total deformation into the inelastic and elastic deformations.

The actual state under the load at any given moment $T$ results from both elastic (with the variable elastic moduli) and inelastic (irreversible) deformations. The “ground state” of the body is characterized by the 3D-metric $g_{t}$. This state is the background to which the elastic deformation is added to reach the actual state ([@SO]).

Transition from the reference state to the “ground state” that manifests in the evolution of the (initial) Euclidean metric $h$ to the metric $g_{t}$ cannot be described, in general, by any point transformation. Transition from the “ground state” to the actual state at the moment $t$ also is not compatible in general. Yet the transition from the reference state to the actual state is represented by a diffeomorphism $\phi _{t}$.

Here we are considering the [**material 4D-metric**]{} $G$ and the [**deformation**]{} $\phi $ (or elastic strain tensor $E^{el}(\phi )$) to be the dynamical variables of the field theory. The reference mass density $\rho _{0}$ is found (for the synchronized deformation $\phi $ and the BD-metric $G$) by the formula (3.4) if its initial value $\rho _{0}(T=0)$ is known. In this study we consider mainly the quasi-static version of 