On the solution of the dynamic Eshelby problem for inclusions of various shapes |
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| Affiliation: | 1. Department of Theory of Mesoscopic Phenomena, Max-Planck Institute for Metals Research, Heisenbergtrasse 3, D-70569 Stuttgart, Germany;2. Department of Civil and Structural Engineering, The University of Sheffield, Sir Frederick Mappin Building, Mappin St. Sheffield S1 3JD, United Kingdom;3. Instituto Mexicano del Petroleo Eje, Central Lazaro Cardenas, No. 152 Col. San Bartolo Atepehuacan, C.P. 07730, Mexico D.F., Mexico;1. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China;2. School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China;3. Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing Normal University, Nanjing 210023, China;1. Department of Mathematics, Quaid-I-Azam University, 45320 Islamabad 44000, Pakistan;2. Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Tarlai Kalan, Islamabad, Pakistan;3. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah 21589, Saudi Arabia;1. Department of Mechanical & Materials Engineering, University of Western Ontario, London, Ontario, N6A 5B9, Canada;2. Department of Physics & Astronomy, University of Western Ontario, London, Ontario, N6A 3K7, Canada |
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| Abstract: | In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions. |
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