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

Adriana Garroni, Robert V. Kohn

Research output: Contribution to journalArticlepeer-review


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
Issue number2038
StatePublished - Oct 8 2003


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

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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