This study investigates the relation between two approaches for modeling fracture propagation. The first one is the classical approach, in which fracture propagates once the stress intensity factor exceeds a critical value, called fracture toughness. In the second approach, the fracture propagates once the tensile stress ahead of the fracture tip exceeds a critical value, called tensile strength. The purpose of this study is to examine the relation between the two approaches and to determine a methodology to make them equivalent. To address the goal, propagation of a radially symmetric fracture is first analyzed. A universal relation between the tensile strength and fracture toughness is obtained, which is then verified via a series of numerical examples. It is found that in order to capture the fracture toughness the tensile strength should be varied with respect to the mesh size and other material parameters. The developments are then applied to a three-dimensional distinct element code, which can be used in various applications involving modeling of a jointed and blocky material. An additional challenge with the distinct element code lies in the fact that the use of uniform value of tensile strength does not lead to a spatially uniform apparent fracture toughness. The latter is caused by mesh distortions and orientation of the elements relative to the fracture front. This problem is successfully addressed by introducing a variation of the tensile strength relative to local geometry of the mesh in the vicinity of the fracture front. The obtained result develop a procedure to accurately model fracture toughness in numerical methods that use tensile strength as a fracture propagation criterion.
Keywords: Hydraulic fracturing, Numerical modeling, Fracture propagation, Distinct element method,