Shear modulus is widely used to represent the stiffness characteristics of geological materials. It is calculated as G = 1/3 x dq/dγ, with dq and dγ representing the changes in deviatoric stress and shear strain, respectively. The small-strain shear modulus, G_(max), specifically refers to a very small strain less than 10^-4%, where behavior is elastic. The G_(max) was calculated for the DEM models using the procedure followed by Gu et al. (2013), Lopera Perez et al. (2017), and Reddy et al. (2022).

3.1.1 Effect of Particle Young's Modulus (E) and Confining Pressure (p′) on G_(max) of Quartz Sand

Since natural sands exhibit varying contact properties due to surface roughness and particle Young's modulus characteristics, a change in surface roughness may lead to changes in Young's modulus, normal contact stiffness, and interparticle friction. This study aims to understand the influence of particle Young's modulus and confining pressure on the DEM model's responses. A numerical exercise was conducted by varying particle Young's modulus and confining pressure while keeping other parameters constant. The representative particle Young's modulus values, i.e., E = 10 GPa, 50 GPa, and 100 GPa, were considered along with a range of confining pressures. The deviatoric stress versus shear strain response and the G_(max)-p′ relationship were investigated and analyzed. The results are presented in Figure 6, showing the G_(max) - p′ relationship and the corresponding power-law fitting based on Hardin and Black (1966).

A chart illustrating the normal contact stiffness (k_(n)) of a DEM model at the G_(max) position for two samples with particle Young’s modulus, E = 50 GPa and 100 GPa, and a confining pressure (p′) of 100 kPa has been generated, as shown in Figure 7. This bar chart depicts the distribution of contact normal stiffness among grain contacts for both samples, with values that are comparable to those derived from micromechanical tests on sands (as shown in Figure 3). The DEM model simulated contact behavior at the grain scale and revealed that the normal contact stiffness was distributed over a wider band as the particle Young’s modulus increased from E = 50 GPa to 100 GPa. This implies that the variation in distribution of normal contact stiffness causes a change in granular matrix formation that affects the contact force network. Therefore, by varying the E value, the DEM model may generate different force transfer networks that affect the stress-strain response of a numerical sample during triaxial testing, ultimately affecting the G_(max) derived from the deviatoric stress versus shear strain curve.

Interparticle friction (μ) is another crucial contact parameter that may influence G_(max) behavior by contributing to the formation of the granular matrix (Fabric). For all DEM simulations, a constant value of μ of 0.23 was considered during the sample preparation and confining stage. Subsequently, the DEM model's response was captured during a triaxial test for five different values of interparticle friction, i.e., μ = 0.2, 0.23, 0.3, 0.4, and 0.6, to assess the influence of interparticle friction on G_(max) behavior. Figure 8(a) shows the G_(max) variation with an increase in interparticle friction, captured at four different values of confining pressure (p′) of 25, 50, 100, and 200 kPa for DEM samples with E = 100 GPa. This assessment can provide an indirect understanding of the effect of roughness on G_(max) behavior of sands.

Lastly, box plots in Figure 8(b) display the contribution of contact shear force with respect to contact normal force from the data obtained at the G_(max) position for DEM samples with different interparticle friction values (μ), E = 100 GPa, and p′ = 50 kPa. This can provide further insight into the effect of interparticle friction on G_(max) behavior.

5. Conclusions

We conducted microscale grain tests and numerical simulations using DEM to investigate the elastic properties of a uniform packing of quartz sand spheres. Our grain-scale tests yielded valuable data on the normal and tangential force-displacement responses of sand grains and the range of contact normal stiffness and coefficient of interparticle friction these grains exhibit. We also observed changes in tangential contact behavior due to micro-slip displacement during shearing.

To develop our Discrete element models, we used contact properties based on the realistic range of particle properties observed in our grain-scale tests. Our models were validated against data obtained through element size (macroscale) tests, and a parametric study was conducted to explore the influence of particle Young’s modulus and interparticle friction on the G_(max) of sands.

We found that an increase in particle Young’s modulus led to an increase in G_(max), indicating the influence of normal contact stiffness distribution on the stress-strain response. This also affected the relationship between the shear modulus and confining pressure. Our observations provide insight into the influence of surface roughness, which can decrease the macroscopic stiffness of granular materials.

We also observed that an increase in interparticle friction from 0.2 to 0.4 led to an increase in G_(max), but this relationship was independent of interparticle friction for higher values. The influence of interparticle friction on G_(max) was most pronounced at lower values of confining pressure. Overall, our findings suggest that normal contact stiffness has a greater influence on G_(max) than interparticle friction.