Investigation of Strength Diversity Characterization in Mineral Materials Using Discrete Element Method

Abstract

Accurate modeling of ore materials is fundamental to high-precision simulations in mineral processing and remains a key research focus. To address the modeling challenges arising from the inherent heterogeneity and strength diversity of ores, this study proposes a novel method based on the bonded particle model (BPM) in the Discrete Element Method (DEM), incorporating multi-sized sub-particle stochastic generation and assembly, as well as bond strength parameter design. The method was applied to model and simulate impact crushing of 30 mm size fraction gold, iron, and copper ores with varying strengths. The resulting particle size distributions of fragmented ores were analyzed. Furthermore, drop weight tests were conducted on ore samples of the same size fraction, and the experimental mass distribution of fragmented particles demonstrated good consistency with simulation results. These findings validate the capability of the proposed method to effectively characterize the strength diversity of natural ores, offering an advanced approach for high-fidelity modeling of mineral materials.

1. Introduction

Mineral processing holds a pivotal position in manufacturing industries. At its core lie ore crushing and grinding, which account for 40%–70% of the total energy consumption in mineral beneficiation, making these processes critical for industrial energy conservation, consumption reduction, and production efficiency enhancement. However, conventional comminution technologies persistently suffer from high energy intensity and low efficiency. Beyond limitations in grinding process optimization, the inherent breakage characteristics of ores—shaped by their structural and mechanical properties—remain one of the most decisive factors influencing grinding efficiency [1,2]. As natural geological products, ores exhibit remarkable structural heterogeneity and strength diversity, stemming from multi-factor coupling during mineralization, prolonged geological evolution, and intrinsic compositional variations among mineral phases [3,4]. This heterogeneity manifests not only in their physical–mechanical behaviors but also directly governs the efficiency and sustainability of resource exploitation. Consequently, research into ore strength characterization and fragmentation efficiency optimization has long been a central theme in mining engineering [5,6].

The structural characteristics of ores exhibit pronounced specificity across micro-to-macroscales, directly governing their strength properties. At the microscale, the spatial distribution of mineral components displays complex topological relationships, including inclusions, zonation structures, and interlocking networks, which lead to significant variations in intergranular bonding modes and interfacial energies [7,8]. The diversity of mineral paragenesis further results in substantial differences in mechanical parameters (e.g., hardness, toughness) among coexisting phases, creating localized stress concentrations and heterogeneous energy dissipation pathways during fragmentation [9,10]. On the mesoscale, the fractal characteristics of pore–fracture systems become dominant. Nanoscale pores, micrometer-scale microfractures, and macroscale structural discontinuities collectively form multi-scale permeable networks. These hierarchical pore structures not only alter fluid diffusion mechanisms but also induce stress redistribution, critically influencing the ore’s global mechanical stability [11,12]. At the macroscale, ore textures (e.g., layered, massive, disseminated) amplify structural heterogeneity. The mechanical coupling between distinct textural units drives nonlinear evolution of failure modes [13,14], while grain size heterogeneity (coarse–fine intergrowths) generates differential strain responses under loading, preferentially initiating crack nucleation and propagation along weak interfaces [15,16]. Furthermore, post-mineralization geological alterations imprint residual stress fields and memory-bearing fracture systems within ores. These latent structural imprints dictate unique fragmentation behaviors during downstream processing, underscoring the necessity of multi-scale characterization for predictive comminution modeling.

The dynamic coupling between ore’s intrinsic structural heterogeneity and external environmental factors gives rise to the multifaceted manifestations of strength diversity. At the mechanical response level, strength characteristics exhibit pronounced anisotropy and nonlinearity. Structural determinants such as the orientation of mineral cleavage planes and spatial distribution of bedding structures can induce over 40% variation in compressive/tensile strengths across different loading directions [17,18]. Such anisotropy renders conventional isotropic strength theories inadequate in describing ore failure behaviors, necessitating modified frameworks like tension–shear coupled failure models or anisotropic constitutive equations [19,20]. Furthermore, confining pressure effects amplify nonlinear strength responses. Deep-seated ores under triaxial stresses demonstrate strength intensification up to several times higher than uniaxial states, fundamentally altering stability assessment criteria for deep resource exploitation [21,22].

