Numerically investigating the crushing of sandstone by a tooth hob

Abstract

To investigate the mechanical process that occurs between rocks and tooth hobs, the crushing of sandstone with a tooth hob was simulated using reconstructed multi-mineral mesoscopic numerical models of various grain-sized sandstone samples. When a piece of sandstone is crushed by the tooth of a hob rolling at a constant speed, the resultant reaction forces of the sandstone on the tooth first hinder and then contribute to the rolling of the hob. The absolute value of the longitudinal reaction force is significantly higher than that of the lateral reaction force. Because the tooth was subjected to reaction forces from the sandstone, forces and moments were applied to the hob in order to keep the hob rolling. The applied forces were equal in value and opposite in direction to the reaction forces of the sandstone on the tooth. Three typical curves of the work done by the applied forces and moment were obtained, and the contribution of the applied lateral force and moment to the total work done for crushing sandstones was variable; however, no work was done by the applied longitudinal force. Moreover, the applied longitudinal force and total work were positively correlated with the strength of sandstone samples. The total work, applied forces, and moment increased with the maximum penetration depth of the tooth in the sandstone.

Graphical abstract

Keywords

sandstone / tooth hob / crushing process / reaction force / numerical simulation

1 Introduction

Drilling holes in rocks is often necessary for constructing pile foundations for large-scale bridge projects. However, the physical and mechanical properties of rocks can affect the drilling rate [1]. Drillability is a parameter that is used to describe the resistance of rocks to being crushed [2], and this parameter could be used to estimate the penetration rate of drilling machines in rocks. Sandstone is a common sedimentary rock in riverbeds and sea beds and is commonly used for the construction of pile foundations [3]. The drillability of sandstone has been a major research focus. For example, Yang et al. [4] analyzed rock properties, including grain size and content, and found that the drillability of sandstone is related to the content of quartz and clay. Ma et al. [5] established a prediction model using a genetic algorithm and back propagation neural network to predict the drillability of sandstone. Zhang et al. [6] tested the drillability of sandstone, and the results showed that drillability was significantly correlated with the compressive strength and tensile strength of the rock. Han et al. [7] developed a model to predict the drillability of rocks, and the average error between the calculation results obtained using the model and laboratory test results obtained from the micro-drilling of sandstone was only 7.41%. Wang et al. [8] proposed a model to evaluate the impact of various factors on the drillability of sandstone and noted that the physical properties of the rock had a significant impact on drillability. The above research shows that drillability is related to the physical and mechanical properties of the rocks.

Laboratory experiments and numerical simulations are commonly used to study the physical and mechanical properties of rocks. Because rocks are natural materials, obtaining identical samples for laboratory experiments is difficult; however, different experiments can be conducted using the same model in numerical simulations [9]. Therefore, the establishment of a digital rock model based on imaging technology has attracted extensive research attention [10]. Lei [11] constructed a three-dimensional (3D) digital rock using X-ray computed tomography (CT) and micro-CT scanning to describe the macroscopic and microscopic pore structures of tight sandstone. You et al. [12] proposed a machine-learning method to reconstruct 3D digital rocks from two-dimensional (2D) cross-sectional images. Yao et al. [13] developed a method to construct a digital rock model using the simulated annealing, Markov Chain Monte Carlo, and superposition methods. Cao et al. [14] constructed a digital rock model by using a few samples to save computational resources and time. Saxena et al. [15] corrected image-derived exponents inferred from micron-scale images to obtain the pore volume of a rock model. Zhu et al. [16] integrated four types of boundary conditions into one boundary condition to improve the computational efficiency of a digital rock model.

