# Damage and Failure in a Statistical Crack Model
## 8 Summary and Discussion  
\"The mistake is thinking that there can be an antidote to the uncertainty.\" [PERSON]  
In this paper, I have taken a close look at a statistical crack model (SCM) with the object of introducing a model of crack interactions (coalescence) and understanding the qualitative changes it induces in the fracture process. Fracture is a statistical process when viewed at the continuum level, which depends on details of microstructure that cannot be known in full. The philosophy of the SCM is to introduce simple microphysical models that contain the appropriate scales of length and time. In Section 2, I summarized the models of a particular program, the BCM [9]. In Section 3, I described a new tool for studying the behavior of BCM; the array code is a computer program in which the microphysical models of the BCM are implemented to simulate the growth a randomly generated array of microcracks. This array code is a realization of a single finite volume computer cell in an engineering scale simulation program. In addition to the physics of BCM, I implemented a new model for microcrack interactions. In Sections 4 and 5, I used the array code to generate statistics of the fracture process, including the evolution of damage and failure. Fracture is initiated by only a few cracks in each realization, namely the largest cracks in the extreme tail of the size distribution. Those largest cracks are like hotspots, perhaps one or two or three in each realization. Thus the statistics are multimodal; however, the ensemble-averaged statistics of damage evolution, damage at time of failure and time to failure are smooth functions of time and applied strain rate, with scaling described by low-order polynomials. The major modification to the SCM is the elimination of the need to evolve the statistical distribution of crack sizes. It should be noted that this simplification is not dependent on the introduction of the crack interaction model, but works as well for the original BCM. Indeed, that was the original motivation for building the array code. The most important modification to the description of the fracture process is that precise definitions of damage and failure lead to a prediction of failure as a function of damage. The relation is not constant and does not depend on identifying the percolation threshold of the effective elastic  
Figure 10: Damage at time of failure is plotted versus time to failure for \(\dot{\epsilon}=1\times 10^{5}\) s\({}^{-1}\), an ensemble size of 50 and the process zone is 10% of crack radius. This plot is a synthesis of the data in Figures 7 and 8 when cell width is eliminated. The curve shows points representing both shapes fall on the same straight line. moduli with failure. Indeed, failure takes place at much lower levels of damage, consistent with experimental data [5, 6]. That result leads to reconsidering aspects of how an SCM is implemented. 1. Time to failure is much shorter: In the avalanche phase, the crack that causes failure grows with the Rayleigh wave speed, cf. Equation (4). Dividing the cell width by that speed produces a time interval characteristic of the fracture process, perhaps even made shorter if several hotspots are present in the row that fails. Independently, that time scale is also the maximum time step allowed by the Courant-Friedrichs-Lewy stability condition for explicit dynamics in a finite volume engineering code. To properly resolve the interactions between computational cells, it will probably be necessary to either subcycle the fracture process or to treat it implicitly. 2. There is residual stress at failure: That failure occurs at a low level of damage also implies that there is residual stress present in the cell at the time of failure. This is shown explicitly in Figure 5. Residual stress means residual strain, which must be distributed to the neighboring cells. 3. Diffusion of damage: Although crack growth ends at the cell boundaries in the array code, it will continue across cell boundaries in the finite volume engineering codes.