We propose to use a geometric description to construct covariant Eulerian models for anisotropic hyper elastic solids. After kinematic considerations we propose a relation between the rate of deformation and a deformation tensor. We obtain covariant state equations and hyper-elastic constitutive models for finite transformations. The covariant context ensures invariance with respect to the changes of observer and enables us to derive these equations and relations on any configuration, without referring to a Lagrangian description; this is of particular interest for the case of anisotropic behaviors to construct hyper-elastic models valid for anisotropic materials in an Eulerian configuration. The geometric treatment, in particular of time derivatives and the rate of deformation ensures that the quantities, operators and equations are genuinely intrinsic.