Modeling Slip Zones with Triangular Dislocation Elements

We derive in algebraic form the displacement and stress fields produced by a triangular element of constant slip by superposing the solution of Comninou and Dundurs (1975) for an angular dislocation in an elastic half-space. Triangular elements are more flexible for simulating complex geometries than the rectangular elements widely used for modeling slip zones as a superposition of constant slip elements. As an example, we use triangular elements to determine the distribution of slip on a planar surface caused by a prescribed stress drop. Because the slip in elements is uniform, the slip does not taper to zero at the edges of the slipping zone. Consequently, the strain energy in volumes containing the slip zone edge and the stress drop averaged over the slip zone are unbounded. To investigate the effects of these features, we compare our results using uniform slip elements with those from the more elaborate procedure of Wu et al. (1991) that takes proper account of stress singularity at the edge of the slipping zone. The comparison indicates that, for a prescribed uniform stress drop, the uniform slip model slightly overestimates the free surface displacements. The predicted slip surface displacements are more severely overestimated, particularly near the edges of the slipping zone. Nevertheless, extrapolation of the slip surface displacements yields values for the stress intensity factors, the coefficients of the singular stresses near the edge of a crack. The values of stress intensity factors are within 10% of those results obtained by Wu et al. (1991) for the same number of elements. INTRODUCTION A prevalent method for investigating subsurface slip is to compare measured surface displacements with those predicted by elasticity solutions for prescribed displacement discontinuities in an elastic half-space. Of these solutions, the most widely used are those for uniform slip in rectangular zones (Chinnery, 1961; Mansinha and Smylie, 1971). The reason for their wide use is tha t the solutions are obtained in algebraic form, and, consequently, quantit ies of interest such as stresses or displacements can be obtained without extensive computation. Furthermore, there is no need for integration when rectangular dislocation surfaces are combined to model a region of nonuniform slip: the contributions of the individual rectangles are simply summed. The availability of these solutions for rectangular dislocation surfaces only is, however, a severe limitation. When a slipping zone with a curved boundary is modeled by dividing the region into rectangles, the result is a staircase edge, and curved surfaces tha t are not cylindrical cannot be simulated easily with rectangles. In this paper, we construct the solution for a t r iangular element of uniform discontinuity in a half-space. The solution is obtained by appropriately superposing the solution for an angular dislocation in an elastic half-space derived by Comninou and Dundurs (1975). Because the solution contains tha t for the rectangular element as a special case, it is inevitably more complicated than 2153 2154 M. JEYAKUMARAN, J. W. RUDNICKI, AND L. M. KEER that for a rectangular dislocation surface. Nevertheless, the solution for the tr iangular element is obtained in algebraic form. The tr iangular element is then used in a procedure to determine the distribution of displacement discontinuities (slip) that result when a stress drop is prescribed on a surface interior to the half-space. In brief, the surface is divided into triangles, the traction is evaluated at the center of each triangle, and the resulting linear equations relating stress at these points to the displacement discontinuity are solved. A limitation of this approach is tha t the assumption of uniform displacement discontinuity in each element precludes the discontinuity from tapering smoothly to zero at the edges of the slipping zone. As a result, the strain energy in volumes containing the edge of the slipping zone is unbounded. Furthermore, because of the strong stress singularity predicted at the edges of an element in which the displacement discontinuity is uniform, the stress drop averaged over the element is unbounded. Consequently, parameters needed to implement simple fracture criteria for the advance of slipping zones, such as stress intensity factors or energy release rates, are difficult to extract from models using such elements. These limitations are common to any approach assuming uniform displacement discontinuity in elements (e.g., Chinnery, 1961). In order to investigate the effects of these features, we compare the results obtained using uniform displacement discontinuity elements with those obtained from a more elaborate numerical method that takes proper account of the stress behavior at the edge of a slipping zone. Wu et al. (1991) have described a numerical method for modeling slipping zones subjected to prescribed stress drop in an elastic half-space. Because the method incorporates the exact asymptotic form of the displacement discontinuity near the edge of the slipping zone, the difficulties jus t mentioned for methods using elements with uniform displacement discontinuities are overcome. In particular, accurate results are obtained for the stress intensity factors. The method is, however, computationally more intensive than tha t using the uniform slip elements. Consequently, par t of our motivation for developing the uniform slip triangle was to determine whether satisfactory results can be obtained with it at less computational cost. The body of the paper begins by describing the construction of the solution for a t r iangular dislocation element in a half-space from the solution for an angular dislocation in a half-space (Comninou and Dundurs, 1975). The remaining sections describe the use of the element to model zones of prescribed stress drop and comparison of the results with the more elaborate method of Wu et al. (1991). ELASTIC FIELD OF A TRIANGULAR ELEMENT An angular dislocation is shown in Figure 1. The origin of the x i coordinate system is on the surface of the half-space with x 3 directed out of the half-space. The dislocation lies in a vertical plane that makes an angle ~o with the x 2 axis and is bounded by the two lines PQ and PR tha t extend to infinity in the negative x 3 direction. The line PQ is vertical (parallel to x 3) and the line PR makes an angle a with the vertical. The coordinates of the intersection point P are ~iThe positive and negative sides of the dislocation surface are defined by the directions of the arrows on QP and PR and the right-hand rule: with the thumb pointing in the direction of the arrows on QP and PR, the fingers of the right-hand curl, without intersecting the surface of discontinuity, from the negative ( ) to positive ( + ) side of the surface. MODELING SLIP ZONES 2155 The elastic fields of the angular dislocation are the displacements and stresses that result when a uniform displacement discontinuity is prescribed on the dislocation surface: U + -U:~ = B i , ( 1 ) where u ÷ and u denote the displacements of the positive and negative sides of the surface. The constants B i a r e the components of the Burgers ' vector relative to the x i coordinate system. The solution to this problem is given by Comninou and Dundurs (1975). They begin with the solution of Yoffe (1960) for an angular dislocation in a full-space. They then use an image dislocation to remove the shear tractions from the surface of the half-space and superposition of the