Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density

Abstract

This paper deals with the construction of materials with arbitrary prescribed positive semi-definite constitutive tensors. The construction problem can be called an inverse problem of finding a material with given homogenized coefficients. The inverse problem is formulated as a topology optimization problem i.e. finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters. Numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies. Numerical results show that arbitrary materials, including materials with Poisson's ratio −1.0 and other extreme materials, can be obtained by modelling the base cell as a truss structure. Furthermore, a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness. Only the two-dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three-dimensional case.