Graeme W. Milton, Marc Briane and Davit Harutyunyan
Vol. 5 (2017), No. 1, 41–94
DOI: 10.2140/memocs.2017.5.41
Abstract
The set
of possible effective elastic tensors of composites built from two materials with elasticity
tensors
and
comprising the
set
and mixed in
proportions
and
is partly characterized.
The material with tensor
corresponds to a material which is void. (For technical reasons
is actually taken to be nonzero and we take the limit
). Specifically,
recalling that
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
-mode material that
is easily compliant to
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.