# Facilitate the introduction of a new constitutive law

*by J.-M. Proix, EDF R&D / AMA*

**Aim : facilitate the development of a new constitutive law, automatically benefiting of the implicit integration from the explicit integration.**

Here is a generic method that applies to the elasto-visco-plastic (standard or nonstandard) constitutive laws, with additive decomposition of deformations (large deformations are then automatically available via ` GDEF_LOG ` ).

One have just to write the two routines necessary for the explicit integration:

- Reading of the material coefficients,
- Calculation of the derivatives of all internal variables with respect to time (the first being the components of the (visco)-plastic deformation).

**Bonus !** We can then take advantage of both the implicit integration and of the very simple programming of the explicit integration : for this, just change `ALGO_INTE ’RUNGE_KUTTA’` in `ALGO_INTE ’NEWTON_PERT’`.

**How is this possible ?** Residues required for the implicit integration by the Newton method are automatically obtained from derivatives of internal variables. The Jacobian matrix for local resolution by Newton’s method is calculated by perturbation, and the tangent matrix is calculated analytically from the Jacobian. All detailsare in the following document D5.04.01 : « Développer un nouveau comportement » .

Residues required for the implicit integration by the Newton method are automatically obtained from derivatives of internal variables. The Jacobian matrix for local resolution by Newton’s method is calculated by perturbation, and the tangent matrix is calculated analytically from the Jacobian. All details are in D5.04.01: "Developing a new constitutive law" .

**Advantages** : This allows to benefit from the stability of the implicit, ie the possibility of larger time steps, and a tangent matrix, interesting for studies, and especially the ease of programming. This allows to easily prototype a new constitutive law.

**Disadvantages** : The calculation of the Jacobian matrix perturbation is slightly more expensive in CPU time that a Jacobian programmed one, but the development effort is much lower ! And the matrix obtained by perturbation has generally a good convergence. Then, when the constitutive law is validated, it is always possible to program the exact Jacobian matrix.

**Example** :

**Conclusion** : The above example shows that with some rapid development (typically 1 to 2 hours), you can benefit from a nonlinear behavior that can be used in implicit as well as in explicit, taking advantage of large deformations. To further gain in performance for studies, one should program the tangent matrix, which requires a development effort much more important ; in the case of the Lemaître behavior used here for the comparison, the formulation and programming of the tangent matrix have required much more than an hour!

*Available in the 11.2.3 version and future releases*