Under conditions of one-dimensional confinement, the compressibility of dry and saturated Aeolian sand—rubber waste mixtures were investigated. The results of oedometer tests are plotted in Figure 4 in the form of void ratio (normal scale) versus the effective normal stress (logarithmic scale). This figure shows the ASRW mixtures’ compressibility curves for the diversity of RW fractions utilized in this work. The compressibility curves become increasingly nonlinear for mixtures with more than 30% RW content and when the applied stresses exceed 400 kPa. On the other hand, at RW content of more than 15%, it can be noted that the response of mixtures reveals distinct rebound curves at higher applied stresses. This is also noted for sandy soil samples tested by other scholars (e.g., Lee et al. 2007; Kim and Santamarina, 2008; Sheikh et al. 2013; Rouhanifar, 2017). The recorded RW content in the literature in which the same behavior was noted is more than 20% ( Edil and Bosscher, 1994; Muir-Wood and Stiffness, 2009; Lee et al. 2014; Liu et al. 2018; Ozkan et al. 2023b).

The Aeolian soil mixed with rubber waste exhibits low sensitivity to soaking. However, high compressibility and further deformation can be seen at higher RW content, at which the void ratio reaches its minimum value (close to 0.2). Moreover, at the end of un-loading stages, higher remnant deformation values can be observed. This is also noted to increase with the addition of the rubber inclusion, which is a compressible elastic material. The linear portion of loading and un-loading curves shown in Figure 4 is used to calculate the compression parameters (compression index, Cc, and rebound index, Cr) with variation of RW content in both dry and saturated conditions as presented in Figure 5. Almost, with the increase in RW content, above 15%, the Cc and Cr are increased with a linear trend; however, the trend is nonlinear with lower RW content. Finally, it can be concluded that the strain of ASRW produced under axially confined loading increases as the samples translate from rigid, pure sand grains, to elastic, deformable pure rubber grains.

Studying the stiffness of ASRW mixtures due to changing the rubber content is one of the main goals of the current paper. To achieve it, the results of the oedometer test have been analyzed and replotted in the form of effective normal stresses (kPa) versus the vertical strain, (%), (both in normal scale), as in Figure 6. It is worth noting that as the RW increased, the non-linearity of the normal stress-axial strain curves increased due to the lower stiffness and elastic deformability of RW grains. The slope of the curves in this figure was adopted in the calculation of the “one-dimensional incremental confined stiffness (SM)” of mixtures using the following equation ( Ozkan et al. 2023b): S M = Δ σ v Δ ε v (2)

Where Δσ _v and Δε _v are the increments in stress and strain for each tested specimen. According to Muir-Wood and Stiffness (2009), the SM, stiffness stress dependency, and effective normal stresses are related in the form shown in Equation 3: S M σ a = b ( σ v σ a ) c (3)

Where σa, b, and c are the reference stress, modulus number, and parameter of stiffness stress dependency, respectively. The value of 100 kPa is conceded for σa ( Ozkan et al. 2023a, 2023b). This value was used to normalize the SM and σv, then the relation between the normalized values was generated and plotted as shown in Figure 7. For both dry and saturated conditions, RW content above 15% has a pronounced effect, with curves in Figure 7 converging near the origin. This behavior indicates that the magnitude and rate of increase in mixture stiffness are at their lowest state. At higher RW content, the mixtures behave more like pure rubber than granular soil. The same trend was observed by scholars such as Madhusudhan et al. (2019) and Ozkan et al. (2023b) on sand mixed with a higher content of shredded tire or rubber material.

According to Muir-Wood and Stiffness (2009), the materials that have a lower value of modulus number are less stiff. Based on the findings of the parameters b and c, an examination of the stiffness properties of the ASRW mixtures was carried out. The variation of the parameters b and c with fractions of RW is plotted in Figure 8. Two behaviors are recognized in this figure: the first is the sand-like, and the second is the rubber-like ( Kim and Santamarina, 2008; Lee et al. 2010; Fonseca et al. 2019). Mixtures with 15% RW exhibit sand-like behavior, while those with 30% or more RW behave similarly to pure rubber. At a lower RW content, the contacts of rubber particles are less dominant; thus, sand-like behavior is dominant, and the value of the parameter b is less, and vice versa. On the other hand, the parameter of stiffness stress dependency reveals a direct proportionality with rubber content. Its value is close to unity with mixtures ASRW45 and ASRW100. This trend would signify the proportionality between the constrained modulus and the level of stress. However, the mixture stiffness increases with decreasing RW, and the soaking has a little impact on the ASRW mixtures’ stiffness.

A final comparison with scholars: the parameters b and c obtained in the current experiments are duplicated on the chart provided by Muir-Wood and Stiffness (2009) for various geomaterials, as in Figure 9. It is clear that the parameters (modulus number and the stiffness exponent) move downwards and upwards, respectively, as the content of the rubber fraction increases. In other words, the mentioned movement indicates a change in mixture behavior from more stiff and sand-like to softer and rubber-like. Nevertheless, all mixtures are within the limits of defined geomaterials.

To understand the collapse behavior of Aeolian soil mixed with rubber waste, the designated mixtures herein were subjected to double oedometer tests. The change in axial strain with effective normal stress was recorded, the calculations were conducted, and the results were plotted in Figure 10a. The effect of replacing the solid AS grains with the compressible soft RW and soaking on their mixtures is obvious in this figure. The dense pure AS exhibits high stress resistance in dry and wet states due to the rigid grains nature. This behavior changes slightly as a low fraction (<30%) of the RW is included. The non-linearity of curves is increased slightly due to the impact of such a soft material inclusion. With further inclusion, this state becomes more pronounced. Meanwhile, in the case of pure RW, even at the low applied stress, large axial strain is produced; this is, however, an indication to the elastic deformability of this material.

