Riemann-Hilbert Formalism in the Study of Crack Propagation in Domains with a Boundary

Abstract

The Wiener-Hopf technique is a powerful tool for constructing analytic solutions for a wide range of problems in physics and engineering. The key step in its application is solution of the Riemann-Hilbert problem, which consists of finding a piece-wise analytic (vector-) function in the complex plane for a specified behavior of its discontinuities. In this dissertation, the applied theory of vector Riemann-Hilbert problems is reviewed. The analytical solution representing the problem on a Riemann surface, and a numerical solution that reduces the problem to singular integral equations, are considered, as well as a combination of the numerical and analytical techniques (partial Wiener-Hopf factorization) is proposed. In this work, we begin with a brief survey of the Riemann-Hilbert problem: constructing solution of the scalar Riemann-Hilbert problem for a class of Holder continuous functions; considering classes of matrices that admit the closed-form solution of the vector Riemann-Hilbert problem; discussing numerical and analytical techniques of constructing solutions of vector Riemann-Hilbert problems. We continue with applications of the Wiener-Hopf technique to problems of Dynamic fracture mechanics: reviewing well-known solutions to problems on propagation of a semi-infinite crack in an unbounded plane in the cases of a stationary crack, a crack propagating at a constant speed, and a crack propagating at a non-uniform arbitrary speed. Based on those, we derive solutions to new problems on a semi-infinite crack propagation in a half-plane (steady-state and transient problems for subsonic speeds) as well as in a composite strip (for intersonic speeds). These latter results are new and were first derived by Y. Antipov and the author.