Experimental assessment of continuum breakage models accounting for mechanical interactions at particle contacts

Abstract

Particle size and shape are major factors in determining the mechanical behavior of granular media. This paper discusses experiments conducted at particle and assembly scales on two materials (i.e., glass beads and quartz sand) and it interprets them in light of fracture mechanics theories. First, diametral compression tests on particles of varying size have been conducted to measure the energy stored in individual grains at the onset of fracture. Then, oedometric compression tests on samples made of the same particles have been performed to measure the yielding pressure, as well as to track the evolution of breakage. These experiments have been used to test the performance of recently proposed scaling laws bridging the energy released by a single particle with the work input required to comminute an assembly. The results show that the variables associated with macroscopic comminution scale with the grain size according to the same power law functions that control the size-dependence of the corresponding particle-scale quantities. Major differences between the scaling laws of glass beads and quartz sands have been found, with the former approaching the size effect law associated with fracture by central splitting and the latter being closer to the trends predicted by fracture at the contacts. These findings emphasize the key role of the particle shape on the energetics of breakage, thus motivating further studies focusing on different shapes, for which even wider ranges of fracture modes and scaling laws may exist.

Appendix

1.1 Fracture models

1.1.1 (a) Surface crack model

The failure load due to the surface crack can be expressed as follows [3]:

$$\begin{aligned} F_{f} =\frac{9}{64}\left( \frac{2\pi b^{\prime }}{1-2v} \right) ^{3}\left( \frac{1-v^{2}}{E}\right) ^{1/2}G_{{\textit{IC}}}^{3/2} d_{p}^{1/2} \end{aligned}$$

(11)

where $b^{\prime }=\frac{1}{1.12}\left( {\frac{\pi \alpha }{2}} \right) ^{-1/2}$ is a coefficient associated with the crack geometry, while $\alpha$ is a dimensionless crack size (i.e., the crack length normalized for the particle diameter). $G_{{\textit{IC}}}$ is the critical energy release rate (here assumed to be $0.95 \hbox { J/m}^{2}$).

1.1.2 (b) Central crack model

The failure load for central split of a particle can be expressed through the following equation [3]:

$$\begin{aligned} {F}_{f} =\frac{\pi a^{\prime }}{2.8} \sqrt{\frac{G_{\textit{IC}}E}{1-v^{2}}} d_{p}^{3/2} \end{aligned}$$

(12)

where $a^{\prime }=\frac{\sqrt{\pi }}{\sqrt{2\alpha }}$ is a coefficient associated with the crack geometry, $G_{IC}$ is the critical energy release rate (here assumed to be $0.95\hbox { J/m}^{2})$ and the coefficients $a^{\prime }$ and $b^{\prime }$ are associated with the dimensionless crack size.

1.2 Contact model

1.2.1 (a) Linear contact model

A linear contact involves a constant contact area (i.e., not altered by the applied force). This type of contact is here modeled through the following force-displacement law:

$$\begin{aligned} F=\frac{ E\pi {a}_{c}^{2} }{R}\delta \end{aligned}$$

(13)

where $a_{c}$ is the contact radius; R is the particle radius; and $\delta$ is the contact deformation ( $\delta = {\Delta }/2$).

By substituting Eq. 13 into Eq. 2, the relation between characteristic particle strength and energy stored at the onset of fracture can be expressed as follows:

$$\begin{aligned} E_{{\textit{pc}}} =\frac{12}{E\pi ^{2}}\left( \frac{a_{c}}{R} \right) ^{-2}\sigma _{{\textit{pc}}}^2 \end{aligned}$$

(14)

1.2.2 (b) Hertzian contact model

In case of the contact between an elastic particle and a rigid loading plate, the force-displacement response for a Hertzian contact can be expressed through the following relations:

$$\begin{aligned} F =\frac{4}{3}R^{1/2}\left( \frac{1-v^{2}}{ E} \right) ^{-1}\delta ^{3/2} \end{aligned}$$

