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Yue’s solution of classical elasticity in n-layered solids: Part 1, mathematical formulation

Abstract

This paper presents the exact and complete fundamental singular solutions for the boundary value problem of a n-layered elastic solid of either transverse isotropy or isotropy subject to body force vector at the interior of the solid. The layer number n is an arbitrary nonnegative integer. The mathematical theory of linear elasticity is one of the most classical field theories in mechanics and physics. It was developed and established by many well-known scientists and mathematicians over 200 years from 1638 to 1838. For more than 150 years from 1838 to present, one of the remaining key tasks in classical elasticity has been the mathematical derivation and formulation of exact solutions for various boundary value problems of interesting in science and engineering. However, exact solutions and/or fundamental singular solutions in closed form are still very limited in literature. The boundary-value problems of classical elasticity in n-layered and graded solids are also one of the classical problems challenging many researchers. Since 1984, the author has analytically and rigorously examined the solutions of such classical problems using the classical mathematical tools such as Fourier integral transforms. In particular, he has derived the exact and complete fundamental singular solutions for elasticity of either isotropic or transversely isotropic layered solids subject to concentrated loadings. The solutions in n-layered or graded solids can be calculated with any controlled accuracy in association with classical numerical integration techniques. Findings of this solution formulation are further used in the companion paper for mathematical verification of the solutions and further applications for exact and complete solutions of other problems in elasticity, elastodynamics, poroelasticty and thermoelasticity. The mathematical formulations and solutions have been named by other researchers as Yue’s approach, Yue’s treatment, Yue’s method and Yue’s solution.

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Correspondence to Zhong-qi Quentin Yue.

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Professor Zhong-qi Quentin Yue obtained his BSc and MSc degrees’ education in earthquake and geology from Peking University in Beijing (1979 to 1986). He obtained his Ph.D. degree’s education in geotechnical engineering from Carleton University in Ottawa (1988 to 1992). Prior to joining HKU on December 1, 1999, Professor Yue had a total of ten years professional working experience in Beijing, Ottawa, and Hong Kong. He chartered as a geotechnical engineer in Ontario in 1995 and in Hong Kong in 1998.

Professor Yue’s research interests include the following six areas: 1) Formulations and applications of new analytical solutions for predicting mechanical behavior of layered elastic and poroelastic solids under various conditions; 2) Quantifying, analyzing and predicting the mechanical behavior of non-homogeneous geomaterials with numerical and digital image processing techniques; 3) Prevention and mitigation of landslide hazards in complex slope grounds from coastal soft soils to mountainous rock masses; 4) Development and invention of automatic drilling process monitor (DPM) for in-situ continuously measuring and recording the strength and distribution of rock masses at depth in real time; 5) Design methods for tunnels and caverns in both saturated soft soils and soft or hard rocks, and their long-term stability and integrity; 6) Gas cause and mechanism of earthquakes, volcanos, landslides, tsunamis, and rock bursts. Most importantly, Professor Yue has discovered the existence of a thin spherical methane gas layer between the crust and the mantle of the Earth and has developed a more realistic model for the Earth.

Professor Yue has published 3 books, 187 journal articles, 133 conference papers and two USA/China patents. He has given more than 430 invited lectures/seminars at more than 100 conferences and more than 100 institutions worldwide. He has received some prestigious awards including the Excellent Contributions Award from International Association for Computer Methods and Advances in Geomechanics in 2008.

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Yue, Zq.Q. Yue’s solution of classical elasticity in n-layered solids: Part 1, mathematical formulation. Front. Struct. Civ. Eng. 9, 215–249 (2015). https://doi.org/10.1007/s11709-015-0298-6

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Keywords

  • elasticity
  • solution
  • layered solid
  • graded material