Abstract
Consider two isotropic, linearly elastic materials with elasticity tensors $C_1 $, $C_2 $ and assume $C_1 < 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.
Keywords
IsotropyMathematicsBounded functionModuliSymmetry (geometry)Invariant (physics)Elasticity (physics)Strain energyRank (graph theory)Mathematical analysisComposite materialCombinatoricsGeometryFinite element methodPhysicsMaterials scienceMathematical physics
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Publication Info
- Year
- 1987
- Type
- article
- Volume
- 47
- Issue
- 6
- Pages
- 1216-1228
- Citations
- 182
- Access
- Closed
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Cite This
M. Avellaneda
(1987).
Optimal Bounds and Microgeometries for Elastic Two-Phase Composites.
SIAM Journal on Applied Mathematics
, 47
(6)
, 1216-1228.
https://doi.org/10.1137/0147082
Identifiers
- DOI
- 10.1137/0147082