Understanding Secondary Fragmentation Characteristics in Cave Mining: A Simulation-Based Analysis of Impact and Compression-Induced Breakage

Abstract

This study investigates the characteristics of secondary fragmentation and fines generation in cave mining through DEM simulations. The objective is not to develop a tool for accurately estimating fragmentation observed at drawpoints. Instead, the research focuses on an improved understanding of the impact of critical parameters (tensile strength, damping coefficients, and micro-defects) on secondary fragmentation characteristics. Attempting to predict outcomes without first comprehending the underlying mechanisms risks oversimplifying complex mine-scale conditions. The analysis shows that tensile failure is the dominant mechanism governing fragmentation. Size-distribution curves of fragmented blocks under impact breakage demonstrate a concave-up exponential relationship between percentage mass passing at 1/10th of the original size ($t_{10}$) and kinetic energy. Furthermore, the analysis of compression-induced breakage highlights the significant role of tensile strength and micro-defects in determining the extent of fragmentation under different conditions. By better understanding these underlying mechanisms, the research establishes a solid foundation for predicting fines generation and ultimately enhancing decision making and operational strategies in mining.

1. Introduction

Fragmentation assessment is critical in cave mine design, as coarse rock fragments can block the drawpoints, while the presence of fines can lead to dry inrush (e.g., rill swell at Cadia East Mine) or mudrush (e.g., Grasberg Block Cave, GBC) if water is involved. Mitigating the risk of dry inrushes and mudrushes impacts production rates, as mines may temporarily close access to drawpoints. Sustained production delays eventually have significant economic implications for operating mines.

There are three stages of fragmentation shown in Figure 1: in situ fragmentation, primary fragmentation, and secondary fragmentation [1,2]. In situ fragmentation refers to the sizes of naturally occurring rock blocks. Primary fragmentation involves the reduction in block sizes due to breakage in the caveback caused by stress-induced and pre-conditioning fractures. Lastly, secondary fragmentation involves further size reduction in blocks travelling toward drawpoints.

While terms such as comminution and attrition have been incorporated in cave mining to describe the fragmentation mechanisms [4,5], it is necessary to reflect upon their definitions, since technical fields often suffer from a tendency to co-opt and adapt specialized terms outside of the conditions for which they were originally developed. Comminution and attrition are terms used in mineral processing to describe the processes to reduce the ore size for separating the clean particles of valuable minerals (i.e., liberation). While comminution often refers to a breakage event that results in a higher degree of fragmentation and liberation, attrition is not strictly a breakage event but rather low-energy particle collisions and shear-induced material abrade off a particle surface [6].

In addition to the semantic problem above, there remain several limitations to the study of fragmentation mechanisms:

Directly assessing and capturing fragmentation mechanisms is challenging because fragmentation processes occur within the inaccessible confines of draw columns.

Observations made at drawpoints represent “known knowns”. However, the processes leading to the resulting size distributions involve “known unknowns” and “unknown unknowns” [7].

For the reasons above, it is impossible to recreate a physical twin (or prototype) of a cave mine at a laboratory scale that replicates realistic stress conditions relative to rock block strength.

There remains the challenge of scientific replicability to determine whether the theories proposed in the literature (comminution and attrition processes) are valid. More work is required to confirm the applicability of the proposed mechanisms and to study the relationships between various factors, such as micro-defects, rock strength, kinetic energy, and their impacts on fragmentation.

Notwithstanding these limitations, laboratory experiments [4,8] and numerical simulations have become increasingly prevalent in block cave fragmentation analysis. Comminution theories developed from laboratory-scale experiments (e.g., drop-weight and piston press tests) have been applied to cave mine design to analyze various types of fragmentation encountered in the draw column. However, it should be noted that the energy levels relative to rock strength inside the cave differ from those in laboratory settings. As an example, a 1 m³ intact rock block falling through a 10 m air gap in the cave generates over 200 kJ of kinetic energy, whereas the maximum kinetic energy a standard JKMRC drop-weight machine can apply to an ore particle is less than 500 J (i.e., 50 kg head). Additionally, the strength of centimetre-scale rock samples can be expected to be much higher than that of metre-scale rock blocks due to defects, veins, and alteration. Therefore, there is a need to investigate the applicability of comminution theory for fragmentation and fines generation analyses at the cave scale.

Techniques such as Discrete Fracture Network (DFN) modelling, along with continuum and discontinuum numerical modelling approaches [9,10,11,12,13], have been extensively used to study in situ and primary fragmentation and caveability. Modelling of the secondary fragmentation process remains complex due to its intricate mechanisms, such as impact breakage, point splitting, corner rounding, cushioning, and abrasion [2,4,5]. The industry has somehow attempted to balance the resulting uncertainty using empirical rules. Elmo and Stead [7] highlighted that forward modelling can sometimes create a misleading sense of certainty. For example, simulations that predict fragmentation progression in cave mining can produce similar outcomes (secondary fragmentation curves) even when different rock block assemblages (but with equivalent size distributions) undergo significantly different breakage processes [11]. This creates a problem when validating fragmentation models solely based on information collected at the drawpoints. The same challenge extends to backward modelling, often used to calibrate empirical parameters to match secondary fragmentation data observed at drawpoints.

