r7.05.01 Mechanical stability criteria

Summary:

This document presents the various mechanical stability criteria available in Code_Aster. They can be classified into two categories:

• The stability criterion associated with conservative systems, which is presented as the generalization of the Euler criterion based on the analysis of the updated global stiffness matrix.

• The stability criterion associated with dissipative systems, which must take into account the irreversibility constraints associated with energy dissipation.

These criteria are used to distinguish, in quasistatic problems, unstable numerical solutions resulting from the equilibrium calculation performed in the finite element method (first derivative of zero energy but negative second derivative) from stable physical solutions for which the second derivative of energy is positive.

The criteria presented in this document are directly transposable to the framework of dynamics, but as they do not take into account either the mass matrix or the damping matrix, one cannot speak of a dynamic stability criterion in the classical sense (for example, of damping becoming negative or zero).

These criteria are called within the operators STAT_NON_LINE and DYNA_NON_LINE, in order to be able to be evaluated at each step of the incremental resolution that is almost static or transient dynamic nonlinear.

• 1. Stability of a conservative system
  • 1.1. Definition of the stability of a conservative system
  • 1.2. General concept of buckling
  • 1.3. Writing the mechanical problem
  • 1.4. Stability study
  • 1.5. Implementation in the code
  • 1.6. Euler criterion
  • 1.7. Nonlinear criterion
  • 1.8. Generalization to dynamics
  • 1.9. Validation of developments
  • 1.10. Extension of the buckling criterion to the treatment of elastoplastic behavior
  • 1.11. Conclusion
• 2. Stability of a dissipative system
  • 2.1. Definition of the stability of a dissipative system
  • 2.2. Writing in the framework of the finite element method
  • 2.3. Optimization algorithm under inequality constraints
  • 2.4. Implementation in the code
  • 2.5. Application example: Case of the bar under uniform tension
  • 2.6. Conclusion
• 3. Bibliography
• 4. Description of document versions