# Damage and Failure in a Statistical Crack Model
## The Array Code
### Lattice Models and the Array Model  
\"Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture \" [21]  
There are many similarities between lattice models and the array model; see [22; 23] for reviews of lattice models. The general idea of the lattice model is to discretize a continuum into a lattice \begin{table}
\begin{tabular}{l c} \hline \hline Shear modulus G & 126. kbar \\ \hline Bulk modulus B & 124. kbar \\ \hline Poisson ratio \(\
u\) & 0.12 \\ \hline Youngs modulus E & 282. kbar \\ \hline First Lamé modulus \(\lambda\) & 40. kbar \\ \hline Rayleigh wave speed v & 10.47 km/s \\ \hline Cell DX & 100 μm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters simulating beryllium. \begin{table}
\begin{tabular}{l c} \hline \hline Cell DX & 100 μm \\ \hline Subcell dx & 1 μm \\ \hline Average crack radius \(\varepsilon\) & 0.125 μm \\ \hline Max crack radius \(c_{max}\) & 0.5 μm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Length scales. in which simple elements, e.g., normal and shear springs or beams, with random properties are connected. A fracture process is initiated by increasing the load until the first element in the lattice fails. To compensate for the failed element, part of the load must be redistributed, increasing the load on most other elements. The redistribution coupled to further increasing load will then cause a second element to fail--and so on. At some point, the process may become self-sustaining, not requiring further increase in load, initiating an avalanche [4]. Beyond the similarities of geometry and randomness, there are also at least two important differences between lattice models and the array model. First, in the lattice model, the constitutive relations are typically linear perfectly elastic. The failure criterion of a lattice element is most often based on either defined critical stress or a critical strain. Simplicity is emphasized in the lattice models [23]. In the array model, crack growth and the associated damage of the elastic moduli is a continuous process. The Griffith criterion [1] associates a different critical stress for each crack size, e.g., the inverse prediction of Equation (1). Significantly, [PERSON]'s law and [PERSON]'s law introduce length and time scales into the array model. Thus, while the constitutive laws and the failure criterion are more complicated in the array model, they allow scaling laws and are easily implemented. A second important difference lies in the way the external loading is felt by the elements. In the lattice models, a detailed calculation determines how the load is distributed to each individual element. In the array model, each crack sees the same stress, namely the unique external stress modified by the self-consistent field corrections [2]. The latter picture is consistent with the idea that the array code is representing one computational cell of a finite volume engineering level code. There is essentially no structure or information calculated in the engineering codes at length scales smaller than \(D\!X\). A more quantitative way to distinguish the various length scales in given in [24]. In the array model, \(dx\) is the microscale and defines the representative volume element where microstructure is resolved. \(D\!X\) is the mesoscale and represents the continuum level where damage and failure are defined. The engineering code, a mesh of finite volume computational cells, represents the macroscale. As noted in the quote from [23], the lattice model is meant for theoretical studies at the level of the microscale. However, there is a class of finite-discrete element codes [25] that has many similarities with lattice codes, while being applicable for engineering scale calculations.