With knowledge of these limitations, this paper studies secondary fragmentation characteristics and fines generation in cave mining through the Distinct Element Method (DEM) simulations. Rather than aiming to create a tool for accurately predicting fragmentation at drawpoints, the study investigates how critical factors, such as tensile strength, damping coefficients, and micro-defects, influence secondary fragmentation. Attempting to make predictions without first grasping the driving mechanisms can lead to overly simplistic models that do not account for the complexities of mine-scale conditions. The simulation results offer valuable insights into connecting comminution theories with secondary fragmentation and fines generation analysis, which can inform the development of improved empirical and semi-empirical methods. Proposing new empirical methods may appear paradoxical when new digital tools and artificial intelligence (AI) are increasingly being proposed in the mining sector. However, they remain reasonable tools when there is a lack of direct measurements and observations of fragmentation mechanisms within the ore columns. AI analyses built on what is visible at the drawpoints cannot overcome the inability to link an input (in situ fragmentation) to an output (secondary fragmentation) without assuming a fragmentation mechanism a priori. A critical review and discussion of widely used secondary fragmentation assessment approaches are provided in the next section.

2. Critical Review of Existing Secondary Fragmentation Assessment Approaches

Current tools for secondary fragmentation include Block Cave Fragmentation (BCF; [14]) and REBOP/MassFlow [5,15]. BCF uses empirical rules and expert judgment [2] to assess the probabilities of fragment splitting and corner rounding within the draw column. In BCF, rock fragments with higher aspect ratios and joint densities are assumed to have higher probabilities of splitting (into halves). The volume removed from the fragments due to corner rounding is estimated based on the average scatter of joint orientations [14]. The limitations of empirical rules used in BCF have been pointed out by Butcher and Thin [16] and Dorador [4], who noted that these methods often produce a coarser size distribution, potentially leading to an underestimation of fines. In the case of cave mining, the term “fines” refers to fragments less than 5 cm in size. This 5 cm threshold represents the smallest material size that personnel can visually map at drawpoints [17]. The definition is, therefore, significantly different from the one (i.e., 0.075 mm) used in soil mechanics.

MassFlow [5,15,18] employs the Representative Element Volume (REV) approach for secondary fragmentation analysis. The concept of REV refers to a volume that includes a sufficient number of heterogeneities to represent the average material properties [19]. REV assumes that the behaviour of a small, representative volume can reflect the entire rock mass. However, one of the key criticisms of using REV is that it oversimplifies the fragmentation process, as the effectiveness of REV in representing bulk rock fragments remains questionable. Additionally, the secondary fragmentation in cave mining is influenced by various factors, including rock heterogeneity, block–block interactions, cave stress conditions, connectivity of the natural fracture network, and micro-defects. As a theoretical model postulated on the notion that rock masses can be treated as equivalent continuum media, the REV approach may not fully capture these complexities, limiting the understanding of secondary fragmentation processes and mechanisms.

The attrition model developed by Bridgwater et al. [20], combined with empirical relationships from Pierce et al. [21], is employed in MassFlow to evaluate the effects of shear strain within the shear band and block travel distances on the degree of rock fragmentation. According to Pierce [5], the shear band is defined as the perimeter of the material flow zone, with a thickness 10 times the average fragment size, based on observations from laboratory-scale gravity flow tests. Note that these tests do not simulate rock breakage mechanisms like those encountered within ore columns. The attrition model suggests that the mass or volume removed from fragments within the shear band is a function of shear band normal stress, shear strain, rock tensile strength, and fragment shape, and it is generally defined according to empirical fitting parameters [20]. The integrated empirical approach [21] adjusts the scale and shape parameters of the Weibull distribution of bulk fragments within the assumed REV by considering volume travel distance calculated using semi-empirical flow rules in MassFlow.

Over the last two decades, BCF and REBOP/MassFlow have become industry-standard tools, providing practical starting points for fragmentation analysis and forecasting in cave mining operations. However, secondary fragmentation mechanisms are inherently complex, and neither tool fully accounts for the range of processes involved. Both tools, as well as more recent studies [22,23], rely on iterative processes to calibrate empirical parameters and align modelling results with historical fragmentation data collected at drawpoints. These calibrated parameters are then input to forecast future fragmentation at drawpoints. Elmo and Stead [7] critically discuss this type of calibration approach in engineering applications, since the accuracy of backward modelling is limited to conditions that existed in the past, and there is no guarantee those conditions will apply to future conditions, particularly when the calibration approach relies on empirical fitting parameters. Quoting Brown [1]:

“It is generally accepted that because of the limited understanding of the mechanisms involved and, for good practical reasons, the lack of availability of sufficient data, the development of a complete, mechanistically based fragmentation model is not currently plausible”.

