The Complex Variable Method For Solving Some Complicated Defect Problems Of The Plane Elasticity

The fracture phenomenon is always in relation to the holes, indentations or cracks in the material. The most obvious feature is that the stress distribution is not even in this material. This phenomenon is called stress concentration. The defects (holes, cracks, dislocation, etc.) and stress concentration are always the principal reasons, which contribute to configuration damage. As one branch of Fracture Mechanics, Linear Elastic Fracture Mechanics once has been great developed. The main job is to find the Stress Intensity Factors for several crack configurations. Westergaard method and Muskhelishvili method are two significant methods for solving such kinds of problems. Muskhelishvili method reduced the problems into solving two complex variable functions ? (z),ψ(z)satisfying the boundary condition.By means of complex variable method, using the technique of conformal mapping, this paper deals with the elastic problem that a plane with an ellipse hole with a straight crack. We get the case of functions ? (ζ),ψ(ζ)and get the analytic solutions of the SIFs at the crack tip. The results can also stimulate two perpendicular cracks.By means of the same method as above, problems of an elastic plane having symmetric power function cracks are discussed in this paper. As solving hole problems, classical complex variable method is followed. Some new conformal mapping formula is proposed so that the exterior part of the symmetric power function cracks mapped into a unit circle. Analytical solutions for the mode I-II stress intensity factors at the crack tips of symmetric power function cracks are thus obtained. These solutions analytically reduced to the classical solutions for the line crack in a special case. It is found that the size and the shape of the symmetric cubic curvier crack affect the mode I-II stress intensity factor at the crack tips.Use the same method, this paper deals with the elastic problem that a plane with an equilateral triangle hole, get the analytic solutions of the SIFs at an tip of the equilateral triangle hole.The polymorphic function ? (z) andψ(z) in a circular or outside of it is able to evolve into Taylor progression or Laurent progression. For the concentric circular ring domain, the method of progression evolution is simpler in getting the solution of the elastic equilibrium problem in elliptic ring domain. In order to give the solution of the elastic equilibrium problem in elliptic ring domain, by means of complex variable method, this paper transforming this problem into the relatively simple problem of circular ring domain, the solution is given.