hyper_elastic_mixte_rivlin#

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 calculation of the strain energy density also includes nine coefficient terms of the strain invariants. The treatment of incompressibility is made by associating this law with the mixed pressure-displacement elements. The coefficients will be declared here under the command **rivlin. 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:

(233)#\[\begin{split}\begin{aligned} W(I_1,I_2)~=~&{\tt rivlin1}(I_1 - 3)+{\tt rivlin2} (I_2 - 3) + \\ &{\tt rivlin3}(I_1 - 3)^2+{\tt rivlin4} (I_1 - 3)(I_2 - 3) + \\ &{\tt rivlin5}(I_2 - 3)^2+{\tt rivlin6}(I_1 - 3)^3 + \\ &{\tt rivlin7}(I_1 - 3)^2(I_2 - 3)+{\tt rivlin8}(I_1 - 3)(I_2 - 3)^2 + \\ &{\tt rivlin9}(I_2 - 3)^3 \\ \end{aligned}\end{split}\]

with \(I_1\), \(I_2\), and, \(I_3\) the first, second and third invariants of the Green-Lagrange strain tensor. rivlin1 to rivlin9 are material coefficients.

The Piola-Kirchhoff stress tensor is written:

(234)#\[\bf S = {\partial W \over\partial \bf E} - p\bf G^{-1}\]

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_rivlin [ **thermal_strain <THERMAL_STRAIN> ] **rivlin rivlin1 COEFFICIENT rivlin2 COEFFICIENT ... rivlin9 COEFFICIENT ***return

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).

The hyperelastic Rivlin model has the following stored variables:

eto & T-2 & Green-Lagrange strain & yes
press & S & hydrostatic pressure & yes
sig & T-2 & Cauchy stress & yes
dvolu & S & volume change & no
welas & S & strain energy density & yes
spk & T-2 & Piola-Kirchoff stress & no
eel & T-2 & “elastic” strain & no

Remark#

Ifsome of the 9 coefficients for this model are zero, it is still necessary to define them in the material file giving a constant value of \(0.0\).

Example#

%
%  zsheet_rivlin
%
***behavior hyper_elastic_mixte_rivlin
  **rivlin
    rivlin1 27.02
    rivlin2 1.42
    rivlin3 -0.27
    rivlin4 0.
    rivlin5 0.
    rivlin6 0.00654
    rivlin7 0.
    rivlin8 0.
    rivlin9 0.
***return