To this day, the mechanisms governing secondary fragmentation remain unknown, and practitioners rely on calibrated rather than validated assumptions for their fragmentation models (back-analysis and engineering judgment are forms of calibration). This emphasizes the need for an approach to analyzing mechanisms, secondary fragmentation characteristics, and fines generation within ore columns to improve current empirical and semi-empirical methods. Mechanisms of secondary fragmentation include impact breakage through the air gap, compression-induced breakage in the stagnant zone, and shear-induced breakage at the boundaries of the movement zone [1,4,5]. These are described below:

Impact breakage marks the initial stage of secondary fragmentation. Numerous researchers [1,4,5,22,24] have discussed the influence of air gap height on rock breakage and fines generation in cave mines. The air gap height refers to the distance between the cave back and the muckpile surface, which varies across different locations within the cave. In fact, a certain air gap is necessary to achieve caving propagation. According to the analysis of Morales et al. [25], the air gap height is suggested to be three times the size of primary fragmentation to ensure caving propagation is met. Impact breakage occurs when disintegrated rock blocks fall through the air gap and impact the muckpile surface. The degree of fragmentation from impact breakage is primarily influenced by the rock block strength and the air gap height (i.e., the kinetic energy). The draw and cave rates typically affect the air gap height during caving operations [2].

The draw column height and rock block strength influence compression-induced breakage. In this paper, draw column height is defined as the distance from the base of the draw column to the muckpile surface, which increases as the cave back propagates upward toward the surface. Draw column heights typically range from several hundred metres (e.g., approximately 500 m in Grasberg’s Deep Ore Zone) to over a kilometre (e.g., more than 1000 m in Cadia East mine). Such substantial draw column heights result in relatively high vertical stresses, leading to a significant degree of secondary fragmentation in the cave.

Shear-induced breakage due to the attrition process along the shear band has been hypothesized by Pierce [5]. The degree of shear-induced breakage is related to normal stress on the shear band, block tensile strength, and block travel distance alone in the shear band.

Despite being extensively studied at laboratory scales, impact and compression-induced breakage mechanisms are generally overlooked in industry practice. Therefore, it is important to bring in different types of secondary fragmentation using more advanced and predictive approaches in the analysis to reduce and prevent inrush and mudrush events.

Analysis of cave-scale secondary fragmentation is challenging due to unknowns discussed previously. However, various laboratory tests have been conducted to investigate the effects of input comminution energy on the fragmentation degree and characteristics across different ore types. Numerous researchers have quantified the size fractions of single-particle breakage in relation to input comminution energy [26,27,28,29], which provides insights into mechanisms similar to those driving secondary fragmentation. Section 3 will discuss comminution concepts and their analysis approaches applied to model data processing in the later sections.

3. Comminution Model for Fragmentation Analysis

The breakage behaviour of various ore types has been extensively studied in terms of comminution processes. The size distribution of broken ore particles has been correlated with the input energy levels. The parameter $t_n$ is commonly used in comminution to characterize single-particle breakage and is defined as the cumulative percentage passing (in percentage) at the nth fraction of the initial particle size (i.e., $x_i/n$). Figure 2 illustrates the determination of $t_n$ (where $n$ = 2, 4, 10, 25, 50, and 75) for characterizing size-distribution curves in comminution. In this figure, $t_2$, $t_4$, $t_{10}$, $t_{25}$, $t_{50}$, $t_{75}$ are the percentage mass passing at the 2nd, 4th, 10th, 25th, 50th, and 75th fractions of the original particle size, respectively.

Narayanan and Whiten [30] found that $t_n$ is uniquely related to $t_{10}$ (i.e., percentage mass passing at 1/10th of initial particle size). According to Kojovic [31], $t_{10}$ can be interpreted as a fitting index for which a larger value represents a finer product size distribution. Figure 3a illustrates a $t_n-t_{10}$ relationship using the spline regression analysis originally proposed by Narayanan [32]. Various researchers have confirmed this relationship across a wide range of ore types [33,34,35]. The universal $t_n-t_{10}$ relationship for single-particle breakage has been further refined by several researchers using non-linear regressions, as shown in Figure 3b.

Davaanyam [26] found that the plateau curve of the form $y=ax/(b+x)$ provides a better fit for the $t_2-t_{10}$ and $t_4-t_{10}$ relationships, while a linear form is more suitable for the $t_{25}-t_{10}$, $t_{50}-t_{10}$, and $t_{75}-t_{10}$ relationships. These relationships are expressed in the following equations:

$$t_n={\displaystyle \frac{{\alpha }_n{·t}_{10}}{{\beta }_n+t_{10}}},n<10$$

(1)

$$t_n={\alpha }_n·t_{10},n\ge 10$$

(2)

where ${\alpha }_n$ and ${\beta }_n$ are fitted constants. Using these relationships, single-particle breakage can be estimated based on the $t_{10}$ value and applied comminution energy.

To estimate the size-distribution curve of particle progenies after breakage, the $t_{10}$ value must be determined in conjunction with the $t_n-t_{10}$ relationships. The widely accepted equation by Napier-Munn et al. [28] for estimating $t_{10}$ based on comminution energy is given by:

$$t_{10}=A\left(1-e^{b{E}_{cs}}\right)$$

(3)

where $A$ and $b$ are fitted breakage parameters, and $E_{cs}$ is the specific comminution energy (kWh/t). The specific energy is the input kinetic energy divided by the sample mass. The fitting parameter $A$ is the limiting value of $t_{10}$ and is related to the texture of the ore [31]. Equation (3) suggests that the higher comminution energy makes the size reduction process less efficient. According to the JKMRC standard, fitting the parameters $A$ and $b$ requires conducting drop-weight tests on 345 samples with sizes ranging from 13.2 mm to 63 mm.

