Energy Estimates, Relaxation, and Existence for Strain-Gradient Plasticity with Cross-Hardening

Anguige, Keith; Dondl, Patrick

doi:10.1007/978-3-319-18242-1_6

Cited by 7 publications

(20 citation statements)

References 37 publications

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“…The relative activity of the two slip systems is fixed by the far-field deformation u 0 , which will be taken affine. We refer to [CO05] for a more detailed physical motivation of the model and to [AD14,AD15] for a discussion of the relaxation of E. The energy scaling for the case of the far-field deformation u 0 (x) = x 1 , corresponding to a halfhalf mixture of the two slip directions e ξ and e η , was studied in [CO05]. The key result was that the optimal energy scales, for small ε, as the minimum of ε 1/2 µ 1/2 and ε 2/3 , similarly to the Kohn-Müller model.…”

Section: The Model For Martensitic Microstructuresmentioning

confidence: 99%

Low volume-fraction microstructures in martensites and crystal plasticity

Conti

Zwicknagl

2016

Math. Models Methods Appl. Sci.

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We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small.

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Section: The Model For Martensitic Microstructuresmentioning

confidence: 99%

Low volume-fraction microstructures in martensites and crystal plasticity

Conti

Zwicknagl

2016

Math. Models Methods Appl. Sci.

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“…It is also interesting to compare the setting discussed above to the setting of minimization problems in crystal plasticity which is discussed in the articles by Anguige & Dondl [AD15] and by Dmitrieva el al. [DRMD15] in this book (see also [AD14b,AD14a]).…”

Section: Introductionmentioning

confidence: 96%

“…In crystal plasticity one considers only those fields γ p which correspond to a superposition of slips across a given set of slip planes and corresponding Burgers vectors, see, e.g., [AD15] or [CGO15]. Similarly one can consider multiple dislocations and their corresponding slip lines.…”

Section: Introductionmentioning

confidence: 98%

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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“…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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Energy Estimates, Relaxation, and Existence for Strain-Gradient Plasticity with Cross-Hardening

Cited by 7 publications

References 37 publications

Low volume-fraction microstructures in martensites and crystal plasticity

Low volume-fraction microstructures in martensites and crystal plasticity

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

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