Theoretical analysis of crack front instability in mode I+III |
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Authors: | Jean-Baptiste Leblond Alain Karma |
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Affiliation: | a UPMC Univ Paris 6, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France b CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France c Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, MA 02115, USA d UPMC Univ Paris 6, UMR 7608, Lab FAST, F-91405 Orsay, France e UPMC Univ Paris Sud, UMR 7608, Lab FAST, F-91405 Orsay, France f CNRS, UMR 7608, Lab FAST, F-91405 Orsay, France |
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Abstract: | This paper focusses on the theoretical prediction of the widely observed crack front instability in mode I+III, that causes both the crack surface and crack front to deviate from planar and straight shapes, respectively. This problem is addressed within the classical framework of fracture mechanics, where the crack front evolution is governed by conditions of constant energy-release-rate (Griffith criterion) and vanishing stress intensity factor of mode II (principle of local symmetry) along the front. The formulation of the linear stability problem for the evolution of small perturbations of the crack front exploits previous results of Movchan et al. (1998) (suitably extended) and Gao and Rice (1986), which are used to derive expressions for the variations of the stress intensity factors along the front resulting from both in-plane and out-of-plane perturbations. We find exact eigenmode solutions to this problem, which correspond to perturbations of the crack front that are shaped as elliptic helices with their axis coinciding with the unperturbed straight front and an amplitude exponentially growing or decaying along the propagation direction. Exponential growth corresponding to unstable propagation occurs when the ratio of the unperturbed mode III to mode I stress intensity factors exceeds some “threshold” depending on Poisson's ratio. Moreover, the growth rate of helical perturbations is inversely proportional to their wavelength along the front. This growth rate therefore diverges when this wavelength goes to zero, which emphasizes the need for some “regularization” of crack propagation laws at very short scales. This divergence also reveals an interesting similarity between crack front instability in mode I+III and well-known growth front instabilities of interfaces governed by a Laplacian or diffusion field. |
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Keywords: | Crack front Mode I+III Instability Elliptic helix Griffith criterion Principle of local symmetry |
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