

Klaus Hackl: Relaxed potentials and microstructures in damage mechanics
Click here for a postscriptversion 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
timeincremental problem. The fundamental numerical problem is now to find the minimum of the
resulting incremental potential which is nonconvex and noncoercive.
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
nonconvex character of the underlying boundaryvalue problem due to softening material
behavior. This leads to localization phenomena and strongly meshdependent finiteelement calculations.
Regular finiteelement results can now be obtained by introducing socalled
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 firstorder laminates. This means we allow
the material to exhibit
microstructers which are modeled in a statistical sense via the volumeration $\lambda$.
Higherorder 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
finiteelement 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 Youngmeasures), will be addressed.