Research on the physical and mechanical properties of rocks has progressed considerably owing to the use of digital rock models. For example, Che et al. [17] numerically simulated the trajectory of discharge particles in hard rock to investigate the factors that influence rock fragmentation. Rojek et al. [18] built 2D and 3D models to simulate rock cutting and subsequently compared the simulation results with the experimental data. Xia and Zhou [19] used a particle simulation method to study the failure process of brittle rock under triaxial compression, and the simulated results agreed well with the experimental results. Vallejos et al. [20] recommended the use of a flat-joint contact model to simulate the mechanical behavior of brittle rock at low confining pressures. Moon and Oh [21] discussed the numerical relationship between the mechanical properties and geometric characteristics of rocks under optimal rock-cutting conditions using simulations. Leine et al. [22] studied the influence of rock geometry on rockfall dynamics using multibody and non-smooth contact dynamics. Moreover, rock crushing can be simulated using a digital rock model.

The interaction between the drill bit and rock causes the rock to be crushed when the rock is drilled, and this process has been successfully simulated in the literature. For example, Saksala et al. [23] employed the finite element method to simulate the drill bit-rock interaction using a damage-viscoplasticity model. Kim et al. [24] explained the mechanism resulting in the crushing of rock and simulated drilling performance by considering the rock type and the dynamic effect of the bit. Zhang et al. [25] researched crushing using a numerical rock model and established a numerical drill bit-rock system. Rock cracking is a major research focus when studying rock crushing. Fakhimi and Lanari [26] simulated crack propagation during rock crushing and identified the crushed zones around the borehole. Menezes [27] predicted the separation of rock fragments from a rock slab using numerical simulation, and the predicted results were in good agreement with the experimental results. Ren et al. [28,29] applied the peridynamic approach developed for fracture analysis and proposed a higher-order nonlocal operator method to solve boundary value problems. Zhuang et al. [30] presented a non-local operator method for dynamic fractures. The aforementioned studies indicate that rock crushing has been extensively numerically simulated. However, the mechanical interaction between the rock and the hob in a rock that is crushed by the insertion of a tooth hob is not well understood.

In this study, different grain-sized sandstone samples were prepared for laboratory experiments, and the differences in the properties of these samples were studied. Based on the experimental results, multi-mineral mesoscopic numerical models of different grain-sized sandstones were constructed using a 2D particle flow code. Furthermore, the motion of the tooth hob was analyzed and simplified. The mechanical interaction between the rock and hob was explored by simulating the crushing of sandstone using the tooth of a rolling hob.

2 Laboratory experiments on sandstone samples

To reconstruct the multi-mineral mesoscopic numerical model of sandstone, the grain size distribution, mineral composition, and mechanical properties of the sandstone samples were determined through laboratory experiments.

2.1 Preparation of sandstone samples

Sandstone samples were obtained from the Nansha Bridge project (once known as the 2nd Humen Bridge project) in Guangdong Province, China. The basic information on the sandstone samples is presented in Tab.1. The sandstone samples were classified into very coarse-, coarse-, medium-, and fine-grained according to grain size. Additionally, the very coarse-, coarse-, and medium-grained samples were grayish-white, whereas the fine-grained samples were reddish-brown and had a rusty smell, indicating high iron content. As shown in Tab.1, the uniaxial compressive strengths of the medium- and fine-grained samples were the highest and lowest, respectively. The higher the strength of the rock, the more difficult it is to crush the rock [31]. Consequently, medium- and fine-grained sandstone are the most difficult and easiest to crush, respectively.

2.2 Grain size distribution of sandstone samples

Fig.1 shows the images and detailed views of the sandstone samples with different grain sizes; the samples are 50 mm in diameter. After processing these images, most grains could be distinguished with a relatively well-defined margin in the images of the very coarse-, coarse-, and medium-grained samples. The margins between the grains were not clear in the images of the fine-grained samples; however, the grains were still distinguishable.

The grain-size distribution of the sandstone samples was obtained by processing the images of the sandstone samples, as shown in Fig.2. According to the Udden-Wenthworth scale, the diameters of the very coarse, coarse, medium, and fine grains are in the ranges of 1.0–2.0, 0.5–1.0, 0.25–0.50, and 0.125–0.250 mm, respectively. The contents of the very coarse, coarse, medium, and fine grains in the very coarse-, coarse-, medium-, and fine-grained samples were 67%, 66%, 55%, and 72%, respectively. The diameters of the grains in all the samples were in the range of 0.125–4.000 mm.

