  A soft solid is said to be initially stressed if it is subjected to a state
of internal stress in its unloaded reference configuration. Developing a sound
mathematical framework to model initially stressed solids in nonlinear
elasticity is key for many applications in engineering and biology. This work
investigates the links between the existence of elastic minimizers and the
constitutive restrictions for initially stressed materials subjected to finite
deformations. In particular, we consider a subclass of constitutive responses
in which the strain energy density is taken as a scalar valued function of both
the deformation gradient and the initial stress tensor. The main advantage of
this approach is that the initial stress tensor belongs to the group of the
divergence-free symmetric tensors satisfying the boundary condition in any
given reference configuration. However, it is still unclear which physical
restrictions must be imposed for the well-posedness of this elastic problem.
Assuming that the constitutive response depends on the choice of the reference
configuration only through the initial stress tensor, under given conditions we
prove the local existence of a relaxed state given by an implicit tensor
function of the initial stress distribution. This tensor function is generally
not unique, and can be transformed accordingly to the symmetry group of the
material at fixed initial stresses. These results allow to extend Ball's
existence theorem of elastic minimizers for the proposed constitutive choice of
initially stressed materials.
