## 4 Mathematical Model for Variation of Elastic Modulus  
It shall be noted that compression is positive in this paper. It is known from the above analyses that different pressures correspond to different elastic moduli. However, the above analysis is based on the assumption that the elastic modulus for the same cylinder diameter is the same. In the real case, the internal stress in SCA increases with the hydration process under constant volume. Correspondingly, the elastic modulus inevitably changes from a low value to a high value and finally a constant value. In order to investigate  
Figure 10: Relationship between Poisson’s ratio and axial expansive pressure.  
Figure 9: Relationship between Poisson’s ratio and amplitude coefficient.  
the mechanical mechanism of the entire hydration process of SCA, a model reflecting the change of elastic modulus with stress shall be established. The relationship between the major principal stress and the elastic modulus under various radial (axial) pressures is plotted in Figure 11.  
In the following section, the relationship between the stress and elastic modulus will be established according to the major principal stress. The major principal stress in Figure 11 is normalized. As the maximum principal stress is 95.09 MPa, the two stresses higher than 75.09 MPa are ignored for normalization. The elastic modulus and the normalized stress are shown in Table 2, which is the model for variation of elastic modulus with the major principal stress.  
The first 3 columns in Table 2 are obtained by extending the curve in Figure 11, instead of the measured data. Limited by the test conditions, the values in the first 3 columns are inaccurate. In order to facilitate its application, the data in Table 2 are fitted by a Boltzman function in Equation (3).  
\[E(\sigma_{N})=a_{2}+\frac{(a_{1}-a_{2})}{1+e^{(\frac{\sigma_{N}-\sigma_{N0}}{ a_{3}})}} \tag{3}\]  
where \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(\sigma_{N0}\) are constants.  
The initial data and the fitting curve are shown in Figure 12. At this moment, the corresponding constants are: \(a_{1}=1.1134\), \(a_{2}=32.885\), \(\sigma_{N0}=0.54644\) and \(a_{3}=0.10989\). Equation (3) is the model for variation of elastic modulus with the major principal stress where the Poisson's ratio is equal to 0.1. In fact the relationship between the Poisson's ratio and the expansion coefficient of SCA is linear. The value of Poisson's ratio doesn't affect the model precision. In order to verify the model, a subroutine module for variation of elastic modulus is compiled based on the USDFLD subroutine module of ABAQUS. The radial and axial pressures for models with various cylinder diameters are simulated and plotted in Figure 13. It can be seen from Figure 13 that the simulation results obtained by the model for variation of elastic modulus are in good agreement with the measured data, indicating that this model is feasible for investigating the expansion process of SCA in a cylinder with volume constraint.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Normalized Stress \(\sigma_{N0}\)** & 0 & 0.233 & 0.33 & 0.427 & 0.51 & 0.578 & 0.677 & 0.871 & 1 \\ \hline
**Elastic modulus (GPa)** & 1.2 & 2.5 & 5.0 & 10.49 & 14.13 & 17.695 & 26.457 & 31.807 & 32.885 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Relationship between normalized stress and elastic modulus.  
Figure 11: Relationship between the major principal stress and elastic modulus.