2.3 Mineral composition analysis test

Mineral composition and content have a significant influence on rock properties. An X-ray diffractometer (Thermo Fisher Scientific) was used to analyze the mineral compositions and contents of the sandstone samples. The samples were ground and dried, and the powdered samples were sieved using a 200-mesh sieve and subsequently tested using an X-ray diffractometer. The results of the X-ray diffraction analysis are shown in Fig.3. All the sandstone samples were mainly composed of quartz. Additionally, the medium- and fine-grained samples had the highest and lowest quartz contents, respectively. The quartz contents of the very coarse- and coarse-grained samples were similar. The fine-grained samples had the lowest feldspar and calcite contents, and the feldspar and calcite contents were similar in the remaining samples. The clay and iron contents of the fine-grained samples were significantly higher than those of the other samples.

2.4 Mechanical tests

2.4.1 Triaxial compression test

Sandstone samples retrieved from the project were processed into cylindrical samples with a diameter of 50 mm and height of 100 mm for the triaxial compression test. Three test samples were prepared for each grain-sized sandstone sample, i.e., a total of 12 test samples. In the triaxial compression test, the confining pressure was calculated as the sampling depth of the sandstone samples multiplied by 0.1 MPa/m. The stress−strain curves of the different grain-sized sandstone samples obtained from the triaxial compression test are shown in Fig.4. The average peak stresses of the medium- and fine-grained samples were the highest and lowest, i.e., 232.06 and 132.93 MPa, respectively. The average peak stresses of the very coarse- and coarse-grained samples were 150.82 and 170.48 MPa, respectively. In addition, the average elastic moduli of the very coarse-, coarse-, medium-, and fine-grained samples were 26.63, 28.45, 32.11, and 16.65 GPa, respectively.

The typical failure modes of the sandstone samples are shown in Fig.5(a). The samples were found to have undergone shear failure, and the cracks in the samples were mainly oblique. The cracks formed in most samples had an X or Y shape formed by two oblique cracks, whereas the cracks formed in some samples contained a single oblique crack. The mesoscopic images of the cracks are shown in Fig.5(b). The damaged grains were observed in the very coarse-, coarse-, and medium-grained samples. The grains in the fine-grained samples were the smallest, and no damaged grains were observed. Therefore, the grains in the fine-grained sample were intact when the sample was damaged. When the very coarse-, coarse-, and medium-grained samples were damaged, a few grains and the cemented substances between the grains were damaged.

2.4.2 Brazilian splitting test

The sandstone samples retrieved from the project were processed into cylindrical samples with a diameter of 50 mm and a height of 30 mm for the Brazilian splitting test. Three samples were prepared for each grain-sized sample, i.e., a total of 12 samples. The Brazilian splitting test results are shown in Fig.6. The average Brazilian splitting strengths of the medium and fine-grained sandstone samples were the highest and lowest, i.e., 16.32 and 6.12 MPa, respectively. The average Brazilian splitting strengths of the very coarse- and coarse-grained sandstone samples were 12.66 and 13.08 MPa, respectively. According to the results of the triaxial compression and Brazilian splitting tests, the ratios of the average Brazilian splitting strength to the average triaxial compressive strength of the very coarse-, coarse-, medium-, and fine-grained samples were 8.4%, 7.7%, 7.0%, and 4.6%, respectively.

3 Numerical reconstruction of the sandstone samples

3.1 Reconstruction of the numerical model

In this study, the rock properties of sandstone samples were numerically simulated using a 2D particle flow code. The numerical model established using this code is an aggregation of multiple particles that is suitable for simulating rock materials composed of mineral grains. The simulation is based on the motion and interaction of the particles, and the slippage and rotation of the particles result in the deformation of the entire model. From a mesoscopic perspective, sandstones are cemented by various mineral grains. When the sandstones were damaged, the cemented substances between the mineral grains were damaged, and some mineral grains may have incurred damage. To study the meso-failure form of sandstone, a grain-based model [32] was used to reconstruct the multi-mineral mesoscopic numerical model of the sandstone samples, as shown in Fig.7.

