To predict fragment separation during rock cutting, previous studies on rock cutting interactions using simulation approaches, experimental tests, and theoretical methods were considered in detail. This study used the numerical code LSDYNA (3D) to numerically simulate fragment separation. In the simulations, a damage material model and erosion criteria were used for the base rock, and the conical pick was designated a rigid material. The conical pick moved at varying linear speeds to cut the fixed base rock. For a given linear speed of the conical pick, numerical studies were performed for various cutting depths and mechanical properties of rock. The numerical simulation results demonstrated that the cutting forces and sizes of the separated fragments increased significantly with increasing cutting depth, compressive strength, and elastic modulus of the base rock. A strong linear relationship was observed between the mean peak cutting forces obtained from the numerical, theoretical, and experimental studies with correlation coefficients of 0.698, 0.8111, 0.868, and 0.768. The simulation results also showed an exponential relationship between the specific energy and cutting depth and a linear relationship between the specific energy and compressive strength. Overall, LSDYNA (3D) is effective and reliable for predicting the cutting performance of a conical pick.
Rock cutting is frequently encountered in some industries, for example, coal mining, tunnel excavation, and oil exploitation, and is the major function of roadheaders and drilling machines. For a given rock formation, the variation in rock morphology and cutting forces in the progress of rock cutting is very important for designing cutting tools [
Many scholars have conducted experimental and theoretical research on mechanical performance and rock behavior. Experimental procedures were performed with a linear cutting machine by Biligin et al. [
Generally, experimental research is the most reliable and effective method but is too costly in terms of both time and money [
In this paper, the explicit finite element method (FEM) code LSDYNA (3D), a computational modeling method that is good at simulating impact, blast, and penetration, was employed to model the interaction between a conical pick and rock. A damage material and erosion criteria were used in the code to dominate the rock failure and to delete the elements. The influences of the speed of the conical pick, the depth of the cut, and the mechanical properties of the base rock on the crack extent, fragment formation, and cutting force were examined. The cutting forces were verified by experimental and theoretical studies, and the relationships among specific energy, compressive strength, and cutting depth were analyzed.
The point attack pick theory was first proposed by Evans [
The cutting parameters of point attack cutting.
Evans’ theory provided a reference for later research. Roxborough and Liu [
Goktan [
One of the most accepted methods for predicting the cutting rate uses specific energy, which is the energy expenditure for cutting per rock volume [
In general, three modeling methods are frequently used to investigate the toolrock interaction: FEM, discrete element method (DEM), and finite difference method (FDM) [
Various numerical methods for fragment separation.
Reference  Numerical method  Code  Crack propagation  Fragment separation  Cutting force verified  

Experimental  Theoretical  
[ 
FEM (3D)  LSDYNA  No  No  Yes  Yes 
[ 
LSDYNA  No  No  No  Yes  
[ 
LSDYNA  No  No  No  No  
[ 
ABAQUS  No  No  No  No  


[ 
FEM (2D)  LSDYNA  Yes  Yes  Yes  No 
[ 
LSDYNA  Yes  Yes  No  No  
[ 
LSDYNA  No  Yes  No  No  
[ 
RFPA  Yes  No  No  No  
[ 
RFPA  Yes  Yes  No  No  


