inelastic strain tensors. In particular we introduce $$E^{in}=\Pi
{\hat{E}}^{in}\Pi=^{s}\frac{1}{2}\left(\begin{array}{cc} 0 & 0 \\
N^{I}-g^{IK}h_{KB}N^{B} & \delta ^{I}_{J}-g^{IK}h_{KJ}
\end{array}\right) ,$$ As a result, we get from (4.11) the corresponding decomposition of “3D total strain tensor” $$\Pi E^{tot}\Pi
=E^{el}+E^{in}.$$ Restriction of these tensors on the 3D slices $B_{\phi ,t}$ leads to the more conventional ($t$-dependent) version of this decomposition.

For a synchronized deformation history, restriction of $E^{in}$ to the slices $B_{t}$ takes the form $$E^{in}\vert
_{B_{t}}=\frac{1}{2}g^{-1}(g-g_{0})=\frac{1}{2}(I-g^{-1}g_{0}),$$ that describes the decline of 3D material metric $g$ from its initial (reference) value $g_{0}=\phi^{*}_{0}h$.

Another way to define strain tensors, more suitable for description of large deformation is to take $${\hat E}^{el}=\frac{1}{2}ln(G^{-1}C_{4}(\phi ))),
{\hat E}^{in}=\frac{1}{2}ln(G^{-1}_{0}G)), {\hat
E}^{tot}=\frac{1}{2}ln(G^{-1}_{0}C_{4}(\phi ))).$$ We can define $E^{el},\ E^{in},\ E^{tot}$ correspondingly, using projector $\Pi$. Strain tensors, defined in such a way will, in some simple cases, enjoy the same additive relations as (4.11),(4.13). On the other case, if elastic deformation happens in the directions different from the principal axes of inelastic deformation, relation between these deformations becomes more complex.

The relationship between these definitions and those of the linear approximation above is established by using the fact that for a couple $A,B$ of (0,2)-tensors such that $A$ is invertible, $ln(A^{-1}B)\approx
A^{-1}(A-B)$ 