A novel stochastic generation method of shale blocks in S-RM considering the particle size effect of morphological features

Abstract

The increasing interest in the effect of block morphological features on shear behaviors of the soil–rock mixture (S-RM) presents a challenge to robust numerical modeling considering the actual block morphology. This paper proposes an improved probabilistic method to generate stochastic virtual blocks considering the particle size effect on morphological features, based on principal component analysis (PCA) and probability density function, for establishing the S-RM discrete element method (DEM) models. The method is more reasonable and efficient compared to the previous study. The results show that the spherical harmonics (SH) can be used to accurately characterize the three-dimensional morphological features of the blocks scanned by computed tomography when the SH coefficient degree n reaches 17 and the mesh accuracy l equals 4. With the increase of the block particle size, the angularity of blocks increases, while the convexity and sphericity decrease. It verified that the morphological characteristics of the virtual blocks generated by the improved method follow those of natural ones at different scales, better than the previous method. The computation efficiency for generating blocks by the improved method was considerably enhanced compared to the previous study. The S-RM DEM models constructed by the generated virtual blocks can be directly used in further simulations to investigate the effect of block shape and size on the mechanical behavior of S-RM.

Appendices

Appendix 1. Abbreviations

3D:

Three-dimensional

CT:

Computed tomography

DEM:

Discrete element method

PCA:

Principal component analysis

SH:

Spherical harmonic

S-RM:

Soil–rock mixture

VBP :

Volumetric block proportion

PDF:

Probability density function

Appendix 2. Symbols

$\overline{a }$  :

Normalization spherical harmonic coefficient

${{a}_{n}}^{m}$ :

SH coefficient

c :

Mean vector of SH coefficient matrix

Cv :

Convexity

d :

Block particle size

EA:

Surface area relative error

EI:

Elongation

EV:

Volume relative error

l :

Mesh accuracy

m,n:

Spherical harmonic order and degree

P_n^m(x):

Associated Legendre function

q :

SH coefficient matrix formed by all blocks

q _c :

Centered SH coefficient matrix

$\widehat{q}$  :

Statistic generated SH coefficient

p :

Number of principal components

r(θ,  φ):

Polar radius of the particle surface at θ and φ

S :

Sphericity

t :

Number of all scanned blocks

u :

Score matrix of principal components

v :

Eigenvectors matrix

w :

Number of vertices contained in the mesh

Y_n^m(θ, φ):

Spherical harmonic series

θ, φ :

Polar angle and azimuthal angle of the spherical coordinates

λ :

Eigenvalue