### Abstract

Consider two isotropic, linearly elastic materials with elasticity tensors C//1, C//2 and assume C//1 less than C//2. We prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite-rank laminates. This implies that for any imposed uniform strain or stress field, optimal bounds on the effective strain or stress energy per unit volume are attained by finite-rank laminates. Explicit bounds on the strain energy, with no symmetry assumptions, are given in dimensions 2 and 3. These bounds are more stringent than the classical Voigt-Reuss bounds. Finally, explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry.

Original language | English (US) |
---|---|

Pages (from-to) | 1216-1228 |

Number of pages | 13 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 47 |

Issue number | 6 |

State | Published - Dec 1987 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*47*(6), 1216-1228.

**Optimal bounds and microgeometries for elastic two-phase composites.** / Avellaneda, Marco.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 47, no. 6, pp. 1216-1228.

}

TY - JOUR

T1 - Optimal bounds and microgeometries for elastic two-phase composites

AU - Avellaneda, Marco

PY - 1987/12

Y1 - 1987/12

N2 - Consider two isotropic, linearly elastic materials with elasticity tensors C//1, C//2 and assume C//1 less than C//2. We prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite-rank laminates. This implies that for any imposed uniform strain or stress field, optimal bounds on the effective strain or stress energy per unit volume are attained by finite-rank laminates. Explicit bounds on the strain energy, with no symmetry assumptions, are given in dimensions 2 and 3. These bounds are more stringent than the classical Voigt-Reuss bounds. Finally, explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry.

AB - Consider two isotropic, linearly elastic materials with elasticity tensors C//1, C//2 and assume C//1 less than C//2. We prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite-rank laminates. This implies that for any imposed uniform strain or stress field, optimal bounds on the effective strain or stress energy per unit volume are attained by finite-rank laminates. Explicit bounds on the strain energy, with no symmetry assumptions, are given in dimensions 2 and 3. These bounds are more stringent than the classical Voigt-Reuss bounds. Finally, explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry.

UR - http://www.scopus.com/inward/record.url?scp=0023592964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0023592964&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0023592964

VL - 47

SP - 1216

EP - 1228

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 6

ER -