# Predicting the [PERSON]'s Modulus of Concrete Using a Particle-Based Movable Cellular Automata Method
## 4 Comparison the MCA Results with Experiments
### Comparison with Mortar  
The mortar simulations were based on data from [67]. The number of automata representing sand was determined based on the volume of sand in 1 m\({}^{3}\). The mass of the sand was divided by its specific gravity of 2.63, assuming the same density. The remaining volume of the sample was occupied by automata representing the cement paste; therefore, only the aggregate volume is reported in Table 2. An exemplary sample of the mortar used in the numerical simulations, designated M60, is presented in Figure 5a. In the experiment, a superplasticizer was added for better workability. It was assumed to be part of the cement paste and to not affect its mechanical properties. The mortars labeled M60, M80, and M110 were made using ASTM type I ordinary Portland cement, the same type as in [52], which was the source of the cement paste elastic moduli used in our simulations. In [67], cylindrical 100 \(\times\) 200 mortar specimens were subjected to uniaxial compression at a strain rate of 2 \(\times\) 10\({}^{-5}\) s\({}^{-1}\) using a Denison universal testing machine according to the ASTM C469/C469M standard [65], with the modulus of elasticity calculated as the slope of the stress-strain curve in the stress range from 0.5 MPa to 40% of the compressive strength of the specimen. The same approach was applied in our numerical simulations, which were conducted only up to 40% of the specimen strength reported in [67]. The [PERSON]'s modulus of sample M60, which was equal to 31.9 GPa, was used to estimate the [PERSON]'s modulus of the sand, which was found to be 52.5 GPa. Simulations with this value were then conducted for samples M80 and M110. For sample M80, very high agreement with the experimental result was achieved; the relative error was only 0.7%. A significantly larger error was observed for the next sample, M110; however, this sample contained 45 kg of silica fumes in 0.65 m\({}^{3}\). Our numerical simulations did not take the influence of the added silica fumes into account, as the [PERSON]'s modulus value for the cement paste with this additive in a given volume is unknown. This probably had a significant influence on the error level, as silica added to concrete mixtures improves the resulting mechanical properties [68; 69; 70]. All results from our numerical simulations and experiments are provided in Table 2, including the mean static elastic modulus \(E_{s}\) and standard deviation evaluated for twenty samples of each mortar. The studies described in [67] were continued in [71] for concrete containing the tested M60, M80, and M110 mortars, to which granite was added in a volume of 35% of the sample volume. The resulting concreteswere designated as C60, C80, and C110, respectively. In [71], uniaxial compression at a strain rate of 2 \(\times\) 10\({}^{-5}\) s\({}^{-1}\) was applied to 100 \(\times\) 200 cylinders \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mixtures** & **w/c Ratio** & **Sand** & \(E_{s}\)[67] & \(E_{s}\)**This Study** & **Relative** \\  & & **V [\%]** & **GPa** & **GPa** & **Error** \\ \hline M60 & 0.43 & 48.61 & 31.9 \(\pm\) 0.7 & 31.86 \(\pm\) 0.06 & 0.1\% \\ M80 & 0.34 & 46.39 & 34.4 \(\pm\) 0.6 & 34.65 \(\pm\) 0.06 & 0.7\% \\ M110 & 0.31 & 46.68 & 38.9 * \(\pm\) 0.6 & 36.43 ** \(\pm\) 0.04 & 6.3\% \\ \hline \hline \end{tabular}  
* This result was obtained for mortar with 45 kg of silica fumes in 0.65 m\({}^{3}\). ** This result was obtained for mortar without silica fumes.  
\end{table} 
Table 2: Numerical simulation results of mortars and comparison with experiments from [67]. Figure 5: Exemplary samples of mortar M60 and concrete C60 used in the numerical simulations. The cement paste is represented by black cells, the fine aggregate by dark grey cells, and the coarse aggregate by light grey cells. using a Denison universal testing machine in order to determine the Young's modulus from the stress-strain response according to the ASTM C469 standard. The only information about the coarse granite aggregate provided in [71] was its maximum size of 10 mm.