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):
\Sigma = \frac{\lambda}{4}(III_C - \ln III_C -1) + \frac{\mu }{4}(I_C - \lnIII_C -3). | (277) |
The stress-strain relation amounts to ( \boldsymbol{S} is the Piola-Kirchoff stress of the second kind) :
\boldsymbol{S}= \frac{\lambda}{2}(\text{det} \boldsymbol{C} -1)\boldsymbol{C^{-1}} + \mu (\boldsymbol{I}-\boldsymbol{C^{-1}}) , det\displaystyle \boldsymbol{C} -1) \boldsymbol{C^{-1}} + \mu (\boldsymbol{I}-\boldsymbol{C^{-1}}) , | (278) |
and the derivative of \boldsymbol{S} with respect to the Green tensor \boldsymbol{E} reads (component notation):
\frac{d S^{IJ} }{d E_{KL} } = \lambda (\text{det} \boldsymbol{C})C^{{-1}^{KL}} C^{{-1}^{IJ}}+[2 \mu - \lambda (\text{det} \boldsymbol{C} -1)] C^{{-1}^{IK}} C^{{-1}^{LJ}}. det\displaystyle \boldsymbol{C}) C^{{-1}^{KL}} C^{{-1}^{IJ}}+[2 \mu - \lambda (det\displaystyle \boldsymbol{C} -1)] C^{{-1}^{IK}} C^{{-1}^{LJ}}. | (279) |
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.
Example: *MATERIAL,NAME=CIARLET_EL *USER MATERIAL,CONSTANTS=2 210000.,.3,400.
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
\mu= \frac{E}{2(1+\nu)} | (280) |
and
\lambda=\frac{\nu E}{(1+\nu)(1-2 \nu)}. | (281) |