Analytic function approach and numerical boundary integral method to the elasticity

Abstract

In this thesis, a new proof of the Muskhelishvili``s formulae for the displacements and stresses are presented and, on the other hand, efficient numerical schemes for the weight function method and the boundary integral method to the crack problems, in plane elasticity, are developed. The Muskhelishvili``s formulae of the displacements and stresses, which are represented by two analytic functions, are available for various plane elasticity problems. On writing the Navier``s displacement equations, by introducing two stress functions which mean volume expansion and rotation of small deformation, we can derive alternative Muskhelishvili``s formulae[21]. It is known that, in application of the boundary integral equation method(BIEM) to the crack problems, there are some serious deficiencies as a consequence of the coincidence of the crack boundaries. to overcome these problems, we propose simplified numerical methods which require less numerical computations than those developed in the previous works. One of the present method is to approximate the weight functions by using the direct formulation, which induces simple evaluation of the stress intensity factor and the crack opening displacement. In addition, we present new approach based on the boundary integral method to attain the approximate solutions for the crack problems. Numerical results of some examples show the effective convergence behavior of the proposed methods in this thesis.