Numerical methods for solving elliptic partial differential equations containing boundary singularities with the application to fracture mechanics

Abstract

The work reported in this thesis concerns the numerical solutions of two crack problems of the tearing mode 111 and opening mode I for Laplace's equation and for the Navier-Cauchy equations respectively. The mode I11 problem is the model Motz's anti-plane crack problem. The mode I problem is the plane strain problem. For the Motz problem, the Motz technique and the Tau-lines method incorporated with the local mesh refinement technique are employed. The results obtained are comparable with those published in the literature where advanced formulations such as integral equation method, finite element method or conformal transformation method were used. For the plane strain problem the Tau-lines method incorporated with the local mesh refinement technique is used The Motz technique consists of dividing the solution domain R of a given elliptic problem into two parts, the neighbourhood N[O] of the singular point 0, and the entire solution domain R except N[O]. In N[O] a truncated series of the analytic solution with unknown coefficients of the given partial differential equation is used, whilst in the rest of R the standard finite difference method is used. The Tau-lines method combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of the given partial differential equation (coupled partial differential equations in the case of plane strain problem). The latter is used to find an approximation of the solution of such a system which involves no further discretization. As determination of the stress intensity factors (SIFs) is central in the solution of crack problems in linear elastic fracture mechanics, approximations to the stress intensity factors are also obtained with good accuracy using extrapolation technique. In the treatment of the Motz problem, the Motz method gives the stress intensity factor as one of the unknowns in the computation, thus avoiding the uncertainties associated with methods which involve plotting and extrapolation techniques. In the Tau-lines method, the extrapolation technique is employed for the determination of the stress intensity factor by using the first singular term of the local analytic solution of the governing partial differential equation. In the treatment of the plane strain problem, approximations to the stress intensity factors with different crack lengths are obtained by using extrapolation techniques. These approximations are based on relations between the stress intensity factors and the near crack tip vertical displacement fields utilizing one-term series expansion of the local analytic solution of the governing coupled partial differential equations.