<HYPERELASTICITY> arruda_boyce

Description

This physically-based model, proposed by Arruda and Boyce [M1], consists in a chain model with a distribution of chains upon eight directions corresponding to the vertices of a cube inscribed in a unit sphere. The implemented model includes an additional coefficient for compressibility treatment. The coefficients for the hyperelastic law are declared under **hyperelasticity arruda_boyce.

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

(412)\[\begin{split}\begin{aligned} W(I_1,I_3)~=~&{\tt nK\theta} [\frac{1}{2}(I_1 - 3) + \frac{1}{20N}(I^2_1 - 9)] + \frac{11}{1050N^2}(I^3_1 - 27)] + \frac{19}{7000N^3}(I^4_1 - 81)] + \\ & \frac{519}{673750N^4}(I^5_1 - 243)] + \frac{{\tt K0}}{2}[(I^2_3-1)/2-\log{I_3}] \end{aligned}\end{split}\]

with \(I_1\) and \(I_3\) the first and third invariants of the Green-Lagrange strain tensor. nK\(\theta\), N and K0 are the three material coefficients. K0 represents the Bulk modulus, while mu = nk\(\theta\) is the material modulus, it corresponds to the slope of the stress-strain curve during loading, and \(\sqrt{lambda}\)= N is the number of connected rigid-links in a chain.

This model of strain energy presents a good agreement with experimental data for equibiaxial extension.

Syntax

**hyperelasticity arruda_boyce **model_coef \(~\,~\,\) mu <COEFFICIENT> \(~\,~\,\) lambda <COEFFICIENT> \(~\,~\,\) K0 <COEFFICIENT>

Example

The following is a simple example of the hyperelastic Arruda-Boyce model ^{1 1}The model parameters have been taken from [M2]:

***behavior hyper_elastic
 **hyperelasticity arruda_boyce
 **model_coef
    mu           2.1
    lambda       3.0
    K0       10000.
***return