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Fundamental solution and the weight functions of the transient problem on a semi‐infinite crack propagating in a half‐plane
Author(s) -
Antipov Y. A.,
Smirnov A. V.
Publication year - 2016
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201400272
Subject(s) - mathematics , mathematical analysis , piecewise , isotropic solid , laplace transform , scalar (mathematics) , boundary value problem , semi infinite , collocation method , plane (geometry) , isotropy , geometry , physics , differential equation , ordinary differential equation , quantum mechanics
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.