Optimizing the bulk modulus of low-density cellular networks.

We present an alternative derivation of upper-bounds for the bulk modulus of both two-dimensional and three-dimensional cellular materials. For two-dimensional materials, we recover exactly the expression of the Hashin-Shtrikman (HS) upper-bound in the low-density limit, while for three-dimensional materials we even improve the HS bound. Furthermore, we establish necessary and sufficient conditions on the cellular structure for maximizing the bulk modulus, for a given solid volume fraction. The conditions are found to be exactly those under which the electrical (or thermal) conductivity of the material reaches its maximal value as well. These results provide a set of straightforward criteria allowing us to address the design of optimized cellular materials, and shed light on recent studies of structures with both maximal bulk modulus and maximal conductivity. Finally, we discuss the specific case of spring networks, and analyze the compatibility of the criteria presented here with the geometrical constraints caused by minimization of surface energy in a real foam.