An Analytical Study On The Half-plane Elasticity Using The Methods Of Bi-polar Coordinates And Complex Variables

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),“Micro mechanical mechanism and experimental investigation of contact fatigue of heterogeneous materials”(106112017CDJQJ328839),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”(cstc2013jcyjA70013).In the plane elasticity,problem with circular boundaries is a classical problem.During the process of mechanical design,manufacturing and geotechnical engineering,one may often meet with the problem of orifice plate.In practical engineering application,the place where orifice exists always occurred the maximum stress concentration,which will seriously impair the normal operation of the machine and mechanical performance,and may even induce the cracks,which will lead to the failure of the components eventually.For these reasons,the stress at the orifice is an important factor to be considered in the mechanical design and engineering implementation.For these problems involving circular boundaries,it is very difficult to be solved in the Cartesian coordinate system,however,it will be very convenient by the method of bipolar coordinate or complex variable function.The bipolar coordinate method can solve the problem of the orifice in the elastic half plane directly by its unique properties,and through conformal mapping,the complex function method can solve the problems which the shape of the orifice plate has complex boundary conditions.They are the most two effective methods in solving the mechanical problem of elastic half plane with circular boundaries and each has its own advantages.Therefore,some typical problems in elastic half plane problems with circular boundaries are studied in this dissertation,and combined with finite element method,the stress field of the problem is analyzed,in the hope of providing theoretical support in improving the anti-fatigue performance of the components in mechanical transmission.The main contents of this paper consist of the following four parts.In the first part,some important properties of bipolar coordinates are studied.By far,the symbol system of bipolar coordinates has different forms,which can easily cause confusion and bring great trouble to the application.Therefore,in this section,the form of bipolar coordinates and its meaning are introduced firstly,then the meaning of coordinate line is analyzed,and finally a series of properties of bipolar coordinates are discussed when the position of focal points on x-axes and y-axes,respectively.This section is a preparation of the application of bipolar coordinates in practical engineering.In the second part,the application of bipolar coordinate method in elastic half plane is studied.In this section,three classical questions are studied: the infinite plate containing two equal circular holes,the semi-infinite plate containing a single circular hole and the eccentric disk.The distribution of stress field is analyzed by means of the method of bipolar coordinates,and the finite element method is used to verify the analytical results.By comparing the results of the two methods,the validity of the analytical results is verified.In the third part,the method of complex variable function in solving the problem of elastic half plane is studied.In this section,a semi-infinite plate containing a single circular hole when a uniform pressure along the circular boundary is considered,through conformal mapping,the solution of complex variable function method in elastic mechanics is discussed in detail,and compare with the results obtained by bipolar coordinates.In the final part,by the method mentioned in the previous chapter,the semi-infinite plate with a single circular hole when a uniform tensile stress acts parallel to the straight line boundary far from the hole is discussed.The solution of the stress field on the elastic half plane and the stress distribution along the boundaries are studied and the analytical results show good agreement with the results gained by bipolar coordinates and finite element method.However,there is a little difference on the convergence of bipolar coordinate method and complex variable method.Therefore,the reason for the difference convergence between the two methods is studied,which can provide useful guidance for the choice of method in solving specific problems.