Distributed Dislocation Fundamentals

The whole of this book is concerned with the exploitation of Bueckner’s theorem and the modelling of cracks by the distribution of strain nuclei of various kinds along crack lines in otherwise perfect bodies. In order to introduce the technique in the simplest possible way we will first consider plane problems. Figure 2.1(a) shows a plane crack opened by a tensile field. From Bueckner’s theorem the solution to this problem can be obtained by a superposition of the problems shown in Figures 2.1(b,c). These are the stresses arising in the uncracked body, as shown in Figure 2.1(b), and the stresses induced in the unloaded body, Figure 2.1(c), due to the application of equal and opposite tractions to those present along the line of the crack in problem Figure 2.1(b). The strategy we will adopt to generate the corrective tractions shown in Figure 2.1(c) is to make a fine slit along the line of the crack; the two sides of the cut are then separated by inserting material to fill ‘the crack’, as shown. The interior of the real open crack is, of course, empty; the inserted material is simply a mathematical device — a means of generating the corrective tractions, and at the same time simulating separation of the crack faces.