16.3.1 Deviatoric Energy Function

A possible way to construct any type of strain energy density function for an incompressible, initially isotropic material is the Rivlin formulation, see for instance Treloar [103]:

$$\sum_{{i=0}}^{{\infty}} \sum_{{j=0}}^{{\infty}} \left(\vphantom{ I_{1} -3 }\right. \left.\vphantom{ I_{1} -3 }\right)^{{i}}_{} \left(\vphantom{ I_{2} -3 }\right. \left.\vphantom{ I_{2} -3 }\right)^{{j}}_{}$$ (16.37)

This formulation can also be used for nearly incompressible materials if the invariants I₁ and I₂ are substituted by the modified invariants J₁ and J₂ .

16.3.1.1 Mooney-Rivlin

A well known model, derived from the Rivlin formulation, is the Mooney-Rivlin model [71,86]. Here only the first order terms in (16.37) are maintained:

$$\left(\vphantom{ J_{1} - 3 }\right. \left.\vphantom{ J_{1} - 3 }\right) \left(\vphantom{ J_{2} - 3 }\right. \left.\vphantom{ J_{2} - 3 }\right)$$ (16.38)

A subset of the Mooney-Rivlin model is the Neo-Hookean model, in which case K₂ = 0 . The Neo-Hookean model can be derived for polymer-like materials from statistical thermodynamics.

16.3.1.2 Besseling

An alternative way to adapt the material model to experimental data is to use a non-integer power $\alpha$ instead of the Rivlin formulation. This model was proposed by Besseling [9]:

$$\left(\vphantom{ J_{1} - 3 }\right. \left.\vphantom{ J_{1} - 3 }\right)^{{\alpha}}_{} \left(\vphantom{ J_{2} - 3 }\right. \left.\vphantom{ J_{2} - 3 }\right)$$ (16.39)

Because of the strict separation of the deviatoric and hydrostatic part, the original model was adapted by using the modified invariants. The model is capable of describing the ascending branch at high strains in a uniaxial test, that cannot be matched by the Mooney-Rivlin model.