## 2 Materials and Methods
### Necessary and Sufficient Conditions of Bale Dimensions Equilibrium and Near-Equilibrium
#### 2.3.2. Bale Volume of Near-Equilibrium  
From Equation (7), provided that inequality from Equation (10) is satisfied, we have  
\[V_{b0}=\frac{\pi}{4}\left[\frac{p_{b}\Omega\Big{(}k_{f}\Big{)}\left(2\delta- b_{fr}k_{fb}\right)\big{(}\sqrt[3]{1+\varepsilon_{0}}\big{)}^{2}}{p_{b} \Omega\Big{(}k_{f}\Big{)}-p_{u}\Big{(}1-k_{fb}\Big{)}-2p_{u}\Big{(}1-k_{fb} \Big{)}\varepsilon_{0}}\right]^{3}, \tag{11}\]  
whence  
\[\frac{\partial V_{b0}}{\partial\varepsilon_{0}}=\frac{\pi p_{b}\Omega\Big{(}k _{f}\Big{)}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}}{2\sqrt[3]{1+\varepsilon_{0}}}V _{b0}{}^{2/3}\frac{\Big{[}p_{b}\Omega\Big{(}k_{f}\Big{)}+2p_{u}\Big{(}1-k_{fb }\Big{)}+p_{u}\Big{(}1-k_{fb}\Big{)}\varepsilon_{0}\Big{]}}{\Big{[}p_{b} \Omega\Big{(}k_{f}\Big{)}-p_{u}\Big{(}1-k_{fb}\Big{)}-2p_{u}\Big{(}1-k_{fb} \Big{)}\varepsilon_{0}\Big{]}^{2}}.\]  
Thus, for given film and wrapping parameters, bale volume resulting in near-equal optimal dimensions is monotonically increasing function of \(\varepsilon_{0}\) whenever  
\[2\delta>b_{fr}k_{fb}, \tag{12}\]  
and decreases with \(\varepsilon_{0}\), if \(2\delta<b_{fr}k_{fb}\). If inequality from Equation (12) holds, then the volume \(V_{b0}\) decreases if the overlap \(k_{fb}\) grows, while in the opposite case the influence of \(k_{fb}\) is not so evident and depends on the sign of the expression in the square brackets of the numerator in the last fraction of the right hand side of the following equation  
\[\frac{\partial V_{b0}}{\partial k_{fb}}=\frac{3\pi p_{b}\Omega\Big{(}k_{f}\Big{)} \big{(}\sqrt[3]{1+\varepsilon_{0}}\big{)}^{2}}{4}V_{b0}{}^{2/3}\Big{[}-b_{fr}p_{ b}\Omega\Big{(}k_{f}\Big{)}+b_{fr}p_{u}(1-2\delta)+2p_{u}\varepsilon_{0} \Big{(}b_{fr}-2\delta\Big{)}\Big{]}}{\Big{[}p_{b}\Omega\Big{(}k_{f}\Big{)}-p_{u }\Big{(}1-k_{fb}\Big{)}-2p_{u}\Big{(}1-k_{fb}\Big{)}\varepsilon_{0}\Big{]}^{2}}.\]If condition from Equation (12) holds, then \(V_{b0}\) given by Equation (11) grows with the increase of the overlap \(\delta\). In the opposite case the larger \(\delta\) is, the smaller is the volume resulting in near-equal optimal dimensions. From the analysis of Equation (11) it also follows that the greater film width is, the smaller is \(V_{b0}\), provided that inequality from Equation (12) holds. The influence of the bottom layers overlaps \(k_{f}\) and \(\delta\) is illustrated in Figure 2a,b, where near-equilibrium volume \(V_{b0}\) is depicted as a function of \(\epsilon_{0}\) for plastic film (e.g., polyethylene, PE) characterized by Poisson's ratio \(v_{f}=0.34\)[38] and unit deformation \(\varepsilon_{lf}=0.7(-)\)[7;12;14] of popular width \(b_{f}=0.75\) m [21; 22; 23; 29; 42]. Four bottom and upper film layers \(p_{b}=p_{u}=4\) and overlap ratio \(k_{f}=0.5\) are assumed. Figure 3a,b illustrate the effect of the film width \(b_{f}\) for commonly used overlap \(\delta=0.2\) m [21; 22; 23] and two overlap ratios \(k_{fb}=0.3,\ 0.35\); other parameters remain unchanged. The volume \(V_{b0}=1.357\) m\({}^{3}\) corresponding to a standard large bale of diameter and height \(D_{b}=H_{b}=1.2\) m [24; 25] is also marked on these pictures.