<HYPERELASTICITY> ogden#

Description#

This model implements the Ogden’s phenomenological theory of elasticity [M13] including an additional coefficient for compressibility treatment. The strain energy density is written in terms of generalized strain. The coefficients for the hyperelastic law are declared under **hyperelasticity ogden.

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

(414)#\[\begin{aligned} W(\lambda_1,\lambda_2,\lambda_3, I_3)~=\sum_{n=1}^{N}\frac{\mu_n}{\alpha_n}\left(\lambda^{\alpha_n}_1+\lambda^{\alpha_n}_2+\lambda^{\alpha_n}_3-3\right)+ \dfrac{{\tt K0}}{2}[(I^2_3-1)/2-\log{I_3}] \end{aligned}\]

where the material parameters \((\mu_n,\alpha_n)_{n=1,N}\) should fulfilled the following stability condition:

(415)#\[\begin{aligned} \mu_n\alpha_n>0 \hspace{0.5cm}\forall n=1,N \end{aligned}\]

with \(\lambda_1\), \(\lambda_2\), and, \(\lambda_3\) the three generalized stretches and \(I_3\) the third invariant of the Green-Lagrange strain tensor. \(\mu_n\), \(\alpha_n\) and K0 are the three material coefficients. K0 represents the Bulk modulus, while \(\mu_n\) and \(\alpha_n\) are material constants.

This model is possibly the most extended model after Mooney-Rivlin model. Even if the determination of material parameters leads to some difficulties, this model is one of the most widely used for large strain problems.

Syntax#

**hyperelasticity ogden **model_coef \(~\,\) *term \(~\,~\,\) mu COEFFICIENT \(~\,~\,\) alpha COEFFICIENT \(~\,~\,\) K0 COEFFICIENT \(~\,\) *term \(~\,~\,\) mu COEFFICIENT \(~\,~\,\)

Example#

The following is a simple example of the hyperelastic Ogden model 1The model parameters have been taken from [M14]:

***behavior hyper_elastic
 **hyperelasticity ogden
 **model_coef
  *term
   mu         0.63
   alpha      1.3
   K0     10000.
  *term
   mu         1.2e-3
   alpha      5.0
  *term
   mu        -1.e-2
   alpha     -2.0
***return