Microstructure in Plasticity, a Comparison between Theory and Experiment

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: 95%

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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“…The overall structure is periodic and has sharp kinks between approximately affine regions. We also refer to [DRMD15] in this volume for an explanation of the experimental techniques and detailed findings.…”

Section: Kink Bands In Stacks Of Paper Under Compressionmentioning

confidence: 98%

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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An assessment of numerical techniques to find energy‐minimizing microstructures associated with nonconvex potentials

Kumar

Vidyasagar

Kochmann

2019

Numerical Meth Engineering

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Summary Microstructural patterns emerge ubiquitously during phase transformations, deformation twinning, or crystal plasticity. Challenges are the prediction of such microstructural patterns and the resulting effective material behavior. Mathematically, the experimentally observed patterns are energy‐minimizing sequences produced by an underlying non‐(quasi)convex strain energy. Therefore, identifying the microstructure and effective response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature, quasiconvexification has traditionally been limited to (semi‐)analytical techniques or has been dealt with by numerical techniques such as the finite element method (FEM). Both have been restricted to primarily simple material models. We here contrast three numerical techniques—FEM, a Fourier‐based spectral formulation, and a meshless maximum‐entropy (max‐ent) method. We demonstrate their performance by minimizing the energy of a representative volume element for both hyperelasticity and finite‐strain phase transformations. Unlike FEM, which fails to converge in most scenarios, the Fourier‐based spectral formulation (FFT) scheme captures microstructures of intriguingly high resolution, whereas max‐ent is superior at approximating the relaxed energy. None of the methods are capable of accurately predicting both microstructures and relaxed energy; yet, both FFT and max‐ent show significant advantages over FEM. Numerical errors are explained by the energy associated with microstructural interfaces in the numerical techniques compared here.

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Microstructure in Plasticity, a Comparison between Theory and Experiment

Cited by 4 publications

References 26 publications

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

An assessment of numerical techniques to find energy‐minimizing microstructures associated with nonconvex potentials

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