Evaluating the particle rolling effect on the characteristic features of granular material under the critical state soil mechanics framework

Abstract

The discrete element method (DEM) has been extensively used to capture the macroscopic and particulate response of granular materials. Although particle rolling (i.e. controlled by rolling resistance) has been acknowledged as a major contributing factor towards micro-mechanical behaviour of idealized spherical granular material, its influence on characteristic behaviour has not been thoroughly investigated within critical state soil mechanics (CSSM) framework. For instance, the influence of particle rolling on characteristic features of undrained and drained behaviour (e.g. phase transformation, characteristic state, instability, dilatancy, critical state) and the state parameter, (ψ) has not been captured. In this study, a series of constant volume (CV) and drained triaxial compression simulations were undertaken using a rolling resistance linear contact model, deployed within a DEM software. The CSSM framework was centrally used to assess the influence of particle rolling tendencies/resistance on CV and drained behaviours from both a macro- and micro-mechanical standpoint. The study advanced the current understanding of the influence of rolling resistance on CS-related behaviour.

Graphic abstract

Appendices

Appendix A: Method of deformability

1.1 Computation of stiffness parameters

Through the method of deformability:

$$k_{n} = \frac{{AE^{*} }}{L}$$

(12)

$$k_{s} = \frac{{k_{n} }}{{k^{*} }}$$

(13)

$$k_{r} = k_{s} R_{r}$$

(14)

here A = πr² represents the contact area. Depending on the type of entities in contact r is computed:

$$r = \left\{ {\begin{array}{*{20}c} {\hbox{min} \left( {R^{\left( 1 \right)} , R^{\left( 2 \right)} } \right), \,for\, \space particle - particle \space \,contacts } \\ {R^{\left( 1 \right)} , \,for \space\, particle - wall \space\, contacts} \\ \end{array} } \right.$$

(15)

where $R^{\left( 1 \right)}$ denotes the radius of contact entity 1. In similar nature the contact length (L) is computed based on contact type:

$$L = \left\{ {\begin{array}{*{20}c} {R^{\left( 1 \right)} + R^{\left( 2 \right)} , \,for \space\, particle - particle \space\, contacts} \\ {R^{\left( 1 \right)} , \,for \space\, particle - wall \space\, contacts} \\ \end{array} } \right.$$

(16)

E^* and k^* = k_n/k_s are input parameters in the study and are discussed and presented in Table 1 in the manuscript. The rolling radius, R_r and its formulation is also presented in the manuscript.

Appendix B: Rolling resistance linear contact model

The rolling resistance linear contact model (RRLCM) utilised in this study adds to the conventional linear contact model commonly used in DEM through the installation of a rolling spring and dashpot at the contact (Fig. 21).

Fig. 21

Rolling resistance linear contact model. After Iwashita and Oda [32]

In a rheological sense, the rolling spring signifies the presence of an elastic resisting moment between contacting pieces ($M_{r}^{k} )$, whilst the dashpot signifies the presence of a viscous moment at the contact $(M_{r}^{d} )$. The overall rolling resistance moment (M_r) may be mathematically defined as:

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

(17)

to effectively utilize the RRLCM in a numerical modelling environment, M_r must be updated incrementally with respect to the time step. At time t + Δt, $M_{r}^{k}$ may be computed via:

$$M_{r, \Delta t + t}^{k} = \left\{ {\begin{array}{*{20}c} {M_{r, t}^{k} + \Delta M_{r}^{k} } \\ { \left| {M_{r, t + \Delta t}^{k} } \right| \le M^{*} } \\ \end{array} } \right.$$

(18)

here M* represents the maximum (limiting) resisting torque ($M^{ *} = \mu_{r} \bar{R}F_{n}$) and $\bar{R} = \left( {r_{i} r_{j} } \right)/\left( {r_{i} + r_{j} } \right)$; where F_n is the normal contact force, $\bar{R}$ represents the effective radius of the contact, $r_{i}$ and $r_{j}$ are the radii of contacting entities. When a boundary or wall element is the contacting entity, r → ∞. Notice, when $\mu_{r} = 0$, $M^{ *} = 0$ and therefore a free-rolling environment is created. $M_{r}^{k}$ is the incremental rolling resistance torque observed at time, t + Δt and is computed through consideration of the rolling stiffness, k_r and the relative rotation between two contacting particles, θ_r,

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

(19)

where k_r = k_sR_r. A limitation of the model is defining a k_r which has strong physical basis. k_r is related to k_s based on an idealized consideration that the moment generated at a contact due to shear displacement is equivalent to the moment generated due to rolling displacement. $M_{r}^{d}$ in Equation 17 is also updated with respect to the time step, t + Δt:

$$M_{r, t + \Delta t}^{d} = \left\{ {\begin{array}{*{20}c} { - C_{rr}\, if \,\left| {M_{r, t + \Delta t}^{k} } \right| < M^{ *} } \\ { 0 \,if \,\left| {M_{r, t + \Delta t}^{k} } \right| = M^{ *} } \\ \end{array} } \right.$$

(20)

where $C_{r} = 2_{r} \sqrt {I_{r} k_{r} }$ is the viscous damping rolling coefficient, η_r is the critical viscous damping ratio and I_r is the equivalent moment of inertia about the contact point between two contacting entities. Instead of an oscillating resisting moment applied at the contact, as applied in some rolling resistance contact models [2], the applied resisting moment at the contact within a quasi-static system is stable and therefore the packing behaviour of the assembly is stabilized. For such reasons, this model is often used in DEM simulation of quasi-static systems. Using various modified versions of this model in DEM along with the tuning of μ_r, some have captured the influence of particle rolling on the behaviour of granular material in triaxial compression. In particular some observed that with the addition of rolling resistance, shear strength and dilatancy increases [32, 46, 54, 94], shear banding and strain localization intensifies [32, 33, 54, 84] and fabric anisotropy intensifies [46].