This article is concerned with the modeling of spongy materials, i.e., linearly elastic materials containing a periodic array of gas bubbles. The size of the cells and that of the bubbles are of the order of the same number \(\epsilon\). The authors consider this mechanical system as a linearly elastic material that is nonhomogeneous in the sense that the shear modulus vanishes inside the bubbles (this represents the gas). The corresponding variational problem for equilibrium is thus non-coercive. Nevertheless, existence of solutions is proved when the body force derives from a potential. By making use of the nonuniqueness of solutions, the authors are able to construct a sequence of solutions that converges weakly in the \(H^ 1\) sense toward the solution of a well- posed homogenized elasticity problem as \(\epsilon\) \(\to 0\). In order to achieve this, they apply the so-called “energy method”, combined with various versions of Poincaré’s inequality and several extension lemmas.
The homogenized elastic moduli are obtained by solving the usual variational problems over the unit cell. However, in these problems, the elasticity tensor is not positive definite (but positive) inside the bubble and existence of minimizers is not immediate. The resulting homogenized elasticity tensor is nevertheless positive definite. More general body forces are considered, in order to allow for loadings in the gas that are different from loadings in the elastic material. Symmetry properties of the homogenized elasticity tensor, inherited from the material symmetry of the elastic material and the geometrical symmetry of the bubbles, are studied in several cases.