Numerical estimation of a Mode III fracture Mechanics parameter for three-dimensional elliptic boundary value problems with a crack singularity

Abstract

In this study two spatial problems of solid bodies with a surface crack singularity (V-notch) are examined. These are problems in the domain of linear elastic fracture Mechanics which are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated by first determining the general solution of the governing equation, in the vicinity of the surface crack and by expressing it as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Very fast convergence and very high accuracy are observed in implementing the method to solve a general 3-D model problem of a solid body having a cross section in the form of an ellipse and a specific spatial model problem of a steel rivet, which connects metal members of a structure. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was used to solve the same problems. The calculated value of Mode III fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is quite interesting since it involves a novel application of this algorithm in fracture Mechanics, a domain in which the method demonstrates its capability to treat problems with crack discontinuities and provide results more efficiently (less computing effort and greater accuracy) than the classical FEM with grid refinement. This advantage of the SFBIM can be exploited for research or design purposes, in several fields of applied Mechanics, by implementing its algorithm with subroutines embedded in engineering packages and aiming to make them more efficient in solving specific problems.