Elsevier

Powder Technology

Volume 360, 15 January 2020, Pages 209-220
Powder Technology

Probabilistic characterization and simulation of realistic particle shape based on sphere harmonic representation and Nataf transformation

https://doi.org/10.1016/j.powtec.2019.10.007Get rights and content

Highlights

  • A new probabilistic method is proposed to characterize realistic particle shape.

  • Realistic particle shape is represented by Sphere Harmonic analysis using CT data.

  • Joint PDF of particle shape descriptors is constructed using Nataf transformation.

  • Virtual particles are simulated by Nataf transformation and used as clump templates.

  • Effects of realistic particle shape on triaxial testing of sand are explored.

Abstract

Significant commitment of cost, man power, and time are often required for characterizing particle morphology of granular materials based on X-ray micro-computed tomography (μCT) scanning and image processing. As a result, only a small portion of particles can be scanned, leading to uncertainty in characterizing realistic particle shape. This paper develops a probabilistic method to characterize and simulate sand particle shape based on Sphere Harmonic (SH) analysis, where Nataf transformation is used to construct the joint probability distribution of SH shape descriptors and to generate their random samples for reconstructing virtual particles. The proposed approach was applied to characterizing and simulating particle shapes of Leighton Buzzard sand (LBS), and these simulated particles were used as clump templates for discrete element method to account for effect of realistic particle shape. It was shown that the proposed approach properly characterizes and simulates the particle shape of LBS in a probabilistic manner.

Introduction

Particle shape is an essential particle characteristic which affects the physical properties and mechanical behaviours of granular materials, such as stiffness, shear strength, permeability, dilatancy, crushability and strain localization [1,6,11,31,34]. Over the past few decades, discrete element method (DEM) proposed by Cundall and Strack [8] has been shown to be a powerful numerical tool for investigating the mechanical responses of granular materials from both macroscopic and microscopic point of view. Although DEM simulation is a promising method to reproduce and predict the mechanical and engineering behaviour of granular materials for the current advance of computation capacity, simplifications on representing real particles (e.g., 2D disks or 3D spheres) are still often adopted in DEM. Such simplifications largely ignore the effect of particle shape on micro and macro mechanical behaviour of granular materials and might result in an unreasonable simulation results.

To consider the effect of particle shape on the bulk behaviour of granular materials, many attempts have been made, such as development of rolling resistance contact models [14,15,45]. The rolling resistance contact models incorporate a rotational friction torque into DEM formulations to account for effect of particle irregularity at the expense of introducing additional fictitious non-physical parameters (e.g., rolling stiffness, rolling friction coefficients), calibration of which remains an intriguing issue in micro-mechanics. Some previous studies also pointed out that using rolling resistance models may cause unrealistic fabric in soil mechanics [38]. Alternatively, effect of particle shape can also be considered in DEM simulation by directly considering a variety of non-sphere particle shapes, such as ellipses [30], polygons [22] and egg-shapes [44] in 2D analysis or ellipsoids [39,46], polyhedrons [21] and super-ellipsoids [40] in 3D analysis. However, all these non-sphere particles are convex for computational convenience of contact detection during the DEM computation, which can only provide an approximation to realistic particles.

Recent development of X-ray micro-computed tomography (μCT) and image process techniques allow extracting the realistic particle shape information of granular materials (e.g., sands) accurately. Based on these processed data, Garboczi [10] and Zhou et al. [41] adopted the Sphere Harmonic (SH) function to characterize and reconstruct the complex particle shape of concrete aggregates and quartz sand particles, respectively. Andrade et al. [2] proposed the non-uniform rational basis-splines method for μCT-based characterization of the complex particle morphology for DEM simulation. However, limited by the significant economic and time costs of X-ray μCT scanning and image processing, only a small portion of particles (e.g., less than 10%) within the assembly are scanned and reconstructed, which unavoidably leads to uncertainty in reconstructed shapes of particles. This is further complicated by inherent variability of the particle shapes due to the complex forming process. To address this issue, different attempts have been made to capture random nature of particle shapes in the literature. For example, SH coefficients-based random field [12,42] and Fourier-based random field [24] have been applied to simulate complex and realistic particle shapes, which involve a large number (e.g., hundreds, if not thousands) of model parameters to simulate detailed morphology of particles. Determining such a large number of model parameters for random simulation of realistic particle shape is not trivial task, particularly when only a limited number of particles can be scanned by X-ray μCT.

