16.3.2 Hydrostatic Energy Function

The hydrostatic part of the strain energy density function, determines the compressibility of the material. Usually this dilatation behavior is much more stiff than the deviatoric behavior. In an analysis, the calculated stresses are a summation of the contribution due to the deviatoric and the hydrostatic part of the strain energy density function.

16.3.2.1 Linear Compressibility

For nearly incompressible rubbery materials a linear relation between volume changes and hydrostatic stresses is valid for pressures up to hundreds of atmospheres, see for instance Van Den Bogert [104]. A linear hydrostatic stress-strain relation is a result of a quadratic strain energy density function

$${\frac{{\kappa}}{{2}}} \left(\vphantom{ J - 1 }\right. \left.\vphantom{ J - 1 }\right)^{{2}}_{}$$ (16.40)

16.3.2.2 Nonlinear Compressibility

For very high compressive stresses in rubber, or for other types of material, nonlinear hydrostatic strain energy density functions have been developed. The two models, implemented in DIANA are presented below.

$${\frac{{\kappa}}{{2}}} \left(\vphantom{ \left( J - 1 \right)^{2} + \left( \ln J \right)^{2} }\right. \left(\vphantom{ J - 1 }\right. \left.\vphantom{ J - 1 }\right)^{{2}}_{} \left(\vphantom{ \ln J }\right. \left.\vphantom{ \ln J }\right)^{{2}}_{} \left.\vphantom{ \left( J - 1 \right)^{2} + \left( \ln J \right)^{2} }\right)$$ Simo-Taylor [97]: (16.41)

$${\frac{{\kappa}}{{\beta}}} \left(\vphantom{ \frac{1}{\beta -1}J^{-\beta} + 1 }\right. {\frac{{1}}{{\beta -1}}} \beta \left.\vphantom{ \frac{1}{\beta -1}J^{-\beta} + 1 }\right)$$ Murnaghan [73]: (16.41a)