Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach |
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| Institution: | 1. School of Engineering, University of Seville, Group of Elasticity and Strength of Materials, Camino de los Descubrimientos s/n, Seville E-41092, Spain;2. Colorado School of Mines, 1500 Illinois Street, Golden, CO 80401, USA;1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China;2. State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, China;1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany;2. Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany;3. Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany;4. Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland;1. Department of Mechanical Engineering, V. R. Siddhartha Engineering College, Vijayawada, Andhra Pradesh 520007, India;2. Department of Mechanical Engineering, JNTU College of Engineering, Hyderabad, Telengana 500085, India |
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| Abstract: | An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented. |
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