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

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

Section: Introductionmentioning

confidence: 94%

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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“…Even with such an additional regularization (certainly also without), one relaxation of the model that can immediately be performed is that of considering a single-plane side condition instead of the single-slip condition above. See [AD14] for more information. After this relaxation, the plastic dissipation can be written as…”

Section: Nj I=1mentioning

confidence: 98%

“…We note that these slip systems are naturally compatible without a lattice rotationthey are coplanar slip systems. Following [AD14], we thus consider coplanar slip systems to be lumped into one.…”

Section: Energy Minimizing Microstructurementioning

confidence: 99%

Microstructure in Plasticity, a Comparison between Theory and Experiment

Dmitrieva

Raabe

Müller³

et al. 2015

Analysis and Computation of Microstructure in Finite Plasticity

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Abstract. We review aspects of pattern formation in plastically deformed single crystals, in particular as described in the investigation of a copper single crystal shear experiment in [DDMR09]. In this experiment, the specimen showed a band-like microstructure consisting of alternating crystal orientations. Such a formation of microstructure is often linked to a lack of convexity in the free energy describing the system. The specific parameters of the observed bands, namely the relative crystal orientation as well as the normal direction of the band layering, are thus compared to the predictions of the theory of kinematically compatible microstructure oscillating between low-energy states of the non-convex energy. We conclude that this theory is suitable to describe the experimentally observed band-like structure. Furthermore, we link these findings to the models used in studies of relaxation and evolution of microstructure.

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“…For a mathematical derivation of this expression from microscopic models, see for example [GM06, GLP10, CGM11] in a geometrically linear setting and [SZ12, MSZ14, MSZ15] in a geometrically nonlinear context. The macroscopic effect of this regularization term was addressed for example in [CO05,AD14,AD15] in various simplified models. In the current geometrically nonlinear single-slip context, a first result was obtained in [Sch14].…”

Section: Higher-order Regularizationsmentioning

confidence: 99%

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

Conti

Dolzmann

Kreisbeck

2015

Analysis and Computation of Microstructure in Finite Plasticity

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Abstract. Slip processes are soft modes of deformation, characteristic of a variety of layered materials. The layers can be at the atomic scale, as in the plastic deformation of crystalline lattices, or on a macroscopic scale, as in stacks of cards or sheets of paper and geological strata. The characteristic deformation processes involve sliding of the layers over one another, leading to a shear deformation with a specific orientation. If the forcing is not parallel to the layers, complex microstructures may form, which have a remarkable similarity over different systems and often consist of alternating shears of different sign. We review here recent results on the detailed analysis of slip processes in crystal plasticity based on the theory of relaxation, discuss the general variational framework for these microstructures, and compare with available experimental results in different systems. We then address the situation in which slip in several different directions may coexist in the same system, as frequently observed in plastically deformed crystals.

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

Cited by 12 publications

References 27 publications

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Microstructure in Plasticity, a Comparison between Theory and Experiment

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

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