# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 1 Introduction  
Concrete is a vital building material that undergoes various types of loads, notably tensile loads during events such as earthquakes or explosions [1]. Despite being commonly treated as a homogeneous material in structural design, concrete exhibits intricate and nonlinear behavior even under simple loading conditions. This complexity arises from its highly heterogeneous internal material structure, with microcracks forming, propagating, accumulating, and coalescing [2, 3]. Moreover, the structural behavior of concrete is profoundly influenced not only by the morphological properties of aggregate particles (e.g., size, shape, volume fraction, spatial distribution) but also by the material properties of each constituent component. Consequently, concrete failure represents a multiscale phenomenon, necessitating a comprehensive consideration of the various material properties of its internal structure [2]. Understanding the mesoscale fracture behavior of concrete materials has significant implications for macroscopic structural performance. Mesoscale simulations reveal crack propagation patterns and interactions, enabling the design of more robust structures with improved resistance to failure. This knowledge enhances load-carrying capacity and strength by optimizing concrete mix designs and reinforcing vulnerable areas. The insightsalso guide material selection and optimization, evaluating different compositions to meet specific performance requirements. Furthermore, understanding mesoscale fracture behavior empowers innovative design concepts, tailoring microstructural features to enhance crack control, strain capacity, and load redistribution. These applications advance concrete engineering practices, ensuring safer and more durable structures in various practical engineering scenarios. Concrete is a complex composite material analyzed at multiple scales: nano, micro, and meso. The mesoscale, the largest length scale, explicitly reveals concrete's heterogeneity. Here, concrete comprises coarse aggregates surrounded by a mortar matrix, cement paste, voids, and the aggregate-mortar interface known as the interface transition zone (ITZ). Morphological and material properties of aggregates and phases at this scale significantly impact concrete's observable behavior at the structural level. In continuum mesoscale modeling of concrete geometries, two main approaches are commonly used. The first approach is the digital-image-based method, where concrete specimens are captured using image processing technology to generate accurate mesostructures [4]. However, this method can be time consuming, expensive, and may result in overlapping aggregates if the boundaries are not properly identified [5]. The second approach is parameterization modeling [3], where concrete mesostructures are generated using algorithms that represent concrete's geometry parametrically. This approach allows for generating various shapes, such as spherical, ellipsoidal, or polyhedral, randomly placed within the mortar matrix [6; 7; 8]. While many publications use simpler shapes such as spheres or ellipsoids [3; 9; 10; 11; 12], others employ polygonal shapes to achieve a more realistic representation of aggregates [13; 14]. It is worth noting that in many mesostructure models, only the aggregates are randomly distributed within the mortar matrix, neglecting the contribution of voids. However, X-ray tomographic analysis reveals the existence of voids in concrete specimens. Therefore, developing algorithms that account for the contribution of voids in the mesostructure of concrete is crucial. As mentioned, the observable behavior of concrete at the structural level is influenced by the morphological and material properties of aggregates and phases at the mesoscale length scale. To gain a deeper understanding of concrete's macroscale behavior, a comprehensive numerical model should consider these properties, as well as the local failure mechanisms within the mesostructure.