J-Integral for Mode I Linear Elastic Fracture Mechanics in h, p, k Mathematical and Computational Framework

This paper presents an infrastructure for computations, and computations of 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 least squares processes. Since the differential operators in this case are self-adjoint, both Galerkin method with weak form and least square processes yield unconditionally stable computational processes. h, p, k framework 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 investigations considered in this paper 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 pointwise sense in the numerical process, (ii) the J-integral path Γ must be continuous and differentiable, (iii) theoretical aspects regarding the uniqueness of the integration path Γ are presented, (iv) the integrand in the J-integral must be continuous along the path as well as normal to the path, (v) influence of the higher order global differentiability approximations on the accuracy of the J-integral is demonstrated, (vi) stress intensity correction factors are computed and compared with published data. The center crack panel under isotropic homogeneous two-dimensional plane strain linear elastic behavior subjected to uniaxial tension (mode I) is used as model problem for all investigations. Some comparisons are made with published results. The work presented here is a straightforward finite element methodology in h, p, k framework 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 Galerkin method with weak form and least square processes perform equally well.

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