: 0.59%; dashed line:1.2%; thick-dotted line: 2%.

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

Fig. 8. Spatial variation of q along the bar from Model 2 for different global strainlevels. Solid line: 0.24%; dot-dashed line: 0.49%; dotted line: 0.59%; dashed line:1.2%; thick-dotted line: 2%.

B. Klusemann et al. / Computational Materials Science 64 (2012) 96–100 99

evident with respect to q (Figs. 7 and 8, respectively). In particular,Model 2 results in slightly thinner boundary layers for c at a given

global deformation state (Fig. 6). These differences are magnifiedwith respect to q and increase with increasing global deformation.In particular, Model 2 predicts a higher value for the dislocationdensity at the boundary and also shows a much smoother transi-tion compared to the results for Model 1. This is due to the fact thatthe boundary condition _cj@B ¼ 0 is modeled differently in the twomodels. In particular, in Model 1, this boundary condition isfulfilled explicitly or strongly. On the other hand, in Model 2, it isfulfilled only implicitly or weakly in the sense that it leads to elim-ination of a boundary integral term in the corresponding weak fieldrelation.

4. Summary

The current work focused on the analysis and comparison oftwo different models for non-convex gradient inelasticity. For con-creteness and the sake of comparing results from the two modelsin a quantitative fashion, the simple Landau-Devonshire form forthe non-convex energy used by [1] had been adopted. The firstmodel considered here is based on the continuum displacementu and the inelastic deformation c as field variables for which weakfield relations are formulated and solved. In Model 1, the disloca-tion density q = �rc is calculated directly from c locally. In Model2, q, rather than c, is treated as additional continuum field forwhich a weak field relation is formulated and solved. In this case,the flow rule for c, which depends in particular on rq related tolengthscale-dependent energetic hardening, is solved locally, suchthat c is treated in this case as a local internal variable. In thecurrent one-dimensional context, it has been shown that most ofthe results from both models for the case of micro-hard boundaryconditions agree quite well with each other as well as with compa-rable results in the literature (e.g., [1]). This includes excellentagreement between implementations of these two models basedon implicit and explicit time integration. Of particular note is thefact that the instability of homogeneous deformation and the tran-sition to inhomogeneous deformation (i.e., inelastic microstructuredevelopment) as a result of non-convexity is predicted equiva-lently by both models. In essence, the only basic differencebetween the two models was found in the boundary-layer devel-opment, something which is related to the way the same physicalboundary condition is fulfilled in each formulation. In work inprogress, this and other aspects of the modeling of boundary con-ditions, as well as other types of boundary conditions (e.g., [29]),are being systematically investigated and will be reported on in fu-ture work.

100 B. Klusemann et al. / Computational Materials Science 64 (2012) 96–100

Acknowledgment

The authors gratefully acknowledge the financial support of theDeutsche Forschungsgemeinschaft (DFG) within the CollaborativeResearch Center (SFB) 761 ‘‘Steel - ab initio.’’

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