ABSTRACT INTRODUCTION
In this paper we derive an expression to predict the onset of growth of a crack in a quasibrittle material. We employ an energy criterion, which was first proposed by Griffith (1920), and is essentially equivalent to the first two laws of thermodynamics. The criterion is applied to the elastic-plastic crack of Olesiak and Wnuk (1968), which is a three-dimensional generalization of the Dugdale (1960) crack.
Our expression for the onset of growth differs in several respects from a criterion based on a crack in a brittle material (i.e., using linear elastic fracture mechanics). Furthermore, the new features are in qualitative agreement with the experimental data. Among these new features are
the dependence of the apparent surface energy, or critical strain energy release rate on crack size (the shape of the R-curve);
the effect of in-plane stresses (i.e., the stresses that do not lead to any traction on the crack faces) on crack growth; and
a brittle-ductile transition.
The energy criterion, as a necessary condition for crack growth, rests on a firm theoretical basis. However, its role as a sufficient condition is less secure. Several alternate criteria have been proposed as necessary and sufficient conditions for the onset of crack growth. We will contrast several of these, including critical strain (0lesiak and Wnuk, 1968; Goodier and Field, 1963) and crack opening displacement ("COD", Burdekin and Stone, 1966) with the energy criterion. We will show that these alternate approaches are inconsistent with the energy criterion, in the sense of being necessary conditions, and so should be rejected as criteria for growth-at least for the elastic-plastic crack.
GROWTH CRITERIA
In this section we briefly review aspects of some of the criteria proposed for crack growth. A more complete treatment can be found in (Knott, 1973; Nichols, 1979). As a statement of the problem, consider a crack, ideally penny-shaped, and embedded in a body whose dimensions are much larger than the crack. A uniform stress is applied to the surface of the body. We require a criterion to predict the onset of crack growth. The criterion, when applied to a specific model of the crack, will yield a critical crack size (for growth) as a function of the applied stress and the material properties.
In 1920, Griffith applied the idea of an energy balance to the problem of predicting crack growth. In general terms, Griffith's energy criterion states that a crack will grow if the potential energy released by that growth exceeds the energy dissipated during growth. Griffith identified the potential energy with the stored elastic strain energy. The energy dissipated by breaking bonds at the crack tip was represented macroscopically by an energy to create new surface. For penny-shaped cracks, the strain energy scales like crack length cubed (L3) whereas the energy to create new surface scales like crack length squared (L2). Application of the energy criterion then leads to the inequality (mathematical equation)(available in full paper) we emphasize that Eq. 1 is only one realization of the energy criterion, based on a particular model of the energies involved in growth.