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Complete Plane Strain Problem of a Nonhomogeneous Elastic Body with a Doubly‐Periodic Set of Cracks
Author(s) -
Li X.
Publication year - 2001
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 - 10.1002/1521-4001(200106)81:6<377::aid-zamm377>3.0.co;2-q
Subject(s) - mathematics , superposition principle , mathematical analysis , plane (geometry) , cauchy distribution , kernel (algebra) , upper half plane , cauchy's integral formula , boundary value problem , complex plane , type (biology) , cauchy problem , initial value problem , geometry , pure mathematics , ecology , biology
In this paper, we wish to use complex potential methods to solve the fundamental complete plane strain (CPS) problems of a three‐dimensional nonhomogeneous elastic body with a doubly‐periodic set of cracks in the x 1 , x 2 plane. We resolve the complete plane strain state, which is a special three‐dimensional elastic system, into two linearly independent two‐dimensional (plane) elastic systems by the superposition principle of force. Based on a suitable modification of Cauchy‐type integrals, which is defined by the replacement of the Cauchy kernel 1/(t — z) by the Weierstrass zeta function ζ (t — z), the general representation for the solution is constructed, under some general restrictions the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel, and the existence of an essentially unique solution is proved.