## 5 Results and Discussion
### Micro-CT Image and Characteristics of the Pore Structure  
The adjusted 3D micro-CT images of the foamed concrete specimens are shown in Figure 9. Here, the 3D images of the specimens are visualized using the binarization and the watershed segmentation, and the generic solid structure and the pore distribution can be identified; the solid structure of the foam specimen becomes denser as the density of the specimen increases, and the pore shape becomes more spherical with increasing specimen density. From these figures, the effect of the density on the material characteristics can be identified; however, it is difficult to examine the detailed pore characteristics, such as the pore size distribution and shape. Therefore, an appropriate method is desired for characterization of the pore structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specimens** & **LithoPore®100** & **LithoPore®200** & **LithoPore®450** & **LithoPore®600** \\ \hline
**Elastic modulus (GPa)** & 0.6 & 0.9 & 2.5 & 2.9 \\
**Poisson’s ratio** & 0.3 & 0.3 & 0.2 & 0.2 \\
**Yield strength (MPa)** & 0.45 & 0.9 & 4.2 & 7.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The input parameters for mechanical simulations. Figure 10 illustrates the probability functions for the pore sizes of the foamed specimens in the \(x\), \(y\) and \(z\) directions. In these functions, \(r\) and \(D\) represent the distance between two random points and the specimen edge length, respectively. In \(P_{pp}(r)\), a large probability for a specific direction indicates that larger pore clusters exist in that direction. For example, in Figure 10a, \(P_{pp}(r)\) in the \(x\) direction is relatively smaller than those of the \(y\) and \(z\) directions; it denotes that the pores are less clustered in the \(x\) direction, and the pore distribution in this specimen is anisotropic. Another probability function, \(L_{p}(r)\), can be used to describe the continuous connectivity of pores. In Figure 10b, \(L_{p}(r)\) of the LithoPore(r)100 sample in the \(z\) direction is slightly larger than the \(x\) and \(y\) directions, and it denotes that the maximum pore size in the \(z\) direction is larger than other directions; the anisotropy of the pore shape can be examined using \(L_{p}(r)\). From \(P_{pp}(r)\) and \(L_{p}(r)\) in Figure 10a,b, it is identified that the LithoPore(r)100 sample contains anisotropic pores. The pore characteristics of other samples are also examined using the probability functions. The functions of each sample in Figure 10c-h are almost the same in the \(x\), \(y\) and \(z\) directions, although small differences are found in Figure 10f,h when \(r/D\) is larger than 0.15; this means that the pores in the LithoPore(r)200, 450 and 600 samples are closer to spherical (isotropic) than those of the LithoPore(r)100 sample. In particular, the relative pore size within the specimen can be described using \(L_{p}(r)\). In Figure 10b, the function values become zero when \(r/D\) is about 0.45, while those in Figure 10f,h become zero when \(r/D\) is about 0.25. The results confirm that the degree of pore clustering, which represents the pore size, becomes smaller as the density of the specimen increases for the foamed samples used in this study. Figure 9: 3D micro-CT images of foamed concrete specimens with different densities: (**a**) LithoPore®100, (**b**) LithoPore®200, (**c**) LithoPore®450, (**d**) LithoPore®600 (note: in each binarized image, the white region represents solid). In addition to the probabilistic descriptions of the pore characteristics, the quantitative investigations are also performed. Figure 11 shows the pore size distributions of the foamed specimens, which are calculated using the 3D images.