A comparison of two algorithms for solving closed crack problems

In a recent paper Elvin and Leung [1] demonstrated the advantages of a new fast boundary element algorithm for solving closed crack problems. This algorithm uses the Displacement Discontinuity Method (DDM) and subdivides the problem into two sub-problems, one for the outer boundaries of the structure and one for the crack. In common with most methods for crack problems, it uses interface-or joint-elements to prevent interpenetration, following the scheme described by Crouch [2] (pp. 208-212). As we shall see, the use of such elements introduces errors of unknown magnitude in the computation of the stress intensity factor, whether the numerical scheme is the DDM, as in the present case, or the finite element method as in [3]. These errors appear not to have been noticed or explained. Thanks to the courtesy of Dr. Elvin, we have been able to compare the results obtained by [1] for their second problem with those obtained with the PATH algorithm [4-6] applied to a complementary formulation of the problem [7]. We will show that this formulation combined with the PATH algorithm eliminates these errors, albeit at the expense of some increase in computer time. We wish to thank Dr. Elvin both for giving us the details of their model and for many fruitful discussions. In the problem here examined a shallow crack EF (see Fig. 1) is subjected to compressive loading from a succession of surface point loads applied between B and C. (Details of the model are given in Table 1.) One wishes to determine the value of the stress intensity factor at one or both crack tips, the parts of the crack which are open, those which are in contact and slipping, and those which are stuck. It is important to note that the effect of each load is considered independently of the effects of the others (Elvin, pers. comm., 1999), see also [3]. In order to make the comparison of the two algorithms as meaningful as possible, we used exactly the same model as [1] with the following exceptions:-in [1] the elastic constants of the joint elements are K S = K N = 2 × 10 17 and Young's modulus E = 10 11 ; there are no joint elements in our model and E = 1.0;-[1] uses quadratic displacement discontinuity elements (QDD) at the surface and constant displacement discontinuity elements (CDD) on the crack (Elvin, pers. com., 1999), while …