# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 1 Introduction  
Brittle fracture is a complex phenomena determined by interaction among several microstructural features of the material under study. These features include grain size, presence of pre-existing cracks or defects and/or pores, frictional characteristics, etc. [1; 2]. When a brittle material containing pre-existing cracks is loaded, stresses concentrate around the crack tips [3; 4]. When these cracks propagate they interact with other defects and cracks in their neighborhood. This results in a complexstate of stress near the crack tips. Especially at lower strain rates, pre-existing cracks act as nucleation points for the generation of new cracks [5; 6]. Nucleation, growth, and coalescence of pre-existing cracks results in the degradation of elastic properties in brittle materials, eventually causing catastrophic failure. This is because the critical amount of damage before rapid and unassisted crack growth can be limited in brittle materials. Moreover, damage accumulation also results in non-linear material behavior, a further challenge for the development of predictive models. Due to the prevalence of brittle materials in several applications in geosciences (e.g., hydraulic fracturing, geothermal, etc.) [7; 8; 9], infrastructure [5] and aerospace industry [10], fast and accurate models of brittle fracturing are needed. Many constitutive models and numerical methods (e.g., seminal works of [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], etc.) have been proposed in the literature [11; 12; 13; 14; 15; 16; 17; 18; 19; 20] that simulate processes such as crack initiation, crack propagation, and crack shielding in brittle materials. Foundational work by [PERSON] [21] and [PERSON] [22] explained how crack growth can be understood as a free energy balance between the surface energy generated by crack formation and the elastic energy released by the crack as it grows, and the crack tip stress intensity factors. However, [PERSON]'s theory over-predicts the fracture toughness as it does not consider the presence of other defects. Shortly thereafter, [PERSON] [23] and [PERSON] [24] proposed the idea of a cohesive zone or fracture process zone (FPZ) preceding a crack tip. These ideas have been extensively expanded on in the past decades (e.g., works by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], etc.) [25; 26; 27; 28; 29; 30]. The current computing power available has allowed for complex fracture models to be applied to a wide range of length scales [31; 32]. Of these, many continuum (or macro-scale) and phase-field modeling approaches [33; 34; 35; 36; 37; 38; 39] have been developed to address crack evolution and the overall material response in large samples (cm-scale and larger) including finite element methods [40; 41; 42] and boundary element methods [43; 44]. Continuum-based approaches [45] assume that the computational domain can be treated as one continuous body. However, the nature of brittle fracture, which often results from growth and coalescence of major flaws, can be extremely localized and therefore not amenable to standard homogenization approaches. This often results in the emergence of a non-physical length scale within the numerical methods used to implement continuum models that can significantly impact the calculated material response [46]. Hence, these numerical methods cannot account for localized strains without further enhancement of the mathematical formulation [46; 47]. Discrete element methods [32; 48] are another class of modeling techniques available at the same length scale as continuum approaches. With these approaches, material is modeled as a collection of discrete blocks or particles that can displace and rotate with respect to each other, and even completely separate or break apart.