49L25 Viscosity solutions
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- 2011 (1)
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The focus of this note lies on the numerical analysis of models describing the propagation
of a single crack in a linearly elastic material. The evolution of the crack is modeled as a
rate-independent process based on the Griffith criterion. We follow two different approaches
for setting up mathematically well defined models: the global energetic approach and an
approach based on a viscous regularization.
We prove the convergence of solutions of fully discretized models (i.e. with respect to time
and space) and derive relations between the discretization parameters (mesh size, time step
size, viscosity parameter, crack increment) which guarantee the convergence of the schemes.
Further, convergence rates are provided for the approximation of energy release rates by
certain discrete energy release rates. Thereby we discuss both, models with self-contact
conditions on the crack faces as well as models with pure Neumann conditions on the crack
faces. The convergence proofs rely on regularity estimates for the elastic fields close to the
crack tip and local and global finite element error estimates. Finally the theoretical results
are illustrated with some numerical calculations.