  A so-called smart material is a material that is the seat of one or more
multiphysical coupling. One of the key points in the development of the
constitutive laws of these materials, either at the local or at the global
scale, is to formulate a free energy density (or enthalpy) from vectors,
tensors, at a given order and for a class of given symmetry, depending on the
symmetry classes of the crystal constituting the material or the symmetry of
the representative volume element. This article takes as a support of study the
stress and magnetization couple ($\sigma$, m) involved in the phenomena of
magnetoelastic coupling in a cubic symmetry medium. Several studies indeed show
a non-monotonic sensitivity of the magnetic susceptibility and magnetostriction
of certain soft magnetic materials under stress. Modeling such a phenomenon
requires the introduction of a second order stress term in the Gibbs free
energy density. A polynomial formulation in the two variables stress and
magnetization is preferred over a tensorial formulation. For a given material
symmetry class, this allows to express more easily the free energy density at
any bi-degree in $\sigma$ and m (i.e. at any constitutive tensors order for the
so-called tensorial formulation). A rigorous and systematic method is essential
to obtain the high-degree magneto-mechanical coupling terms and to build a free
energy density function at any order which is invariant by the action of the
cubic (octahedral) group. For that aim, theoretical and computer tools in
Invariant Theory, that allow for the mathematical description of cubic
nonlinear magneto-elasticity, are introduced. Minimal integrity bases of the
invariant algebra for the pair (m, $\sigma$), under the proper
(orientation-preserving) and the full cubic groups, are then proposed. The
minimal integrity basis for the proper cubic group is constituted of 60
invariants, while the minimal integrity basis for the full cubic group (the one
of interest for magneto-elasticity) is made up of 30 invariants. These
invariants are formulated in a (coordinate free) intrinsic manner, using a
generalized cross product to write some of them. The counting of independent
invariants of a given multi-degree in (m, $\sigma$) is performed. It is shown
accordingly that it is possible to list without error all the material
parameters useful for the description of the coupled magnetoelastic behavior
from the integrity basis. The technique is applied to derive general
expressions $\Psi$ $\star$ ($\sigma$, m) of the free energy density at the
magnetic domains scale exhibiting cubic symmetry. The classic results for an
isotropic medium are recovered.
