Identifier

etd-03212016-125717

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

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.

Date

2016

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Antipov, Yuri

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