Linear elastic materials for large strains (Ciarlet model)

In [19] it is explained that substituting the infinitesimal strains in the classical Hooke law by the Lagrangian strain and the stress by the Piola-Kirchoff stress of the second kind does not lead to a physically sensible material law. In particular, such a model (also called St-Venant-Kirchoff material) does not exhibit large stresses when compressing the volume of the material to nearly zero. An alternative for isotropic materials is the following stored-energy function developed by Ciarlet [17] ($ \mu$ and $ \lambda$ are Lamé's constants):

$\displaystyle \Sigma = \frac{\lambda}{4}(III_C - \ln III_C -1) + \frac{\mu }{4}(I_C - \ln III_C -3).$ (295)

The stress-strain relation amounts to ( $ \boldsymbol{S}$ is the Piola-Kirchoff stress of the second kind) :

$\displaystyle \boldsymbol{S}= \frac{\lambda}{2}($det$\displaystyle \boldsymbol{C} -1) \boldsymbol{C^{-1}} + \mu (\boldsymbol{I}-\boldsymbol{C^{-1}}) ,$ (296)

and the derivative of $ \boldsymbol{S}$ with respect to the Green tensor $ \boldsymbol{E}$ reads (component notation):

$\displaystyle \frac{d S^{IJ} }{d E_{KL} } = \lambda ($det$\displaystyle \boldsymbol{C}) C^{{-1}^{KL}} C^{{-1}^{IJ}}+[2 \mu - \lambda ($det$\displaystyle \boldsymbol{C} -1)] C^{{-1}^{IK}} C^{{-1}^{LJ}}.$ (297)

This model was implemented into CalculiX by Sven Kaßbohm. The definition consists of a *MATERIAL card defining the name of the material. This name HAS TO START WITH ”CIARLET_EL” but can be up to 80 characters long. Thus, the last 70 characters can be freely chosen by the user. Within the material definition a *USER MATERIAL card has to be used satisfying:

First line:

Following line:

Repeat this line if needed to define complete temperature dependence.

For this model, there are no internal state variables.



defines an isotropic material with elastic constants $ E$=210000. and $ \nu$=0.3 for a temperature of 400 (units appropriately chosen by the user). Recall that

$\displaystyle \mu= \frac{E}{2(1+\nu)}$ (298)


$\displaystyle \lambda=\frac{\nu E}{(1+\nu)(1-2 \nu)}.$ (299)