### 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] ( and are Lamé's constants):

 (295)

The stress-strain relation amounts to ( is the Piola-Kirchoff stress of the second kind) :

 det (296)

and the derivative of with respect to the Green tensor reads (component notation):

 detdet (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:

• *USER MATERIAL
• Enter the CONSTANTS parameter and its value, i.e. 2.

Following line:

• (Young's modulus).
• (Poisson's coefficient).
• Temperature.

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 =210000. and =0.3 for a temperature of 400 (units appropriately chosen by the user). Recall that

 (298)

and

 (299)