Prediction of 3D size and shape descriptors of irregular granular particles from projected 2D images

Abstract

Macroscopic mechanical properties of granular materials are closely related to particle morphology. For practical reasons, the morphological features are commonly examined from projected two-dimensional (2D) images of the three-dimensional (3D) particles. This brings forward the need for quantifying the correlations between the 2D and 3D particle descriptors. This paper addresses these correlations for irregular granular particles. Three-dimensional images of sand particles were acquired through microfocus X-ray computed tomography, based on which 3D surfaces of the particles were reconstructed using spherical harmonic analyses. The 3D particle size and shape descriptors were subsequently evaluated. All-around projection and random projection of the particles onto 2D planes were undertaken numerically to obtain the projected 2D images and thus the corresponding 2D size and shape descriptors. The results indicate that there are close correlations between 3D and 2D size descriptors averaged from the all-around projections. 2D and 3D shape descriptors can be approximately fitted with a linear relationship. The mean value of 2D descriptors of the tested sands obtained from a series of independent random-projection tests is essentially identical to that from the all-around projections; except that the data from the random-projection tests show a larger scatter. In light of the relationships among the descriptors, a novel and promising approach to predict the cumulative distribution of 3D descriptors from that of 2D descriptors evaluated from a random-projection test is proposed.

Appendices

Appendix A: Spherical harmonic functions and their derivatives

The real-form expressions of the spherical harmonic functions $Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})$ are

$$ Y_{n}^{m} (\theta^{\prime},\varphi^{\prime}) = \sqrt {\frac{{\left( {2n + 1} \right)\left( {n - m} \right)!}}{{4\pi \left( {n + m} \right)!}}} P_{n}^{m} (\cos \theta^{\prime})\cos (m\varphi^{\prime})\quad {\text{for}}\;m \ge 0 $$

(26a)

$$ Y_{n}^{m} (\theta^{\prime},\varphi^{\prime}) = \sqrt {\frac{{\left( {2n + 1} \right)\left( {n - \left| m \right|} \right)!}}{{4\pi \left( {n + \left| m \right|} \right)!}}} P_{n}^{\left| m \right|} (\cos \theta^{\prime})\sin (\left| m \right|\varphi^{\prime})\quad {\text{for}}\;m < 0 $$

(26b)

In the above equations, $P_{n}^{m} (\cos \theta^{\prime})$ is the associated Legendre function with degree $n$ and order $m$, which is given by

$$ P_{n}^{m} (x) = \left( { - 1} \right)^{m} \left( {1 - x^{2} } \right)^{m/2} \frac{{{\text{d}}^{m} }}{{{\text{d}}x^{m} }}p_{n} (x) $$

(27)

In Eq. (27), $p_{n} (x)$ is the Legendre polynomial of degree $n$, given by

$$ p_{n} (x) = \frac{1}{{2^{n} n!}}\frac{{{\text{d}}^{n} }}{{{\text{d}}x^{n} }}\left( {x^{2} - 1} \right)^{n} $$

(28)

Similar to [15], differentiating both sides of Eq. (1a) with respect to $\theta^{\prime}$ gives,

$$ x^{\prime}_{{\theta^{\prime}}} (\theta^{\prime} ,\varphi^{\prime}) = \sum\limits_{n = 0}^{\infty } {\sum\limits_{m = - n}^{n} {c_{xn}^{m} \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \theta^{\prime}}}} } $$

(29)

where

$$ \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \theta^{\prime}}} = \sqrt {\frac{{\left( {2n + 1} \right)\left( {n - m} \right)!}}{{4\pi \left( {n + m} \right)!}}} \frac{{\partial P_{n}^{m} (\cos \theta^{\prime})}}{{\partial \theta^{\prime}}}\cos (m\varphi^{\prime})\quad {\text{for}}\;m \ge 0 $$

(30a)

$$ \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \theta^{\prime}}} = \sqrt {\frac{{\left( {2n + 1} \right)\left( {n - \left| m \right|} \right)!}}{{4\pi \left( {n + \left| m \right|} \right)!}}} \frac{{\partial P_{n}^{\left| m \right|} (\cos \theta^{\prime})}}{{\partial \theta^{\prime}}}\sin (\left| m \right|\varphi^{\prime})\quad {\text{for}}\;m < 0 $$