[ 
DEM (3D)  EDEM  No  Yes  Yes  No 
[ 
PFC  No  Yes  No  Yes  


[ 
DEM (2D)  PFC  Yes  Yes  No  No 


[ 
FDM (2D)  FLAC  Yes  No  Yes  No 
[ 
FLAC  No  No  Yes  No  
[ 
FLAC  Yes  No  No  No  
[ 
FLAC  Yes  No  No  No 
FEM is a very practical numerical procedure for simulating engineering physics problems. Both 2D and 3D FEM are widely used to solve 2D and 3D dimensional problems. In the field of rock mechanics, various numerical simulation codes, such as LSDYNA [
Results simulated in the references.
Rock cutting using FEM (3D) [
Rock cutting using FEM (2D) [
Rock cutting using PFC (3D) [
Rock cutting using PFC (2D) [
DEM is a numerical method for simulating a discontinuous medium, and it is widely used in slope stability, tunnel excavation, and rock dynamic behavior simulations. The representative codes of DEM are EDEM and PFC, and the method includes both 2D and 3D modes. Dai et al. [
FLAC is a fast Lagrangian analysis program based on FDM and is an international geotechnical engineering analysis software. Stavropoulou [
All the simulations in this study were carried out using the code LSDYNA (3D). In the simulation, a conical pick and a cuboid rock were modeled, as shown in Figure
The rock cutting model.
The conical pick had an impact angle
Mechanical properties of the base rock [
Properties  Compressive strength 
Tensile strength 
Shear strength 
Elastic modulus 
Poisson’s ratio  Density 

Sandstone1  173.7  11.6  22.4  28  0.29  2670 
Sandstone2  113.6  6.6  13.7  17.0  0.20  2650 
Sandstone3  87.4  8.3  14.0  33.3  0.25  2670 
Limestone  121.0  7.8  15.4  57  0.20  2720 
The numerical simulation results of the chipping progress between the conical pick and the rock for a cutting depth of 9 mm, a speed 3 m/s, and a base rock of sandstone1 are shown in Figure
The cutting process of the rock for a cutting depth of 9 mm, a cutter speed of 3 m/s, and sandstone1: (a) initial crack generation at 0.01318 s, (b) crack propagation at 0.1344 s, (c) crack connection at 0.01370 s, (d) fragment separation from the base rock at 0.1472 s, (e) cutting result at 0.02667 s, and (f) variation in the cutting force.
The variation in cutting force with distance in this numerical simulation is shown in Figure
To study the base rock failure mode, two test points at different positions were selected to analyze the variation in the stress, as shown in Figures
The failure mode of the base rock: (a) the variation in the stress at test point 1 and (b) the variation in the stress at test point 2.
The cutting results for the simulation that were carried out at a cutting depth of 9 mm and cutter speeds of 1 m/s and 2 m/s with sandstone1 are shown in Figure
The cutting results with a cutting depth of 9 mm for sandstone1: (a) the cutting results with a cutting speed 1 m/s and (b) the cutting results with a cutting speed of 2 m/s.
The cutting process and the variation in the cutting forces were simulated at a cutting depth of 3 mm and a cutting speed of 3 m/s with sandstone1, as shown in Figures
The cutting results with a cutter speed of 3 m/s for sandstone1: (a) cutting results with a cutting depth of 3 mm, (b) the variation in the cutting force with a cutting depth of 3 mm, (c) the cutting results at a cutting depth of 4 mm, and (d) the cutting results at a cutting depth of 5 mm.
To obtain the cutting depths at which the rock transitions from ductile to brittle regimes, simulations were carried out at cutting depths of 4 mm and 5 mm and at a cutting speed of 3 m/s for the four types of rock material. The cutting results at a cutting depth of 4 mm of sandstone1 as shown in Figure
The variation in the forces is influenced by the cutting depth, as shown by comparing Figures
Analyzing the shape of the cutting groove shows that compared with the other three rock materials shown in Figures
The cutting results for a cutting depth of 6 mm and a cutter speed of 3 m/s: (a), (c), (e), and (g) are the cutting results for sandstone1, sandstone2, sandstone3, and limestone, respectively, and (b), (d), (f), and (h) are the variation in the cutting force with distance for sandstone1, sandstone2, sandstone3, and limestone, respectively.
In addition, the peak cutting force decreases with decreasing elastic modulus of the rock, as shown by comparing the results in Figure
The simulation results of the cutting forces at different cutting depths, speeds, and rock properties are given in Table
The results of the cutting forces obtained from numerical studies.
Rock name  Depth of cut = 3 mm  Depth of cut = 6 mm  Depth of cut = 9 mm  Depth of cut = 12 mm  

