Setting the Free Material Design problem through the methods of optimal mass distribution

The paper deals with the Free Material Design (FMD) problem aimed at constructing the least compliant structures from an elastic material the constitutive field of which play the role of the design variable in the form of a tensor valued measure $\lambda$ supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class $\mathscr{H}$ while the integral of a cost $c(\lambda)$ is bounded from above. The convex $p$-homogeneous elastic potential $j$ is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (LCP) known from the theory of optimal mass distribution by G. Bouchitt\'{e} and G. Buttazzo. A theorem linking solutions of (FMD) and (LCP) allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new optimization problems including design of structures made of a material whose elastic response is dissymmetric in tension and compression. By employing the explicitly derived optimality conditions we give several analytical examples of optimal designs.