we have $$C_{4}(\phi )_{IJ}=\left(\begin{array}{cc} h_{ij}\phi
^{i}_{,0}\phi ^{j}_{,0}=\Vert \mathbf{V}\Vert
^{2}_{h} & h_{ij}\phi ^{i}_{,0}\phi ^{j}_{,J}\\
 h_{ij}\phi ^{i}_{,I}\phi ^{j}_{,0} & \sum ^{3}_{i,j=1} h_{ij}\phi
^{i}_{,I}\phi ^{j}_{J}
\end{array}\right) .$$ The spatial part of this tensor is the conventional Cauchy-Green strain tensor $C_{3}(\phi)$ of the Elasticity Theory. Components of this tensor with indices (0J) and (I0), $I,J=1,2,3$ have the form [*velocity $\times $  deformation covector*]{} (see [@E2]). (00)-component of $C_{4}(\phi )$ is the square of the material velocity $\mathbf{V}=\phi_{3*}(\frac{\partial}{\partial
T})=^{s}\frac{\partial \phi^{i}}{\partial t}\frac{\partial
}{\partial x^{i}}$.

0.5cm

Elastic Strain Tensor
---------------------

Here we [*define*]{} the 4D [*(1,1)-elastic strain tensor*]{} in $P$. We will do it first in linear approximation and then, using logarithm of a (1,1)-tensor function, in another way, more suitable for large deformations.

We start with the following, conventional definition: $$\begin{gathered}
{\hat E}^{el\ \cdot }_{\cdot }=\frac{1}{2}G^{-1}(C_{4}(\phi
)-\gamma )=^{s}\\
 =\frac{1}{2} \begin{pmatrix} S^{-2}\Vert
\mathbf{V}\Vert^{2}_{h}-S^{-2}N^{I}\langle
\mathbf{V},\phi^{.}_{,I}\rangle_{h} & S^{-2}\langle
\mathbf{V},\phi^{.}_{,J}\rangle_{h}-S^{-2}N^{K}\langle
\phi^{.}_{,K}\phi^{.}_{,J}\rangle_{h} \\
 - S^{-2} N^{I} \Vert \mathbf{V}\Vert^{2}_{h}-N^{I}+ & -S^{-2}N^{I}\langle
\mathbf{V},\phi^{.}_{,J}\rangle_{h}+g^{IK}C_{3}(\phi)_{KJ}-\\
+(S^{-2}N^{I}N^{K}+g^{IK})\langle \phi^{.}_{,K},\mathbf{V}
\rangle_{h}  & -\delta ^{I}_{J}-S^{-2}N^{I}N^{K}\langle
\phi^{.}_{,K}\phi^{.}_{,J}\rangle_{h}
\end{pmatrix}.\end{gathered}$$

This tensor contains the square of material velocity and the shift vector field. Having in mind the general, dynamical situation it is more appropriate to use the following tensor as the proper Elastic Strain Tensor $$E^{el\ .}_{.}(\phi )=\Pi {\hat E}^{el}\Pi = \frac{1}{2}\Pi
