On The Elementary Solution Of Computational Contact Micromechanics: Analytical Study And Application

The work in this dissertation is a part of the projects: “Numerical and Experimental Investigations of Microstructural Evolution in Bearing Steels under Rolling Contact Fatigue”(51475057), sponsored by the National Natural Science Foundation of China; “Micro-mechanisms of White Etching Bands under High Cycle Contact Fatigue”(CDJZR14285501), sponsored by the Fundamental Research Funds for Central Universities; and Chongqing City Science and Technology Program of “Monte Carlo Simulation of the Influence of Nonmetallic Inclusions on Fatigue Life of Bearing Steels”(cstc2013jcyj A70013). The titled contact and inclusion problems are related to many fields of mechanical transmission, including bearing rotation, gear transmission, frictional transmission and so on. The above two problems are also of fundamental importance in computational contact micromechanics, which is an interdisciplinary research relies heavily on micromechanics, contact analyses and computational techniques. The current work mainly deals with three–dimensional half-space models that have more practical value and applications. Based on the principles of micromechanics, we perform analytical investigations on the elementary solutions of the elastic field due to contact loads and inclusions. The analytical solutions are also programmed on a personal computer, with numerical results compared with finite element method, so as to provide theoretical references for the mechanical transmissions and related parts with enhanced fatigue and wear properties.The main contents of this paper are listed as follows.First, based on the mechanisms of excitation-response, a novel notation is introduced to express the relationship between excitation and response. The notation represents the constant excitation distributed over rectangular-type regions whose boundaries are parallel to the coordinate axes: specifically segment(in the one dimensional case), rectangular(in the two dimensional case) or cuboidal(in the three dimensional case). The response point can be any point in the system for all the cases. The formulae of the notation, also called elementary solutions, can be obtained by integrating the Green's function in the area of excitation. The notation can be expressed simply as a combination of response primitive functions and shape parameters of the element. For responses caused any distributed excitation in an arbitrary shaped domain, the results may be approximately obtained by superposing a series of elementary solutions, each of which represents the contribution from constant excitation distributed in a basic shape, i.e. such line, rectangular or cuboidal element domain. Typical elementary solutions of the present notation in such fields as contact analysis, inclusion problem and electromagnetics are also listed for the readers' convenience. The above work may provide convenient references to the subsequent research.Secondly, for the half-space contact problem, the influence coefficient matrices and the response primitive functions of the subsurface solution are represented by the proposed notation method, as well as the elementary solutions on the surface plane of half-space. It should be noted that, although the model of the problem is three dimensional, the excitation domain is a two-dimensional rectangular plane, and accordingly the derivation of response primitive functions only involves two-dimensional integration. The solution and expression of elementary solution can be simplified by setting the center of the rectangular basic element as the coordinate origin while the response primitive functions will remain identical. Benchmark comparisons of the elementary solution with the finite element method, including respectively normal and tangential constant tractions over a square region, demonstrate the accuracy and effectiveness of the present method.Lastly, the influence coefficient matrices and the response primitive functions of the half-space inclusion problem are further obtained via the proposed notation, as well as the stress field produced by a thermal elastic inclusion and the surface displacement field. In derivation, we employ the direct method based on the Galerkin vectors and potential theory, which represent the results of response in terms of four influence coefficient matrices with the first one corresponding to the full space solution. Benchmark comparisons of the elementary displacements and stresses solution with the results yield by the finite element analysis demonstrate the accuracy and effectiveness of the present method.