This paper develops a new probabilistic method for characterizing and simulating the realistic particle shape of sands based on X-ray μCT data and Nataf transformation. The proposed approach makes use of the amplitude, Ln, n = 0, 1, 2, …, N, of SH frequencies at different SH degrees as particle shape descriptors, dimensions of which are much lower than that of SH coefficients (e.g., one order of magnitude less for N > 9). Then, Nataf transformation (e.g. Refs. [3,17,25]) is applied to construct the joint probability distribution function (PDF) of Ln, n = 0, 1, 2, …, N. By this means, determining the joint PDF of Ln, n = 0, 1, 2, …, N is divided into two sub-tasks: determining the marginal PDF of Ln, n = 0, 1, 2, …, N, and quantifying their dependence. This makes development of joint probability distribution of particle shape descriptors more tractable, particularly for the case with a limited number of sand particles are scanned. Based on joint probability distribution of Ln, n = 0, 1, 2, …, N, virtual particles with realistic particle shapes are generated, which inherit the particle shape characteristics of the scanned parental particles (including particle size distribution characteristics). These virtual particles can be used as templates to generate clumps in DEM simulation. For illustration, the proposed approach is applied to probabilistically characterize and simulate particle shapes of Leighton Buzzard sand (LBS) according to 106 μCT-scanned LBS particles. The effect of the particle shape on the triaxial compression test results of LBS is also discussed in DEM.

Section snippets

Sphere harmonic analysis

3D X-ray μCT is able to provide detailed information of surface micromorphology of the scanned sand particles through a sequence of procedures, including high-resolution μCT scanning, specimen volume reconstruction, and image processing [10,41]. The micromorphology information can be presented as coordinates of a set of vertices and faces in the 3D Cartesian coordinate space. A vertex V(x, y, z) can be specified either by X-, Y-, and Z-coordinates in Cartesian space or by the polar radius r(θ,ϕ)

Probabilistic characterization of realistic sand shape using joint PDF of Ln

In the context of Nataf transformation, construction of joint PDF f(L1, L2, …, LN) of Ln, n = 0, 1, 2, …, N, is given by (e.g. Refs. [17,25,29]):fL1,L2,...,LN=fL1fL2...fLNϕz1ϕz2...ϕzNϕNz;ρzwhere ϕ(z1), ϕ(z2), …, ϕ(zN) are the marginal PDFs of N standard normal random variables denoted by z=[z1,z2,...,zN]T; ϕN(z;ρz) is the joint normal PDF of z with mean values of zero, standard deviations of unity, and correlation coefficient matrix ρz. Completely defining the joint PDF of Ln, n = 1, 2, …, N

Illustrative and application example

In this section, the proposed approach is applied to probabilistically characterize and simulate realistic particle shapes of Leighton Buzzard sand (LBS) based on 106 μCT-scanned particles. And then as an application example, these virtual particles are used as clump templates to generate clumps in DEM simulation of triaxial compression test, and the effect of the particle shape on the triaxial compression behaviour of LBS are discussed.

Summary and conclusions

This paper develops a new probabilistic method to characterize and simulate the realistic particle shape of sands based on X-ray μCT data. The proposed approach makes use of the sphere harmonic (SH) function to represent realistic particle morphology and deliberately chooses the amplitude (i.e., Ln, n = 0, 1, 2, …, N) of SH frequencies at different degrees, instead of a large number SH coefficients, as particle shape descriptors, which reduces the dimensionality of uncertain parameters involved

Acknowledgement

This work was supported by the National Key R&D Program of China (Project No. 2017YFC1501300), the National Natural Science Foundation of China (Project Nos. 51579190, 51679174, and 51779189), and the Shenzhen Key Technology R&D Program (Project No. 20170324). The financial support is gratefully acknowledged.

References (46)

  • X.,S. Tang et al.

    Impact of copula selection on geotechnical reliability under incomplete probability information

    Comput. Geotech.

    (2013)
  • T. Wichtmann et al.

    On the influence of grain shape on the cumulative deformations in sand under drained high-cyclic loading

    Soils Found

    (2019)
  • D.H. Wei et al.

    Generation of realistic sand particles with fractal nature using an improved spherical harmonic analysis

    Comput. Geotech.

    (2018)
  • J. Yang et al.

    Exploring the relationship between critical state and particle shape for granular materials

    J. Mech. Phys. Solids

    (2015)
  • B.D. Zhao et al.

    3D quantitative shape analysis on form, roundness, and compactness with μCT

    Powder Technol.

    (2016)
  • S.W. Zhao et al.

    Shear-induced anisotropy of granular materials with rolling resistance and particle shape effects

    Int. J. Solids Struct.

    (2018)
  • S.W. Zhao et al.

    Three-dimensional Voronoi analysis of monodisperse ellipsoids during triaxial shear

    Powder Technol.

    (2018)
  • B. Zhou et al.

    Micromorphology characterization and reconstruction of sand particles using micro X-ray tomography and spherical harmonics

    Eng. Geol.

    (2015)
  • W. Zhou et al.

    A geometric potential-based contact detection algorithm for egg-shaped particles in discrete element modeling

    Powder Technol.

    (2018)
  • Z.Y. Zhou et al.

    Discrete particle simulation of gas fluidization of ellipsoidal particles

    Chem. Eng. Sci.

    (2011)
  • A.H.S. Ang et al.
    (2007)
  • Z.J. Cao et al.

    Bayesian approach for probabilistic site characterization using cone penetration tests

    J. Geotech. Geoenviron. Eng.

    (2013)
  • G.C. Cho et al.

    Particle shape effects on packing density, stiffness, and strength: natural and crushed sands

    J. Geotech. Geoenviron. Eng.

    (2006)
  • Cited by (44)

    View all citing articles on Scopus
    View full text