A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Please always quote using this URN:urn:nbn:de:0296-matheon-12605
  • This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable. Hence, for the viscous problem we conclude the existence of weak solutions satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.

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Metadaten
Author:Dorothee Knees, Riccarda Rossi, Chiara Zanini
URN:urn:nbn:de:0296-matheon-12605
Referee:Dietmar Hömberg
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2014/01/13
Release Date:2014/01/13
Tag:arclenght reparameterization; rate-independent damage evolution; regularity estimates; time discretization; vanishing viscosity method
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Dxx Generalized solutions / 35D40 Viscosity solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K86 Nonlinear parabolic unilateral problems and nonlinear 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]
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Cxx Plastic materials, materials of stress-rate and internal-variable type / 74C05 Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials)
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Rxx Fracture and damage / 74R05 Brittle damage
Preprint Number:1050
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