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

Please always quote using this URN:urn:nbn:de:0296-matheon-11933
  • In this work, we introduce a degenerating PDE system with a time-depending domain for complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field which are strongly nonlinearly coupled. In our proposed model, the material may completely disintegrate which is indispensable for a realistic modeling of damage processes in elastic materials. Complete damage theories lead to several mathematical problems since, for instance, coercivity properties of the free energy are lost and, therefore, several difficulties arise. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. The main aim is to prove existence of weak solutions for the introduced degenerating model. In addition, we show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions which is a novelty in the theory of complete damage models of this type. For the existence results, we had to handle the following problem: During the damage process it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a time-depending domain. In this context, two major challenges arise: Firstly, the time-dependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $\Gamma$-convergence techniques. Secondly, the time-depending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown.

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Metadaten
Author:Christian Heinemann, Christiane Kraus
URN:urn:nbn:de:0296-matheon-11933
Referee:Ralf Kornhuber
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2013/01/03
Release Date:2013/01/03
Tag:complete damage; doubly nonlinear differential inclusions; elliptic-parabolic systems; energetic solution; existence results; linear elasticity; rate-dependent systems; weak solution
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Axx General theory / 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J50 Variational methods for elliptic systems
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K55 Nonlinear parabolic equations
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K85 Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J40 Variational methods including variational inequalities [See also 47J20]
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Sxx Variational principles of physics (should also be assigned at least one other classification number in this section) / 49S05 Variational principles of physics (should also be assigned at least one other classification number in section 49)
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Axx Generalities, axiomatics, foundations of continuum mechanics of solids / 74A45 Theories of fracture and damage
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Cxx Plastic materials, materials of stress-rate and internal-variable type / 74C10 Small-strain, rate-dependent theories (including theories of viscoplasticity)
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Gxx Equilibrium (steady-state) problems / 74G25 Global existence of solutions
Preprint Number:994
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