the theory, i.e. inertia forces and kinetic energy are assumed to be negligible.

Parameters of Material Evolution, metric Lagrangian
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In examining the processes of deformation and aging of a solid with a synchronized deformation history $\phi $ we use both general $(G_{IJ})$ and ADM ($S,\vec{N},g$) notations for the 4D material metric $G$.

Following the framework of Classical Field Theory ([@BFS]) we take a Lagrangian density ${\mathcal L}(G,\phi )$ referred to the [**volume form**]{} $dV$ as a function of 4D-material metric $G$, its invariants (with respect to the group of $Diff^{+}(B)$ of the orientation preserving diffeomorphisms of the base $B$, see above) and the elastic strain tensor: $ L(G,E^{el})$.

The Lagrangian $L(G,E^{el})dV$ is represented as a sum of the two parts: the metric part $L_{m}(G)$ and, as a perturbation of the ground state, the elastic part $L_{e}(G,E^{el})$ associated with elastic deformation $$L=L_{m}(G)+L_{e}(E^{el},G).$$

Metric Lagrangian $L_{m}(G)$ in (5.1) is introduced to account for the “cohesive energy” or strength of the solid state, the strain energy of “residual strain” and the energy of the change associated with a evolution of material properties in time, for instance material aging processes of phase transitions.

Metric Lagrangian $L_{m}$ is the sum of several terms with 