<HYPERELASTICITY> st_venant_kirchhoff#

Description#

The St. Venant-Kirchhoff model is a simple hyperelastic model, which is defined by a strain energy \(W\) as

\[W(\ten E)=\dfrac{\lambda}{2}(trace~\ten E)^2+\mu \ten E:\ten E\]

where \(\lambda\) and \(\mu\) are Lamé coefficients and \(\ten E\) is the Green-Lagrange strain tensor. The second Piola-Kirchhoff stress tensor can be obtained as

\[S=\dfrac{dW}{d \ten E}=\lambda(trace~ \ten E) I+2\mu \ten E ~~ \text{or} ~~ S=C:\ten E\]

where \(I\) is the identity tensor of order 2 and C is the elasticity tensor defined as

\[C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\]

This model has some limitations, especially in compression loading conditions (see Fig. 18). Let’s consider the following loading

\[\begin{split}\hspace{5cm} \begin{pmatrix} F_{11}&0&0\\0&1&0\\0&0&1 \end{pmatrix}\end{split}\]

the value of Cauchy stress component \(\sigma_{11}\) tends to zero as \(F_{11}\) approaches 0 and it goes through a minimum at \(F_{11}=\sqrt{1/3}\).

../../_images/hyper_mat.svg

Fig. 18 Simple extension response for various hyperelastic materials.#

Syntax#

***behavior hyper_elastic \(~\,\) **hyperelasticity st_venant_kirchhoff \(~\,~\,\) *elasticity isotropic \(~\,~\,~\,\) young COEFFICIENT \(~\,~\,~\,\) poisson COEFFICIENT

Example#

The following is a simple example of the hyperelastic Saint Venant-Kirchhoff model.

***behavior hyper_elastic
 **hyperelasticity st_venant_kirchhoff
  *elasticity isotropic
    mu           80000.
    lambda       121000.
***return