Effect of particle rolling resistance on drained and undrained behaviour of silty sand

Abstract

This paper explores the effect of particle rolling resistance on the mechanical behaviour of silty sand under various shearing conditions. The elastic–plastic spring–dashpot rolling resistance model was employed, and drained and undrained triaxial tests were conducted on silty sand with various fines contents and rolling resistance. It was found that the rolling resistance of fines contents had a strong impact on the critical state, peak state, phase transformation state as well as zero-dilatancy state of silty sands. Moreover, depending on the correlation between rolling resistance of coarse and fine particles, fines can positively or negatively contribute to the overall structure of silty sand. Increasing the rolling resistance of fines enhances the liquefaction resistance of silty sands. Thus, the presence of fines of high rolling resistance in the sand can result in a marked decrease in collapsibility or liquefaction susceptibility compared with the presence of fines of lower rolling resistance.

Appendix

1.1 Appendix A: DEM modelling

In a conventional discrete element formulation proposed by Cundall and Strack [15], the motion of a solid particle i can be described by Newton’s second law, which consists of the force and moment balance equations as follows:

$$m_{i} \frac{{d\overrightarrow {{u_{i} }} }}{dt} = \mathop \sum \limits_{{j \in CL_{i} }} \vec{f}_{ij}^{c} + \vec{f}_{i}^{g}$$

(8)

$$I_{i} \frac{{d\overrightarrow {{\omega_{i} }} }}{dt} = \mathop \sum \limits_{{j \in CL_{i} }} \vec{M}_{ij}^{c}$$

(9)

where $m_{i} , I_{i}$ are the mass and moment of inertia of particle i; $\overrightarrow {{u_{i} }} , \overrightarrow {{\omega_{i} }}$ are the translational and angular velocities of particles i; $\vec{f}_{ij}^{c}$ and $\vec{M}_{ij}^{c}$ are the contact force and moment acting on particle i by particle j and/or wall; and $\vec{f}_{i}^{g}$ is the force due to gravity. The interactions between two particles and between particles and boundaries consist of contact spring forces and damping forces in both the normal ($\vec{n}$) and tangential ($\vec{t}$) directions given by:

$$\overrightarrow {{f_{ij}^{c} }} = F_{nij} \vec{n} + F_{tij} \vec{t}$$

(10)

where $F_{nij}$ and $F_{tij}$ are the normal and tangential components of the total force, which can be computed based on Hertz–Mindlin contact theory as follows [65], respectively:

$$F_{nij} = - \kappa_{n} \delta_{n}^{{\frac{2}{3}}} - \eta_{n} \vec{v} \cdot \vec{n}$$

(11)

and

$$F_{tij} = - \kappa_{t} \delta_{t} - \eta_{t} \left[ {\vec{v} - \left( {\vec{v} \cdot \vec{n}} \right)\vec{n} + r\left( {\overrightarrow {{\omega_{i} }} + \overrightarrow {{\omega_{j} }} } \right) \times \vec{n}} \right] \cdot \vec{t}$$

(12)

In these equations, $\vec{v} = \overrightarrow {{u_{i} }} - \overrightarrow {{u_{j} }}$ is the relative velocity between two particles; $\vec{\omega }$ is their rotation speed; $\delta$ is the overlap distance; $\kappa$ is the equivalent stiffness which is calculated using Young’s modulus E and Poison ratio $\nu$ [65]; and $\eta$ is the damping coefficient calculated based on restitution coefficient $e$. The tangential force $F_{tij}$ is limited by Coulomb friction $\mu_{s} F_{nij}$ where $\mu_{s}$ is the coefficient of static sliding friction [65].

The moment $\vec{M}_{ij}^{c}$ is calculated as follows:

$$\vec{M}_{ij}^{c} = r_{s} \vec{n} \times F_{tij} \vec{t}$$

(13)

where $r_{s}$ is the radius of particle $i$.

The elastic–plastic spring–dashpot (EPSD) model is employed to provide a stable resistance torque in quasi-static condition [1]. This model is illustrated in Fig. 19.

