# Sand Rubber Mixtures under Oedometric Loading: Sand-like vs. Rubber-like Behavior
## 3 Results and Discussion
### General One-Dimensional Response
Figure 2: Vertical strain as a function of the vertical stress for samples with 0% to 50% rubber fractions, loading, and unloading during oedometer tests: (**a**) SR-D tests; (**b**) SR-L tests (see Table 1). also witnessed by [40]. The unrecovered strains follow the same trends, with lower and higher strains for sR and Sr cases. The loading behavior in one-dimensional testing is generally represented by the evolution of the current void ratio, \(e\), with the vertical stress in the log scale (\(\log\sigma_{v}\)). The current void ratio is related to the vertical strain, \(\varepsilon_{v}\), by the equation:  
\[e=e_{0}-\varepsilon_{v}(1+e_{0}) \tag{1}\]  
where \(e_{0}\) is the initial sample void ratio. The one-dimensional loading behavior of the pure sand is mostly linear in the e--(\(\log\sigma_{v}\)) plane, slightly concave for large normal stresses (i.e., negative second derivative, see Figure 2). When rubber is added to the mixture, the stiffness progressively decreases, and the concavity becomes more pronounced as the behavior asymptotically converges to a normal compression line [22]. However, for high rubber fractions and at high vertical stresses (generally larger than 1000 kPa), one can observe a change of curvature in the void ratio--vertical stress semi-logarithmic plane--which becomes slightly convex, see for instance [22; 38; 41]. This is the effect of the saturation of the voids filled by the rubber grains and, thus, a clear sign of the influence of the rubber grains on the macroscopic behavior of the system. We, firstly, propose here to estimate the mean curvature of the e--(\(\log\sigma_{v}\)) curve between 1 and 2 MPa vertical stress. This property leads to a method that can be used to determine the behavior or the mixture: \"sand-like\" if the concavity increases with the rubber content and \"rubber-like\" if an increase in the rubber content induces decreasing concavity (suggesting, as mentioned above, that the pores are being saturated progressively). Then, the sign of the following curvature evolution with the rubber fraction:  
\[\widetilde{C}(\sigma_{v},x_{r})=\frac{\partial}{\partial x_{r}}\frac{\partial ^{2}e(\sigma_{v},x_{r})}{\partial(log\sigma_{v})2}=\frac{\partial^{3}e(\sigma _{v},x_{r})}{\partial x_{r}\partial(log\sigma_{v})2} \tag{2}\]  
indicates whether the mixture is sand (negative \(\widetilde{C}\)) or rubber type (positive \(\widetilde{C}\)) at the corresponding stress level. In the following, for practical reasons, the sign is determined by the mean of the curvature over a given vertical stress interval. Figure 4 shows the evolution of the mean curvature between 1 and 2 MPa of vertical stress with the rubber fraction for the two densities of the mixture and for the two particle sizes. It can be observed that the concavity of the e--(\(\log\sigma_{v}\)) curve increased with the rubber fraction, up to about 25-30% of rubber, and then decreased. The transition point between sand and rubber-type behaviors seems fairly similar for the two densities and the two particle sizes. Figure 5 presents the evolution of the mean curvature for five vertical stress intervals. For one rubber content, one point in the figure represents the mean curvature value for the SR-L, SR-D, and sr-D tests. This averaging is justified by the weak dependence of this curvature on the mixture type (see Figure 3). For vertical stresses higher than 600 kPa, the transition point seems to be fairly independent of the stress level. On the contrary, at vertical stresses lower than 600 kPa, the one-dimensional response of the mixture seems to be a sand-like type in all the tested ranges of rubber fractions. It is also interesting to observe that the curvature of the e--(\(\log\sigma_{v}\)) curve became convex for a rubber fraction of 50%, a phenomenon that clearly reflects a behavior closer to pure rubber packing. The overall information is schematically presented in the phase diagram (see Figure 6).