## 2 Materials and Method
### Numerical Experiments  
In order to back analyze the mechanical parameters of SCA, a finite element model (1/4 model) is established according to the geometric parameters of steel cylinder and SCA, as shown in Figure 3. The contact relations between the steel cylinder and SCA, piston and SCA are face-to-face smooth contact, and the mean value of the static friction coefficient between the steel cylinder and SCA was determined experimentally as 0.3. It is assumed that there is no relative displacement at the contact surface between the steel cylinder and the SCA. The steel cylinder and SCA are discretized by 8-node hexahedral elements and 20-node hexahedral elements. Both symmetry surfaces are constrained symmetrically, i.e., the out-of-plane displacements and out-of-plane rotation are constrained. The vertical displacement of the bottom of the steel cylinder, the upper and lower pistons are constrained. The elastic modulus and Poisson's ratio of steel cylinder and piston are set  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Cylinder No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\
**Inner diameter of** & 30 & 42 & 50 & 65 & 90 & 113 & 130 & 150 \\
**cylinder (mm)** & 7.065 & 13.847 & 19.625 & 33.166 & 63.585 & 100.237 & 132.665 & 176.625 \\
**Slurry height (cm)** & 1.1 & 1.6 & 1.9 & 2.4 & 3.4 & 4.2 & 4.9 & 5.6 \\
**Slurry volume (cm\({}^{3}\))** & 7.772 & 22.155 & 37.288 & 79.598 & 214.599 & 420.995 & 650.059 & 989.100 \\
**Slurry mass (g)** & 18 & 52 & 87 & 185 & 499 & 980 & 1510 & 2300 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters for cylinder and SCA.  
Figure 1: The test principle for the axial-output method.  
Figure 2: Time histories of radial and axial expansive pressures obtained by experiments: (**a**) radial expansive pressure, (**b**) axial expansive pressure.  
as \(E=205\) GPa and \(\mu=0.3\). During numerical simulation, thermal expansion is adopted to simulate the volume expansion during the hydration process of SCA.  
To facilitate numerical modeling, the normalized deviation method is adopted to normalize the time histories of radial expansive pressure for steel cylinders with varying diameters. The normalized deviation method can be expressed as:  
\[f_{i}(t)=\frac{\sigma_{i}(t)-\sigma_{i,\min}}{\sigma_{i,\max}-\sigma_{i,\min}}, \ \ (i=1,2,3\cdots 8) \tag{1}\]  
where \(i=1\)-\(8\) denotes the number of steel cylinder, \(f_{i}(t)\) is the normalized radial pressure, \(\sigma_{i}(t)\) is the radial pressure at different moments, \(\sigma_{i,\max}\) is the maximum radial pressure, which is taken as \(\sigma_{\max}=\sigma(65\) h) in this study, \(\sigma_{\min}\) is the minimum radial pressure, taken as \(\sigma_{\min}=0\) in this study.  
The normalized curves are shown in Figure 4. Each curve represents the degree of hydration at different moments for SCA in the cylinder with varying diameter, and also shows the variation of temperature for SCA driven by thermal expansion. The normalized temperature-time curves can be expressed as:  
\[T_{i}(t)=k_{i}f_{i}(t) \tag{2}\]  
where \(k_{i}\) is the amplitude coefficient for the model with varying cylinder diameters and \(T_{i}(t)\) is the normalized temperature.  
Figure 4: Time histories of amplitude coefficient for radial pressure and temperature for SCA in steel cylinders with varying diameters.  
Figure 3: The finite element model.