Simple criteria for brittle and ductile crack extension are applied to the stress and strain fields adjacent to the tip of a crack. They are applied at a specified distance from the crack tip, which should be related to the material’s microstructure. The basic approach is to examine each criterion and find which is satisfied first, as the external loading is increased; the predicted fracture is classified either brittle or ductile accordingly. The stress and strain fields depend upon temperature, principally through the variation of flow stress σ
0
with temperature and, to avoid excessive computation, a constitutive relation is constructed which allows stresses and strains both to be scaled in terms of σ
0
, so that major computations need to be done only at a reference temperature, for a range of applied loads. For any given crack configuration, the result of the calculation is a theoretical prediction of fracture toughness as a function of temperature. At low temperatures, the fracture toughness is low and rises rapidly with temperature, corresponding to satisfaction of the criterion for brittle failure. Above a transition temperature,
T
T
, the ductile criterion is satisfied first, and the toughness variation thereafter falls slowly as temperature increases, corresponding to failure ‘on the upper shelf’. Both the absolute level of the toughness at a given temperature and the transition temperature
T
T
are sensitive to crack size as well as specimen geometry. Although this is self-evident for cracks of microstructural dimensions, the striking feature of this work is the prediction that substantial sensitivity to size and geometry may well be displayed for cracks as large as 1 cm in materials of significance for major engineering structures. Generally, toughness increases and transition temperature decreases as crack size decreases, but these beneficial effects can be nullified by stress triaxiality. Detailed calculations are performed for a buried crack and an edge crack under conditions of plane strain and for a penny-shaped crack loaded axisymmetrically. The plane strain calculations are supplemented by ‘boundary layer’ calculations, in which the effect of specimen geometry appears through a single parameter. The close agreement of the ‘boundary layer’ calculations with the full specimen calculations offers the prospect of a simple characterization of specimen geometry and loading, without the need for geometry-specific computations. The calculations that are reported are, of course, based upon a particular model, chosen in part for computational convenience. Thus, their status is that they display possible trends which may be considered to merit further investigation, both theoretical and experimental.
CORRELATION FUNCTIONS OF STRESS AND STRAIN FIELDS IN MICRO-HETEROGENEOUS MEDIA
