# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths
As the surrogate models are trained based on HOSS simulations, they contain crack interaction effects. Such interaction will ultimately impact the evolution of crack lengths and their orientations during tensile loading. Hence, this micro-crack interaction information can be incorporated into the effective moduli through the surrogates during the upscaling process. This implies that the material response we obtain at macro-scale scale includes crack interactions at micro-scale, which are typically lost if one employs traditional mechanistic or homogenization approaches [57; 66; 67; 90]. An example of an upscaling method, which could make use of the proposed surrogate models to calculate effective elastic moduli, is described in reference [91]. ## 6 Conclusions

Predicting damage evolution and when failure occurs is important to accurately predict the overall material response. This includes accounting for degraded material properties as damage accumulates and when and where failure of the specimen or part occurs, for example, which elements or cells will fail first. Providing such information is of great interest to the fracture and infrastructure maintenance communities. In addition, estimating likely failure paths and accumulated damage along the failure path is an important aspect for upscaling micro-scale models to macro-scale models. FDEM models and multi-physics numerical tools like HOSS are highly-accurate to simulate various stages of brittle fracture propagation. However, HOSS is data-intensive and computationally expensive. Moreover, scaling to larger and more complex problems is often computationally prohibitive with computational tools such as HOSS. This is due to the difference in length scales associated with small fractures and bulk material sizes. In this paper, we provided algorithms to address these aspects that are \(\approx\mathcal{O}(10^{6})\) faster than high-fidelity simulations to estimate key QoIs for brittle fracture such as failure paths, failure zones, and damage along failure. Hence, the proposed methodology is ideal for usage in comprehensive uncertainty quantification studies. The proposed failure path method was compared against high-fidelity HOSS simulations. From this comparison, we found that out of 190 simulations, the proposed method predicted failure paths of 118 simulations with reasonable accuracy (greater than 50% of initial cracks in the failure path) and zone of failure of 143 simulations with 100% accuracy. Damage along the failure path was estimated using a non-parametric approach. The damage evolution model was obtained by constructing a 95% confidence interval over time. 150 HOSS simulations were used to develop the damage evolution model and the remaining 40 simulations are used for testing. Bootstrapping was used to estimate lower and upper bounds for damage. Empirical coverage was used as a metric to understand the accuracy of the proposed damage evolution model. Empirical coverage tells us the fraction of test cases that falls within our estimated confidence interval at anytime. From empirical coverage results, it can be concluded that our confidence interval captures the damage from the test dataset with an accuracy greater than 90%. Damage is the key QoI for upscaling HOSS information to continuum codes such as FLAG. This is because continuum-scale models need effective moduli, which is a function of damage accumulated over time at micro-scale. Since the discrete crack network cannot be accounted for in continuum models, exact locations of the failure pathways becomes less important. The proposed damage model which takes into account the crack interaction effects provides such micro-scale information with high accuracy for simulating damage at larger-scales. The limitation of the proposed methods is that it is applicable only for Mode I failure at low-strain rates. Extensions to higher-strain rates and other modes of failure will be considered in our future works.