An evaluation of a new crack tip element?the distorted 8-node isoparametric element

Abstract

In late 1974 a report [i] came to the attention of the author that the 8-node isoparametric, planar element can be slightly modified to incorporate the required singularity of I/Jr for calculating stress intensity factors in linear elastic fracture mechanics.In addition, further references [2,3] have been made to this new singularity element. The author feels that these references [1,2,3] have not adequately presented enough data to show the powerful potential of this element.The 8-node or quadratic isoparametric element can be made to simulate the i//r singularity in stress by placing the mid-side nodal coordinates on any side (connected to the crack tip) at the quarter point of that side adjacent to the crack tip. Figure 1 illustrates this concept for 8-node isoparametric elements surrounding a crack tip. Reference [2] gives an excellent discussion of the reasons this element behaves in its unique way; therefore, no mention of the theory behind this element need be made here.Several example problems were run to investigate this unique singular element.The single edge crack problem was run to check convergence of the solution, and the results are given in Table I. Additional examples of the center-cracked plate, double edge-cracked plate, and three-point bend specimen were run. Figure 2 illustrates the grid used for the 32 element problems.The 8 and 128 element grids were proportionally spaced in a manner similar to Figure 2. The 32-element problems had typical CPU times of 2 seconds on the CDC 7600 computer.The procedure used to determine the stress intensity factor, KI, was the displacement method [4]. This method involves extrapolatIng a displacement parameter at nodal points along the free surface of the crack back to the crack tip. A similar approach using nodal stresses instead of displacements was found to produce inferior results. All results given in Table I were compared to accepted values of stress intensity factors found in [5]. Based on the present evaluation, it appears that the 8-node distorted isoparametric element is extremely cost effective in calculating stress intensity factors for two-dimensional bodies when used in conjunction with the displacement method. Acknowledgement:The author wishes to express his thanks to

Computational method for thermoviscoelasticity with application to rock mechanics. [Ph. D. Thesis]

“…For the standard constant·strain element, the convergence of the solution is no longer valid since by definition, the strain within each element is constant and as such can not adequately represent the true strain which approaches infinity near the tip. The treatment of the strain singularity in the finite element method entails use of special elements which embed the singularity for the crack tip region [66,115] or use of the conventional isoparametric element modified for the crack tip singularity [5,18,12,49].…”

Crack extension modeling with singular quadratic isoparametric elements

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Quadratic isoparametric elements with the mid-side nodes placed at the quarter position have been employed [1,2,3] for obtaining stress intensity factors K I for elastic crack problems.Several aspects of these elements that were not discussed in those studies are treated here.The displacements along the edge containing the quarter position node indicated in Figs. la and ib are given by u = u A + (-3u A + 4u B -Uc)~(r/L ) + (2u A -4u B +2UC)r/L where UA, UB, and u C are the displacements at the nodes A, B, and C respectively.

The origin of the x-y and the r-@ coordinate systems is located at the tip of the crack.

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“…Quadratic isoparametric elements with the mid-side nodes placed at the quarter position have been employed [1, 2,3] for obtaining stress intensity factors K I for elastic crack problems.…”

Comparison of methods for calculating stress intensity factors with quarter-point elements

“…Indeed, there have been many articles written about the quarter-point element with demonstrations of its efficacy [1][2][3][10][11][12][13][14][15]. But a comparison of the various methods for computing stress intensity factors with this element is lacking.…”