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Author: Knees, Dorothee × Author: Mielke, Alexander × End date: 2009 × Start date: 2006 × Type: Text ×

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Now showing 1 - 3 of 3
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    Crack growth in polyconvex materials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Knees, Dorothee; Zanini, Chiara; Mielke, Alexander
    We discuss a model for crack propagation in an elastic body, where the crack path is described a-priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed.
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    Energy release rate for cracks in finite-strain elasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Knees, Dorothee; Mielke, Alexander
    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 derive rigorously 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.
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    On the inviscid limit of a model for crack propagation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Knees, Dorothee; Mielke, Alexander; Zanini, Chiara
    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.