  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.
