On the effect of quarter‐point element size on fracture criteria

Abstract

SUMMARYThe effect of the quarter-point element size on the solution parameters of fracture criteria is investigated in view of its effect on the computed stress intensity factors. An analytical relationship between the error in calculating crack propagation increment and stress intensity for single mode problems is obtained while, for mixed mode problems, the investigation is based on numerical experimentation. The singular element size is found to have a substantial effect on both crack propagation angle and increment for the geometries considered. INTRODIJCTIONFor the past two decades, the finite element method has been used extensively in numerical studies of fracture mechanics in both the elastic and plastic ranges. Since the determination of stress intensity factors at a crack tip is a key point for any fracture analysis, it has to be evaluated as accurately as possible. A number of finite element based procedures,'-' yielding varying accuracy, have been developed and used for this evaluation. In early studies, a highly refined mesh near a crack tip was used to piecewise continuously simulate the stress gradient' 3 2 and demonstrated the ability to achieve results within the acceptable engineering accuracies of 5 per cent in a single mode stress intensity factor.2 Tracey's special crack tip clement3 were shown to producc 5 per cent accuracy with an order of magnitude fewer degrees-of-freedom than was required using a refined mesh of low order polynomial elements. Barsoum4 and Henshell and Shaw ' havc independently shown that the collapsed quarter-point isoparametric element allows a highly accurate computation of mode I stress intensity factor for linear fracture problems.A good estimate of the separate values of K , and K,, was obtaincd by Chang et aL2 by extrapolating the near tip displacement (or stress) field to the crack tip. The sum of the mixed mode stress intensity factors can be obtaincd with the energy release rate6 and the j-integralapproaches. Procedures exist to calculate the separate values of mixed mode stress intensity factors with the last two approaches. However, multiple finite element analyses, with the same mesh and different boundary conditions, are required for this separation.'A displacement correlation technique, proposed by Shih et aL9, was used for calculating mode 'Assi\tant Profe\sor, Civil Fngineering Department tAssociate Dircctor of CICG and A w m a t e Professor, Civil bngincering Department

Generalized displacement correlation method for estimating stress intensity factors

“…This limitation of the quarter-point method was described by Tracey [10] but has largely been neglected, as it does not apply to the typical loading conditions in mechanical engineering, where crack surface tractions are absent. Although this limitation of the quarter-point displacement method does not lead to inaccuracies in many studies comparing these two methods in the context of mechanical engineering [12,13,19,22], it is highly deleterious if the method is to be used for hydraulic fracturing modeling or similar problems. The displacement extrapolation method suffers similarly since the loading scenario shown in case (c) of Figure 1 is not supported in the assumptions underlying that method.…”

“…The theoretical solutions for the stress intensity factors in this crack configuration are (15) and (16) or (18) and (19). We then seek an enhancement measure in the form of a "correction multiplier" to be added to equations (15), (16), (18), and (19).…”

“…We then seek an enhancement measure in the form of a "correction multiplier" to be added to equations (15), (16), (18), and (19). We will test the performance of the corrected/enhanced formulation on a number of more complex crack systems in next section for Methods A and B.…”

“…We will test the performance of the corrected/enhanced formulation on a number of more complex crack systems in next section for Methods A and B. The values of C I =K I /K' I and C II =K II /K' II , which are the multipliers that need to be applied to equations (15) and (16) or (18) and (19), respectively to correct the numerical solutions are shown in Figure 6 as functions of a/l E . The correction factors are significantly larger than unity, since the 6-node triangular finite element cannot accurately represent the near-tip displacement field.…”

“…5,14]. Many of the displacement-based methods were developed in the 1970's and 1980's in tandem with various special "quarter-point" finite element types [15][16][17][18][19] used in these methods. Though few new developments have been made for the displacement-based methods in the intervening decades [20], they continue to be widely used.…”