I have performed a finite element simulation which corresponds to the example in https://de.wikipedia.org/wiki/Deformationsgradient#Beispiel .
The result from the theory in wikipedia is the following (2x2) stretch tensor:
$$
\mathbf{U}=\left[\begin{array}{cc}
0.92307692 & 0.38461538 \\
0.38461538 & 1.24358974
\end{array}\right]
$$
The eigenvalues of this matrix are: 0.44444 and 2.25.
The finite element simulation yields the following (2x2)true strain tensor for the given transformation:
$$\begin{aligned}
\mathbf{\varepsilon_{true}}=
\left[\begin{array}{cc}
-0.3698638 & 0.16354624 \\
0.16354624 & 0.3698638
\end{array}\right]
\end{aligned}$$
The eigenvalues of this matrix are -0.4044 and 0.4044. With
$$\exp(\varepsilon_{true}\cdot2) =
\left[\begin{array}{cc}
0.4453 & 1.0000 \\
1.0000 & 2.2453
\end{array}\right]$$.
This seems close enough to the theoretical result (0.44444 and 2.25).
However what I would like to do is to recover the deformation gradient from the finite element true strain tensor. The deformation gradient according to the theory is
$$
\mathbf{F}=\left[\begin{array}{cc}
1.0 & \frac{5}{6} \\
0. & 1.0
\end{array}\right]
$$
I am trying to find the necessary calculations to get there. $$\varepsilon_{true} \rightarrow \mathbf{F}$$
I have made a jupyter notebook with what I got so far: https://github.com/pytunia/deformation-gradient
