## 1 Introduction
[PERSON] et al. [8] investigated the relationship between the degree of decomposition and solid particle density. It was found that an increase in the degree of decomposition increased the solid particle density. This was caused by the loss of organic matter, which had high volume and low mass. [PERSON] et al. [4] also studied the correlation between the solid particle density and decomposition. They found that the studied quantities showed a strong positive linear correlation. [PERSON] et al. [3] investigated the influence of particle size, compaction, and the degree of decomposition on solid particle density. The results they obtained showed that the solid particle density increased with a decrease in particle size, compaction, and increasing decomposition (aged waste). The authors attributed the increase in the solid particle density (with reduced particle size and compaction) to the potential exposure of previously occluded intraparticle voids and the access to these voids. For aged waste, the increasing solid particle density was attributed to the loss of biodegradable components (with low solid particle density values). Therefore, the solid particle density of waste is a variable quantity, which depends on waste composition, the decomposition stage of organic content, the size of individual particles, the increase in vertical stress, and the moisture content. Another physical quantity that has a significant impact on the available landfill capacity is the dry density (\(\rho_{d}\)). By definition, dry density is the ratio between the dry mass (\(m_{s}\)) and the total specimen volume (\(V\)). It is commonly associated with compaction/compression processes and the densification of waste material in response to the applied vertical stress [9; 10]. The variation in the dry density associated with vertical stress can be established by Equation (2) [2]:  
\[\rho_{d}=\frac{\rho_{d0}}{1-\frac{\Delta e}{1+e_{0}}} \tag{2}\]  
where \(\rho_{d0}\) is the initial dry density, \(e_{0}\) is the initial void ratio, and \(\Delta e\) is the change in the void ratio caused by the vertical load. It is worth noting that the dry density of landfilled waste material is also a variable quantity, which depends on mass and volume changes caused by the vertical load and the decomposition process. Finally, the theoretical correlation between the solid particle density, dry density, and the corresponding measure of volume change (i.e., void ratio \(e\)) is commonly expressed using:  
\[e=\frac{\rho_{s}}{\rho_{d}}-1 \tag{3}\]  
where \(e\) is the current void ratio, defined as the ratio between the volume of pores (\(V_{\text{\tiny D}}\)) and the volume of solid particles (\(V_{s}\)). A closer inspection of Equation (3) reveals that, in the case of the constant solid particle density and increasing dry density, a hyperbolic function with an asymptote parallel to the horizontal axis at \(e=-1\) is obtained. Such a case can be typically observed in soils for which the solid particle density is commonly assumed to have a constant value. Thus, for soils, the void ratio is a function of only one variable: dry density. However, the strong heterogeneous nature of MSW prevents the approximation of the solid particle density of waste materials with a constant value. Therefore, it is reasonable to assume that a set of hyperbolic curves for waste materials should exist, wherein each curve is defined with a unique solid particle density value. Therefore, for waste materials, the void ratio is a function of two variables: solid particle density and dry density. The second case that arises from Equation (3) is that, although the solid particle density varies, the dry density remains constant. In such a scenario, Equation (3) reveals that a linear relationship between the solid particle density and void ratio should exist. In many cases, the validity of Equation (3) for waste materials is tacitly assumed.