## 1 Introduction
### Scale Effect  
[PERSON] (1972) assumed that every solid contains imperfections or defects (microracks) that diminish the strength of the material due to rupture taking place \"as soon as they [weak places] fall within a volume subjected to the stress\". This means that, for two cylinders of different size (but identical shape), the probability of the failure of the large specimen is greater than that of the small one, because it contains more defects, and the strength of the large sample is less than that of the small one ([PERSON] and [PERSON], 2017). Equation (25) is the generalized modification by which [PERSON] (1972) introduced the positive constant \(m\) for a better statistical fit, where \(\sigma_{1}\) and \(\sigma_{2}\) are the strength of the two specimens and \(V_{1}\) and \(V_{2}\) the respective volumes, according to [PERSON] (1973). \[m\log\left(\frac{\sigma_{1}}{\sigma_{2}}\right)=\log\left(\frac{V_{2}}{V_{1}}\right) \tag{25}\]  
In the case of cylindrical specimens with identical shapes and a constant \(H/d\) ratio, the volume can be replaced by \(d\), hence Equations (26) and (27), proposed by [PERSON] and [PERSON] (1972, 1973) and [PERSON] (1973), respectively. It can be seen that the constant \(1/m\) is now called \(k\) (Figure 1), so in the case of [PERSON] and [PERSON] (1972, 1973) the value of \(k_{1}\) is 0.18 and, in that of [PERSON] (1972), \(k_{2}\) can be approximated to 0.45. [PERSON] et al. (2018) reported \(k_{1}\) values between 0.1 and 0.3 for hard and homogeneous rocks, while for weathered rocks they are between 0.3 and 0.9. Both correlations refer to the relationship between the strength of the specimen at any diameter (\(\sigma_{c}\), \(I_{s}\)) and another at a 50 mm diameter (\(\sigma_{c(50)}\), \(I_{s(50)}\)). \[\frac{\sigma_{c}}{\sigma_{c(50)}}=\left(\frac{50}{d}\right)^{k_{1}} \tag{26}\]  
\[\frac{I_{s}}{I_{s(50)}}=\left(\frac{50}{d}\right)^{k_{2}} \tag{27}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Correlation** & \(\mathbf{R}^{2}\) & **Rock Type** & **Geometry \(\sigma_{c}\)** & **Geometry R \({}^{*}\)** & **Reference** & **Equation** \\ \hline \(\sigma_{c(50)}=9.97\beta^{(202R_{c},\alpha_{p}}\) & - &
\begin{tabular}{c} Basalt, diabase, dolomite, genesis, granite, \\ limestone, marble, quartz, rock salt, \\ sandstone, schelt  
The models of [PERSON] [PERSON] [33; 34] and [PERSON] [16] are considered descending, since \(\sigma_{c}\) decreases with increasing \(d\). However, the model of [PERSON] and [PERSON] [33; 34] only took into account one sedimentary rock (limestone) of the 10 that it used to generate the experimental model, and also diameters greater than 50 mm. This fact did not go unnoticed by [PERSON] [44], who pointed out that the model of [PERSON] and [PERSON] [33; 34] did not adapt to their experimental results obtained from five types of limestone and two of sandstone. To determine \(\sigma_{c}\) he tested cylindrical specimens with \(d\) values ranging from 12.5 to 150 mm. [PERSON]'s [44] discovery was no longer so much that his results did not fit a descending model with a \(k_{1}\) of approximately 0.18, but that for a diameter less than 40 mm and greater than 60 mm, the results of \(\sigma_{c}\) were inferior to those obtained within that interval (mainly 54 mm). In a \(\sigma_{c}\)-\(d\) graph, this means that for values of diameter less than 40 mm the model is ascending, and the model is descending for values greater than 60 mm. This phenomenon showed that the model of [PERSON] and [PERSON] [33; 34] was not applicable to all types of rock, being able to follow an ascending, descending, or mixed model.