# Sand Rubber Mixtures under Oedometric Loading: Sand-like vs. Rubber-like Behavior
## 3 Results and Discussion
### One-Dimensional Confined Stiffness  
The one-dimensional compression of the sand-rubber mixtures is non-linear and irreversible. The incremental confined stiffness, M, is given by the following relation:  
\[M{=}\delta\sigma_{v}/\delta\varepsilon_{v} \tag{3}\]  
where \(\delta\sigma_{v}\) is the increment of vertical effective stress and \(\delta\varepsilon_{v}\) is the increment of vertical strain, which is, therefore, not constant during the one-dimensional loading. A power law has been suggested to describe the link between the one-dimensional incremental confined stiffness, M, and the vertical effective stress, \(\sigma_{v}\), [49]:  
\[M=m\sigma_{ref}\left(\frac{\sigma_{v}}{\sigma_{ref}}\right)^{\alpha} \tag{4}\]  
Figure 4: Mean curvature of the e—log (\(\sigma_{v}\)) evolution between 1 and 2 MPa of vertical stress for equally sized particle mixtures of SR-L, SR-D, and sr-D (see Table 1). Figure 5: Mean curvature of the e—log \(\sigma_{v}\) evolution in several ranges of vertical stress; the mean of the values is determined for the equally sized SR-D, SR-L, and sr-D mixtures at the same rubber fraction. where \(\sigma_{ref}\) is a reference pressure, introduced for dimensional consistency, taken as 100 kPa, and \(m\) and \(\alpha\) are material parametersf. The exponent \(\alpha\) controls the way in which the incremental stiffness changes with the level of vertical stress. The m, material parameter, (known also as modulus number) is the magnitude of the confined stiffness at the reference pressure. Thus, the confined stiffness was analyzed for vertical stresses ranging from 100 kPa to 500 kPa. This choice was made to avoid the influence of particle crushing, which may occur at higher levels of compression stresses [50]. Post-test particle size analysis showed considerable sand grain crushing. The parameters m and \(\alpha\) can be easily obtained by plotting M/\(\sigma_{ref}\) versus the \(\sigma_{v}/\sigma_{ref}\) ratio. Figure 7 reports the corresponding curves for the equally sized sand-rubber particles, SR-D and SR-L test series. The parameter \(\alpha\) seems to be independent of the rubber fraction and slightly influenced by the initial void ratio, with averages of 0.69 for SR-D and 0.71 for SR-L. However, the observed difference of \(\alpha\)-values between the two packing densities is not statistically relevant given the standard errors, 0.012 and 0.022 respectively. The sr-D test series having smaller particles (D\({}_{50}\) = 1.4 mm) showed identical trends and similar \(\alpha\) values. In all, the observed values of the power index \(\alpha\) are close to the results of [30]. Figure 6: Schematic phase diagram showing the typical behavior of the equally sized sand–rubber mixtures as a function of the stress level and the rubber fraction. Figure 7: Confined modulus, M, versus vertical stress, \(\sigma_{v}\), relation for equally sized particles (D\({}_{50}\) = 1.8 mm): (**a**) SR-D and (**b**) SR-L for different rubber fractions \(\mathrm{x_{r}}\) (low-pass filter was applied on the confined modulus). The values of the modulus number, m, with the rubber content for all SR-L, SR-D, and sr-D test series are presented in Figure 8. A power law decrease of the modulus number, m, with the rubber fraction is observed for both densities. The ratio between the m-values for loose and dense packings is relatively constant, around 75% (standard deviation of 7%), in the ranges of all rubber fractions. The influence of the particle size ratio on \(\alpha\) and m parameters is analyzed in Figure 9. It can be observed that \(\alpha\) is not significantly affected by the particle size ratio. The slopes of the one-dimensional confined stiffness versus vertical stress for both sR-D and Sr-D tests are because of the SR-D for both the 10% and 50% rubber contents. However, the influence of the sand-rubber particle size ratio on the modulus number is non-negligible, when the rubber particles are bigger than the sand particles (sR-D-10 and sR-D-50 tests).