Vol. 5, No. 1, 2017

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On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

Graeme W. Milton, Marc Briane and Davit Harutyunyan

Vol. 5 (2017), No. 1, 41–94
DOI: 10.2140/memocs.2017.5.41

The set GUf of possible effective elastic tensors of composites built from two materials with elasticity tensors C1 > 0 and C2 = 0 comprising the set U = {C1,C2} and mixed in proportions f and 1 f is partly characterized. The material with tensor C2 = 0 corresponds to a material which is void. (For technical reasons C2 is actually taken to be nonzero and we take the limit C2 0). Specifically, 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 easily compliant to p 5 independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains “complementary Avellaneda material”, which is a hierarchical laminate that minimizes the sum of complementary energies.

printed materials, elastic $G$-closures, metamaterials
Mathematical Subject Classification 2010
Primary: 74Q20, 35Q74
Received: 10 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
Marc Briane
Institut de Recherche Mathématique de Rennes
INSA de Rennes
20 Avenue des Buttes de Coësmes
CS 70839
35708 Rennes Cedex 7
Davit Harutyunyan
Department of Mathematics
University of Utah
155 South 1400 East Room 233
Salt Lake City, UT 84112-0090
United States