The multi-mineral mesoscopic numerical model of the sandstone samples was reconstructed using the following steps.

1) An initial model filled with disks was generated, and the disk size distribution of the initial model was consistent with that of the sandstone samples.

2) Voronoi cells were constructed from the disks such that each Voronoi cell contained a disk. The size of each Voronoi cell closely followed that of the disk.

3) All disks in the model were deleted, and the model was then filled with smaller disks. The disks in the same Voronoi cell were bonded to a mineral grain. The mineral grains changed slightly in shape and position during the equilibration of the final model.

From a mesoscopic perspective, when the fine-grained samples were damaged, the grains were intact. The diameters of most grains in the fine-grained were in the range of 0.125–0.250 mm. The computational efficiency of the numerical model built using the particle flow code is related to the number of particles. If the disks bonded into the mineral grains are too small, the consumption of computing resources and computational time will be high. Therefore, the diameters of the disks bonded to the mineral grains were set in the range of 0.125–0.250 mm.

Fig.8 shows the numerical models of the sandstone samples. The size of the numerical model was 50 mm × 100 mm, and each model contained approximately 140000 disks. The contacts in the numerical models were activated with a contact gap of 1e−5, and each model contained approximately 330000 contacts. Two types of contacts were present between two adjacent disks in the numerical models: 1) both disks were in the same grain and 2) the two disks were in different grains. The mineral type of the ball (disk) was the same as the mineral type of the grain into which it was bonded. When reconstructing the numerical models, they were simplified to consist of quartz, feldspar, and debris grains. The quartz content of the models was consistent with that of the sandstone samples. Additionally, the feldspar content of the models was consistent with the feldspar and calcite contents of the sandstone samples. Finally, the debris content of the models was consistent with the clay and iron contents of the real sandstone samples.

The mineral grains in the numerical models were randomly determined, and a mineral grain may be adjacent to grains of the same or different mineral types. Furthermore, only the balls (disks) were colored differently according to the mineral type, as shown in Fig.8, to avoid assigning different colors to too many mineral grains. When adjacent grains in the numerical models were of the same mineral type, they visually appeared to have fused into one grain; however, they were still separate grains.

3.2 Calibration of meso parameters

In this study, all the contacts between disks in the numerical model were assigned as parallel-bonded contacts. To investigate the mechanical properties of the different mineral grains in the sandstone samples, the mechanical properties of the quartz grains were set to be higher than those of the feldspar grains, and the mechanical properties of the feldspar grains were set to be higher than those of the debris grains. The meso-parameters of the numerical models were calibrated according to the calibration process reported in Ref. [33], and the calibration results are shown in Tab.2 and Tab.3. The moment contribution factor (Pb_mcf) of all contacts was set to 0.7. Tab.2 shows the meso-parameter values obtained from the contact between two adjacent disks when the two disks were in the same grain. Tab.3 shows the calculation method for obtaining the meso-parameter values of the contact between two adjacent disks when the two disks are in different grains.

3.3 Comparison between the test and simulation results

Fig.9 shows a comparison between the simulation results of the biaxial compression test and the test results of the triaxial compression test. The stress–strain curves of the numerical models were found to be consistent with those of the sandstone samples. The peak stress values of the very coarse-, coarse-, medium-, and fine-grained sandstone models were 150.75, 170.94, 233.16, and 132.26 MPa, respectively. In addition, the elastic moduli of the very coarse-, coarse-, medium-, and fine-grained sandstone models were 26.64, 28.65, 31.65, and 16.57 GPa, respectively. For each sandstone sample, the error between the peak stress of the numerical model and the average peak stress of the sandstone samples, as well as the error between the elastic modulus of the numerical model and the average elastic modulus of the sandstone samples, was less than 5%.

