Applications of the Theory of Critical Distances to the Prediction of Brittle Fracture in Metals and Non-metals

The theory of critical distances (TCD) proposes that the failure of a body containing a stress concentration (e.g. a crack or notch) can be predicted using elastic stress information in a critical region close to the notch tip. This paper investigates the use of TCD for predicting brittle fracture. The critical region is defined in terms of a characteristic material length constant, L, which is a function of the fracture toughness Kc and a failure stress, σo. For very brittle materials (ceramics), σo is equal to the plain-specimen strength but for polymers and metals σo has a larger value. Two complications arise: (i) there exist non-damaging notches whose strength is equal to the plain-specimen strength, and; (ii) strength varies with the degree of constraint. These effects can be incorporated into TCD allowing predictions of experimental data for many types of materials and stress concentration features. Introduction This paper is concerned with the prediction of brittle fracture, defined as any failure which occurs by crack initiation and/or propagation in a rapid, unstable manner. Stress concentrations such as notches and cracks frequently cause brittle fracture even in relatively ductile materials. Traditional methods for the prediction of failure in materials use one of two approaches: (i) failure occurs when the maximum stress (or strain) in the body reaches some critical value, say the ultimate tensile strength σu; (ii) failure occurs when the stress intensity associated with a crack reaches some critical value: the fracture toughness Kc. Unfortunately these approaches only work in a limited number of cases; approach (i) works only for plain (i.e. unnotched) specimens in simple tension, or for notches which are so large that the local gradient of stress near the notch is negligible. Even mild stress gradients (e.g. bending loads applied to plain specimens) cause difficulties for this approach. Approach (ii), on the other hand, only works for long, sharp cracks; it is known to break down if the crack length is physically short (sub-millimetre) or if applied to notches having a significant root radius. In practice, then, many industrial components have stress-concentration features which neither of these approaches can handle, and several different theories have been suggested to solve this problem. It has been recognised for some time that the theory of linear elastic fracture mechanics (LEFM), in order to be strictly valid, requires another length constant. Irwin [1] recognised this in his use of the plastic zone size, ry, as an addition to the physical crack length. Broberg [2] made the following argument: if we assume that failure of material near the crack tip occurs at some critical stress, σc, then, given the evidence that the nominal fracture stress, σf, is a function of the crack length, a, it follows on dimensional grounds that another length constant is required, which will be a material parameter, L, so that we can write σf = σcf(a/L). Over the years, various theories have arisen which use this length constant, explicitly or