# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 3 Automatic Generation of Cohesive Elements
The corresponding point of each stage on the mean stress–strain curve is illustrated in \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]4

\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Aggregate Diameter (mm)** & **Volume Fraction 50\%** & **Volume Fraction 35\%** & **Volume Fraction 20\%** \\ \hline \([15-13]\) & 6.68\% & 4.68\% & 2.67\% \\ \([13-11]\) & 7.2\% & 5\% & 2.9\% \\ \([11-9]\) & 8\% & 5.5\% & 3.16\% \\ \([9-6]\)



### Effect of Volume Porosity

To study the effects of porosity volume fraction, circular voids are created and randomly distributed within the concrete mesostructure specimens, representing 2.5% and 5% of the total volume fraction of the concrete specimen. The circular voids have diameters between 1 mm and 2 mm. Figure 14 shows a concrete mesostructure containing smoothed polyhedral aggregates with the aggregate volume ratio of 50% and porosity contents of 2.5% and 5%. The corresponding average stress-strain curve is shown in \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]5 For each porosity volume fraction, a set of ten concrete specimens is generated and simulated under the uniaxial tensile state, and the averaged mean stress-strain curve is obtained by statistical analysis on the simulation results for each porosity content. As Figure 15 shows, the mean tensile strength decreases from 3.2 MPa to 2.5 MPa when the volume porosity increases from 0% to 5%. The reason for this is that the voids reduce the surface area available to support the imposed load. As a consequence, they provide easier paths for crack propagation and, thus, encourage the formation of small cracks initiating within the ITZ zone. From Figure 14, it can be seen that the macrocrack leading to specimen failure propagates through the voids when possible and it takes the path of least resistance to fracture the concrete specimen. \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]6 Concrete mesostructure with different levels of coarse aggregate volume fraction. \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]7 Influence of aggregate volume fraction. All mesoscale samples are subjected to a uniform displacement of 0.15 mm in the x-direction. Strain is determined by dividing the displacement at each step frame by the initial length of the specimen. \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]8 Simulated crack pattern for samples with different volume fractions of porosity in 5 times magnification. \[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]9 Effect of volume fraction of porosity. All mesoscale samples are subjected to a uniform displacement of 0.15 mm in the x-direction. Strain is determined by dividing the displacement at each step frame by the initial length of the specimen. ### Effect of Aggregate Shape

In order to analyze the effect of the geometric shape of the aggregate on the maximum tensile strength and load capacity immediately after the onset of the softening zone, concrete specimens are generated with circular, elliptical, smoothed polyhedral, and polyhedral aggregates with sharp edges. It is worth noting that 12 random points are used to generate smoothed polyhedral aggregates, while only 8 random points are used for polyhedral aggregates with sharper edges. For each aggregate shape, five concrete specimens are generated, and after simulation under uniaxial tensile loading, the corresponding average stress-strain curve is obtained from the simulation results. In this study, the total volume fraction of coarse aggregate is 50%, and there is no void in the generated specimens.