<ISOTROPIC_HYPERELASTICITY> rivlin#

Description#

This model implements the Rivlin hyperelastic behavior [M22] as a polynomial series of \((I_1-3)\) and \((I_2-3)\) including an additional coefficient for compressibility treatment. The coefficients for the hyperelastic law are declared under **isotropic_hyperelasticity rivlin.

Hyperelastic behavior here defines the strain energy density with the following form:

\[\begin{aligned} W(I_1,I_2,I_3)~=~&\sum_{i=0,j=0}^{\infty}{\tt C_{ij}} (I_1 - 1)^i (I_2 - 1)^j + \dfrac{{\tt K0}}{2}[(I^2_3-1)/2-\log{I_3}] \end{aligned}\]

with \(I_1\), \(I_2\), and, \(I_3\) the first, second, and third invariants of the Green-Lagrange strain tensor. C\(_{ij}\) and K0 are material parameters. K0 represents the Bulk modulus, while C\(_{00}=0\). The series is often truncated to terms of the second or third order.

This model of strain energy is classically used for very large strain problems.

Syntax#

**isotropic_hyperelasticity rivlin **model_coef \(~\,~\,\) C10 <COEFFICIENT> \(~\,~\,\) C01 <COEFFICIENT> \(~\,~\,\) C20 <COEFFICIENT> \(~\,~\,\) C11 <COEFFICIENT> \(~\,~\,\) C02 <COEFFICIENT> \(~\,~\,\) ... \(~\,~\,\) K0 <COEFFICIENT>

Example#

The following is a simple example of the hyperelastic Rivlin model 1The model parameters have been taken from [M17]:

***behavior hyper_elastic
 **isotropic_hyperelasticity rivlin
 **model_coef
    C10            0.208
    C01            0.0233
    C20           -0.0024
    C11            0.0
    C02            0.0
    C30            0.0005
    C21            0.0
    C12            0.0
    C03            0.0
    K0          1000.
***return