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Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Please always quote using this URN:urn:nbn:de:0296-matheon-9544
  • A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are $B^{3/2}_{2,\infty}$ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

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Metadaten
Author:Dorothee Knees, Andreas Schröder
URN:urn:nbn:de:0296-matheon-9544
Referee:Dietmar Hömberg
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/01/02
Release Date:2012/01/02
Tag:Signorini contact; Tresca friction; a priori error analysis; cracks with selfcontact; difference quotient; finite elements; global spatial regularity
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B65 Smoothness and regularity of solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J88 Systems of elliptic variational inequalities [See also 35R35, 49J40]
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
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 / 74Mxx Special kinds of problems / 74M15 Contact
Preprint Number:841
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