# Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes
## 4 Mechanical Model
### Lattice Model
#### 4.1.4 Definition of Beam Properties  
Mechanical properties of Beam 1 are defined by the location of the two nodes at the extremities. If the nodes are positioned in the same phase (two voxels with the same material phase), Beam 1 has  
Figure 6: (a) 3D regular lattice model, (b) FE model—grey: unreacted cement, red: C-S-H inner, yellow: C-S-H outer, green: CH, black: all interface beams, blue: pore phases. the mechanical properties of the respective phase; if not, an interface beam is considered between the two connected phases. Definition of the elastic properties of the interface beam should be based on relevant material tests. However, due to lacking data from such experiments, Young's modulus of the interface beam can be calculated based on the serial/parallel rule of mixtures for composite materials, as pursued by [PERSON] et al. [14]. Based on this theory, an interface element is considered as a two-phase material with a lower-bound Young's modulus of:  
\[\frac{1}{E_{l}^{I}}=\frac{v_{A}}{E_{l}^{A}}+\frac{(1-v_{A})}{E_{l}^{B}} \tag{3}\]  
where \(E_{l}^{I}\), \(E_{l}^{A}\) and \(E_{l}^{B}\) are, respectively, the local Young's moduli of the interface and phase A and phase B beams. \(v_{A}\) is the volumetric fraction of phase A, taken as 0.5, assuming an equal contribution of phases A and B.  
Alternatively, the effective Young's modulus of the interface material can be calculated using the classical Hashin-Shtrikman bounds method [31]. According to this method, the local bulk modulus of a two-phase composite material can be described by a matrix of phase A (a material with a lower bulk modulus), within which spherical inclusions of phase B (a material with higher bulk modulus) are distributed in a particular way [31]. This is mathematically expressed as follows:  
\[K_{l}^{I}=K_{l}^{A}+\frac{(1-v_{A})}{\frac{1}{K_{l}^{B}-K_{l}^{A}}+\frac{3v_{ A}}{3K_{l}^{A}+4G_{l}^{A}}}\quad(K_{l}^{B}>K_{l}^{A}) \tag{4}\]  
where \(K_{l}^{I}\), \(K_{l}^{A}\) and \(K_{l}^{B}\) are the local bulk moduli of the interface and phase A and B beams, respectively. \(G_{l}^{A}\) is the local shear modulus of material phase A. As the independent elastic properties of a material are [PERSON]'s modulus \(E_{l}\) and the Poisson's ratio \(v_{l}\), the local bulk and shear moduli for each phase are:  
\[K_{l}=\frac{E_{l}}{3(1-2v_{l})} \tag{5}\]  
\[G_{l}=\frac{E_{l}}{2(1+v_{l})} \tag{6}\]  
The difference between using Equations (3) and(4) is shown in Figure 7 by plotting the dimensionless ratio \(E_{l}^{I}/E_{l}^{A}\) versus \(E_{l}^{B}/E_{l}^{A}\). By increasing the difference between Young's modulus of material phases A and B, both methods give higher values for the interface Young's modulus, being the value from Hashin-Shtrikman theory always higher than that from the serial/parallel rule. In Figure 7, the vertical dotted lines indicate the theoretical range of \(E_{l}^{B}/E_{l}^{A}\) for the cement paste microstructure of present study. Both theories are used for calculation of the effective Young's modulus of the interface elements, and the respective results will be discussed in Section 6.1. The local tensile strength of the interface beam is defined as the minimum value between the tensile strength of phases A and B (i.e., \(\min\{f_{TI}^{A},f_{TI}^{B}\}\)). Although it has been previously reported that the interface strength may not be the minimum of the corresponding phases [25], the local tensile strength is not relevant for the present study, as fracture behaviour is not addressed here. On the other hand, the mechanical properties of Beam 2 (diagonal) are defined only based on the properties of the middle node. This means that none of the Beam 2 counted as an interface beam. The intrinsic elastic properties of the individual phases of the cement paste are defined based on the values reported in the literature. There are ranges of values for the elastic properties of C-S-H inner and outer products, depending on their porosity.