# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths  
In this section, we provide a detailed description of our proposed method to estimate failure paths. The goal is to find the possible pathways to failure using weighted shortest path algorithms [Chapter-3][72] based on initial crack configuration. Figure 3 provides a graph based pictorial description of failure paths and Algorithm 1 summarizes the proposed method to estimate failure paths. In our method, we assume graph nodes to be crack tips and graph edges are line segments connecting these tips. At initial times, when the system is not loaded, our premise is that pre-existing cracks are not connected to each other. In the graph theory representation, this implies that there is no connectivity between edges. As the system is being loaded, pre-existing cracks grow and intersect with other cracks leading to the formation of new edges. The schematic of all the possible paths connecting one side of the domain to another side at the time of failure for an initial pre-existing crack configuration is shown in Figure 3. In fracture mechanics, there are three different modes of fracture: Mode I, Mode II, and Mode III [1; 73; 74]. Mode I corresponds to crack opening mode, which is the most common mode in fracture toughness testing of brittle materials [5; 6; 75]. Mode II is in-plane shear and Mode III is out-of-plane shear or tearing mode. The proposed method focuses on Mode I failure and while it is possible to extend the algorithm to consider other failure modes, it is outside the scope of this study. Failure of brittle and quasi-brittle materials under Mode I typically starts with the development of a FPZ around the crack tip [2; 3; 76]. The FPZ is a region of high stress around the crack tip where damage accumulates as the crack propagates over time. At a given instant of time, length of the FPZ is directly proportional to the length of the crack [77]. Very small micro-cracks are formed in the vicinity of the crack tip as stresses are the highest in this region. As the crack advances over time, the micro-cracks within the FPZ coalesce to become a single entity. This results in the formation of larger cracks. Larger cracks are bound to have a stable crack growth compared to smaller cracks [73]. Once these long cracks are formed, they grow rapidly with minimal additional loading. Therefore, the possible failure paths correlates directly with FPZs, which is the basis for our proposed method. Various crack interaction and coalescence rules (e.g., alignment rules, combination rules) have been proposed in literature [78; 79; 80; 81] based on fracture mechanics analysis. [PERSON] and co-authors [77] provide FPZ-based rules for multiple crack interaction and coalescence. In these rules, crack interaction and coalescence are based on the Euclidean distance between cracks tips (for e.g., see [Table-1][81]). We note that these rules have been widely adopted in brittle and quasi-brittle failure analysis (for e.g., see [Equation-1 and Sec-1][77]) and fitness-for-service codes [Table-1 and Introduction][81]. In this paper, we take advantage of these crack interaction and coalescence rules that are built on sound fracture mechanics underpinnings to develop FPZ-based graph surrogate model to estimate failure paths. The first step of the proposed method is to identify the cracks which are likely to coalesce. In order to achieve this, we find pre-existing cracks that are orthogonal to the tensile loading. These correspond to all 0-degree angle cracks in the domain (as the orientation of these cracks is best suited for Mode I opening). It is expected that they will have the fastest growth and we assume that the failure path will include one or more 0-degree angle cracks. For each tip of these 0-degree angle cracks, we calculate ten nearest crack tip neighbors using a \(k\)NN algorithm [82] with \(k=10\) and their respective Euclidean distances. These nearest neighbor crack tips can belong to either 0-degree, 60-degree or a 120-degree crack.