^{*}_{0}(H)$ (here and below we denote by $\phi ^{*}Q$ the pullback of a covariant tensor $Q$ by the differentiable mapping $\phi $). In the coordinates $(X,T)$ the matrix of the metric $G_{0}$ is $\left(\begin{array}{cc} 1& 0\\
0&h_{IJ}
\end{array}\right) $. Denote by $dV_{0}=\sqrt{\vert G_{0}\vert}d^{4}X=\phi
_{0}^{*}(dv^{4})$ the 4D-volume element defined by the metric $G_{0}$.

Projection $\pi : P \longmapsto B$ along $T$-axes plays the same role in the construction below as in the relativistic elasticity theory ([@CQ],[@KM]). In particular, we require invariance of Lagrangian theory with respect to the automorphisms of the bundle $(P, \pi , B)$ (diffeomorphisms of material space-time $P$ onto itself, projecting to $B$, so that material points retains their identity during the material evolution) preserving the direction of the flow of the “intrinsic” time (see below), but not with regard to the whole group of diffeomorphisms of $P$ as in Gravity Theory.

Deformation History 
--------------------

The [**history of the deformation**]{} of the body $B$ is represented by a [**diffeomorphic embedding**]{} $\phi :P \longmapsto M$ of the material space-time $P$ into the physical (Newtonian) space-time $M$ (see Fig.1).

A deformation history $\phi$ for which $t=\phi ^{0}(X)=T$ will be called [**“synchronized”**]{}. The synchronization can practically performed for relatively slow deformation 