# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 2 Two-Dimensional Heterogeneous Mesostructures of Concrete
### Aggregate Size Distribution  
The aggregates used in concrete can be divided into two different groups, namely, fine and coarse aggregates. Fine aggregates consist mainly of sands or crushed stones with a diameter of less than 4.75 mm, and all particles with a diameter of more than 4.75 mm are considered coarse aggregates in concrete samples [44]. In normal concrete, the coarse aggregates account for about 40% to 50% of the concrete volume [27]. Many researchers use the Fuller curve as a grading curve to illustrate the size distribution of aggregate in a mesoscale numerical model for concrete. The Fuller curve illustrates the gradation of aggregate in the concrete sample, and its implementation would result in a concrete mix with optimum aggregate compaction and strength. The Fuller curve can be expressed as follows:  
\[P(d)=100\left(\frac{d}{d_{max}}\right)^{n}, \tag{1}\]  
where \(P(d)\) represents the cumulative percentage of aggregates that is below the size of the aperture diameter of a sieve (\(d\)), \(d_{max}\) is the maximum aggregate particle size, and \(n\), which is a constant parameter, is the exponent of the equation and varies from 0.45 to 0.70.  
To obtain the volumetric ratio of coarse aggregates in a concrete sample, it is possible to divide the total weight of coarse aggregates per volume of concrete by the density of the aggregate particles. This can be expressed as [14]:  
\[v_{p}=\frac{\omega_{p}}{\rho_{p}V}, \tag{2}\]  
where \(v_{p}\) is the total aggregate's volume fraction, \(\omega_{p}\) represents the total weight of aggregate particles in the concrete specimen, \(\rho_{p}\) is the specific weight, and \(V\) is the total volume of the concrete specimen. To numerically implement the Fuller curve from Equation (1), it can be divided into any number of segments, and for each segment the volume fraction of aggregates can be calculated as follows [27]:  
\[V_{R}[d_{s},d_{s+1}]=\frac{V_{agg}[d_{s},d_{s+1}]}{V}=\frac{P(d_{s+1})-P(d_{s })}{P(d_{max})-P(d_{min})}v_{p}, \tag{3}\]  
where \(V_{agg}\) represents the volume percentage of aggregates within the discretized aggregate size range and \(d_{max}\) and \(d_{min}\) denote the maximum and minimum sizes of the aggregate particles, respectively. The reference case of concrete considered in this paper is a sample with medium and coarse aggregates varying in size from 3.5 mm to 15 mm. For the numerical implementation of the Fuller curve, the aggregates are divided into five aggregate segments, namely, \([6-3.5]\) mm, \([9-6]\) mm, \([11-9]\) mm, \([13-11]\) mm, and \([15-13]\) mm. To calculate the cumulative volume fraction of each aggregate, \(n\) is assumed to be 0.5, and the target aggregate volume ratio in the 2D mesoscale model of concrete is assumed to be 50%. Table 1 illustrates the aggregate size distribution of coarse aggregate used in this study. According to Table 1, aggregates smaller than 3.5 mm are not considered in the model and form the mortar together with the cement matrix. Since small elements are required to discretize the fine aggregates, eliminating them does not require a large number of elements to discretize the entire mesoscale model, and consequently avoids high computational costs. It should be mentioned that there is no restriction on the number of aggregate size intervals. Furthermore, the implementation of the segmented size distribution of aggregates in the developed algorithm distributes the aggregate sizes uniformly within the upper and lower bounds of each interval segment. More precisely, the aggregate size is increased uniformly within each gradation segment, resulting in a mesoscale model where the actual Fuller curve is perfectly approximated. In this study, circular, elliptical, and polygonal aggregates are used to model the mesostructure of concrete in 2D. The equations used to generate the different aggregate shapes are described in detail below.