The $t_n-t_{10}$ and $t_{10}-$ energy relationships have been extended to apply to compression-induced breakage. Gong et al. [27] compared the breakage relationships for both impact and compressive breakage, demonstrating that the $t_n-t_{10}$ and $t_{10}-$ energy relationships show similar trends for both types of breakage (Figure 4). This paper uses these relationships to quantify impact and compression-induced breakage simulation results.

The established comminution models and their associated parameters $t_n$ and $t_{10}$ provide a solid foundation for understanding fragmentation characteristics under controlled conditions (i.e., laboratory setting). However, applying these principles to field-scale scenarios in cave mining requires more advanced simulation techniques to capture real-world rock breakage complexities.

4. DEM Simulations of Rock Breakage

Numerical modelling tools such as the DEM offer powerful capabilities for simulating the dynamic and quasi-static processes of rock fragmentation. This will be explored in detail in the following sub-sections.

4.1. Distinct Element Method

The particle-based DEM, initially developed by Cundall [37], has proven to be a powerful tool for simulating various engineering problems. This method explicitly models the movement and interaction of rigid circular disks within a defined domain. Newton’s laws of motion govern the forces and motions of these disks, while disk–disk interactions at contact points follow the force-displacement law. A soft-contact approach allows overlaps of rigid disks at their contact points. The dynamic behaviour of the model is numerically represented using a time-stepping algorithm, similar to the solution scheme of the explicit finite-difference method used in continuum modelling [38]. In recent years, DEM has been widely used to simulate particle breakage and rock fragmentation under complex boundary conditions.

Three main modelling techniques based on particle-based DEM have been developed for simulating particle breakage and fragmentation:

• Bonded Particle Model (BPM).
• Fast Breakage Model (FBM).
• Particle Replacement Model (PRM).

The Bonded Particle Model (BPM) consists of an assembly of non-uniformly sized, densely packed rigid disks bonded at their contact points. Various contact models can be applied to the bonds based on the material’s microscopic structures and constitutive behaviours. The deformation of BPM under different boundary conditions is simulated through the normal and tangential displacements of contact bonds. Full or partial bond breakage events represent stress-induced fractures within the BPM. This method has been widely used to investigate various materials’ fragmentation and fracturing mechanisms at laboratory and field scales.

The Fast Breakage Model (FBM), initially proposed by Potapov and Campbell [39], discretizes a particle into polygonal (2D) or polyhedral (3D) sub-blocks. The sub-blocks can be separated once the failure criterion is met. The failure criterion for both models can be based on stress or energy. The $t_{10}$ of the sub-block size-distribution curve follows the mathematical model proposed by Shi and Kojovic [29], while the size distribution of progeny (i.e., child particles/blocks) in FBM can be predefined using different mathematical functions (e.g., Gaussian–Schumann and incomplete beta function) based on laboratory observations. The size of each progeny can be further reduced until the minimum size is reached.

The Particle Replacement Model (PRM), initially proposed by Cleary [40], investigates the effect of power consumption on particle size reduction. In PRM, each parent model representing a particle is replaced with several child particles/clumps (i.e., progeny) once the failure criteria (e.g., energy and stress) are met. Similar to FBM, the size distribution of progeny can be predefined using the incomplete beta function [41] based on laboratory observation.

FBM and PRM are computationally efficient compared to BPM and well suited for laboratory samples where the size of progeny and breakage probability can be derived from laboratory tests. However, for larger rock block impact breakage scenarios, where obtaining direct data on fragmentation is challenging, BPM is more suitable for fragmentation analysis using the forward modelling approach [7]. BPM can explicitly simulate detailed breakage mechanisms and produce realistic fracture patterns if strength and deformation properties are correctly calibrated. This makes BPM the preferred choice for forward analysis in large-scale applications such as impact breakage and compression-induced breakage in cave mines.

4.2. Flat-Jointed BPM

Traditional Bonded Particle Models (BPM) utilize linear or parallel contact bonds to simulate brittle materials. However, Diederichs [42] and Potyondy and Cundall [38] identified limitations in using traditional BPM for simulating brittle rocks. Specifically, when the (unconfined and confined) compressive strength of BPM is calibrated to match that of brittle rock, the tensile strength tends to be overestimated. Potyondy [43] explained that in BPMs employing linear and parallel bonds, failure at peak load under compression is triggered by grain rolling due to the lack of rolling resistance after bond failure, leading to a significantly lower strength than typical compact rock.

Potyondy [43,44] proposed the Flat-Jointed Bonded Particle Model (FJ-BPM) to address this limitation. In FJ-BPM, disks are bonded by a notional flat-joint contact at their contact points. Figure 5a illustrates the schematic of two disks bonded by a flat-joint contact when the distance between them is less than or equal to the installation gap ($g_o$). Figure 5b shows that the flat-joint contact can be discretized into several elements (line segments in 2D and disks in 3D), with cracks represented by full or partial breakage events of the flat-joint elements.

