  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. These
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 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 compatibility of the criteria presented here with the geometrical
constraints caused by minimization of surface energy in a real foam.
