Alternative formulation of the energy density

[andreaantonelli1]

For a particular study case of a slightly compressible hyperelastic disk subjected to uniform radial traction, using the hyperplasticity code with the Neo-Hookean model we encountered some instability phenomena. After some research, we discovered that this is due to the form of the volumetric part of the energy density, that shows a softening phase for a specific range of values of the third invariant (the determinant of the gradient tensor of the transformation).
The paper “A new constitutive model for the compressibility of elastomers at finite dimensions” by Bishoff, Arruda, Grosh [1] explains that this instability phenomenon is due to the non respected necessary condition of strong ellipticity of the energy density. This causes the non convergence of the Newton solver because of the non-existence of a unique solution.
For this reason, we propose to substitute the volumetric part of the energy density used in the code with the one referred to as $W_3(J)$ in [1]:

$$W_3 (J) = {B (J - 1)^2 \over 2}$$

This energy always satisfies the necessary condition to guarantee the existence of the solution. This change would allow the use of this code also for different hyperelastic configurations, without encountering instability issues.

Togheter with @giuliacomand, @cmaurini , @riccobelli
Source paper: [1]