The bond properties of a bonded flat-joint contact include (i) effective modulus ($E^{\ast }$); (ii) normal-to-shear stiffness ratio ($\kappa =k_n/k_s$); (iii) contact tensile strength (${\sigma }_t$); (iv) contact shear strength parameters: peak and residual cohesion ($c_p$ and $c_r$), peak friction angle (${\phi }_p$) and residual friction coefficient ($\mu$); and (v) radius multiplier ($\lambda$). Note that this paper adopts the terminology “bond properties” instead of the more common “micro-properties” when referring to the properties of bonded particles. The term micro-property adds a dimensionality that is not always respected in numerical simulations (i.e., micro suggests particles less than 1 μm); furthermore, micro-properties may incorrectly indicate that the bond properties are equivalent to actual physical properties and not just modelling properties.

The flat-joint contact normal stiffness ($k_n$) as a function of effective modulus ($E^{\ast }$) is defined as follows:

$$k_n={\displaystyle \frac{E^{\ast }}{R_1+R_2}}$$

(4)

where $R_1$ and $R_2$ are the radii of the two disks bonded at the flat-joint contact (Figure 5a). The radius multiplier ($\lambda$) controls the length/radius of the flat-joint contact ($\overline{R}$), which is defined as:

$$\overline{R}=\lambda \mathrm{m}\mathrm{i}\mathrm{n}(R_1,R_2)$$

(5)

Figure 6 illustrates the constitutive behaviours and strength envelope of bonded flat-joint elements under tension and shear loads. Figure 6a,b show that the bond breaks when the local tensile or shear stresses exceed the contact tensile strength (${\sigma }_t$) or shear strength (${\tau }_p$). In Figure 6b, the bond’s residual shear stress can either be reduced to zero, followed by the mobilization of frictional strength (i.e., shear-drop residual mode, $M_r$ = 0), or directly reduced to the residual level with simultaneous mobilization of both residual cohesive and frictional strengths (i.e., shear-drop residual mode, $M_r$ = 1). The flat-joint contact is deleted when the relative displacement of the unbonded flat-joint contact exceeds $2\overline{R}$. A linear frictional contact is then assigned to the newly detected disk–disk contact. The deformation of the linear contact is controlled by $E^{\ast }$ and $\kappa$, which are equivalent to the deformation properties of the flat-joint contact. The peak and residual strength envelopes of the flat-joint elements follow the Coulomb slip envelope, as shown in Figure 6c.

The Flat-Jointed Bonded Particle Model (FJ-BPM) offers distinct advantages over the FBM and PRM for exploring fragmentation mechanisms and estimating progeny sizes for rock-like materials. While no model can perfectly replicate real-world conditions, FJ-BPM allows valuable insights into the fragmentation process to be acquired through a forward modelling approach. By simulating the detailed fracture development and interactions within the material, FJ-BPM provides a more nuanced understanding of how fragmentation occurs under various conditions.

5. DEM Simulation of Impact and Compression-Induced Breakages

This section illustrates using FJ-BPM in the DEM program Particle Flow Code 3D (PFC^3D, [45]) to simulate the impact and compression-induced breakage of rock blocks at a field scale, enabling a more realistic and informative analysis of fragmentation characteristics.

5.1. FJ-BPM Calibration

A representative FJ-BPM model for simulating rock material requires iterative calibration of the properties at the flat-joint contacts to ensure that the macroscopic properties (i.e., Young’s modulus, Poisson’s ratio, unconfined and confined strengths) match those of the rock material. In this study, unconfined and confined compression tests were simulated on a cuboidal-shaped FJ-BPM model with dimensions of 1 m × 1 m × 1.5 m to calibrate the bonds’ properties as shown in Figure 7. The size of the FJ-BPM model for the compression test simulations was chosen to match the major and minor axes of the irregular FJ-BPM block that will later be used in simulations of impact and compression-induced breakages. The minimum ball diameter in the FJ-BPM was set to 0.05 m, aligning with the caving industry standard where fine material is classified as ≤5 cm. An axial velocity of 0.015 m/s (corresponding to a strain rate of 0.02 s⁻¹) was applied to the walls in PFC^3D to compress the FJ-BPM model, mimicking the incremental compressive loads applied by steel platens in a piston machine. Note that the velocity used in the simulation is numerical and may not be related to the actual velocity in the laboratory experiment.

Figure 8 illustrates the systematic calibration process of FJ-BPM properties, following the approach proposed by Potyondy [43,46]. The calibration begins by adjusting the effective modulus $E^{\ast }$ and $\kappa$ of FJ contact to calibrate the model’s Young’s modulus and Poisson’s ratio. According to Potyondy [46], $E^{\ast }$ primarily influences the model’s Young’s modulus, while $\kappa$ controls the Poisson’s ratio. During this phase, all flat-joint strength parameters were assumed to be infinite (i.e., 1e + 20 Pa) to focus solely on calibrating the deformation properties.

