J-integral Computations for Linear Elastic Fracture Mechanics in h,p,k Mathematical and Computational Framework

Abstract

This thesis presents an infrastructure for computations of the J-integral for mode I linear elastic fracture mechanics in h,p,k mathematical and computational framework using finite element formulations based on the Galerkin method with weak form and the least squares process. Since the differential operators in this case are self-adjoint, both the Galerkin method with weak form and the least square processes yield unconditionally stable computational processes. The use of h,p,k frameworks permits higher order global differentiability approximations in the finite element processes which are necessitated by physics, calculus of continuous and differentiable functions and higher order global differentiability features of the theoretical solutions. The significant aspect of this research is that with the proposed methodology very accurate J-integral computations are possible for all paths including those in very close proximity of the crack without use of special crack tip or quarter point elements at the crack tip. A center crack panel under isotropic homogeneous plane strain linear elastic behavior, subjected to uniaxial tension (mode I) is used as model problem for all numerical studies. The investigations presented in this thesis are summarized here: (i) J-integral expression is derived and it is shown that its path independence requires the governing differential equations (GDEs) to be satisfied in the numerical process used for its computations (ii) It has been shown that the J-integral path must be continuous and differentiable (iii) The integrand in the J-integral must be continuous along the path as well as normal to the path (iv) Influence of the higher order global differentiability approximations on the accuracy of the J-integral is demonstrated (v) Stress intensity correction factors are computed and compared with published data. The work presented here is a straight-forward finite element methodology in h,p,k framework is presented in which all mathematical requirements for J-integral computations are satisfied in the computational process and as a result very accurate computations of J-integral are possible for any path surrounding the crack tip without using any special treatments. Both the Galerkin method with weak form and the least square processes perform equally well.