The environmental sensitivity of ore strength constitutes another critical dimension of its diversity. Temperature fluctuations can markedly alter ore’s brittle–ductile behavior through mineral phase transitions or thermal stress reorganization, suggesting that dynamic optimization of crushing temperature parameters could enhance energy utilization efficiency [23,24]. Meanwhile, moisture content exhibits threshold effects on strength: pore water films synergistically degrade interparticle bonding via physical lubrication and chemical corrosion mechanisms, with this phenomenon being particularly pronounced in clay-cemented ores [25,26]. Crucially, ore’s structural heterogeneity and strength diversity are not isolated phenomena but engage in cascading effects through multiphysics coupling. Microscopic heterogeneity complicates stress wave propagation paths, potentially triggering localized over-crushing or incomplete mineral liberation under impact loading [27,28]. The multi-scale nature of pore–fracture systems enables interactive permeation pressure–mechanical stress coupling—a fluid–solid interaction mechanism pivotal in leaching operations and hydraulic fracturing, ultimately governing fragmented product size distributions and mineral liberation degrees [29].

Given these multifaceted factors, research on ore strength characteristics has been extensively pursued. However, studies focusing on single influencing factors lack practical relevance, whereas investigations into multi-factor coupled strength mechanisms have emerged as a critical frontier, typically employing integrated experimental and simulation modeling approaches. The Discrete Element Method (DEM) is one of the commonly used simulation methods for studying ore strength. As a numerical modeling technique for discontinuous media mechanics, it was first proposed by Cundall in the 1970s. The core idea is to discretize the research object into independent units (such as particles, blocks, or polyhedrons), each of which follows Newton’s second law (translation) and Euler’s rotation equation (rotation). The motion trajectory (displacement, velocity, and acceleration) of each unit is iteratively solved through explicit time integration. Based on the contact overlap or potential contact distance, the contact force or bonding force between units is calculated using different mechanical models (such as elastic, plastic, or bonded contact models), and the system state is dynamically updated. This method strictly satisfies the conservation of momentum and energy and is suitable for the detailed simulation of complex mechanical processes such as large deformation, dynamic collision, particle flow, and fracture of discontinuous media [30,31,32]. DEM inherently excels in ore fragmentation studies. Current research emphasizes optimizing fragmentation efficiency and product size control under varied operational conditions [33,34]. However, the predictive accuracy of DEM simulations critically depends on the fidelity of ore strength models. A persistent challenge lies in accurately characterizing actual ore fragmentation strengths during modeling, constrained by structural heterogeneity and strength diversity. This limitation hinders reliable predictions of energy consumption and liberation patterns in industrial comminution processes.

In the grinding production process, determining the strength characteristics of materials is the basis for establishing the process flow. However, due to the heterogeneity and complexity of the ore strength, each batch of material requires pre-production strength testing, which involves tedious and time-consuming procedures. Therefore, in order to improve the efficiency of determining the material strength, this study addresses the challenge of accurately modeling variable ore strength by proposing a bonded particle model-based approach. This paper establishes the crushing simulation of anisotropic internal mechanics of ores, and adopts the characterization method of ore strength diversity for modeling to accurately characterize the strength of different ores. The study procedures accurately simulated the crushing results of ores by using simulation methods. Taking 30 mm size fraction gold, iron, and copper ores as examples, we simulate the fragmentation process under varying impact energies. The simulations precisely capture the fragmentation behavior of ore particles, with results analyzed and validated against drop weight test data. This methodology achieves accurate representation of ore strength diversity, enhances the precision of fragmentation simulations, and lays the foundation for simplifying the process design flow, increasing grinding production efficiency, and applying crushing simulations to real-world industrial practices.

