# How can I recover the deformation gradient from the finite element strain result

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

• you're welcome.. you can delete the comment ... it clutters up the post .. I'll delete this one May 12 at 21:40