Fig.10 shows the simulation results of the Brazilian splitting tests. The Brazilian splitting strengths of the very coarse-, coarse-, medium-, and fine-grained sandstone models were 12.33, 12.92, 16.71, and 6.13 MPa, respectively. For each sample, the error between the Brazilian splitting strength of the numerical model and the average Brazilian splitting strength of the sandstone samples was less than 5%.

Fig.11 shows the typical failure modes obtained from the numerical models in the biaxial compression and Brazilian splitting tests. In the biaxial compression test, the sandstone model underwent shear failure, and a single oblique crack was formed. In the Brazilian splitting test, the sandstone model was split in half.

The simulation results obtained using the numerical models were in good agreement with the test results obtained using the sandstone samples. Therefore, the numerical models of different grain-sized sandstones were considered to be accurately reconstructed.

4 Numerical simulations of sandstone crushed by a tooth hob

The motion of a hob when the rock was being crushed by the insertion of a rolling tooth hob was analyzed and simplified in this study. Subsequently, based on the size of the tooth hob used in the Nansha Bridge project, a simplified hob was built into the numerical models. The mechanical interaction between the sandstone and the hob was explored by simulating the process of crushing the sandstone using the tooth of a rolling hob. Furthermore, the energy consumption of the rock-crushing process was studied.

4.1 Numerical model

Fig.12(a) shows the dimensions of the tooth hob used in the Nansha Bridge project. The length of the tooth mentioned in this paper refers to the length of the tooth exposed outside the hob without considering the length of the part embedded in the hob. Fig.12(b) shows a schematic of the rock-crushing process by the tooth hob. The teeth continuously crush the rock when the tooth hob rolls. Rock crushing using a tooth hob can be regarded as a repetitive process. The process of crushing rock using a single tooth was used as an example to explore the mechanical interaction between the rock and the hob. As shown in Fig.12, the hob is not displaced in the longitudinal direction when the hob rolls. From the point at which the tooth tip starts to invade the rock to the point at which the tooth penetrates the rock at its maximum depth, the hob rolls at an angle of α. Subsequently, from the point at which the tooth penetrates the rock at its maximum depth to the point at which the tooth tip begins to leave the rock, the hob also rolls at an angle of α. When the maximum penetration depth of the tooth into the rock was 15 mm, the entire tooth was considered to have penetrated the rock. The rolling angle α of the hob increased with the increase in the maximum penetration depth of the tooth into the rock and can be expressed as follows:

(1)\alpha =\frac{{180}^{\circ }}{\text{π}}\times \arccos \frac{R+L-h}{R+L},

where R denotes the radius of the tooth hob, L denotes the tooth length, and h denotes the maximum penetration depth of the tooth into the rock.

Numerical models with a width of 200 mm and a height of 80 mm were constructed to simulate the process of crushing sandstone using the tooth of a rolling hob, as shown in Fig.13. The numerical model contained approximately 445000 balls (disks) and 1060000 contacts. The displacements of the left, right, and bottom boundaries of the numerical model were fixed. The hob rolled at a speed of 5 m/s. The contacts between the sandstone and hob models were assigned as linear contacts. The effective modulus (emod) of the hob model was set to be 500 GPa, and the friction coefficient (fric) between the hob and sandstone was set to be the same as that between the disks in the sandstone model.

4.2 Simulation results

To explore the mechanical interaction between the rock and hob, the rock-crushing process by the tooth of a rolling hob was simulated when the maximum penetration depth of the tooth in the sandstone was 3, 5, 10, and 15 mm. In the numerical simulations, the medium- and fine-grained sandstone models required the most and least computational time, respectively. The coarse- and very coarse-grained sandstone models consumed almost the same amount of computational time. The above results show that the calculation time is positively correlated with the current stiffnesses and masses of all the objects in the numerical model.