(15)

By substituting Eq. 15 into Eq. 2, the relation between characteristic particle strength and the energy stored at the onset of fracture can be expressed as follows:

$$\begin{aligned} E_{{\textit{pc}}} =4.989\frac{1}{\pi }\left( {\frac{1-v^{2}}{E}} \right) ^{2/3}\sigma _{{\textit{pc}}}^{5/3} \end{aligned}$$

(16)

1.2.3 (c) Conical contact model

The load-displacement relation for a conical contact model derived by Zhang et al. [3] can be expressed as follows:

$$\begin{aligned} F=\frac{E}{1-v^{2}}\left( \frac{2}{\pi A_{s}} \right) ^{1/2}\delta ^{2} \end{aligned}$$

(17)

By substituting Eq. 17 into Eq. 2, the relation between characteristic particle strength and the energy stored at the onset of fracture is:

$$\begin{aligned} E_{{\textit{pc}}} =3.360A_{s}^{1/4} \pi ^{-(3/4)}\left( {\frac{1-v^{2}}{E}} \right) ^{1/2}\sigma _{{\textit{pc}}}^{3/2} \end{aligned}$$

(18)

where $A_s$ is the tangent of the contact angle between the particle and a horizontal surface [3].

1.3 Combination of failure model and contact model

It is possible to express the characteristic particle strength, $\sigma _{{\textit{pc}}}$, in Eqs. 14, 16, and  18 in terms of grain size by combining Eqs. 11 and  12 with Eq. 1. While the failure model based on central splitting can be combined with any of three contact models, the failure criterion based on surface fracture can only be used in combination with a Hertzian contact law. Therefore, four expressions of $E_{{\textit{pc}}}$ are obtained as a function of the grain size, d.

1.3.1 (a) Surface crack

• (i) Hertzian contact model

Equation 11 can be expressed as a function of $\sigma _{{\textit{pc}}}$ by using Eq. 1. By substituting the resulting expression into Eq. 16 gives:

$$\begin{aligned} E_{{\textit{pc}}} =6.071\pi ^{-1}{} { b}^{{\prime }^{5}}\left( {\frac{1-v^{2}}{{\textit{E}}}} \right) ^{3/2}\left( {\frac{\pi }{1-2v}} \right) ^{5}G_{{\textit{IC}}}^{5/2} d_{p}^{-5/2}\nonumber \\ \end{aligned}$$

(19)

1.3.2 (b) Central crack

Equation 12 can be expressed as a function of $\sigma _{{\textit{pc}}}$ by using Eq. 1. Then, by substituting the resulting expression into Eqs. 14, 16 and  18 the following relations are obtained:

• (i) Linear contact model

  $$\begin{aligned} E_{{\textit{pc}}} =1.531a^{\prime }\left( {\frac{ a_{c}}{{\textit{R}}}} \right) ^{-2}\left( {\frac{G_{{\textit{IC}}} }{1-v^{2}}} \right) d_{p}^{-1} \end{aligned}$$

  (20)

• (ii) Hertzian contact model

  $$\begin{aligned} E_{{\textit{pc}}}=0.897\pi ^{2/3} a^{{\prime }^{{5/3}}}\left( {\frac{1-v^{2}}{{\textit{E}}}} \right) ^{-1/6}G_{{\textit{IC}}}^{5/6} d_{p}^{-5/6} \end{aligned}$$

  (21)

• (iii) Conical contact model

  $$\begin{aligned} E_{{\textit{pc}}}=0.717A_{s}^{1/4} \pi ^{3/4} a^{{\prime }^{{3/2}}}\left( {\frac{1-v^{2}}{{\textit{E}}}} \right) ^{-1/4}G_{{\textit{IC}}}^{3/4} d_{p}^{-3/4}\nonumber \\ \end{aligned}$$

  (22)