The VolumicPotential material law regroups all the functions $\mathcal{f}(J)$ such that the volumetric part of the strain-energy density function $W_{vol}$ can be expressed as $$ W_{vol} = k_0\mathcal{f}(J) $$ with the compression modulus $k_0$.

Quadratic volumetric strain density (default for FunctionBasedHyperMaterial) $$ \mathcal{f}(J) = \frac{1}{2}\left(J-1\right)^2 $$

Logarithmic volumetric strain density $$ \mathcal{f}(J) = \frac{1}{2}\left(\text{ln}J\right)^2 $$

Quadratic-Logarithmic volumetric strain density (same as NeoHookeanHyperMaterial and MooneyRivlinHyperMaterial) $$ \mathcal{f}(J) = \frac{1}{2}\left(J-1\right)^2 + \frac{1}{2}\left(\text{ln}J\right)^2 $$

Volumetric strain density from Hartmann S.,Neff P., 2003 Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility, Int. J. Solids Struct., 40, 2767–2791. $$ \mathcal{f}(J) = \frac{1}{50}\left(J^5+J^{-5}-2\right) $$

Volumetric strain density from Miehe C., 1994, Aspects of the formulation and finite element implementation of large strain isotropic elasticity, Int. J. Numer. Meth. Engng., 37, 1981–2004. $$ \mathcal{f}(J) = J - \text{ln}J -1 $$

Volumetric strain density from Simo J., Taylor R., 1991, Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Eng., 85, 273–310. $$ \mathcal{f}(J) = \frac{1}{4}\left( J^2 - 2\text{ln}J - 1 \right) $$

Volumetric strain density from Ogden R. W., 1972, Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond., 326, 565–584. $$ \mathcal{f}(J) = \frac{1}{\beta^2}\left( \beta\text{ln}J + J^{-\beta} - 1 \right) $$ where $\beta$ is an experimentally determined material parameter.