WIAS Preprint No. 1722, (2012)

Complete damage in linear elastic materials -- Modeling, weak formulation and existence results


  • Heinemann, Christian
  • Kraus, Christiane

2010 Mathematics Subject Classification

  • 35K85 35K55 49J40 49S05 74C10 35J50 74A45 74G25 34A12


  • complete damage, linear elasticity, elliptic-parabolic systems, energetic solution, weak solution, doubly nonlinear differential inclusions, existence results, rate-dependent systems




The analysis of material models which allow for complete damage is of major interest in material sciences and has received an increasing attraction in the recent years. In this work, we study a degenerating evolution inclusion describing complete damage processes coupled with a quasi-static force balance equation and mixed boundary conditions. For a realistic description, the inclusion is considered on a time-dependent domain and degenerates when the material undergoes maximal damage. We propose a weak formulation where the differential inclusion is translated into a variational inequality in combination with a total energy inequality. The damage variable is proven to be in a suitable SBV-space and the displacement field in a local Sobolev space. We show that the classical differential inclusion and the boundary conditions can be regained from the notion of weak solutions under additional regularity assumptions.

The main aim is to prove global-in-time existence of weak solutions for the degenerating system by performing a degenerate limit. The variational inequality in the limit is recaptured by suitable approximation techniques whereas the energy inequality is gained via Gamma-convergence techniques. To establish a displacement field for the elastic behavior in the limit, a rather technical representation result of nonsmooth domains by Lipschitz domains, which keep track of the Dirichlet boundary, is proven.

Appeared in

  • Calc. Var. Partial Differ. Equ., 54 (2015) pp. 217--250.

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