Optimal energy scaling for a shear experiment in single-crystal plasticity with cross-hardening

Anguige, Keith; Dondl, Patrick

doi:10.1007/s00033-013-0379-0

Cited by 4 publications

(30 citation statements)

References 19 publications

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“…Note that, for the simple case where a slip plane connects free surfaces, one can achieve E rel = 0 with a simple, relaxed shear (see, for example, the L > 2 result from [19])-hence our theorem shows that the required envelope is in fact zero in this case. Remark 4.3.…”

Section: The Full Relaxation Statementmentioning

confidence: 81%

“…The main open question remaining is whether our (at least partially) relaxed model does in fact admit a minimizer. For some simple-but experimentally relevant-situations, it is easy to show that a minimizer for our relaxed model exists, since a test function with vanishing energy can be explicitly constructed [19]. More generally, it is possible to see that fine oscillations between potential wells of our non-convex relaxed slip condition are penalized by the convex curl-term in the relaxed energy.…”

Section: Conclusion and Open Problemsmentioning

confidence: 95%

“…Relaxation of the energy functional (1.1) was only briefly alluded to in [19], in the course of proving an energy upper bound for a shear experiment on a B2 crystal. Here, we wish to derive the relaxed model rigorously for a class of slip systems which includes the case of B2 symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Relaxation of the single-slip condition in strain-gradient plasticity

Anguige

Dondl

2014

Proc. R. Soc. A.

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We consider the variational formulation of both geometrically linear and geometrically nonlinear elasto-plasticity subject to a class of hard singleslip conditions. Such side conditions typically render the associated boundary-value problems non-convex. We show that, for a large class of non-smooth plastic distortions, a given single-slip condition (specification of Burgers vectors) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. The relaxed model can be thought of as an aid to simulating macroscopic plastic behaviour without the need to resolve arbitrarily fine spatial scales.

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Section: The Full Relaxation Statementmentioning

confidence: 81%

Section: Conclusion and Open Problemsmentioning

confidence: 95%

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Relaxation of the single-slip condition in strain-gradient plasticity

Anguige

Dondl

2014

Proc. R. Soc. A.

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“…[DRMD15] in this book (see also [AD14b,AD14a]). For this comparison consider first a single dislocation in the geometrically linear setting.…”

Section: Introductionmentioning

confidence: 96%

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Müller

Scardia

Zeppieri

2015

Analysis and Computation of Microstructure in Finite Plasticity

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Abstract. This article gives a short description and a slight refinement of recent work [MSZ15], [SZ12] on the derivation of gradient plasticity models from discrete dislocations models. We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in the regime where the number of dislocation N ε is of the order log 1 ε (where ε is the ratio of the lattice spacing and the macroscopic dimensions of the body) the contributions of the self-energy of the dislocations and their interaction energy balance. Upon suitable rescaling one obtains a continuum limit which contains an elastic energy term and a term which depends on the homogenized dislocation density. The main novelty is that our model allows for microscopic energies which are not quadratic and reflect the invariance under rotations. A key mathematical ingredient is a rigidity estimate in the presence of dislocations which combines the nonlinear Korn inequality of IntroductionClassical theories of plasticity are scale independent. Nonetheless experiments show a notable size dependence of plastic behaviour in the micron and submicron range.

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“…Here, we continue our previous investigations , which focused on optimal energy scalings and relaxation of the single‐slip condition to a (still non‐convex) single‐plane condition, by looking at the existence question for a class of incremental minimisation problems which arise in the way described above. Our main goal is to extend the existence result of Mielke and Müller for finite, multiplicative strain‐gradient elasto‐plasticity to the single‐crystal, single‐plane case.…”

Section: Introduction and Main Resultsmentioning

confidence: 82%

On the existence of minimisers for strain‐gradient single‐crystal plasticity

2017

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We prove the existence of minimisers for a family of models related to the single‐slip‐to‐single‐plane relaxation of single‐crystal, strain‐gradient elastoplasticity with Lp‐hardening penalty. In these relaxed models, where only one slip‐plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower‐semicontinuous along bounded‐energy sequences which satisfy the single‐plane condition, meaning precisely that this side condition should be preserved in the weak Lp‐limit. This is done with the aid of an ‘exclusion’ lemma of Conti & Ortiz, which essentially allows one to put a lower bound on the dislocation energy at interfaces of (single‐plane) slip patches, thus precluding fine phase‐mixing in the limit. Furthermore, using div‐curl techniques in the spirit of Mielke & Müller, we are able to show that the usual multiplicative decomposition of the deformation gradient into plastic and elastic parts interacts with weak convergence and the single‐plane constraint in such a way as to guarantee lower‐semicontinuity of the (polyconvex) elastic energy, and hence the total elasto‐plastic energy, given sufficient (p>2) hardening, thus delivering the desired result.

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Optimal energy scaling for a shear experiment in single-crystal plasticity with cross-hardening

Cited by 4 publications

References 19 publications

Relaxation of the single-slip condition in strain-gradient plasticity

Relaxation of the single-slip condition in strain-gradient plasticity

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

On the existence of minimisers for strain‐gradient single‐crystal plasticity

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