Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity

Adriana Garroni, Robert Kohn

Research output: Contribution to journalArticle

Abstract

The well-known Sachs and Taylor bounds provide easy inner and outer estimates for the effective yield set of a polycrystal. It is natural to ask whether they can be improved. We examine this question for two model problems, involving three-dimensional gradients and divergence-free vector fields. For three-dimensional gradients, the Taylor bound is far from optimal: we derive an improved estimate that scales differently when the yield set of the basic crystal is highly eccentric. For three-dimensional divergence-free vector fields, the Taylor bound may not be optimal, but it has the optimal scaling law. In both settings, the Sachs bound is optimal.

Original languageEnglish (US)
Pages (from-to)2613-2625
Number of pages13
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume459
Issue number2038
DOIs
StatePublished - Oct 8 2003

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Polycrystal
Polycrystals
polycrystals
Electric breakdown
plastic properties
Plasticity
Breakdown
divergence
breakdown
Divergence-free Vector Fields
gradients
Three-dimensional
Scaling laws
eccentrics
estimates
scaling laws
Optimal Scaling
Gradient
Crystals
Scaling Laws

Keywords

  • Dielectric breakdown
  • Nonlinear homogenization
  • Polycrystal plasticity
  • Sachs bound
  • Taylor bound

ASJC Scopus subject areas

  • General

Cite this

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AB - The well-known Sachs and Taylor bounds provide easy inner and outer estimates for the effective yield set of a polycrystal. It is natural to ask whether they can be improved. We examine this question for two model problems, involving three-dimensional gradients and divergence-free vector fields. For three-dimensional gradients, the Taylor bound is far from optimal: we derive an improved estimate that scales differently when the yield set of the basic crystal is highly eccentric. For three-dimensional divergence-free vector fields, the Taylor bound may not be optimal, but it has the optimal scaling law. In both settings, the Sachs bound is optimal.

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