2. DEM-Based Theory of Ore Fragmentation Model

The Discrete Element Method (DEM) effectively bridges microscopic contact mechanics and macroscopic fragmentation behaviors by simulating interparticle interactions based on Newtonian dynamics and contact mechanics models [35]. The core of DEM in simulating mineral crushing lies in choosing the appropriate crushing model. Different crushing models directly affect the simulation accuracy, computational efficiency, and application scenarios. This study adopts EDEM V2022.0 software to implement the crushing simulation. In EDEM, common crushing models include the following:

Replacement Method: Comprising the traditional API (Application Programming Interface) replacement method and Tavares UFRJ model [36], this approach replaces parent particles with sub-particle clusters when forces exceed critical thresholds. While suitable for large-scale rapid simulations, it suffers from manual sub-particle distribution rules, error susceptibility, and complex parameter calibration [37,38].

Dynamic Fragmentation Model: This model dynamically calculates fragment size and distribution using energy thresholds and stress profiles. It supports complex-shaped particles, making it ideal for detailed simulations of non-spherical particles [39].

Bonded Particle Model (BPM): Virtual “bonds” connect small particles into aggregates. Bonds break when external forces (tension, shear, or torsion) exceed strength thresholds, causing aggregate fragmentation [31]. BPM enables precise fracture path simulation, crack propagation analysis, and energy dissipation studies, particularly suited for heterogeneous materials. However, its high computational cost limits applications in large-scale dynamic fragmentation [40].

Model selection requires balancing simulation goals (accuracy vs. efficiency), particle characteristics (shape/size), and operational complexity (multi-field coupling). Given the structural heterogeneity and strength diversity of ores, the bonded particle model (BPM) aligns best with the research objectives. BPM’s sub-particle clusters allow detailed simulation of bond failures, facilitating studies on internal crack propagation and strength variability during ore fragmentation.

2.1. Mechanical Behavior of Bonds

The bonded particle model (BPM) simulates material mechanical behavior and fracture processes through bond breakage, governed by principles such as bond mechanical response and failure criteria [41]. In EDEM, it is specifically the Hertz–Mindlin with Bonding model. The Hertz–Mindlin with Bonding contact model can be used to bond particles with a finite-sized “glue” bond. This bond can resist tangential and normal movement up to a maximum normal and tangential shear stress, at which point the bond breaks. Thereafter the particles interact as hard spheres. This model is particularly useful in modeling concrete and rock structures. The forces acting on the bond are shown in Figure 1.

Particles are bonded at the bond formation time tBOND. After bonding, the forces ($F_n,F_t$)/torques ($T_n,T_t$) on the particle are set to zero and are adjusted incrementally every timestep according to

$${\delta F}_n=-v_n{s}_{n}A\delta t$$

(1)

$${\delta F}_t=-v_t{s}_{t}A\delta t$$

(2)

$${\delta M}_n=-{\omega }_n{s}_{t}J\delta t$$

(3)

$${\delta M}_t=-{\omega }_t{s}_n{\displaystyle \frac{J}{2}}\delta t$$

(4)

where

$$A=\pi {R_B}^2$$

$$J={\displaystyle \frac{1}{2}}\pi {R_B}^4$$

$R_B$ is the radius of the “glue”, $S_n$, $S_t$ are the normal and shear stiffness, respectively, and $\delta t$ is the timestep. $v_n$, $v_t$ are the normal and tangential velocities of the particles and ${\omega }_n$, ${\omega }_t$ are the normal and tangential angular velocities.

The basis of the Hertz–Mindlin with Bonding model is the Hertz–Mindlin (no slip) model. This contact model is the default model used in EDEM due to its accurate and efficient force calculation. In this model the normal force component is based on Hertzian contact theory. The tangential force model is based on Mindlin–Deresiewicz work. Both normal and tangential forces have damping components where the damping coefficient is related to the coefficient of restitution. The tangential friction force follows the Coulomb law of friction model. The rolling friction is implemented as the contact independent directional constant torque model [42,43,44,45].

