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  Klaus Hackl: Relaxed potentials and microstructures in damage mechanics

Click here for a postscript-version of the abstract.
In a new approach we try to formulate a specific class of damage materials in terms of a variational principle for the time-incremental problem. The fundamental numerical problem is now to find the minimum of the resulting incremental potential which is non-convex and non-coercive. An interesting feature of the approach presented is, that it can be formulated without assuming any evolution equations for the damage parameters. Instead the evolution is governed by a selforganization process via the minimization procedure explained below. One fundamental difficulty concerning the numerical simulation of damage phenomena is the non-convex character of the underlying boundary--value problem due to softening material behavior. This leads to localization phenomena and strongly mesh-dependent finite-element calculations. Regular finite--element results can now be obtained by introducing so--called relaxed energies, the simplest one given by the $R_1$--convexification $$ R_1 \Psi(\bfF) = \inf\{(1-\lambda) \Psi(\bfF-\lambda \, \bfa\otimes\bfb) +\lambda \, \Psi(\bfF+(1-\lambda) \, \bfa\otimes\bfb) | \, 0 \leq \lambda \leq 1, \, \bfa, \, |\,\bfb|=1 \}, $$ which constitutes a relaxation with respect to all first--order laminates. This means we allow the material to exhibit microstructers which are modeled in a statistical sense via the volume--ration $\lambda$. Higher--order relaxations can be introduced by iteration of the procedure above as $R_k \Psi(\bfF) = R_1^k \Psi(\bfF)$. The relaxed energies usually have to be calculated numerically by solving a global minimization problem. Because this procedure has to be carried out in every integration point of a finite--element approximation efficient optimization algorithms have to be used. Numerical results will be given and the question of the evolution of the internal variables, which now are only given in a statistical sense (as Young--measures), will be addressed.
Impressum
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