Damage separation model: A replaceable particle method based on strain energy field

We present a realistic model for simulating particle fragmentation in granular assemblies, the damage separation model (DSM), that addresses the limitations of previous methods by replacing the particle with smaller ones after fragmentation. The method is based on the calculation of the strain energy field inside the particle, and it solves the two major issues of the existing replaceable particle methods: the oversimplification of particle stress, and the unrealistic geometrical constraints needed in postbreakage replacements. Our model is formulated with three modules: (i) a boundary element calculation of stress and strain fields inside the spheropolygons that represent individual particles; (ii) a strain-energy-based theoretical framework to determine the onset of fragmentation; and (iii) an advanced geometrical algorithm, the subset separation method (SSM), to handle the postbreakage replacements in the discrete element simulations. Especially, the SSM effectively calculates the fragments required by the replacement with no geometrical limitation on the number, location, and orientation of the fracture planes. A uniaxial compression test based on laboratory setups is used to validate the method. A comparison is further conducted to study the performance of four different replaceable irregular particle methods. Results indicate that our method overcomes most of the existing issues, including stability, accuracy, and artificial constraints on the number and shape of fragments. The DSM has great potential for capturing the morphological changes of particle breakage and comminution with an unprecedented numerical resolution.

Figure 1

The three modules of the computational framework of inner particle stress and fragmentation calculation. (a) A loaded particle is in equilibrium with a pair of contact forces ${\mathit{f}}_{\alpha }$ and ${\mathit{f}}_{\beta }$; (b) the particle is represented as a continuum domain with a local coordinate system; (c) the particle boundary is discretized, and the boundary tractions ${\mathit{t}}_{\alpha }$ and ${\mathit{t}}_{\beta }$ are assigned to the corresponding elements (red lines); (d) the elastic strain energy field is calculated; (e) the damage domains (in lime) and a connection point ${\mathit{c}}_i$ (red dot) are calculated; (f) the fracture plane (red dashed line) is determined; (g) the subset separation method is applied to the polygon; (h) the polygon is separated into two polygonal fragments.

Figure 2

A polygon with a different number of fragmenting lines and the resultant set/subsets generated by the fracturing lines. The blue nodes denote the vertices of the original polygon; the green dots are at the end points of the fragmenting lines; the red dots represent the intersection points of the fragmenting lines. (a) $n=1$, (b) $n=2$, and (c) $n=3$.

Figure 3

The new polygons created by the fragmenting lines are calculated from SSM. The workflow for the generation of the polygon subsets is presented. The polygon $P_5$ does not exist since it corresponds to the intersection subset $S_5$, which is empty.

Figure 4

The situation in which the intersection subset only contains one point and does not construct a new polygon.

Figure 5

Examples of $n=4$ and 5 calculated by the subset separation method.

Figure 6

The contact layout of the original particle and the new fragment. (a) The vertex is in contact with the boundary before the breakage; (b) the breakage occurs and the new particles are inserted and eroded as spheropolygons; loss of mass is marked out with the dashed red square.

Figure 7

The particles and fragments obtained by the DSM (right) at different times and the corresponding comparison with the experimental (left) and split-cell (middle) results obtained from Cantor et al. [15]. (a) $t=0.0$ s, (b) $t=0.9$ s, (c) $t=1.1$ s, and (d) $t=2.4$ s.

Figure 8

(a) The variation of the total mass of the granular sample; (b) the variation of average velocity and angular velocity of the granular sample.

Figure 9

The replacement method used for each replaceable particle method. The actual contact forces are shown in (d) along with the results of DSM. (a) The split-cell method, (b) the cross-cut method, (c) the Mohr-Coulomb method, and (d) the damaged-separation method.

Figure 10

The scheme of the “locking effect” generated by the replacement mode in the split-cell method. Black arrows are loading forces; black dots denote the centroid of the polygon; blue lines and green lines are the fracture planes and connection lines between contact points, respectively.

Figure 11

The three breakage types: chipping, splitting, and fragmentation. (a) chipping, two fragments have extremely different sizes generated from the concentration of the contact forces near the boundary. (b) splitting, the particle is separated into two roughly equal parts. (c) fragmentation, the particle is separated into three or more roughly equal parts.

Figure 12

The breakage types generated with different replacement methods under various loading conditions and coordination numbers. The first column is the loading configuration with black arrows. The second column shows the results of the split-cell method. The third column is the results of the cross-cut method. The fourth column is the Mohr-Coulomb method. Results in the last column are produced by the DSM. The subscripts $s$, $f$, and $c$ denote the breakage types of splitting, fragmentation, and chipping. (a) $N=2$, (b) $N=3$, and (c) $N=4$.

Figure 13

Final breakage patterns of particles simulated with each replaceable particle method. The black color represents the intact and original particles; the red color represents the new replacements. (a) The split-cell method, (b) the cross-cut method, (c) the Mohr-Coulomb method, and (d) the DSM method.

Figure 14

Spatial distribution of the fragments generated by each method: (a) the split-cell method, (b) the cross-cut method, (c) the Mohr-Coulomb method, and (d) the DSM method.

Figure 15

The directional frequency of fracture planes in each replaceable particle method: (a) the split-cell method, (b) the cross-cut method, (c) the Mohr-Coulomb method, and (d) the DSM method.

Figure 16

The particle size distribution provided by the replacement methods. The black dashed circle marks out the lump that is numerically caused by the cross-cut method.

Figure 17

(a) The force chain network $I=({\sigma }_1-{\sigma }_2)$ of the granular sample and (b) zoomed areas. The calculation is conducted at $t=7.00$ s in the test of the DSM.