2.2. Failure Criteria

In the bonded particle model, bond failure conditions are determined by stress or energy thresholds [31]. Common criteria include the following:

• Maximum Stress Criterion

The bond is broken when the normal and tangential shear stresses exceed some predefined value:

$${\sigma }_{max}<-{\displaystyle \frac{F_n}{A}}+{\displaystyle \frac{2M_t}{J}}{R}_B$$

(5)

$${\tau }_{max}=-{\displaystyle \frac{F_t}{A}}+{\displaystyle \frac{M_n}{J}}{R}_B$$

(6)

• Energy Criterion

The energy criterion accounts for multi-directional loading effects and applies to complex stress states. Fracture occurs when the strain energy density exceeds the critical fracture energy:

$$U={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{F_n}^2}{k_n}}+{\displaystyle \frac{{F_t}^2}{k_t}}+{\displaystyle \frac{{M_b}^2}{k_b}}+{\displaystyle \frac{{M_t}^2}{k_t}}\right)\ge U_c$$

(7)

where $U_c$ is the material-specific fracture energy.

To address the structural heterogeneity and strength anisotropy of ores, we design multi-sized sub-particle clusters to represent bulk material strength. Sub-particles of varying diameters simulate distinct internal mineral compositions, while bond strengths between sub-particles scale with their sizes, thereby characterizing the ore’s heterogeneous strength properties. A stochastic generation and assembly algorithm is employed for sub-particle initialization, eliminating artificial structural biases and ensuring the model faithfully replicates real ore’s structural–strength relationships.

3. Ore Strength Heterogeneity Modeling

Sub-particle parameters and interparticle bond parameters are two critical factors determining the microscopic performance of the bonded particle model (BPM). Sub-particle parameters include the size distribution (types and quantities of sub-particle diameters). Bond parameters are defined by five key variables: normal stiffness per unit area ($k_n$), shear stiffness per unit area ($k_t$), tensile strength (${\sigma }_c$), shear strength (${\tau }_c$), and bond radius (λ). Rational design of these parameters enables the BPM to authentically replicate the strength characteristics of heterogeneous ores.

3.1. Simulation of Ore Heterogeneity

This study focuses on 30 mm size fraction ores for drop weight test simulations. The cubic model measures 30 mm × 30 mm × 30 mm. The bonded particle model (BPM) consists of irreducible sub-particles (smallest units, typically spherical), whose diameters determine the minimum fragment size post-fragmentation due to inherent model limitations. Consequently, the simulated particle size distribution (PSD) can only characterize fragments larger than sub-particle sizes. Coarser fragments are represented by clusters of bonded sub-particles, with varying cluster sizes corresponding to fragmented ore products.

To replicate real ore heterogeneity, we employ multi-sized sub-particles stochastically generated and assembled, where different diameters represent distinct internal mineral components. Bonds between sub-particle types are assigned varying strengths, effectively simulating complex compositional structures and strength heterogeneity.

Excessive sub-particle size types increase model complexity, computational cost, and uncertainty [46,47]. Balancing efficiency and accuracy, three sub-particle sizes are selected: small (P₁): diameter d₁; medium (P₂): diameter d₂; large (P₃): diameter d₃. Under identical model packing density (α = 0.6), different size combinations yield varying particle counts (n₁, n₂, n₃). Drop weight tests on 30 mm ores reveal that fragments ≥3.35 mm account for >60% of total mass, justifying our focus on simulating PSDs above 3.35 mm. To ensure clusters represent ≥3.35 mm fragments, the maximum sub-particle diameter is set below 3.35 mm. Key parameters are listed in

Table 1. Parameter design for sub-particles in BPM of 30 mm ore.

Type	Diameter/mm	Number
P1	1	2900
P2	1.5	2400
P3	2	2500

.

Based on the model parameters listed in the table, the ore model was constructed. The shape of the ore is simplified to a cuboid, with Figure 1 illustrating the bonded particle model (BPM) construction process. In Figure 2a, an open shell in the shape of ore was established. Three kinds of particles were randomly generated inside the shell, and a pressure plate was set above the opening of the shell. Figure 2b shows the completion of particle generation, which naturally accumulates in the shell, and the pressure plate moves downward at a constant speed slowly. In Figure 2c, the pressure plate slowly squeezes the sub-particles to make the sub-particles arranged closely, facilitating the subsequent formation of bonds. After the bonds are stably formed, the shell and the pressure plate are removed, and the ore bonded particle model can be obtained, as shown in Figure 3a. In this model, magenta spheres represent P₁ particles, gray spheres denote P₂ particles, and brown spheres correspond to P₃ particles. In Figure 3b, the display of sub-particles is hidden, allowing the bond structure between particles to be clearly seen. Different colors represent different types and strengths of bonds. Interparticle bonds are color-coded as follows: blue lines for P₁–P₁ bonds, red lines for P₂–P₂ bonds, gray lines for P₃–P₃ bonds, crimson lines for P₁–P₂ bonds, dark green lines for P₁–P₃ bonds, and green lines for P₂–P₃ bonds