Fig. 19

Contact model with rolling resistance [27]

This model generates an additional torque contribution to the momentum equation. The torque resulting from the tangential force and the relative rotation of particles consists of two components corresponding to the spring and dashpot:

$$M_{r} = M_{r}^{k} + M_{r}^{d}$$

(14)

The torque component due to spring $M_{r}^{k}$ is calculated by rolling stiffness $k_{r}$ and the incremental relative rotation $\Delta \theta_{r}$ between 2 particles given by:

$$M_{r}^{k} = - k_{r} \Delta \theta_{r}$$

(15)

Rolling stiffness $k_{r}$ is proportional to the stiffness of normal contact law $k_{n}$, the effective radius $R$ and the coefficient of rolling resistance $\mu_{r}$ [1]:

$$k_{r} = 2.25k_{n} \mu_{r}^{2} R^{2}$$

(16)

The torque component due to viscous damping $M_{r}^{d}$ is mathematically determined by damping coefficient $\eta_{r}$ and relative rotation speed $\omega_{r}$:

$$M_{r}^{d} = - \eta_{r} \omega_{r}$$

(17)

where the damping coefficient $\eta_{r}$ can be inferred from the rolling stiffness $k_{r}$, the effective inertia moment $I_{r},$ and the rolling viscous damping coefficient $\alpha_{r}$:

$$\eta_{r} = 2\alpha_{r} \sqrt {k_{r} I_{r} }$$

(18)

1.2 Appendix B: Comparisons between rolling resistance of spherical particles and particle clump

There several studies using both experimental and numerical approaches indicated that particle shape strongly affects the mechanical behaviour of the granular materials [23, 68]. Therefore, numerical simulations idealising soil grains with spherical particles cannot be thoroughly captured the behaviour of soil [13]. However, it is too cumbersome and time-consuming to replicate the real particle shape by scanning the particle surface. Numerical techniques such as the granular element method (GEM) [26] or the level set discrete element method (LS-DEM) [29, 30, 40] require a lot of experimental and numerical effort. Therefore, researchers often deal with super-quadric or multi-spherical particles [23, 51]. As mentioned by Gong et al. [23], the sphere clump or multi-sphere particle approach that requires a significant number of prime spheres will lead to simulation with an extremely high computational cost for high accuracy of the shape approximation. Similarly, the superquadric approach, which can only model ellipsoidal, box-like and cylinder-like particles, is also computationally expensive due to the contact detection procedure [51].

Particles with rolling resistance, in the view of reducing computational cost, can be considered as an alternative since it mitigates the rolling effect while keeping the particle shape as sphere according to Wensrich and Katterfeld [70]. Using spherical particles with the constraint of rotation is somewhat unrealistic because the rotating moment is always against the particle motion. Nevertheless, the study of Wensrich and Katterfeld [70] showed that some similarities were observed between particles with rolling resistance and sphere clumps, but, quantitatively, there was a significant discrepancy. To make an appropriate link between particle rolling resistance and the particle aspect ratio and to establish how particle rolling friction can approximate the particle’s shape, different 3D simulations using DEM were conducted. In these simulations, drained triaxial tests of two different grain types: (1) particle clumps (multi-sphere particles) vs (2) spherical particles with rolling resistance, are performed to compare the mechanical response under shear.

The clump of particles consisting of 4 overlapping spheres of the same diameter represents the non-spherical particle that can constrain the free-rolling. The volume, density, and mass of the clump are similar to the spherical particle to produce the equivalent grain size distribution. Samples of clump are generated using the same method used for preparing samples described in the previous section. The final samples used for triaxial drained tests can be seen in Fig. 20.

Fig. 20

Sample containing clumps of 4 spheres: a clump configuration; b DEM assembly

Simulation results for drained triaxial tests are shown in Fig. 21. The behaviour of the sample made of clumping particles is quite close to that of a sample composed of spherical particles with a rolling resistance coefficient of 0.2. Higher rolling resistance increases the shear strength of the materials, evidenced by the increase in the peak strength and residual strength of the samples as the rolling coefficient increases. In addition, peak state and zero-dilatancy state can be achieved at a similar shear strain. Without rolling resistance effect, even further increase in sliding friction coefficient higher than 0.5 [4, 5, 18], the residual stress ratio cannot reach the value around 1.25 which is often observed in experimental studies [5, 32].

Fig. 21

Comparison of shear behaviour among clump and sphere with and without rolling resistance at confining pressure of 100 kPa and void ratio ~ 0.6

Furthermore, the similarity in critical states can be found in Fig. 22 between samples using clump and sphere with a rolling resistance coefficient of 0.2. However, clump particles show more interlocking effect and stiffen the material causing the change in the slope of the critical state in volumetric space, especially when effective stress is higher than 800 kPa. At low effective stress, the effect of particle shape can be approximated by the effect of particle rolling resistance. Each combination between rolling resistance and sliding resistance can identify a certain grain morphology from well-rounded to high angular. This result supports the use of rolling resistance as a shape parameter accounting for particle angularity and shows that the partial hindrance of rotations as a result of the particle shape is one of the main influencing factors behind the mechanical behaviour of granular systems composed of noncircular particles [19].

Fig. 22

Rolling resistance effect on critical state: a critical state in stress space; b critical state in volumetric space