# Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes
## 4 Mechanical Model
### Lattice Model
#### 4.1.5 Definition of Beam Geometry  
Both lattice Beams 1 and 2 are assumed as cylindrical bars, with a constant circular cross-section. Lengths \(l_{1}\) for Beam 1 and \(l_{2}\) for Beam 2 can be directly calculated from the coordinates at their nodal extremities. In the proposed regular model, these values are constant and given by:  
\[l_{1}=l,\,l_{2}=\Big{(}\sqrt{3}/2\Big{)}l \tag{7}\]  
where \(l\) is the length of the voxels, adopted as 1.0 \(\upmu\)m in this study. However, calculation of diameters \(d_{1}\) for Beam 1 and \(d_{2}\) for Beam 2 is not straightforward. Let us assume an isotropic and homogenous cubic REV of 20 \(\times\) 20 \(\times\) 20 \(\upmu\)m\({}^{3}\), discretised with voxels of 1 \(\times\) 1 \(\times\) 1 \(\upmu\)m\({}^{3}\). Each voxel represents a single material phase, with a local Young's modulus \(E_{l}\). A 3D lattice model constructed with this REV-based unit cell is reproduced in Figure 5a, where \(L_{eff}=20\) and \(l=1.0\). The uniaxial tensile boundary conditions described in Section 4.1.2 are assigned to this lattice model, as shown in Figure 8a. For \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\) ranging from 0.2 to 2.0, the results of the lattice model in terms of \(E_{eff}/E_{l}\) are reproduced in Figure 9a as a surface. An example of the deformation field obtained from such direct tensile test is illustrated in Figure 8a. By increasing both \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\), the value of \(E_{eff}/E_{l}\) increases significantly. Since the model is homogenous (a single phase REV is considered), theoretically the following condition has to be satisfied:  
\[E_{eff}=E_{l} \tag{8}\]  
meaning that the calculated effective Young's modulus must be equal to Young's modulus of the individual phase. If the \(E_{eff}/E_{l}\) surface intersects the surface \(E_{eff}/E_{l}=1\), the corresponding values \begin{table}
\begin{tabular}{c c c} \hline \hline
**Hydration Product** & **Elastic Modulus (Average Value) (GPa)** & **Poisson’s Ratio** \\ \hline C-S-H Inner (high-dense) & 26–32 (29) & 0.25 \\ C-S-H Outer (low-dense) & 13–26 (19.5) & 0.25 \\ CH & 35 & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The elastic properties of hydration products adopted in the modelling. Figure 7: Effective Young’s modulus of an interface element calculated with two different methods: the serial/parallel and the Hashin–Shtrikman classical bounds theoryfor \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\) meets the condition in Equation (8). Since the REV is elastic and isotropic, the best set of values for \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\) can be selected based on the most representative lateral deformation of the lattice model due to the [PERSON]'s effect, that is:  
\[v_{t}=v_{s} \tag{9}\]  
being  
\[v_{s}=\frac{E_{eff}}{2G_{eff}}-1 \tag{10}\]  
where \(G_{eff}\) is the effective shear modulus that can be obtained by modelling a direct shear test as shown in Figure 8b. A prescribed shear deformation along direction \(Y\) is applied to the nodes at the top surface of REV (plane \(Z=n\cdot l\)), while the nodes at the bottom surface (plane \(Z=0\)) are fixed in the \(Z\) and \(Y\) directions. The obtained shear deformation field is illustrated in Figure 8b. The value of \(G_{eff}\) is calculated similarly as for \(E_{eff}\) using Equation (1), where \(\Delta_{long}\) is replaced by the transversal deformation along direction \(Y\) for the nodes at \(Z=n\cdot l\). Figure 8: The configuration of a homogeneous lattice model of a 20 \(\times\) 20 \(\times\) 20 \(\upmu\)m\({}^{3}\) REV and deformation fields: (**a**) direct tensile model; (**b**) direct shear model (DtX, DtY, and DtZ are, respectively, deformation in \(X\), \(Y\) and \(Z\) directions). Predictions for \(v_{t}\) obtained with Equation (2), and for \(v_{s}\) using Equation (10) (with \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\) ranging from 0.2 to 2.0), are plotted in Figure 9b. As shown, the intersection of the two obtained surfaces for \(v_{t}\) and \(v_{s}\) corresponds to values of \(d_{1}/l_{1}\) and \(d_{2}/l_{2}\), where \(v_{t}=v_{s}\).