we get the classical local mass conservation law $\frac{\partial \rho _{0}}{\partial T}=0.$

Elastic, Inelastic and Total Strain Tensors
===========================================

In this section we introduce the principal quantities characterizing both elastic and inelastic deformation processes. Total deformation is presented as a composition of elastic and inelastic ones and is integrable. Its elastic and inelastic “components” are non-integrable, in general, but might be such in special situations (see Sec.11). We recall that the presentation of total deformation as a composition of this type was studied in different forms in many works ([@K; @Lee; @SO], to name a few). What is new here is the 4D-approach to this decomposition and direct definition of elastic, inelastic and total strain tensors in terms of material metric $g_{t}$ as an independent dynamical variable, reference (undeformed) Cauchy metric $g_{0}$ and the current Cauchy metric $C_{3}(\phi )$ rather then using the “deformation gradients” (integrable or not) of elastic and inelastic (plastic) deformations.

Slicing $B_{\phi \ ,t}$ of $P$ defines the covariant tensor $\gamma
=G-u_{\phi }\otimes u_{\phi }=^{s}\left(\begin{array}{cc} N_{A}N^{A} & N_{J}\\
N_{I} &g_{IJ} \end{array}\right) $ ([@CQ],[@MTW]). Here and later the sign $s$ over $=$ means that this equality is true in synchronized case. Tensor $\gamma $ induces the time 