Optimizing the bulk modulus of 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. 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. Cellular solids appear widely in nature and are manufactured on a large scale by man. Examples include wood, cancellous bone, cork, foams for insulation and packaging, or sandwich panels in aircraft. Material density, or solid volume fraction, φ, is a predominant parameter for the mechanical properties of cellular materials. Various theoretical studies on the mechanical properties of both two-dimensional (2D) and three-dimensional (3D) structures have been attempted [1]. Unfortunately, exact calculations can be achieved for cellular materials with simple geometry only [2], and numerical simulations [2][3] or semi-empirical models [4][5][6] are required in order to study the mechanical properties of more complex structures. However, expression of bounds on the effective moduli can be established. Perhaps the most famous bounds are those given by Z. Hashin and S. Shtrikman for isotropic heterogeneous media [7][8]. In particular, the Hashin-Shtrikman bounds for the effective bulk modulus in the low-density asymptotic limit (φ ≪ 1) read: