International Journal of Fracture 83: 191–206, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Crack growth criteria incorporating non-singular stresses: Size effect in apparent fracture toughness A.V. DYSKIN The University of Western Australia, Department of Civil Engineering Nedlands, WA 6907, Australia Received 29 November 1995; accepted in revised form 19 December 1996 Abstract. Simple criteria accounting for the non-singular stresses at the crack tip are considered. They are based on the comparison of the local stress concentration with the material microstrength. The local stress concentration is estimated either as the magnitude of the conventional elastic stress ahead of the process zone, or by its averaging over the process zone length. When a criterion of this type is used to find the critical load and then the conventional fracture toughness, the latter will be dependent of the crack length. This size effect in fracture toughness manifests itself as an increase (if the non-singular part of the near-tip stress field is positive) or decrease (if the non-singular part is negative) in the apparent fracture toughness as the crack length increases. The obtained dependence is compared with available experimental data. It is also shown that when the load can be resolved into a superposition of elementary loads, the size effect can asymptotically (for long cracks) be presented as a weighted sum of the elementary size effects (i.e. the size effects associated with the elementary loads) with the weights equal to the relative contributions of the elementary loads into the total stress intensity factor. 1. Introduction Propagation of macroscopic cracks in brittle materials such as rocks, cements, concretes and composites is associated with different types of local fracture, microscopic with respect to the crack in question, but still macroscopic compared to the atomic or dislocation scales. The criteria of brittle crack propagation adopted in Linear Elastic Fracture Mechanics (LEFM) do not consider details and scales of the local fracture, since these criteria are based on comparison of the corresponding stress intensity factor (or a function of stress intensity factors) with a critical value, the fracture toughness (e.g., [1-4]). The main assumption underlying this approach is that the size of the process zone (or a zone of inelastic deformation) is negligibly small compared to the crack length (e.g., [2]; or the concept of small-scale yielding [5]). This allows, first, to use only singular terms of the stress distribution at the crack tip and, second, to combine all the material parameters governing microfracture into one, the fracture toughness, which is supposed to be a constant of the material and be determined empirically. There are some cases however, first of all in Rock Mechanics, where the experimental determination of fracture toughness is not feasible (e.g., because of large scales of the cracks involved) and therefore it should be evaluated by modelling fracture processes at the crack tip. It is also very well known that the fracture toughness is not a constant: in particular it might depend on the crack length (the size effect in fracture toughness); see for instance [6–11]. Rigorous analysis of crack propagation would require explicit modelling of the process zone. A number of models of the process zone were proposed starting from models of cohesive forces of a certain distribution (e.g., [6, 12–15]) to models specifying a certain type of post peak behaviour of the material in the process zone (e.g., [16, 17]) or models of near-tip microcracking and bridging (e.g., [18–26]); see also overview [27]. These models usually involve complex calculations, especially in the case when the crack does not propagate in its