## 3 Comparisons  
For the purpose of a comparison, \(K_{eq}/K_{I}\) was plotted as a function of \(K_{I}/K_{II}\) and of \(K_{II}/K_{I}\). Equation (10) was adopted as representative of the [PERSON] approach, and the previous Equations (3)-(10) were rewritten as follows:  
_Energy:_  
\[\frac{K_{eq}}{K_{I}}=\left[1+\left(\frac{K_{II}}{K_{I}}\right)^{2}\right]^{ \frac{1}{2}} \tag{13}\]  
_[PERSON]:_  
\[\frac{K_{eq}}{K_{I}}=\left[1+8\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{4}\right] ^{\frac{1}{4}} \tag{14}\]  
_[PERSON]:_  
\[\frac{K_{eq}}{K_{I}}=\left[1+2\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{2}\right] ^{\frac{1}{2}} \tag{15}\]  
_Lardner B:_  
\[\frac{K_{eq}}{K_{I}}=\left[1+4\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{4}\right] ^{\frac{1}{4}} \tag{16}\]  
_[PERSON] and Keer:_  
\[\frac{K_{eq}}{K_{I}}=\left\{\left[1+3\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{2 }\right]^{3}\left[1+\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{2}\right]\right\}^{ \frac{1}{8}} \tag{17}\]  
_[PERSON]:_  
\[\frac{K_{eq}}{K_{I}}=\left\{\left[1+3\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{2} \right]^{3}\left[1+\bigg{(}\frac{K_{II}}{K_{I}}\bigg{)}^{2}\right]\right\}^{ \frac{1}{8}} \tag{18}\]Demir:_  
\[\frac{K_{eq}}{K_{I}}=\frac{K_{II}}{K_{I}}\left(1.0519\left(\frac{K_{I}}{K_{II}} \right)^{2}-0.035\left(\frac{K_{II}}{K_{I}}\right)^{2}+2.3056\right)^{\frac{1}{2}} \tag{19}\]  
Richard:  
\[\frac{K_{eq}}{K_{I}}=\frac{1}{2}+\frac{1}{2}\left[1+6\left(\frac{K_{II}}{K_{I}} \right)^{2}\right]^{\frac{1}{2}} \tag{20}\]  
Figure 1 plots the non-dimensional \(K_{eq}/K_{I}\) as a function of \(K_{I}/K_{II}\) for the approaches considered, in a region \(K_{I}/K_{II}\) < 2. Figure 2 plots the non-dimensional \(K_{eq}/K_{I}\) as a function of \(K_{II}/K_{I}\) for the approaches considered, in the region \(K_{II}/K_{I}\) < 5.  
Within the range of \(K_{I}\) and \(K_{II}\) values of Figures 1 and 2, [PERSON]'s approach [27; 28] leads to \(K_{eq}\) values that approximately follow the trend of other \(K_{eq}\) approaches. Nevertheless, the format of Equation (19) and its resulting curve in Figure 2 raised an interest in analyzing its behavior over a wider range of \(K_{I}\) and \(K_{II}\) values. Notice that, unlike the others, [PERSON]'s curve shows a noticeable decrease of the derivative as \(K_{II}/K_{I}\) increases. Indeed, Equation (19) shows \(K_{eq}/K_{I}=0\) for \(K_{I}/K_{II}=0.178\), which results from:  
\[\begin{array}{c}1.0519\left(\frac{K_{I}}{K_{II}}\right)^{2}-0.035\left(\frac {K_{II}}{K_{I}}\right)^{2}+2.3056=0\\ 1.0519\left(\frac{K_{I}}{K_{II}}\right)^{2}-\frac{0.035}{\left(\frac{K_{I}}{K _{II}}\right)^{2}}+2.3056=0\end{array} \tag{21}\]  
Discarding the complex region (square root of a negative number), Figure 3 shows \(K_{eq}/K_{I}\) as a function of \(K_{I}/K_{II}\) and of \(K_{II}/K_{I}\). Clearly, Equation (19) does not apply to all domains of \(K_{I}\) and \(K_{II}\); it presents zeroes for \(K_{I}/K_{II}\) = 0.178 and \(K_{II}/K_{I}\) = 8.144. Nevertheless, it follows the expected trend of the solution for \(K_{I}/K_{II}>\) -0.4 and for \(K_{II}/K_{I}\) < -4, as shown in Figure 3, where for comparison it is plotted together with [PERSON]'s equation.  
Two conditions that candidate definitions of \(K_{\alpha\beta}\) should satisfy are:  
\[\begin{array}{l}K_{\alpha\beta}/K_{I}\to 1\ \text{for}\ K_{I}/K_{II}\to\infty\\ K_{\alpha\beta}/K_{I}\to\infty\ \text{for}\ K_{I}/K_{II}\to 0\end{array} \tag{22}\]  
Figure 3: [PERSON]’s \(K_{\alpha\beta}\) definition: illustration of the limitations of the approach. For comparison, [PERSON]’s approach is also included.  
Since the present analyses are concerned with solutions that are as general as possible, the \(K_{\alpha\!q}\) definition of Equations (9) and (19) was no longer used, given its limited application.