A novel multi-scale large deformation approach for modelling of granular collapse

Abstract

Collapse of granular material is usually accompanied by long run-out granular flows in natural hazards, e.g. rock/debris flow and snow avalanches. This paper presents a novel multi-scale approach for modelling granular column collapse with large deformation. This approach employs the smoothed particle hydrodynamics (SPH) method to solve large deformation boundary value problems, while using a micromechanical model to derive the nonlinear material response required by the SPH method. After examining the effect of initial cell size, the proposed approach is subsequently applied to simulate the flow of granular column in a rectangular channel at a low water content by varying the initial aspect ratio. The numerical results show good agreement with various experimental observations on both collapse process and final deposit morphology. Furthermore, the meso-scale behaviour is also captured owing to the advantages of the micromechanical model. Finally, it was demonstrated that the novel multi-scale approach is helpful in improving the understanding of granular collapse and should be an effective computational tool for the analysis of real-scale granular flow.

Appendix 1: Contact law

This elastic-perfect plastic model includes a Mohr–Coulomb criterion and can be expressed under the following incremental formalism:

$$\begin{aligned} \left\{ \begin{array}{l} \delta {N_i} = {k_n}\delta u_n^i\\ \delta {T_i} = \min \left\{ {\left\| {{T_i} + {k_t}\delta u_t^i} \right\| ,\tan {\varphi _g}\left( {{N_i} + {k_n}\delta u_n^i} \right) } \right\} \times \frac{{{T_i} + {k_t}\delta u_t^i}}{{\left\| {{T_i} + {k_t}\delta u_t^i} \right\| }} - {T_i} \end{array} \right. \end{aligned}$$

(38)

where: $i=1,2,3,4$ denotes the identifier of contact number.

According to Eqs. (8), (38) can be rewritten as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \delta {N_i} = - {k_n}\delta {d_i}&{}\\ \delta {T_i} = {k_t}{d_i}\delta {\alpha _j} &{}\text {Elastic regime}\\ \delta {T_i}= \tan \varphi _g \left( {N_i} - {k_n}\delta {d_i}\right) \xi _i- {T_i} &{} \text {Plastic regime} \end{array}\right. \end{aligned}$$

(39)

where: $\xi _i$ is the sign of ${T_i} + {k_t}{d_i}\delta {\alpha _j}$; $j=1$ when $i=1,2$; $j=2$ when $i=3,4$; plastic regime is reached when $\parallel {k_t}{d_i}\delta {\alpha _j}+T_i\parallel \geqslant \tan \varphi _g \left( {N_i} - {k_n}\delta {d_i}\right)$, otherwise it is in elastic regime.

To facilitate the derivation, $I_i^p$ and $I_i^e$ are introduced as indicator functions of the contact state, expressed as follow:

$$\begin{aligned} I_i^p = \left\{ {\begin{array}{ll} 1 &{}\quad \text {in plastic regime}\\ 0 &{}\quad \text {in elastic regime} \end{array}} \right. ; \quad I_i^e = 1 - I_i^p \end{aligned}$$

(40)

Thus, the constitutive relations can be expressed as:

$$\begin{aligned} \left\{ \begin{array}{l} \delta {N_i} = - {k_n}\delta {d_i}\\ \delta {T_i} = {B_i}\delta {\alpha _j} - {A_i}\delta {d_i} + C_i \end{array}\right. \end{aligned}$$

(41)

where:$\left\{ \begin{array}{l} {A_i} = I_i^p {k_n}{\xi _i}\tan {\varphi _g}\\ {B_i} = I_i^e {k_t}{d_i}\\ C_i=I_i^p(\xi _i \tan \varphi _g N_i - T_i) \end{array}\right.$