On the other hand, the collapse potential (CP) was calculated at different effective stresses, as shown in Figure 10b. Limited collapse potential occurs in pure AS, mixtures with lower RW fractions, and pure RW. While moderate to moderately severe collapse (ASTM D5333) occurs for higher RW mixtures. Limited collapse is noted for wetted AS due to the rigid skeleton, high stiffness, and fewer voids. The CP is increased as the RW inclusion increases, allowing the re-arrangement of grains and causing a slight collapse. Higher rubber content replaces more of the solid skeleton, forming hybrid mixtures with increased collapsibility. The skeleton of these mixtures is weaker, allowing more rearrangements of grains, and as a result, a greater collapse (this is pronounced for mixture ASRW45). On the other hand, the pure RW samples do not seem to experience structural collapse due to the nature of the rubber material, i.e., the hydrophobicity and resistance to water ingress.

In light of the rubber energy dissipation property ( Liu et al. 2020; Han et al. 2020), the mixtures of ASRW were tested under cycles of loading—unloading-reloading confined compression conditions in both dry and wetted states, as shown in Figure 11. From a geotechnical perspective, the loops in this figure can provide insights into stiffness and energy dissipation of the dry and saturated Aeolian soil—rubber waste mixtures under the above condition. Regarding the dissipated energy, Fonseca et al. (2019) stated that the energy dissipation can be obtained by measuring the area between the loading curve and the reloading curve for each of the loops of the loading—unloading-reloading test like those in Figure 11. These authors emphasized that energy dissipation measured from the cyclic oedometer test is applicable to provide comments regarding the nature of energy.

However, to provide a precise measurement of the energy dissipation, each test sample, the curves in a specific loop of loading and unloading in Figure 11 are subjected to nonlinear regression. The fitting using a saturating hyperbolic function ( Equation 4) was applied to capture a smooth transition and the curvature of the progressive loading and unloading stages. The values of the constants (a1 and a2) and the statistical coefficients for Equation 4 were determined. e v = s v n 1 + n 2 s v (4)

The obtained differentiable hyperbolic function enables the calculation of the closed area of each loop. According to the literature ( Jiang et al. 2019; Al-Taie, 2025a), the analytical integration of the saturated function of fitted stress-strain curves is the energy dissipated (ED). Therefore, the area for each loop in Figure 11 is calculated using the following expressions: ED = ∫ σ 1 σ 2 e v ( σ ) dσ (5)

The loading curve represents the absorbed energy, while the unloading curve is the released energy; both can be calculated exactly from Equation 5. Therefore, the loop area was computed using this equation to represent the dissipated energy. The values calculated herein is applicable to provide comments regarding the nature of energy.

Table 2 presents the variation of the ED with different rubber fractions for all loops. It is proven in the previous section that the stiffness of RW is much lower than that of AS; also, its deformation tendency is higher. The RW has a tendency to recover the compressed volume, while the AS may undergo crushing, thus, it has a lower volume recovery. Therefore, it is expected that the behavior of the mixtures of these materials differs from that of the pure materials.

The internal friction and irreversible deformation of the geomaterials under loading cycles are reflected in the form of dissipated energy. In each cycle of loading, the material loses work, and this work is represented by the dissipated energy. The energy is related to the ability of the geomaterials to mitigate the vibrations from dynamic loading (i.e., the damping). These quantities are directly proportional in the mixtures of granular geomaterials and rubber materials. The mechanisms controlling the damping and energy dissipation of such mixtures are related to the interfacial friction, viscoelastic hysteresis, and particle rearrangement of the grains of rubber. A direct assessment of the ED is possible from the loading—unloading curves of the cyclic odometer’s loops. More energy dissipation ability is expected in mixtures with a higher loop area. From Figure 11 and Table 2, it is clear that the inclusion of RW grains in the mixture of AS augments the energy dissipated. Such inclusion introduces additional viscoelastic resistance and deformability to the produced mixtures. This is, however, dependent on the level of the applied stresses. Both the AS skeleton and the RW particles undergo compression and rearrangement simultaneously when this level is low to moderate (less than 400 kPa). At higher stresses, a strong compression occurs in RW grains, causing enlargement in the area of the loading—unloading loop, thereby increasing the energy dissipation. Also, at very high stresses, the densification of the mixtures is increased, where both AS and RW grains compact considerably; as a result, more energy is dissipated.

The effect of saturation on the dispersion of energy of ASRW mixtures is investigated in this work. As shown in Figure 11, the loading—unloading loops were generated for both dry and soaked mixtures. In general, the soaking impact on the energy dissipation is found to be slight. At a stress level of 50 to 100 kPa, a slight shrinkage in the area of the loading—unloading loops is noted due to soaking. However, the reduction is less pronounced at 100 kPa. Furthermore, a narrow difference in ED, compared to the unsoaked state, was recorded for mixtures loaded to 200 and 400 kPa. Meanwhile, a negligible impact on energy dissipation has been noted in saturated mixtures tested at 800 kPa where the viscoelastic behavior of the RW mostly controls the dissipated energy ( Fonseca et al. 2019; Dai et al. 2023, 2024, Li et al. 2024).