Vol. 5, No. 1, 2017

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Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

Graeme W. Milton, Davit Harutyunyan and Marc Briane

Vol. 5 (2017), No. 1, 95–113
DOI: 10.2140/memocs.2017.5.95

The set GUf of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors C1 and C2 = δC0 comprising the set U = {C1,δC0} and mixed in proportions f and 1 f is partly characterized in the limit δ . The material with tensor C2 corresponds to a material which (for technical reasons) is almost rigid in the limit δ . This paper, and the underlying microgeometries, has many aspects in common with the companion paper “On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials”. The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that GUf is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite-dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate p-mode material that is almost rigid to 6 p 5 independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains “Avellaneda material”, which is a hierarchical laminate that minimizes an appropriate sum of elastic energies.

elasticity $G$-closure, composites, metamaterials
Mathematical Subject Classification 2010
Primary: 74Q20, 35Q74
Received: 12 June 2016
Revised: 11 October 2016
Accepted: 14 November 2016
Published: 6 March 2017

Communicated by Robert P. Lipton
Graeme W. Milton
Department of Mathematics
University of Utah
155 South 1400 East Room 233
Salt Lake City, UT 84112-0090
United States
Davit Harutyunyan
Department of Mathematics
University of Utah
155 South 1400 East Room 233
Salt Lake City, UT 84112-0090
United States
Marc Briane
Institut de Recherche Mathématique de Rennes
INSA de Rennes
20 Avenue des Buttes de Coësmes
CS 70839
35708 Rennes Cedex 7