On the inviscid limit of a model for crack propagation

Dorothee Knees, Alexander Mielke, and Chiara Zanini

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Submission date: 12. Nov. 2007
Pages: 40
published in: Mathematical models and methods in applied sciences, 18 (2008) 9, p. 1529-1569 
DOI number (of the published article): 10.1142/S0218202508003121
Bibtex
MSC-Numbers: 49J40, 49L25, 35K90, 74B20, 74G65, 74R10
Keywords and phrases: Rate-independent problems, energy release rate, Griffith fracture criterion, vanishing viscosity method
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Abstract:
We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.