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
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
the simplest one given by the $R_1$--convexification
\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 $\lambda$.
can be introduced by iteration of the procedure above as $R_k \Psi(\bfF) = R_1^k \Psi(\bfF)$.
relaxed energies usually have to be calculated numerically by solving a global
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.