duality of material and physical relations of the Doyle-Erikson type in article [@SM], the thermodynamics of a continuum in [@CF], and in [@Ep] where the curvature of material metric $g$ defined by a uniformity mapping of a uniform material was employed as the driving force of the material evolution. We use the 4-D material metric $G$ as an additional state parameter that reflects the aging process. G is introduce with the largest covariance group allowed by the condition that a small vicinity of each point of the material preserves its topology during the aging process (see Sec. 2 below). We consider the 3-D material metric $g$ on the slices $B_{t}$ of constant physical time as one of the main dynamical variables following the ADM (Arnovitt, Deser, Misner) presentation of $G$ (see [@MTW; @ADM] or Sec.2.3 below). In that respect, we follow the tradition of the cited works. What is new in our work is that the smaller (in comparison to the General Relativity) covariance group of the Lagrangian allows us to use the lapse function $S$ and the shift vector $\vec N$ as independent dynamical variables reflecting the proper material time scale and the intrinsic material flows respectively.

This aging 