hyper_elastic_mixte_mooney#

Description#

This model 1this behavior is Z-set specific, and therefore does not apply for Z-mat for other codes provides a modification of the Mooney-Rivlin hyperelasticity described above to maintain incompressibility conditions. The treatment of incompressibility is made by associating this law with the mixed pressure-displacement elements. As was the case for the compressible Mooney-Rivlin hyperelasticity model, the coefficients are declared under the command **mooney. Thermal deformations are also permitted by using using the **thermal_strain option.

The strain energy density is re-defined from the previous case to the following expression:

(231)#\[W(I_1,I_2,I_3)={\tt moon1} \left(\frac{I_1}{{(I_3)}^{1/3}}\right) + {\tt moon2}\left(\frac{I_2}{{(I_3)}^{2/3}}\right)\]

with \(I_1\), \(I_2\), and, \(I_3\) the first, second and third invariants of the Green-Lagrange strain tensor, and moon1 et moon2 are the material coefficients. The Piola-Kirchhoff stress tensor is calculated from the strain energy:

(232)#\[\bf S = \pder{W}{\bf E} - p\bf G^{-1}\]

where : \(\bf S\) : second Piola-Kirchhoff stress tensor \(\bf E\) : Green-Lagrange strain tensor \(p\) : hydrostatic pressure \(\bf G\) : metric tensor

Syntax#

***behavior hyper_elastic_mixte_mooney [ **thermal_strain <THERMAL_STRAIN> ] **mooney moon1 COEFFICIENT moon2 COEFFICIENT

Compatible elements#

This material model is programmed to be used with all versions of the total Lagrangian mixed pressure-displacement elements (element types starting with total_lagrangian and ending in mixte_u_p). The behavior must therefore be used in conjunction with a mesh type declaration of these elements in the .inp file (see ***mesh under ****calcul).

This material law has the following variables stored during the calculation:

eto & T-2 & Green-Lagrange strain & yes
sig & T-2 & Cauchy stress & yes
welas & S & strain energy density & yes
spk & T-2 & Piola-Kirchoff stress & no

Example#

%
%  rubbert3m
%
***behavior hyper_elastic_mixte_mooney
  **mooney
   moon1 0.04
   moon2 0.00
  **thermal_strain
    alpha temperature
        1.e-4  0.
        1.e-4  1000.
    ref_temperature 0.0
***return