scheme presented here is essentially different from relativistic elasticity theory ([@CQ],[@KM]) as well as from 4D version of conventional elasticity theory.

We see from (4.6) that $\el =0$ if and only if the following two conditions are fulfilled:

1) g\_[ij]{}=C\_[3]{}()\_[IJ]{}=\_[3]{}\^[\*]{}(h)\_[IJ]{},\
2) N\^[I]{}=g\^[IK]{}\^[.]{}\_[,0]{},\^[.]{}\_[,K]{}.

In particular, metric $g$ coincide with the Cauchy metric induced by deformation $\phi$ and is flat.

If $\el=$ then ${\hat E}^{el}=0$ if and only if in addition to the conditions (4.9) the following conditio is fulfilled 3)  V\^[2]{}\_[h]{}=g\^[IJ]{}\^[.]{}\_[,0]{},\^[.]{}\_[,I]{}\^[.]{}\_[,0]{},\^[.]{}\_[,J]{}.

0.4cm

Inelastic Strain Tensor
-----------------------

Now we introduce the [*inelastic strain tensor*]{} in linear approximation $${\hat{E}}^{in}=\frac{1}{2}G^{-1}(\gamma -\gamma
_{0})=^{s}\frac{1}{2}\left(\begin{array}{cc} 0 & S^{-2}N^{K}h_{KJ}\\
N^{I} & \delta ^{I}_{J}-g^{IK}h_{KJ}-S^{-2}N^{I}N^{K}h_{KJ}
 \end{array}\right),$$ (last equality being true in synchronized case) and the [*total strain tensor*]{} ${\hat{E}}^{tot}$ of the body at each given moment $T$ to characterize the deviation of the deformed Euclidean metric $\phi ^{*}h\vert _{B_{\phi ,t}}$ from the initial (Euclidean) 3D-metric $h$ ($h$ being the restriction of $G_{0}$ to the slices $B_{\phi ,t}$) $${\hat{E}}^{tot}=\frac{1}{2} G^{-1}(C_{4}(\phi )-\gamma _{0}).$$ Tensor ${\hat{E}}^{tot}$ can be represented as the sum of the elastic strain tensor ${\hat{E}}^{el}$ and of inelastic strain tensor ${\hat{E}}^{in}$: $${\hat{E}}^{tot}={\hat{E}}^{el}+{\hat{E}}^{in}.$$ To obtain the corresponding decomposition for the 3D strain tensors we apply projector $\Pi$ to the total and 