# Damage and Failure in a Statistical Crack Model
## 1 Introduction  
\"The whole is simpler than its parts.\" [PERSON]. [PERSON]  
In this paper I will consider the evolution of damage and failure as described by a statistical crack model (SCM) designed for use in an engineering scale simulation program. In particular, I will explore how the introduction of local crack interactions modify the qualitative description of fracture and failure of a simple SCM. My analysis will be based on data generated in a numerical laboratory, an array code that I have built which performs stochastic integrations of the fracture process. I will focus on the idealized case of horizontal fracture under vertical tension in two spatial dimensions. An SCM is designed to provide a numerical description of the fracture process based on a simplified set of idealized microphysical processes. The SCM operates in the context of a finite volume computer program and postulates that within each computational cell, a brittle material contains a large number of microcracks. Those microcracks vary in size and orientation; the initial size and orientation probability distribution is considered to be a material property. The initial locations of the cracks are assumed to be uncorrelated in space. Under load, the largest of the cracks (favorably oriented) begin to grow and the distribution evolves. All pertinent information concerning damage and failure of the cell is contained in the distribution. SCMs are used to describe fracture in high-strain rate processes for a variety of brittle and quasi-brittle solids including many metals, rocks, ceramics and concrete. A brief history of the origin and motivation of SCMs is included in Section 2. Cracks within the finite-volume computational cell are randomly distributed and there is no information in the SCM as to where the cracks are located. Cracks are assumed to grow as if isolated from each other, i.e., under [PERSON] theory [1] modified by self-consistent field corrections [2] that represent the averaged effect of all the other cracks within the cell. Interactions of a crack with its nearest neighbors were ignored as being too difficult to model. A statistical model of crack interactions would require a knowledge of the two-crack correlation function, which is not attempted in this paper. The details of crack interactions as determined by their relative locations and orientations are still not well understood. In order to proceed to model the simple case of horizontal cracks subjected to vertical tension, I wrote a new \"array\" code which serves as a realization of a single 2D finite volume computational cell in a large scale engineering code. I seeded the array code with a lattice of horizontal microcracks, randomly filled from a known size distribution. Thus in the array model, each crack has explicit knowledge of the size and distance to its nearest neighbors. I supplemented the modified Griffith growth with a simple model of crack coalescence. Under high strain rate loading, the crack distribution evolves until a single crack spans the entire computational cell; this is interpreted as a failure of the computational cell. The array code is described in Section 3 where a detailed description of the coalescence model is given. Various elements of the array code are common to previous work in statistical fracture mechanics, and the relation/differences of the array model compared to lattice models are briefly discussed at the end of Section 3. **Remark**: To avoid possible confusion, I want to emphasize that the array code is not meant to be incorporated into the engineering code. Rather, the array code is a tool to numerically study the SCM modified by crack coalescence. The array code will produce failure criteria and scaling relations that will considerably simplify the SCM implementation in the engineering code. The array code is like a numerical laboratory in which to do computational experiments. The introduction of the interaction/coalescence model significantly changes the behavior of the fracture process.