Following the calibration of deformation properties, the strength parameters of the FJ contact were adjusted to match the unconfined and confined compressive strengths of the FJ-BPM model with those of the rock block. An iterative calibration process was employed to fine-tune the cohesion and friction angle of the flat joint. Throughout this process, the residual friction coefficient was assumed to be 0.4, and the flat-joint tensile strength was set equal to the material’s tensile strength. The properties of intact volcaniclastic sediments [47,48] from the Cadia East mine in Australia were used to calibrate the properties of the FJ-BPM model, ffollowing the procedures above. The strength envelope of the calibrated FJ-BPM is shown in Figure 9, and the input bond properties are listed in

Table 1. Calibrated input FJ-BPM properties.

	Bond Properties	Value	Unit
Ball	$\mathrm{Minimum} \mathrm{ball} \mathrm{diameter} d_{min}$	0.05	m
	$\mathrm{Max}-\mathrm{to}-\min \mathrm{diameter} \mathrm{ratio} d_{max}/d_{min}$	1.66	-
	$\mathrm{Density} \rho$	2700	Kg/m3
Flat-joint contact	$\mathrm{Effective} \mathrm{modulus}, E^{\ast }$	90	GPa
	$\mathrm{Normal}-\mathrm{to}-\mathrm{shear} \mathrm{stiffness} \mathrm{ratio}, \kappa$	2.5	-
	$\mathrm{Peak} \mathrm{friction} \mathrm{angle}, {\phi }_p$	10	°
	$\mathrm{Peak} \mathrm{cohesion}, c_p$	200	MPa
	$\mathrm{Tensile} \mathrm{strength}, {\sigma }_t$	10	MPa
	$\mathrm{Residual} \mathrm{friction} \mathrm{coefficient}, \mu$	0.4	-
	$\mathrm{Shear} \mathrm{residual} \mathrm{mode}, M_r$	0	-
	Number of elements on the contact	3	-
Linear contact (ball–facet and ball–ball)	$\mathrm{Effective} \mathrm{modulus}, E^{\ast }$	90	GPa
	$\mathrm{Normal}-\mathrm{to}-\mathrm{shear} \mathrm{stiffness} \mathrm{ratio}, \kappa$	2.5	-
	$\mathrm{Friction} \mathrm{coefficient}, \mu$	0.4	-

.

As discussed in many studies that calibrated input bond properties of discontinuum models to match physical material properties, the calibrated bond properties are one of the possible combinations that could lead to similar results [49,50]. Furthermore, the input bond properties require re-calibration if different disk sizes and packing approaches are used to generate FJ-BPM. The calibrated FJ-BPM model based on input parameters listed in Table 1 was used to simulate impact and compression-induced breakages, as presented in the later subsections.

5.2. Simulations of Impact and Compression-Induced Breakages

An irregularly shaped FJ-BPM model, with calibrated flat-joint bond properties, was used to study impact and compression-induced breakages. Figure 10 illustrates the geometry of the irregular block designed to simulate a caved rock fragment (primary fragmentation block). The block geometry was assumed to have an aspect ratio of 1.5 × 1 × 1 with an average diameter of 1 m. This geometry was then used to carve irregular-shaped FJ-BPM out of the cuboid model in Figure 7 for the simulations.

Figure 11a,b present the models used for simulating impact and compression-induced breakages, respectively. In the simulations of impact breakage, it is crucial to justify the damping coefficient for the ball–wall contact. The damping coefficient in the contact model is related to kinetic energy dissipation, which significantly influences the degree of impact breakage. This parameter is typically associated with the stiffness of the impact surface, reflecting a cushioning effect. Since this parameter exists only numerically and is practically impossible to measure in the field, many studies have proposed educated hypotheses and assumptions based on field experience.

This study assumed that the fine materials at the muckpile surface provide cushioning to blocks falling from the cave back. According to Brown [1] and Laubscher [2], this cushioning effect is influenced by the presence of fines between larger blocks, which prevents direct contact between the blocks and reduces the probability of block splitting. Brown [1] also noted that the fines generated from primary fragmentation play an active role in cushioning. Consequently, linear normal and shear critical damping ratios of 0.5, 0.6, and 0.7 were assumed in the sensitivity analysis to simulate varying levels of cushioning and their impact on the degree of fragmentation resulting from impact breakage. Note that the model simulates the normal impact of rock blocks, meaning that the normal damping ratio primarily influenced the results, while the shear damping ratio had a minimal effect. For simplicity, the shear damping ratio was assumed to be equal to the normal damping ratio.

In the simulation of compression-induced breakage, the loading conditions were maintained consistent with those used during the calibration process (as illustrated in Figure 7). Because the simulation involves quasi-static analysis, a sensitivity analysis on the viscous damping coefficients was considered unnecessary.

Initial simulations were conducted to investigate the failure modes of FJ bonds under impact and compression conditions. Figure 12 shows that tensile failure is the predominant failure mode in both simulated conditions. The captured failure mode agrees with tensile dominant fragmentation under rapid loading described by [6]. Therefore, a sensitivity analysis was performed by varying the FJ tensile strength by ±20% (i.e., 8 MPa and 12 MPa) to assess its influence on the degree of fragmentation for both types of breakages. The objective was to understand how changes in tensile strength affect the fragmentation outcomes under different loading scenarios.

