Bonded-cell model for particle fracture

Particle degradation and fracture play an important role in natural granular flows and in many applications of granular materials. We analyze the fracture properties of two-dimensional disklike particles modeled as aggregates of rigid cells bonded along their sides by a cohesive Mohr-Coulomb law and simulated by the contact dynamics method. We show that the compressive strength scales with tensile strength between cells but depends also on the friction coefficient and a parameter describing cell shape distribution. The statistical scatter of compressive strength is well described by the Weibull distribution function with a shape parameter varying from 6 to 10 depending on cell shape distribution. We show that this distribution may be understood in terms of percolating critical intercellular contacts. We propose a random-walk model of critical contacts that leads to particle size dependence of the compressive strength in good agreement with our simulation data.

Figure 1

Definition of the degree of regularity of the mesh $\lambda$ (a) and two examples of the discretization for $\lambda =0$ (b) and $\lambda =0.8$ (c).

Figure 2

Contact dynamics model of a side-side contact between two particles by two contact points with their normals ${\vec{n}}_1$ and ${\vec{n}}_2$ and forces ${\vec{f}}_1$ and ${\vec{f}}_2$.

Figure 3

Boundary conditions for Brazilian crushing of a particle.

Figure 4

A single circular particle subjected to diametral compression (Brazilian test) for $\lambda =0$ (a) and $\lambda =0.8$ (b). The red and green lines represent compressive and tensile contacts, respectively.

Figure 5

Axial stress versus axial strain for several values of tensile strength ${\sigma }_c$. The inset shows the axial stress normalized by ${\sigma }_c$. The parameters are ${\mu }_s=0.3,\lambda =0.8$, and $n_v=50$.

Figure 6

Evolution of compressive strength ${\sigma }_p$ as a function of local friction coefficient ${\mu }_s$ for ${\sigma }_c=10$ MPa, $\lambda =0.8$, and $n_v=500$. The error bars represent standard deviation calculated from nine independent simulations with different tessellations.

Figure 7

(a) Survival probability of a particle as a function of compressive stress ${\sigma }_a$ in a Brazilian test for $n_v=500$ and two values of mesh regularity parameter $\lambda$ in log-log scale. The lines represent fits to the Weibull function with two different values of the Weibull modulus $m$. (b) The Weilbull modulus $m$ as function of $\lambda$. (c) Compressive strength ${\sigma }_p$ and scale stress ${\sigma }_w$ as a function of $\lambda$. The open symbols are predicted values of ${\sigma }_p$ by Eq. (4).

Figure 8

Normal force network before rupture. The dashed lines represent the contours of the stress concentration zone.

Figure 9

(a) Compressive strength of a particle normalized by internal cohesion as a function of the number of cells for constant particle diameter $d$ and varying cell size $d_0$ (green circle) and for constant $d_0$ and varying $d$ (red square). The error bars represent standard variation for 11 independent simulations. (b) Compressive strength as a function of the number of cells for two extreme values of mesh variability parameter $\lambda$.