4.2.1 Rock-crushing process

When the maximum penetration depth of the tooth in the sandstone was 10 mm, the simulation results of the coarse-grained sandstone model crushed by the tooth of a rolling hob were used as an example to illustrate the rock-crushing process, as shown in Fig.14. The rolling angle α of hob was 19.91°. To intuitively observe the changes in the area of crushed sandstone, balls (disks) with displacements greater than 5e−4 mm in the sandstone model were considered to be crushed. When the tooth tip started to invade the sandstone model, the rolling angle of the hob was zero, and tension cracks were present under the tooth tip. As the hob rolled from 0 to 0.5α, the cracks increased significantly, extending from where the tooth and sandstone were in contact. Moreover, the sandstone around the tooth was crushed, and the area of crushed sandstone on the front of the tooth was larger than that on the back of the tooth. When the tooth arrived at the position in which it penetrated the sandstone with maximum depth, the rolling angle of the hob was 1.0α. As the hob rolled from 0.5α to 1.0α, the lengths of the transverse cracks on the left side of the tooth increased, resulting in an increase in the area of crushed sandstone on the back of the tooth. Meanwhile, the rest of the cracks remained unchanged, and the area of crushed sandstone on the front of the tooth and under the tooth showed little change. As the hob rolled from 1.0α to 1.5α, the cracks slightly increased, and the area of the crushed sandstone on the back of the tooth increased. When the tooth tip started to leave the sandstone model, the rolling angle of the hob was 2.0α. As the hob rolled from 1.5α to 2.0α, almost no changes were observed in the sandstone model.

The above simulation results show that in the process of tooth entering the sandstone, the sandstone under and on either side of the tooth were crushed owing to downward compression and by being pushed to either side, respectively. Subsequently, in the process of tooth leaving the sandstone, only the sandstone on the back of the tooth was crushed by being pushed back.

4.2.2 Mechanical interaction between sandstone and hob

When the rock is crushed by the tooth of a rolling hob, the tooth is subjected to reaction forces from the rock. To maintain the hob rolling at a constant speed, the corresponding forces and moment should be applied to the hob, as shown in Fig.15. The applied forces are equal in value and opposite in direction to the reaction forces of the rock on the tooth. In 2D space, the reaction forces and applied forces can be decomposed into lateral and longitudinal forces. When the hob rolled from 0 to 2.0α, the lateral force, longitudinal force, and moments applied to the hob were recorded to explore the mechanical interaction between the sandstone and hob. When the rolling angle of the hob was zero, the tooth tip started to invade the sandstone. When the rolling angle of the hob was 1.0α, the tooth arrived at the position in which it penetrated the sandstone at its maximum depth. When the rolling angle of the hob was 2.0α, the tooth tip started to leave the sandstone.

Fig.16 illustrates the variation pattern of the lateral force applied to the hob during rolling. The value of the applied lateral force to the right was set to be positive, and the negative value of the applied lateral force indicated that the lateral reaction force of the sandstone on the tooth contributed to the rolling of the hob. As shown in Fig.16, as the hob rolled from 0 to 0.5α, the lateral forces fluctuated sharply, and their values could be positive or negative. As the hob rolled from 0.5α to 1.0α, almost all the lateral forces increased to positive values, whereas some of them remained negative, regardless of whether they were originally positive or negative. As the hob rolled from 1.0α to 2.0α, the lateral forces decreased to 0. When the rolling angle of the hob was 1.5α, the lateral forces decreased to zero at the maximum penetration depths of 3 and 5 mm. No obvious relationship between the lateral forces and the sandstone models was found. As the maximum penetration depth of the tooth in sandstone increased, the lateral forces increased. The above results indicate that from the point at which the tooth tip started to invade the sandstone to the point at which the tooth penetrated the sandstone at its maximum depth, the lateral reaction force of the sandstone on the tooth may help or hinder the rolling of the hob. From the point at which the tooth penetrated the sandstone at its maximum depth to the point at which the tooth tip started to leave the sandstone, the lateral reaction force of the sandstone on the tooth hindered the rolling of the hob.

