ABSTRACT
The first analytical result for fracture mechanics is the stress concentration at crack tips and sharp corners obtained by Inglis (1913). The main result of Inglis (1913) is that the stress singularity near the crack tip is 1/r1/2, where r is the distance from the crack tip. Based upon this result, Griffith (1920) presented his celebrated fracture criteria for ideally brittle solids using the concept of “minimum potential energy”, in which surface energy was incorporated. In this classic paper, Griffith used his results to explain why glass and quartz crystals have tensile strength that is much smaller than that for the perfect solids containing no cracks. Although Griffith’s classic paper was originally motivated by its applications to brittle materials, most of the developments in fracture mechanics were, however, fostered by its application to ductile metallic solids. For example, the concept of energy release rate, G, was originally proposed by Irwin (1956) for steel and aluminum alloys. The application of fracture mechanics to rock-like solids remains a relatively new area of research; and many fundamental issues remain to be resolved. The main difficulty in applying classical fracture mechanics to rocks is due to the fact that rock masses are normally subjected to all-round compressions when tensile fractures start to grow. It is usually believed that the deviatoric stress and stress difference will cause a local tensile stress field near the crack tip. However, the relation between the local tensile stress and the far field compressions remains uncertain. Although various wing crack models have been commonly adopted to account for the micromechanics of tensile cracks under compression (e.g., Nemat-Nasser and Obata, 1988; Ashby and Hallam, 1986), actual wing cracks are seldom observed in real rocks (e.g., Tapponnier and Brace, 1976; Kranz, 1979). We will start the chapter by following the tradition approach. We will first consider the stress concentration at an elliptical hole subject to tensile stress, then specialize the solution to crack geometry (i.e., the size of the minor axis is very small compared to the major axis). This consideration then extends to shear crack and tearing crack. The universality of the order of stress singularity at the crack tip by Williams (1957, 1959) will be discussed for all mode I, mode II, and mode III. The idea of energy release rate is then introduced, followed by the discussion on the J-integral. The method of superposition using the Westergaard stress function is summarized. The concept of cohesive crack is introduced through the growth of slip surface in slopes and in fault zones. The wing crack model is demonstrated by considering the local tensile stress induced by compression. Bažant’s model of size effect on compressive strength is formulated via the application of the J-integral.
