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 nlayered 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.

Yue’s solution of classical elasticity in n-layered solids: Part 2, mathematical verification

“…(55c), (60c), (73c) and (78c) as well as the Appendices A to D in Ref. [1] do not have any zero value for any in 0£ < þ 1 [7,8]. i.e.,…”

“…Similarly, they can be used to show the solutions given in the Sections 4 and 5 for the layered solids in Ref. [1] because they are also systematically expressed in matrix forms in terms of either inverse 2-D Fourier integral transforms or inverse Hankel integral transforms. The integrals are improper integrals with depending parameters and 2D-infinite or semi-infinite integration intervals.…”

“…[1], it can be shown that the determinants of the four coefficient matrices M Ap , M Aq , M Qp and M Qq respectively in Eqs. (55c), (60c), (73c) and (78c) as well as the Appendices A to D in Ref.…”

“…In the companion paper [1], details of the Yue's approach, Yue's treatment, Yue's method and Yue's solution have been presented for the mathematical formulation of the solutions in n-layered solids in both transform and physical domains and in both Cartesian and cylindrical coordinate systems, where n is an arbitrary nonnegative integer. This paper presents a detailed and rigorous mathematical verification of the approach, the treatment, the method and the solution.…”

“…(1) is in the forms of improper integrals of infinite intervals either over the entire horizontal plane ð -1 < ξ, η <þ 1Þ or along the radial axis 0£ <þ 1. The improper integrals have many depending parameters including 5 Â ðn þ 2Þ elastic constants ðc ij , i ¼ 1,:2,3,4,5; j ¼ 0,1,:::, n, n þ 1Þ, n layer thicknesses ðh j , j ¼ 1,2,:::, nÞ, the three independent variables ðx, y, zÞ and the applied loading vectors f ðξ, ηÞ.…”