3.2. Simulation of Ore Anisotropy in Different Strengths

The ore model is constructed by filling and bonding three types of sub-particles, with adjacent particles connected via bonds whose parameters are determined by the ore’s fragmentation properties. Variations in bond strength parameters reflect the intrinsic strength differences among ore types, enabling the simulation of mechanical anisotropy within heterogeneous ores. Specifically, distinct bond strengths between particle pairs dictate varying energy thresholds for bond rupture, thereby replicating anisotropic fracture behaviors. This study compares the strengths of 30 mm size fraction gold, iron, and copper ores. Interparticle bonds are categorized as K₁₁ (P₁–P₁), K₂₂ (P₂–P₂), K₃₃ (P₃–P₃), K₁₂ (P₁–P₂), K₂₃ (P₂–P₃), and K₁₃ (P₁–P₁). Firstly, the variation range of the anti-crushing strength of the actual ore was obtained based on the ore crushing test. Then, the bond parameters were adjusted within the range, and the preliminary simulation debugging and result analysis were carried out [48]. Finally, three sets of bond parameters were, respectively, designed to represent the fragmentation strengths of each ore type, as detailed in

Table 2. Bond parameters of BPM for gold ore sample.

Bond	kn/GPa/m	σc/MPa	kt GPa/m	τc/MPa
K11	25.84	0.46	24.34	0.46
K22	25.00	0.46	23.59	0.46
K33	24.34	0.46	22.77	0.46
K12	32.17	0.46	30.07	0.46
K23	31.51	0.46	29.49	0.46
K13	30.87	0.46	28.75	0.46

,

Table 3. Bond parameters of BPM for iron ore sample.

Bond	kn/GPa/m	σc/MPa	kt GPa/m	τc/MPa
K11	26.43	0.42	24.48	0.42
K22	25.37	0.42	23.50	0.42
K33	24.39	0.42	22.59	0.42
K12	32.67	0.42	30.68	0.42
K23	31.59	0.42	29.66	0.42
K13	30.58	0.42	28.71	0.42

and

Table 4. Bond parameters of BPM for copper ore sample.

Bond	kn/GPa/m	σc/MPa	kt/GPa/m	τc/MPa
K11	25.60	0.48	23.90	0.48
K22	24.40	0.48	22.80	0.48
K33	23.40	0.48	21.90	0.48
K12	31.40	0.48	29.20	0.48
K23	30.30	0.48	28.20	0.48
K13	29.30	0.48	27.20	0.48

.

3.3. Modeling and Simulation of Ore Crushing and Result Analysis

Following the completion of BPM, impact fragmentation simulations were conducted on the ore models. Then, a simulation model based on the parameters of the drop weight test must be established. The tests and simulation were conducted on three ore types (gold, iron, and copper), respectively, under three impact energies (E = 0.25 kWh/t, E = 1.00 kWh/t, E= 2.50 kWh/t). Due to the limitation of the article’s length, the crushing results under the median energy of E = 1.00 kWh/t were selected for presentation. Their fragmentation results are shown in Figure 4. Figure 4 shows the particle clusters of different sizes formed after the crushing of each ore. Under impact, the bonds of the ore model break to varying degrees. When the bonds are completely broken, the sub-particles detach from the model, representing small-size ore debris. The unbroken parts formed particle clusters, representing the fragments after crushing of the actual ore. The bond strengths between different types of sub-particles are different, resulting in irregular fracture of the bonds and a relatively diverse shape of the formed clusters.

