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### Linear elastic materials for large strains (Ciarlet model)

In [18] 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 is the following stored-energy function developed by Ciarlet [17]:

| (146)

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

det | (147)

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

detdet | (148)

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:

- .
- .
- 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
    80769.23,121153.85,400.

defines a single crystal with elastic constants =121153.85 and =80769.23 for a temperature of 400. Recall that

| (149)

and

| (150)

where E is Young's modulus and is Poisson's coefficient.

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