## 2 Literature Review on the Relation between Young and Apparent Compression Moduli  
One of the first models was proposed by [PERSON] and [PERSON] (1965). Considering a disk sample and assuming a condition of total incompressibility, the resultant apparent compression modulus can be determined by Equation (3), considering the geometry of the specimen ([PERSON], 1965):  
\[E_{c}=E_{0}\Big{(}1+2f^{2}\Big{)} \tag{3}\]  
where \(f\) is the shape factor. Other cross-section geometries and loading modes are studied in ([PERSON] and [PERSON], 1965). Another aspect of the relation between moduli noticed, and accounted for in the work of [PERSON] and [PERSON] (1965), was the influence of bulk modulus for blocks with large shape factors.  
The study presented by [PERSON] et al. (2019) did not follow the same assumption of a parabolic deformation of lateral surfaces considered in ([PERSON], 1965). Thus, another approach to the problem of compression loading was performed based on a superposition method. The expression determined by the authors also included the effect of bulk modulus (\(K\)), as presented in Equation (4), for the case of circular cross-section blocks:  
\[\frac{1}{E_{c}}=\frac{1}{E_{0}}\Bigg{[}1-f\sqrt{\frac{2}{3}}tanh\Bigg{(} \frac{1}{f}\sqrt{\frac{3}{2}}\Bigg{)}\Bigg{]}+\frac{1}{K}. \tag{4}\]  
[PERSON] (1965) developed theoretical relations for the compression moduli of blocks with circular and cross-section for soft elastic materials, following the same assumptions used for incompressible materials in the [PERSON] and [PERSON]'s study ([PERSON], 1965). The approach was validated by the results' agreement with finite element analysis. For circular cross-section blocks, the resultant analytical expressions for the determination of apparent compression modulus are presented:  
\[E_{c}=2G+\frac{\lambda G}{E_{3}}\Bigg{[}1+\frac{3}{8}\frac{\lambda}{E_{3}} \Bigg{(}\frac{w}{h}\Bigg{)}^{2}\Bigg{(}1-\frac{G\Big{(}\frac{w}{h}\Big{)}^{2} }{2E_{3}+\frac{33}{32}G\Big{(}\frac{w}{h}\Big{)}^{2}}\Bigg{)}\Bigg{]},\quad w<w _{2} \tag{5}\]  
\[E_{c}=\lambda+2G-\frac{\lambda^{2}}{15E_{3}}\frac{w_{2}}{w}\Big{(}8-\frac{w_{ 2}}{w}\Big{)}\quad w\geq w_{2} \tag{6}\]  
where \(G\) is shear modulus, \(\lambda\) is second Lame constant, \(w\) is the width of the cross-section and \(h\) is the thickness of the block. The parameters \(E_{3}\) and \(w_{2}\) are calculated using Equations (7) and (8):  
\[E_{3}=\lambda+G \tag{7}\]  
\[w_{2}=\sqrt{\frac{64E_{3}h^{2}}{15G}} \tag{8}\]  
A simpler model to determine the relationship between apparent compression and [PERSON]'s moduli was proposed by [PERSON] and [PERSON] (2000) for disks. The resulting equation is only dependent on [PERSON]'s ratio (\(v\)) and aspect ratio (\(S\))--ratio between the radius and thickness--and is presented in Equation (9):  
\[\frac{E_{c}}{E_{0}}=\frac{1+3\
u\Big{(}\frac{1-\
u}{1+\
u}\Big{)}S^{2}}{1+3 \
u(1-2\
u)S^{2}} \tag{9}\]  
Discussion about the compression behaviour of elastomer materials continues as other conditions are considered to the problem solving, and mathematical models' performance is compared against experimental testing data ([PERSON] and [PERSON], 1965; [PERSON] and [PERSON], 1965). Other conditions studied include the influence of the boundary condition type, such as the existence of frictional contact between sample and plates ([PERSON] and [PERSON], 1965; [PERSON] and [PERSON], 1965; [PERSON] and [PERSON], 1965; [PERSON] and [PERSON], 1965) and the application of non-linear models for large strains ([PERSON] and [PERSON], 1965), for example.