The simulations reveal distinct fragmentation strengths among ore types, evidenced by variations in particle cluster sizes under identical impact energy. The irregular shapes of particle clusters closely matched actual ore fragmentation outcomes, validating the model’s capability to simulate ore heterogeneity and strength diversity.

Statistical analysis of fragmented particle size distributions (

Table 5. Mass fraction of crushing simulation for various ore types (%).

Type of Ore	Size Fraction/mm
	<3.35	3.35	4.75	6.70	9.50	13.20	19.00	26.50
Gold ore	39.98	11.90	14.07	19.14	12.25	2.66	0.00	0.00
Iron ore	43.33	7.03	12.76	24.08	10.37	2.44	0.00	0.00
Copper ore	38.31	7.20	10.78	17.48	14.56	6.96	4.71	0.00

) demonstrates ore-specific fragmentation patterns. At this energy level, all three ores exhibited maximum mass fractions in the ≤3.35 mm size fraction, yet with differing mass distributions across other fractions. Notably, the largest clusters reached 13.2 mm for gold and iron ores, compared to 19 mm for copper ore, quantitatively reflecting their relative strength disparities.

4. Experimental Validation with Drop Weight Tests

To validate the accuracy of the bonded particle model (BPM) in replicating actual ore strength, drop weight tests were conducted on three ore types (gold, iron, and copper) of the same size fraction. During the test, ten pieces of ore with 30 mm size were selected first. The falling height was designed based on the impact energy, the mass of each piece of ore, and the weight of the drop hammer, and the drop weight test was completed one by one. All the fragments of the same kind of ore were collected. Post-fragmentation ore fragments were sieved to quantify mass fractions of size fractions, which were then compared with simulation results.

The drop weight tester and ore samples are shown in Figure 5a and Figure 5b, respectively, with experimental data summarized in

Table 6. Mass fraction of drop weight test for various ore types (%).

Type of Ore	Size Fraction/mm
	<3.35	3.35	4.75	6.70	9.50	13.20	19.00	26.50
Gold ore	37.76	12.42	15.26	20.88	11.20	2.49	0.00	0.00
Iron ore	41.47	11.63	13.67	22.55	9.65	1.03	0.00	0.00
Copper ore	33.29	10.93	12.38	20.95	15.39	6.13	0.93	0.00

.

Comparative analysis between experimental and simulated mass fractions revealed high model fidelity. Figure 6 displays the alignment of experimental and simulated size fraction distributions for all ores. The average errors in mass fractions were 0.86% (gold), 1.38% (iron), and 2.41% (copper), demonstrating the model’s precision. Notably, the largest fragments from gold and iron ores measured 13.2 mm, while copper ore fragments reached 19 mm, corroborating strength-dependent fragmentation patterns observed in simulations.

5. Conclusions

This study addresses the challenges in modeling ore materials caused by their heterogeneity and strength diversity by proposing a novel bonded particle model (BPM) framework, validated through drop weight tests. Key conclusions are as follows:

The heterogeneous characteristics of ores are effectively represented by freely combining sub-particle sizes and quantities. A high-precision ore model is achieved by rationally selecting sub-particle types and numbers. For 30 mm size fraction ores, the optimal configuration includes sub-particle diameters of d₁ = 1 mm, d₂ = 1.5 mm, and d₃ = 2 mm, with corresponding counts of n₁ = 2900, n₂ = 2400, and n₃ = 2500. Strength diversity is characterized by varying bond strengths between sub-particles. Random generation and arrangement of sub-particles produce diverse bond configurations, realistically simulating mechanical anisotropy and strength variability within ores. Tailoring bond parameters enables the construction of ore models with distinct strength profiles. Validation via drop weight tests confirms high accuracy: mass fraction errors across size fractions for gold, iron, and copper ores remain within 3%, demonstrating robust alignment between simulations and experimental results.

When modeling ores in grinding and crushing simulations, the ore modeling method described in this paper can be used to obtain a crushing model that conforms to the actual strength of the ore relatively conveniently and quickly, thereby enhancing the modeling speed and reliability of grinding simulations.