<HYPERELASTICITY> van_der_waals#

Description#

This hyperelastic model, taking into account the van der Waals forces, is based in the works of Kilian [M15]. The implemented model includes an additional coefficient for compressibility treatment. The coefficients for the hyperelastic law are declared under **hyperelasticity van_der_waals.

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

(416)#\[\begin{split}\begin{aligned} W(I_1,I_2,I_3)~=~&{\tt \mu} [-(\lambda^2_m-3)(\ln(1-\Theta)+\Theta)-\frac{2}{3}a(\frac{\tilde{I}-3}{2})^{\frac{3}{2}}] + \\ & \dfrac{{\tt K0}}{2}[(I^2_3-1)/2- \log{I_3}] \end{aligned}\end{split}\]

where

(417)#\[\begin{split}\begin{aligned} \Theta = \sqrt{(\tilde{I}-3)/(\lambda^2_m-3)}\\ \tilde{I}=(1-\beta)I_1+\beta I_2 \end{aligned}\end{split}\]

with \(I_1\), \(I_2\), and, \(I_3\) the first, second, and third invariants of the Green-Lagrange strain tensor. \(\mu\), \(\lambda_m\), a and \(\beta\) are the four material coefficients. \(\mu\) is the initial shear modulus; \(\lambda_m\) is the locking stretch; a is a global interaction parameter and \(\beta\) has no physical meaning, conferring an empirical nature to this model. K0 represents the Bulk modulus.

Syntax#

**hyperelasticity van_der_waals **model_coef \(~\,~\,\) mu <COEFFICIENT> \(~\,~\,\) lambda_m <COEFFICIENT> \(~\,~\,\) a <COEFFICIENT> \(~\,~\,\) beta <COEFFICIENT> \(~\,~\,\) K0 <COEFFICIENT>

Example#

The following is a simple example of the hyperelastic van der Waals model 1The model parameters have been taken from [M16]:

***behavior hyper_elastic
 **hyperelasticity mooney
 **model_coef
    mu            0.434
    lambda_m     10.24
    a             0.32
    beta          0.958
    K0         1000.
***return