In reality, rock blocks are not perfect materials, and they contain micro-defects likely to impact fragmentation characteristics. Additional sensitivity analyses were conducted to investigate the impact of micro-defect percentages on fragmentation behaviour to realistically simulate rock block conditions at field scales. This was achieved by randomly adjusting some of the bonded FJ contacts to frictional contacts (i.e., with zero cohesion and tensile strength). Figure 13 shows the defected FJ-BPMs with various percentages of defect. For the sensitivity analysis, 5% to 20% of the total number of bonded FJ contacts were altered to frictional (unbonded), allowing for the examination of how varying levels of micro-defects affect the overall fragmentation process.

6. Simulation Results

Figure 14 illustrates the general trend of size-distribution curves at varying energy levels for both impact and compression-induced breakage simulations. The fragment sizes were computed using the “fragment compute” command in PFC. This command computes the ball assembly of a fragment based on the contact bond status (i.e., bonded/broken). Figure 14a illustrates that smaller fragments are attritted from larger blocks at lower energy levels, indicating an increased generation of fines. As the energy input increases (Figure 14b,c), the size of the blocks at higher mass passing percentages decreases. It is important to note that there is no standardized sieve test for large blocks, so the fragmentation curves were plotted using the raw simulation data without any smoothing process. The size-distribution curves at different kinetic energy levels agree with Wills and Finch [6], who stated that no fracture but chipping (i.e., broken edges) occurs when the impact kinetic energy is lower than the particle yield stress.

The universal $t_n-t_{10}$ relationships, as described in Equations (1) and (2), were applied to analyze the fragmentation results. Figure 14b and Figure 15a demonstrate that the $t_n-t_{10}$ curves are in good agreement with the fragmentation data across various damping coefficients and FJ tensile strengths for both impact and compression-induced breakage. The best-fit functions for $t_n-t_{10}$ relationships with ±95% confidence are listed in

Table 2. Fitted $t_n-t_{10}$ functions in Figure 15 with ±95% confidence.

Impact Breakage	Compression-Induced Breakage
$t_2={\displaystyle \frac{124.36 (\pm 4.81)\ast t_{10}}{16.14 (\pm 1.86)+t_{10}}}$	$t_2={\displaystyle \frac{123.31 (\pm 2.78)\ast t_{10}}{16.19 (\pm 1.21)+t_{10}}}$
$t_4={\displaystyle \frac{199.21 (\pm 19.99)\ast t_{10}}{82.55 (\pm 13.58)+t_{10}}}$	$t_4={\displaystyle \frac{184.73 (\pm 12.76)\ast t_{10}}{67.61 (\pm 8.34)+t_{10}}}$
$t_{25}=0.75 (\pm 0.019)\ast t_{10}$	$t_{25}=0.67 (\pm 0.009)\ast t_{10}$
$t_{50}=0.62 (\pm 0.019)\ast t_{10}$	$t_{50}=0.51 (\pm 0.006)\ast t_{10}$
$t_{75}=0.55 (\pm 0.018)\ast t_{10}$	$t_{75}=0.47 (\pm 0.006)\ast t_{10}$

. The analysis revealed that neither tensile strength nor damping coefficients significantly affect the $t_n-t_{10}$ relationships.

Figure 16 illustrates the $t_{10}$—energy (air gap height) relationships fitted with concave-up exponential curves for cases with varying damping coefficients and tensile strengths. The results show that the slope of the exponential curve increases as the damping coefficient and FJ tensile strengths decrease. This is reasonable, as it indicates that blocks with lower tensile strength and stiffer ground conditions require less kinetic energy to achieve a higher degree of fragmentation.

The general trend of $t_{10}$—displacement curves in Figure 17 shows a slow increment in $t_{10}$ with wall-relative displacement up to about 0.2 m, followed by a more rapid increase beyond this point. The differences in $t_{10}$ among cases with varying tensile strengths become more pronounced after 0.2 m of displacement. For an equivalent displacement, blocks with lower tensile strength exhibit compressive comminution energy to achieve higher $t_{10}$ values, indicating that weaker blocks require a less high degree of fragmentation.

Figure 18 demonstrates that the percentage of defects has little influence on the overall $t_n-t_{10}$ relationships, independently of whether impact or compression-induced breakages are considered. This agrees with the findings presented in Figure 15 and suggests that the fragmentation characteristics are largely independent of variations in micro-defect percentages. Fitted $t_n-t_{10}$ functions are listed in

Table 3. Fitted $t_n-t_{10}$ functions in Figure 18 with ±95% confidence.

Impact Breakage	Compression-Induced Breakage
$t_2={\displaystyle \frac{121.85 (\pm 4.63)·t_{10}}{16.05 (\pm 1.88)+t_{10}}}$	$t_2={\displaystyle \frac{125.74 (\pm 1.32)·t_{10}}{18.66 (\pm 0.65)+t_{10}}}$
$t_4={\displaystyle \frac{208.03 (\pm 28.78)·t_{10}}{91.82 (\pm 20.26)+t_{10}}}$	$t_4={\displaystyle \frac{212.93 (\pm 7.31)·t_{10}}{92.38 (\pm 5.32)+t_{10}}}$
$t_{25}=0.77 (\pm 0.022)·t_{10}$	$t_{25}=0.68 (\pm 0.007)·t_{10}$
$t_{50}=0.64 (\pm 0.019)·t_{10}$	$t_{50}=0.51 (\pm 0.007)·t_{10}$
$t_{75}=0.60 (\pm 0.018)·t_{10}$	$t_{75}=0.46 (\pm 0.006)·t_{10}$

.

