Fundamental solution and the weight functions of the transient problem on a semi-infinite crack propagating in a half-plane

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 96; no. 10; pp. 1156 - 1174
• Main Authors: Antipov, Y. A., Smirnov, A. V.
• Format: Journal Article
• Language: English
• Published: Weinheim Blackwell Publishing Ltd 01.10.2016; Wiley Subscription Services, Inc
• Subjects: Boundaries; Constants; Crack propagation; Cracks; Exact solutions; Fourier transforms; Mathematical analysis; Mathematical models; system of singular integral equations; transient problem; Weight function; Wiener-Hopf method
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201400272

The two‐dimensional transient problem that is studied concerns a semi‐infinite crack in an isotropic solid comprising an infinite strip and a half‐plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time‐independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann‐Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed‐form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi‐infinite crack beneath the half‐plane boundary at piecewise constant speed. The two‐dimensional transient problem that is studied concerns a semi‐infinite crack in an isotropic solid comprising an infinite strip and a half‐plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time‐independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann‐Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed‐form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi‐infinite crack beneath the half‐plane boundary at piecewise constant speed.