# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 2 HOSS Simulations and FDEM Model for Dynamic Fracture  
HOSS is a FDEM code that was specifically designed to simulate problems involving fracture and fragmentation processes in a diverse range of materials [54]. Within the FDEM framework [50], finite element method solutions are combined with the discrete element method to simulate dynamic fracture. In this setting, solid domains are discretized using finite elements in order to describe their deformation and stresses. Finite rotations and finite displacements are assumed _a priori_. That is, we do not make prior assumptions of small deformations or small rotations in our FDEM numerical formulation. To resolve fracture and contact processes between different parts, we allow material interaction along the boundaries of finite elements. The finite element deformation kinematics is handled via a multiplicative decomposition-based finite strain formulation [42]. The fracture model used in this present work is the combined single and smeared crack model introduced by [PERSON] [49]. The contact interaction is resolved using the triangle-versus-point algorithm, which is the 2D extension of the tetrahedron-versus-point algorithm [51]. In the FDEM formulation, the interface between any two finite elements consists of non-linear springs that model tensile and shear behavior. These springs can hold a maximum tensile stress equal to the tensile strength of the material \(\sigma_{n}^{max}\) and a maximum shear stress that is based on a combination of the cohesion (Mohr-Coulomb model) and the frictional strength [68]. Figure 1 presents a schematic representation of stress-displacement curve, which shows the response of the springs between the elements accounting for opening (i.e., the normal response). We note that the HOSS simulations have both normal springs and shear springs present between all elements, but in Figure 1 we have only shown normal springs for convenience. Hence the finite elements can move apart or open (via normal springs), and slide against each other (via shear springs). The non-linear response of the shear springs is similar in shape to that of the normal springs, although parameterized slightly differently to properly account for degradation in shear directions. In the region \(0\leq\delta_{n}\leq\delta_{n}^{\varepsilon}\) and \(0\leq\sigma\leq\sigma_{n}^{max}\), the springs undergo non-linear elastic behavior without any damage. Beyond the elastic limit, \(\delta_{n}^{\varepsilon}<\delta_{n}<\delta_{n}^{max}\), strain softening is assumed that mimics degradation in strength. When \(\delta_{n}>\delta_{n}^{max}\), these springs are broken and cannot bear any load. In this work, HOSS has been used to simulate brittle fracturing under uniaxial tensile loading conditions in concrete samples containing 20 randomly placed pre-existing cracks. The loading is provided by displacement control. The sample size was set to 2 m wide and 3 m high. A schematic of the numerical simulation setup is shown in Figure 2. The bottom edge of the sample is fixed and the top edge of the sample moves upwards with a constant velocity of 0.3 m/s. This boundary condition corresponds to a strain rate of 0.1 s\({}^{-1}\). The lengths of pre-existing cracks was set to 0.3 m. Random sampling for crack placement and crack orientation selection is based on a discrete uniform distribution. The model domain is divided in to a grid of size 0.5 m \(\times\) 0.5 m resulting in 24 grid cells. A grid cell is chosen without replacement to randomly place a crack. This is done to ensure that the pre-existing defects do not overlap. This step is repeated until we place all the 20 cracks of size 0.3 m in the model domain. The center of the crack coincides with the center of the grid. The crack orientation was randomly selected between 0, 60, and 120 degrees with respect to horizontal. Other parameters used include: sample density of 2500 kg/m\({}^{3}\), Young's modulus of 22.6 GPa, shear modulus of 9.1 GPa, and a Poisson's ratio of 0.242.