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

Abstract

This paper presents a detailed and rigorous mathematical verification of Yue's approach, Yue's treatment, Yue's method and Yue's solution in the companion paper for the classical theory of elasticity in n-layered solid. It involves three levels of the mathematical verifications. The first level is to show that Yue's solution can be automatically and uniformly degenerated into these classical solutions in closed-form such as Kelvin's, Boussinesq's, Mindlin's and bimaterial's solutions when the material properties and boundary conditions are the same. This mathematical verification also gives and serves a clear and concrete understanding on the mathematical properties and singularities of Yue's solution in n-layered solids. The second level is to analytically and rigorously show the convergence and singularity of the solution and the satisfaction of the solution to the governing partial differential equations, the interface conditions, the external boundary conditions and the body force loading conditions. This verification also provides the easy and executable means and results for the solutions in n-layered or graded solids to be calculated with any controlled accuracy in association with classical numerical integration techniques. The third level is to demonstrate the applicability and suitability of Yue's approach, Yue's treatment, Yue's method and Yue's solution to uniformly and systematically derive and formulate exact and complete solutions for other boundary-value problems, mixed-boundary value problems, and initial-boundary value problems in layered solids in the frameworks of classical elasticity, boundary element methods, elastodynamics, Biot's theory of poroelasticity and thermoelasticity. All of such applications are substantiated by peerreviewed journal publications made by the author and his collaborators over the past 30 years.

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

“…This paper and the companion paper [24] have three objectives: 1) to give a step by step mathematical formulation process of the approach, treatment, method and solutions developed by the author for elasticity in nlayered solids; 2) to present a detailed and rigorous mathematical verification to the questions on the convergence, singularity and satisfaction of the solution; 3) to show the approach, treatment and method applicable to transversely isotropic layered solids, mixed-boundary value problems, boundary element method, and initialboundary value problems in the framework of elastodynamics, thermoelasticity and Biot's theory of poroelasticity.…”

“…Do they satisfy the governing partial differential equations and the boundary and interface conditions? These questions are analytically and rigorously examined and verified in the companion paper [24].…”

“…In the companion paper [24], the fundamental singular solutions in exact closed-form are presented for the basic and classical boundary-value problems in either homogeneous or bi-homogeneous solids and show their mathematical properties and singularities. Furthermore, the mathematical properties of the solutions for elasticity in n-layered solids are examined and presented to analytically show their convergence, singularity and satisfaction.…”

“…His closed-form fundamental singular solutions can automatically and analytically degenerate as Kelvin solution, Boussinesq solution and Mindlin's solution once the material properties become homogeneous and isotropic. Details of his mathematical approach, treatment, method and solutions are presented in this paper and the companion paper [24] using the model of n-layered solid with both transverse isotropy and isotropy.…”

Response of bimaterials with interfacial tension subjected to axisymmetric body force

“…The interfacial tension is modeled as the residual stress in the pre-tensed interfacial atom membrane on the basis of Gurtin and Murdoch's surface/interface theory [5,57]. Similar to the generalized Kelvin's solutions and/or Yue's solutions [58][59][60], the problem under consideration is addressed in a systematic and concise manner, and the final solutions are uniformly given in terms of semi-infinite line integrals. The general applicability of the present model is justified by its absolute convergences to classical bimaterials with bonded interface [39,41] and unsinkable interface.…”