Lower bound on complementary energy in shape optimization of shallow shells

The paper deals with the problem of optimal design of shallow shells. More precisely, we tackle the case of shape optimization, understood here as the question of optimal distribution of one isotropic material, with the total compliance of the shell chosen as the merit function. Roughly speaking, the solution of this problem requires bounding the complementary energy of the shell from below. It is established by applying the well-known translation method of Gibianski and Cherkaev. This technique was successfully used in case-by-case analysis of the optimization problems of thin plates in bending and stretching. In the present paper, following the idea of extension of the translation method introduced by Milton, both cases are coupled and treated as a more general issue. Explicit formulae for the lower estimate of the complementary energy functional allow defining the inverse constitutive equations for the nonlinear equilibrium problem of the shallow shell made of an optimal, nonlinear isotropic material.