On solving the cracked‐beam problem by block method

Abstract

SUMMARYAn extremely accurate solution is obtained for the cracked-beam problem by one-block version of the block method (BM). The obtained numerical results demonstrate the exponential convergence of the BM with respect to the number of quadrature nodes. A simple and high accurate formula to compute the stress intensity factor is given. The comparisons with other high accurate results in the literature have been carried out.

One-block method for computing the generalized stress intensity factors for Laplace’s equation on a square with a slit and on an L-shaped domain

“…Moreover, the classic Jacobi or Gauss-Seidel iterations converge as geometrical progression with the ratio independent of n (see [12,13]). Let u ε m be an approximate value of the solution u m of system (21), with an accuracy of ε = 5 × 10 −16 in double, and ε = 5 × 10 −34 in quadruple precisions.…”

“…Finally, we mention the papers [11][12][13], in which by using the one block version of the block method (BM) given in [25,26] the highly accurate results are obtained for the solution of the Motz problem, cracked-beam problem and problems in L shaped domains respectively.…”

“…These coefficients have great importance in many applications, especially in fracture mechanics. The methods for the approximation of the coefficients a j can be divided into two groups: (i) the post-processing approach in the finite element or finite difference methods (see [5][6][7][8][9][10]), in the block method (see [11][12][13]) and in the block-grid method [14][15][16] (ii) directly calculated methods, without finding approximate solution of the boundary value problem (see [4,1,[17][18][19][20] and the references therein).…”

Analysis of the Block-Grid Method for the Solution of Laplace's Equation on Polygons with a Slit

“…In engineering problems a very important constant is the so-called stress intensity factor . This constant gives a measure of "the amount of torsion the beam can endure before fracture occurs" [10,13]. On the basis in the "singular" part for ℎ −1 = 64, = 140. of (31) we give a simple and highly accurate formula for the stress intensity factor denoting by for = 2, 6:…”