### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- General

### Cite this

**Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity.** / Garroni, Adriana; Kohn, Robert.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 459, no. 2038, pp. 2613-2625. https://doi.org/10.1098/rspa.2003.1152

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TY - JOUR

T1 - Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity

AU - Garroni, Adriana

AU - Kohn, Robert

PY - 2003/10/8

Y1 - 2003/10/8

N2 - 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.

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.

KW - Dielectric breakdown

KW - Nonlinear homogenization

KW - Polycrystal plasticity

KW - Sachs bound

KW - Taylor bound

UR - http://www.scopus.com/inward/record.url?scp=1542722156&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1542722156&partnerID=8YFLogxK

U2 - 10.1098/rspa.2003.1152

DO - 10.1098/rspa.2003.1152

M3 - Article

VL - 459

SP - 2613

EP - 2625

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2038

ER -