# Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes
## 6 Results and Discussion  
### Lattice vs. FEM Results  
The [PERSON]'s modulus and the Poisson's ratio predicted by the lattice and FEM models at different hydration ages for the white cement paste with \(w/c=0.40\), and some computational details about the numerical models, are provided in Table 5. The number of nodes and elements in the lattice model is much higher than the FEM model. Therefore, the computational time that is required to solve the lattice model is significantly higher than the FEM model. However, as it was addressed earlier, the advantages of using lattice models become visible when solving a fracture behaviour, transport problems, or using combined problems in cement paste is intended.  
Performances of the lattice and FEM models are illustrated in Figure 12a, by comparing their predictions for the effective Young's modulus of the cement pastes. Results from the lattice model fit nicely to those obtained by the FEM, with an error less than 2%. The error becomes less than 0.62% when the Hashin-Shrittkman theory is used to define [PERSON]'s modulus for the interface elements. Young's modulus of the interface elements has no significant effect on the effective elastic properties because only 8% of all lattice elements are interface elements (see Table 5). Figure 12b compares the two numerical approaches as far as the Poisson's ratio of the cement pastes is concerned, the conclusion is that the lattice model nicely captures the [PERSON]'s ratio from the FEM model. Therefore, the improved lattice model for predicting [PERSON]'s modulus and the Poisson's ratio of the cement pastes performs much better than the lat  
Figure 11: Experimental setup for the Elasticity Modulus Measurement through Ambient Response Method (EMM-ARM) testing of cement paste and data processing.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Hydration Time** & **Model** & **Number of** & **Number of** & **Number of** & **Interface Percentage** & **Solving Time \({}^{*}\)** & **Young’s Modulus** & **Poisson’s Ratio** \\
**(Hours)** & **Element** & **Nodes** & **Interfaces** & **(x)** & **(sec)** & **(GFa)** & **(s)** \\ \hline  & Lattice & 5,199,531 & 1,366,992 & 222,459 & 4.28 & 2641 & 3.60 & 0.248 \\
5 & FE & 515,200 & 897,518 & - & 0.0 & 242 & 3.62 & 0.237 \\ \hline  & Lattice & 6,042,059 & 1486,326 & 431,944 & 7.15 & 2932 & 6.32 & 0.252 \\
15 & FE & 593,551 & 942,238 & - & 0.0 & 242 & 6.32 & 0.239 \\ \hline  & Lattice & 6,688,144 & 1,564,456 & 543,773 & 8.13 & 3145 & 9.56 & 0.253 \\
28 & FE & 651,815 & 964,251 & - & 0.0 & 263 & 9.53 & 0.241 \\ \hline  & Lattice & 7,278,024 & 1,630,933 & 578,575 & 7.95 & 4086 & 12.31 & 0.255 \\
50 & FE & 704,181 & 979,932 & - & 0.0 & 271 & 12.34 & 0.229 \\ \hline  & Lattice & 7,857,111 & 1,660,055 & 572,667 & 7.58 & 4592 & 13.84 & 0.257 \\
84 & FE & 728,502 & 985,321 & - & 0.0 & 274 & 13.86 & 0.241 \\ \hline  & Lattice & 7,725,450 & 1,677,884 & 574,797 & 7.44 & 5123 & 14.72 & 0.261 \\
111 & FE & 743,217 & 988,786 & - & 0.0 & 281 & 14.68 & 0.243 \\ \hline  & Lattice & 7,904,593 & 1,696,642 & 577,694 & 7.31 & 5637 & 15.32 & 0.263 \\
156 & FE & 758,822 & 992,322 & - & 0.0 & 296 & 15.32 & 0.246 \\ \hline \end{tabular} \({}^{*}\)As reported by DIANA FEA for the whole process including solving and output writing, and obtained using a desktop computer.  
\end{table}
Table 5: Computational details for the lattice and Finite Element Method (FEM) models for \(w/c=0.40\).  
Figure 12: Elastic results of lattice and FEM models regarding **(a)** [PERSON]’s modulus; **(b)** [PERSON]’s ratio.