## 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 language | English (US) |
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Pages (from-to) | 2613-2625 |

Number of pages | 13 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 459 |

Issue number | 2038 |

DOIs | |

State | Published - Oct 8 2003 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- General Engineering
- General Physics and Astronomy