# Damage and Failure in a Statistical Crack Model
## 2 Statistical Crack Models  
Statistical crack models (SCM) were introduced more than forty years ago to simulate the growth of damage and the failure of brittle solids in finite volume continuum computer programs [8, 9, 10]. The motivation was to relate the continuum concepts of damage and failure more closely to a mesoscale description based on microstructure, specifically microcracks. A basic the assumption is that there are many microcracks within each computational cell. Such a description necessarily must be statistical in nature since the detail of where the individual microcracks are located within a computational cell is both unknown and assumed to be unimportant. Instead, the SCM postulates the existence of a statistical distribution of tiny cracks that initially are a material property of a brittle solid. The SCM evolves that distribution within each computational cell as a function of loading. All the relevant information defining damage of the cell and ultimate failure is contained in that distribution. The two fundamental questions that the SCM must address are (1) what criteria to use to determine whether an individual crack in the distribution will grow and at what velocity? (2) what is the effect of the distribution of cracks on the effective elastic properties of the computational cell? An important sub-question of the latter is: \"what is the relation between damage and failure\"? Clarifying that relation could be considered the principal result of this paper. Given the central importance of the concept of the evolution of a statistical distribution of cracks, I suggest that the first SCM was the BFRACT model of [PERSON], [PERSON] at the Stanford Research Institute [10; 11]. The detailed micromechanical models of BFRACT were somewhat ad hoc, reflecting a compromise between microphysics and computability. For example, crack growth in BFRACT was postulated to be self-similar, i.e., that when a principal component of stress exceeded some tensile threshold, all cracks in the spectrum grew in proportion to their radius [12]. BFRACT defined a scalar damage that was the ratio of the crack opening strain to the total strain; this damage was used to degrade the elastic moduli. Failure was defined to occur when that scalar damage equaled one [11]. BFRACT was successfully applied to diverse problems of fracture and failure in metals, e.g., armor penetration, and in rocks, e.g., mining. The Bedded Crack Model (BCM) [9] was originally designed to calculate fragment size distributions resulting from explosive fracturing in mining operations. Although its structure is closely modeled on BFRACT, BCM incorporated models for crack growth and damage that are more theoretically based in micromechanics. For example, crack growth in BCM is predicted based on a generalized Griffith criterion [13] that uses thermodynamic arguments to determine when a crack of a given size is unstable to growth. Bigger cracks are found to be more unstable to growth than smaller cracks. Energy balance leads to prediction of the speed of growth, which has asymptotic limits of the order of the wave speeds of the solid [3; 14; 15]. Cracks may grow under the combined influence of normal and shear stresses, and the effect of friction on closed cracks is taken into account. There is a price to pay for this increased physicality. In BFRACT the self-similar growth law ensures that the form of the size distribution remains simple and can be described by a single parameter, e.g., the total time of growth. The situation is much more complicated in BCM where the critical crack size for growth implies that the spectrum of crack sizes will become discontinuous and where growth speed is a complicated function of loading, crack size and orientation. BCM also replaced the scalar damage model with effective modulus theory [16] in which tensor damage is defined as the dimensionless moment of the crack distribution function.