From finite to linear elastic fracture mechanics by scaling

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• 作者：M. Negri (1)
  C. Zanini (2)
  
• 关键词：Finite elasticity ; Quasi ; static crack propagation ; Linearized elasticity ; $$\Gamma $$ Γ ; convergence ; 49S05 ; 74A45
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：50
• 期：3-4
• 页码：525-548
• 全文大小：
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• 作者单位：M. Negri (1)
  C. Zanini (2)
  
  1. Department of Mathematics, University of Pavia, Via A. Ferrata 1, 27100, Pavia, Italy
  2. Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy
  
• ISSN：1432-0835

文摘

In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution with finite elasticity converges to the evolution with linearized elasticity. In the proof the crucial step is the (locally uniform) convergence of the non-linear to the linear energy release rate, which follows from the combination of several ingredients: the \(\Gamma \) -convergence of re-scaled energies, the strong convergence of minimizers, the Euler–Lagrange equation for non-linear elasticity and the volume integral representation of the energy release.