New option DDL_STAB : Stabilized version of damage calculus

by J. Beaurain, EDF R&D / AMA

To study the durability of nuclear facilities and to represent the irreversible damage of materials, softening constitutive laws are generally used. For that kind of laws, the uniqueness of the solution is lost so that multiple solutions which respect the first order mechanical evolution of the structures could emerge.

These solutions can however be found using alternative solvers or continuation methods. Taking into account that possible existence of multiple solutions, the stability of numerical solutions is studied by means of the second order mechanical evolution to assure their physical observance.

Stability diagram of the homogeneous solution in case of uniform traction of a beam, discretized with 75000 degrees of freedom

The criterion we choose to define the stability is the positivity of the second derivative of the total energy considering only evolutions which increase damage, to respect the unilateral constraint. This leads to the minimization of a quadratic quantity, subjected to inequality constraints of positivity.

Direction of instability

Several algorithms of inequality constraints minimization exist in the literature but their efficiency is limited in the case of important number of degrees of freedom. We overcome that limitation in Code_Aster by using the power method with projection combined with the Sorensen model reduction based on sub-space decomposition which contains the most relevant eigenvectors. To use that feature, users have to declared the option DDL_STAB into the analysis operator CRIT_STAB of STAT_NON_LINE.

Homogeneous solution direction of instability for a biaxial problem in case of uniform loads, 36000 degrees of freedom

So that it is now possible to:

• Follow the state of stability of the solution we find at each time step, whatever the solving method we use;
• Track stable bifurcated solutions using the direction of instability given by the analysis method in case of negative second derivative.

Traction of a fibre reinforced matrix (Displacements of the Disc Dr are blocked)	Unstable symmetric solution discretized with 46000 degrees of freedom	Stable non-symmetrical solution discretized with 53000 degrees of freedom

This work has resulted in three scientific papers in national and international conferences :

• "Méthode numérique pour l’étude de stabilité de modèles d’endommagement à gradient" , J. Beaurain, J.-J. Marigo, K. Kazymyrenko, Colloque National en Calcul des Structures, CSMA 2011, 9-13 mai 2011, Giens.

• "Numerical Algorithm to Study the Stability of Damage Problems", J. Beaurain, J.-J. Marigo, K. Kazymyrenko, International Conference on Computational Modeling of Fracture and Failure of Materials and Structures, CFRAC 2011, 6-8 June 2011, Barcelona, Spain.

• "Méthode numérique pour l’étude de stabilité de modèles d’endommagement à gradient" , J. Beaurain, J.-J. Marigo, K. Kazymyrenko, Congrès Français de Mécanique, CFM 2011, 28 août-02 septembre 2011, Besançon.