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IMA Journal of Applied Mathematics Advance Access published online on August 24, 2006
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxl014
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© The Author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received August 12, 2004
Accepted July 20, 2005

Article

Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration

V. A. Kovtunenko 1 *

1 Lavrent'ev Institute of Hydrodynamics, 630090 Novosibirsk, Russia and Institute for Mathematics, University of Graz, 8010 Graz, Austria

* To whom correspondence should be addressed.
V. A. Kovtunenko, E-mail: kovtunen{at}kfunigraz.ac.at


  Abstract

Based on a level-set description of a crack moving with a given velocity, the problem of shape perturbation of the crack is considered. Nonpenetration conditions are imposed between opposite crack surfaces which result in a constrained minimization problem describing equilibrium of a solid with the crack. We suggest a minimax formulation of the state problem thus allowing curvilinear (nonplanar) cracks for the consideration. Utilizing primal-dual methods of shape sensitivity analysis we obtain the general formula for a shape derivative of the potential energy, which describes an energy-release rate for the curvilinear cracks. The conditions sufficient to rewrite it in the form of a path-independent integral (J-integral) are derived.

Keywords: curvilinear crack; nonpenetration; energy release; constrained minimization; shape sensitivity analysis; primal-dual methods.
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