Fig.17 illustrates the variation pattern of the longitudinal force applied to the hob during the rolling of the hob. The value of the applied longitudinal force in the downward direction was set to be positive. As the hob rolled from 0 to 1.0α, the tooth was displaced downward in the longitudinal direction, and the positive value of the applied longitudinal force indicated that the longitudinal reaction force of the sandstone on the tooth hindered the rolling of the hob. As the hob rolled from 1.0α to 2.0α, the tooth was displaced upward in the longitudinal direction, and the positive value of the applied longitudinal force indicated that the longitudinal reaction force of the sandstone on the tooth contributed to the rolling of the hob. As shown in Fig.17, as the hob rolled from zero to 0.5α, the longitudinal force generally increased. As the hob rolled from 0.5α to 2.0α, the longitudinal force generally decreased. When the rolling angle of the hob was 1.5α, the longitudinal force decreased to zero at the maximum penetration depths of 3, 5, and 10 mm. The longitudinal force curves of the medium-grained sandstone model were significantly higher than those of the other sandstone models. The longitudinal force curves of the very coarse-, coarse-, and fine-grained sandstone models were similar, whereas the average longitudinal force of the fine-grained sandstone model was slightly lower than that of the other two sandstone models, and the average longitudinal force of the coarse-grained sandstone model was slightly higher than that of the other two sandstone models. This indicates a positive correlation between the longitudinal reaction force of sandstone on the tooth and the strength of the sandstone. As the maximum penetration depth of the tooth in sandstone increased, the longitudinal force increased. Moreover, the absolute values of the longitudinal forces were significantly higher than those of the lateral forces when the hob rolled. The above results indicate that from the point at which the tooth tip started to invade the sandstone to the point at which the tooth penetrated the sandstone at its maximum depth, the longitudinal reaction force of the sandstone on the tooth hindered the rolling of the hob. From the point at which the tooth penetrated the sandstone at its maximum depth to the point at which the tooth tip started to leave the sandstone, the longitudinal reaction force of the sandstone on the tooth contributed to the rolling of the hob.

Fig.18 illustrates the variation pattern of the moment applied to the hob during rolling. The value of the moment in the clockwise direction was set to be positive, and the negative value of the moment indicated that the resultant reaction forces of the sandstone on the tooth contributed to the rolling of the hob. As shown in Fig.18, as the hob rolled from 0 to 0.5α, the moments fluctuated sharply, and their values were positive. As the hob rolled from 0.5α to 1.0α, the moment decreased from positive to negative values. As the hob rolled from 1.0α to 2.0α, the moment increased to zero. When the rolling angle of the hob was 1.5α, the moments increased to zero at the maximum penetration depths of 3 and 5 mm. No obvious relationship between the moments and the sandstone models was found. As the maximum penetration depth of the tooth in the sandstone increased, the moment increased. The above results indicate that from the point at which the tooth tip started to invade the sandstone to the point at which the tooth penetrated the sandstone with maximum depth, the resultant reaction forces of the sandstone on the tooth first hindered and then contributed to the rolling of the hob. From the point at which the tooth penetrated the sandstone with maximum depth to the point at which the tooth tip started to leave the sandstone, the resultant reaction forces of the sandstone on the tooth contributed to the rolling of the hob.

The simulation results above show that in the process of the tooth entering and subsequently leaving the sandstone, the resultant reaction forces of the sandstone on the tooth first hindered and then contributed to the rolling of the hob. The lateral and longitudinal reaction forces of the sandstone on the tooth increased with the maximum penetration depth of the tooth in the sandstone. The longitudinal reaction force of the sandstone on the tooth was positively correlated with the strength of the sandstone. Moreover, the absolute value of the longitudinal reaction force was significantly higher than the absolute value of the lateral reaction force.

