hyper_elastic_mooney#

Description#

This model 1this behavior is Z-set specific, and therefore does not apply for Z-mat for other codes implements a two coefficient Mooney-Rivlin hyperelastic behavior including an additional coefficient for compressibility treatment. The coefficients for the hyperelastic law are declared under **mooney.

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

\[\begin{split}\begin{aligned} W(I_1,I_2,I_3)~=~&{\tt moon1} (I_1 - 2I_3 - 1) + {\tt moon2}\times(I_2 - 2I_3 - 1) + \\ &0.5\times{\tt moon3}\times({\tt moon1}+{\tt moon2})\times(I_3-1)^2 \end{aligned}\end{split}\]

with \(I_1\), \(I_2\), and, \(I_3\) the first, second, and third invariants of the Green-Lagrange strain tensor. moon1, moon2 and moon3 are the three material coefficients. moon3 represents a dimensionless compressibility coefficient, while moon1 and moon2 have the dimension of a stress.

The Piola-Kirchhoff stress tensor is written:

\[\bf S = \pder{W}{\bf E}\]

Thermal deformations may also be accounted for with the addition of a **thermal_strain option within the behavior definition to declare thermal strain coefficients.

Syntax#

***behavior hyper_elastic_mooney [ **thermal_strain <THERMAL_STRAIN> ] **mooney moon1 COEFFICIENT moon2 COEFFICIENT moon3 COEFFICIENT [ **save_principle_strain ]

If the option **save_principle_strain is given, the principle values of the Green strain tensor will be stored in addition to the standard variables. These are in order from the greatest (most positive) to the smallest (biggest negative or smallest positive) in the variables e1, e2, and e3.

Compatible elements#

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

The stored variables for this model include the Green-Lagrange total strain tensor, the Cauchy stress tensor, and the strain energy density \(W\). As an auxiliary output variable, the Piola-Kirchhoff stress tensor is stored but not saved during the calculation (may be requested as output). The post-processor prefixes used for these variables are the following:

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
e# & S & principle strain values & opt

Example#

The following is a simple example of the hyperelastic Mooney model:

***behavior hyper_elastic_mooney
  **mooney
    moon1            0.34
    moon2            0.0
    moon3            500.
  **save_principle_strain
***coefficient
    masvol 0.95e-9
***return