  Polyconvexity is one of the known conditions which guarantee existence of
solutions of boundary value problems in finite elasticity. In this work we
propose a framework for development of polyconvex strain energy functions for
hyperelastic materials with cubic anisotropy. The anisotropy is captured by a
single fourth order structural tensor for which the minimal polynomial basis is
identified and used for the formulation of the strain energy functions. After
proving the polyconvexity of some polynomial terms, we proceed to propose a
model based on a simple strain energy function. We investigate the behavior of
the model analytically in one dimension and numerically in two and three
dimensions. These investigations allow us conclude that the model possesses the
physically relevant directional properties, in particular, strains in different
directions are different and the lack of any loading symmetries leads to
development of ``shear'' stresses and displacements.
