## 3 Results and Discussion
### Neighborhood Test  
The neighborhood test consists, in essence, of assigning to each pore a numerical value \(d\) that corresponds to its distance to the nearest pore. This value depends not only on the size of the specimen but also on the number of pores.  
In case the pores are evenly distributed in the specimen, the distance to the nearest pore would be equal in all pores and its value can be obtained according to the following expression (Equation (5)):  
\[n=\frac{1}{6}\cdot\left(\frac{\sqrt{2}\cdot\sqrt[3]{3}\cdot\sqrt[3]{V}}{d} \right)\cdot\left(\frac{\sqrt{2}\cdot\sqrt[3]{3}\cdot\sqrt[3]{V}}{d}+1\right) \cdot\left(\frac{\sqrt{2}\cdot\sqrt[3]{3}\cdot\sqrt[3]{V}}{d}+2\right) \tag{5}\]  
where \(n\) is the number of pores, \(V\) is the volume of the specimen and \(d\) is the distance to the nearest pore.  
This expression was obtained considering an analogy of the hexagonal close-packing of spheres.  
As mentioned, if the pores were uniformly distributed, all pores would show the same distance value to the nearest pore. However, in real specimens, each pore shows a different distance value. Figure 16 shows the histograms of the nearest pore distance distribution for each of the specimens, as well as its theoretical value corresponding to a uniform pore distribution.  
The results reveal that the pores are non-uniformly distributed in the specimens. There are regions (which this test is not able to identify) where they are closer together and others where they are farther apart.  
In this case study, the distribution closely resembles a uniform distribution. In the majority of the specimens, the most frequent value coincides with the theoretical value. Only in specimens CUB-01 and CUB-04 is the most frequent value lower than the theoretical value.  
From the data shown by the histogram it is possible to determine the \"pore concentration factor\" \(pcf\), according to the following expression (Equation (6)):  
\[pcf=\sum_{i=1}^{n}f_{i}\cdot\frac{d_{T}-\overline{d}_{i}}{d_{T}} \tag{6}\]  
where \(d_{T}\) is the theoretical distance, \(\overline{d}_{i}\) is the mean value of each range of distances in the real histogram, \(f_{i}\) is its frequency and \(n\) is the number of bars in the histogram.  
The pore concentration factor takes values between zero and one, with zero being when the pores are uniformly distributed throughout the specimen and one being when the pores are fully concentrated. The values of the pore concentration factors of the specimens in this study are shown below (Table 2):  
The results show that there are three specimens (CUB-02, CUB-03 and CUB-05) in which the pores are quite evenly distributed. In the rest of the specimens the distribution is far from uniform.  
The neighborhood test can be applied to correlate porosity distribution with concrete failure mechanisms. For example, it can be tested whether pore concentrations act as preferential pathways for cracks, and consequently specimens with a higher \(pcf\) have lower strength.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Specimen** & **CUB-01** & **CUB-02** & **CUB-03** & **CUB-04** & **CUB-05** & **CUB-06** \\ \hline Pore Concentration Factor & 0.024 & 0.008 & 0.008 & 0.037 & 0.014 & 0.031 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pore concentration factors.