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Energy release rate for cracks in finite‐strain elasticity
Author(s) -
Knees Dorothee,
Mielke Alexander
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.922
Subject(s) - quasistatic process , mathematics , elasticity (physics) , elastic energy , strain energy release rate , smoothness , nonlinear system , mathematical proof , mathematical analysis , constant (computer programming) , linear elasticity , convergence (economics) , energy (signal processing) , fracture mechanics , geometry , finite element method , physics , composite material , materials science , computer science , quantum mechanics , economics , economic growth , statistics , thermodynamics , programming language
Griffith's fracture criterion describes in a quasistatic setting whether or not a pre‐existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: if the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we rigorously derive the well‐known Griffith formula and the J ‐integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. Copyright © 2007 John Wiley & Sons, Ltd.