(30b)

In above equations, the derivate of $P_{n}^{m} (\cos \theta^{\prime})$ with respect to $\theta^{\prime}$ can be obtained from the recurrence relation of the associated Legendre functions,

$$ \frac{{\partial P_{n}^{m} (\cos \theta^{\prime})}}{{\partial \theta^{\prime}}} = - \frac{1}{{\sin \theta^{\prime}}}\left[ {\left( {n + 1} \right)\cos \theta^{\prime}P_{n}^{m} (\cos \theta^{\prime}) - \left( {n - m + 1} \right)P_{n + 1}^{m} (\cos \theta^{\prime})} \right] $$

(31)

Differentiating both sides of Eq. (1a) with respect to $\varphi^{\prime}$ gives,

$$ x^{\prime}_{{\varphi^{\prime}}} (\theta^{\prime} ,\varphi^{\prime}) = \sum\limits_{n = 0}^{\infty } {\sum\limits_{m = - n}^{n} {c_{xn}^{m} \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \varphi^{\prime}}}} } $$

(32)

where

$$ \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \varphi^{\prime}}} = - m\sqrt {\frac{{\left( {2n + 1} \right)\left( {n - m} \right)!}}{{4\pi \left( {n + m} \right)!}}} P_{n}^{m} (\cos \theta^{\prime})\sin (m\varphi^{\prime})\quad {\text{for}}\;m \ge 0 $$

(33a)

$$ \frac{{\partial Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})}}{{\partial \varphi^{\prime}}} = \left| m \right|\sqrt {\frac{{\left( {2n + 1} \right)\left( {n - \left| m \right|} \right)!}}{{4\pi \left( {n + \left| m \right|} \right)!}}} P_{n}^{\left| m \right|} (\cos \theta^{\prime})\cos (\left| m \right|\varphi^{\prime})\quad {\text{for}}\;m < 0 $$

(33b)

The derivations of $y (\theta^{\prime} ,\varphi^{\prime})$ and $z (\theta^{\prime} ,\varphi^{\prime})$ with respect to $\theta^{\prime}$ and $\varphi^{\prime}$ can be obtained by the same token.

Appendix B: Symbols

a, b, c:

3D principal dimensions

$a_{xn}$, $a_{yn}$, $a_{zn}$:

Amplitude of the Fourier/SH coefficients of degree $n$

$A$ :

Area of 2D images

${\text{AI}}_{\text{r}}$ :

Radius angularity index

${\text{AI}}^{{ 2 {\text{D}}}}$, ${\text{AI}}^{{ 3 {\text{D}}}}$:

2D, 3D angularity indices

${\text{AR}}^{{ 2 {\text{D}}}}$, ${\text{AR}}^{{ 3 {\text{D}}}}$:

2D, 3D aspect ratios

$C_{x}^{{ 2 {\text{D}}}}$, $C_{x}^{{ 3 {\text{D}}}}$:

2D, 3D convexity

d₁, d₂:

2D principal dimensions

$d_{\text{m}}^{{ 2 {\text{D}}}}$, $d_{\text{m}}^{{ 3 {\text{D}}}}$:

2D, 3D mean dimensions

EI:

Elongation index

FI:

Flatness index

$K$ :

Total number of points on the hemisphere

n :

Projection direction vector

n_x, n_y, n_z:

Cartesian coordinates of a point on the surface of the unit sphere

N :

Total number of harmonics/the maximum degree of SH series

$P$ :

Perimeter of 2D images

$S^{{2{\text{D}}}}$, $S^{{ 3 {\text{D}}}}$:

2D, 3D sphericity

$S_{\text{p}}$ :

Surface area of a particle

$V_{\text{p}}$ :

The volume of a particle

$Y_{n}^{m} (\theta^{\prime},\varphi^{\prime})$ :

Spherical harmonic functions

$\varphi$, $\varphi^{\prime}$:

Azimuthal angle

$\theta$, $\theta^{\prime}$:

Polar angle