Figure 19 illustrates the impact of defects on the degree of fragmentation (represented by $t_{10}$) under both impact (Figure 19a) and compression-induced breakage (Figure 19b) conditions. In Figure 19a, the $t_{10}$ value shows a similar rate of increase with kinetic energy when the percentage of defects is below 10%. However, the $t_{10}$ value increases more rapidly when the percentage of defects exceeds 10%. Figure 19b shows that the percentage of defects does not have a significant impact on the $t_{10}$ increment under compression-induced breakage conditions. The $t_{10}$ values increase with vertical displacement, but the presence of defects does not cause a marked deviation in the fragmentation pattern.

7. Discussion

This section discusses the differences between the exponential $t_n-t_{10}$ relationships obtained from the simulation results presented in Section 6, and those commonly observed in comminution laboratory tests. These differences are interpreted in the context of the conditions between field-scale fragmentation and laboratory-scale experiments. The DEM simulations conducted in this study provide insights into the fragmentation characteristics of rock blocks under both impact and compression-induced conditions. The fragmentation characteristics of single-particle breakage were well illustrated in $t_n-t_{10}$ and $t_{10}$—energy plots. The fragmentation curves also showed the generation of fine materials at low energy levels. One of the key findings from these simulations is the contrast between the shape of the $t_n-t_{10}$ curves typically observed in comminution studies and those identified in this study.

In traditional comminution studies (laboratory scale), the relationship between the degree of fragmentation (represented by $t_{10}$) and the specific comminution energy is typically described by a concave-down exponential curve (Equation (3)). This relationship suggests that, as the input energy increases, the fragmentation rate initially rises rapidly but then plateaus; the rate at which these changes occur is controlled by fitted breakage parameter $A$. The plateau phase indicates a constant degree of fragmentation as more energy is applied.

Contrary to what is typically observed at the laboratory scale, the DEM simulations, which refer to large-scale conditions, exhibit a concave-up exponential relationship between $t_{10}$ and kinetic energy (or displacement in the case of compression). This finding suggests that at larger scales and with higher energy levels, the degree of fragmentation continues to increase at an accelerating rate. To explain the difference, it must be recognised that the energy-to-strength ratio in the DEM simulations is relatively high due to the larger dimensions of the rock blocks compared to the ones of typical laboratory experiments. In a larger rock block in the field, the presence of veins, alterations, and micro-defects can significantly reduce the overall block’s strength, making it more susceptible to fragmentation. At the same time, rock fragments are reduced to much smaller (finer) sizes at the laboratory scale than the 5 cm threshold used in this study. Therefore, the fragmentation efficiency (i.e., increment of $t_{10}$—energy curve) decreases with energy because it is impossible to break strong mineral grains further.

The identification of a concave-up exponential $t_{10}-t_n$ relationship has significant implications for cave mining operations. It suggests that traditional models based on laboratory-derived concave-down $t_{10}$—energy relationships may underestimate the degree of fragmentation at the field scale, particularly in scenarios involving high energy inputs.

8. Conclusions

This paper started with a critical review of existing fragmentation assessment methods and highlighted the overlooked impact and compression-induced fragmentations in these tools. This paper then provided new insights into the secondary fragmentation characteristics and fines generation in impact and compression-induced fragmentation processes using DEM simulations. The challenges associated with traditional approaches using backward modelling highlight the need for more scientifically grounded methodologies that account for the complexities of rock behaviour in cave mining environments. The universal $t_n-t_{10}$ relationship from comminution offers a framework for improving fragmentation analysis.

The DEM simulations yielded a concave-up exponential relationship between $t_{10}$ and kinetic energy. Sensitivity analyses further highlighted the critical influence of parameters such as tensile strength and micro-defects. The analyses demonstrated that lower tensile strengths and higher defect densities are associated with an increased degree of fragmentation. However, the overall $t_n-t_{10}$ relationships remained largely unaffected by these variations.

The study also examined compression-induced breakage, demonstrating how vertical displacement and tensile strength influence the degree of fragmentation. The results showed that rock blocks with lower tensile strength are more prone to fragmentation under compressive forces. This emphasizes the importance of considering material properties when assessing the extent of secondary fragmentation, as variations in tensile strength can significantly alter the fragmentation patterns, especially under the quasi-static loading conditions typical of compression-induced breakage.

Sensitivity analyses further highlighted the critical impact of tensile strength, damping coefficients, and micro-defects for impact conditions and compression-induced breakage. Lower tensile strengths and higher defect densities were consistently associated with increased fragmentation degree, indicating the need for detailed consideration of these parameters in practical mining operations. These findings provide valuable insights for refining current empirical models, which may not fully capture these complex interactions. More importantly, this paper demonstrates the feasibility of adopting comminution theory originally developed for mineral processing to improve existing analyses of secondary fragmentation and fines generation.