4.2.3 Work done by the hob to crush sandstone

Fig.19 illustrates the three typical work curves with the changes in the rolling angle of the hob when the sandstone model was crushed by the tooth of a rolling hob. In all three curves, the total work done for crushing sandstone initially increased and then stabilized when the hob rolled from 0 to 1.0α, and the total work was essentially unchanged when the hob rolled from 1.0α to 2.0α. Because no displacement of the hob was observed in the longitudinal direction when the hob rolled, the work done by the longitudinal force applied to the hob was zero. As shown in Fig.19(a), the lateral force contributed the most to the total work. In Fig.19(b), both the lateral force and the moment contributed to the total work, and the work done by the moment was slightly higher than the work done by the lateral force. As shown in Fig.19(c), the moment contributed the most to the total work. As the hob rolled from zero to 0.5α, the work done by the moment increased, whereas that done by the lateral force may have increased or decreased. As the hob rolled from 0.5α to 1.0α, the work done by both the lateral force and moment may increase or decrease. As the hob rolled from 1.0α to 1.5α, the work done by the lateral force increased at first and then stabilized, whereas the work done by the moment decreased initially before stabilizing. As the hob rolled from 1.5α to 2.0α, the work done by both the lateral force and moment remained unchanged. The above results indicate that in the process of the tooth invading and then leaving the sandstone, the total work done for crushing the sandstone initially increased and then stabilized in the middle of the process. The longitudinal force applied to the hob does not work. The lateral force applied to the hob might perform positive or negative work before definitely doing positive work, and finally no work. The moment applied to the hob initially did positive work, then negative work, and finally no work. Although the sum of the work done by the lateral force and moment was equal to the total work, the contribution of the lateral force and moment to the total work varied.

Fig.20 illustrates the work curves with the maximum penetration depth of the tooth in sandstone. As the maximum penetration depth increased, the total work done for crushing sandstone increased. The work curve of the medium-grained sandstone model was higher than those of the other sandstone models, and the work curve of the fine-grained sandstone model was lower than those of the other sandstone models. The work curve of the coarse-grained sandstone model was slightly higher than that of the coarse-grained sandstone model. The above results indicate that when sandstone was crushed by the tooth of a rolling hob, the work done for crushing sandstone was positively correlated with the strength of the sandstone.

5 Conclusions

In this study, multi-mineral mesoscopic numerical models of sandstones with different grain sizes were reconstructed using a 2D particle flow code based on the results of laboratory experiments. The process of crushing sandstone using the tooth of a rolling hob was simulated to explore the mechanical interaction between the rock and the hob. The main conclusions drawn are as follows.

1) After obtaining the grain size distribution, mineral composition, and mechanical properties of sandstone samples through laboratory experiments, a grain-based model was used to reconstruct the multi-mineral mesoscopic numerical models of the sandstone samples of different grain sizes. The simulation results show that the mechanical properties of the sandstone models are in good agreement with the mechanical test results obtained using the sandstone samples.

2) The process of crushing rock by inserting a tooth hob can be regarded as a repetitive process. When the rock was crushed by the tooth of a rolling hob, the tooth initially entered the sandstone and then exited. Initially, the sandstone around the tooth was crushed, and subsequently, the sandstone on the back of the tooth was crushed.

3) In the sandstone crushing process, the resultant reaction forces of sandstone on the tooth initially hindered and then contributed to the rolling of the hob. The lateral and longitudinal reaction forces of sandstone on the tooth increased with the maximum penetration depth of the tooth in sandstone. A positive relationship existed between the longitudinal reaction force of the sandstone on the tooth and the strength of the sandstone. Moreover, the absolute value of the longitudinal reaction force was significantly higher than that of the lateral reaction force.

4) Three typical curves of the work done by the forces and moments applied to the hob were obtained. In the sandstone crushing process, the total work first increased and then stabilized in the middle of the process. Before the forces and moments applied to the hob returned to zero, the longitudinal force did not perform any work, the lateral force may perform positive or negative work before definitely performing positive work, and the moment initially performed positive work and then negative work. Although the sum of the work done by the lateral force and moment was equal to the total work, the contribution of the lateral force and moment to the total work varied. As the maximum penetration depth increased, the total work done for crushing sandstone increased. Moreover, the total work was positively correlated with the strength of the sandstone.

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