Speed of cutter (m/s)  Speed of cutter (m/s)  Speed of cutter (m/s)  Speed of cutter (m/s)  
1  2  3  5  10  1  2  3  5  10  1  2  3  5  10  1  2  3  5  10  
Mean peak cutting force (kN)  Mean peak cutting force (kN)  Mean peak cutting force (kN)  Mean peak cutting force (kN)  
Sandstone1  3.4  3.6  3.0  3.2  3.6  7.2  7.7  7.3  8.0  9.2  11.4  10.7  11.8  13.9  15.4  14.7  13.2  13.1  15.9  18.4 
Sandstone2  2.4  2.7  2.8  2.2  2.6  4.7  4.3  4.3  4.7  5.8  7.3  7.3  7.9  9.0  11.2  10.4  10.1  10.4  12.1  14.4 
Sandstone3  2.8  2.7  2.4  2.6  2.9  5.2  5.9  5.6  6.7  7.3  7.6  7.6  7.7  8.2  10.9  11.8  11.0  12.6  13.7  15.1 
Limestone  3.1  3.5  2.9  2.9  3.2  7.1  7.7  7.3  8.1  9.2  10.9  10.8  9.6  12.8  14.6  13.4  14.7  13.9  15.7  17.4 
The variation in the mean peak cutting force with the speed of the conical pick for deferent rock types: (a) sandstone1, (b) sandstone2, (c) sandstone3, and (d) limestone.
The mean peak cutting forces vary nonsignificantly with all the simulated speeds for a cutting depth of 3 mm. However, the mean peak forces did not vary significantly with cutting speed up to 3 m/s at cutting depths of 6 mm, 9 mm, and 12 mm. The influence of the conical pick speed on the mean peak cutting force was significant only at higher cutting speeds, particularly 10 m/s, and at cutting depths of 6 mm, 9 mm, and 12 mm. Specifically, the influence of conical pick speed on the mean peak cutting force is significant only at higher cutting depths and higher cutting speeds. This outcome can be attributed to the strain rate, which is the strain variation over time. The dynamic strength of the rock had a significant dependence on the strain rate. The change in the rock strength is not obvious at a lower strain rate, but when the strain rate increased to a certain value, the strength of the rock increased significantly [
The relationships between the mean peak cutting force and cutting depth for the different rock types of sandstone1, sandstone2, sandstone3, and limestone are shown in Figure
The variation in the mean peak cutting force with the cutting depth for different rock types: (a) sandstone1, (b) sandstone2, (c) sandstone3, and (d) limestone.
On the basis of the rock properties presented in Table
The mean peak cutting forces obtained from experimental and theoretical studies [
Rock name  Depth of cut = 3 mm  Depth of cut = 6 mm  Depth of cut = 9 mm  

Mean peak cutting force (kN)  Mean peak cutting force (kN)  Mean peak cutting force (kN)  











 
Sandstone1  0.6  0.9  1.1  9.2  2.4  3.5  4.4  23.3  5.4  7.9  10.0  48.7 
Sandstone2  0.3  0.5  0.6  9.1  1.2  1.8  2.5  18.2  2.7  4.1  5.7  28.1 
Sandstone3  0.6  0.8  0.8  4.5  2.4  3.0  3.2  9.1  5.5  6.8  7.2  15.9 
Limestone  0.4  0.6  0.8  11.8  1.6  2.3  3.0  21.5  3.5  5.5  6.7  29.4 
The numerical, theoretical, and experimental values were significantly different from the results shown in Tables
The relationship between numerical and theoretical studies and the relationship between numerical and experimental studies: (a) LSDYNA (3D) and Evans’ theory, (b) LSDYNA (3D) and Roxborough’s theory, (c) LSDYNA (3D) and Goktan’s theory, and (d) LSDYNA (3D) and experimental results.
A linear correlation coefficient of 0.698 was obtained from Figure
The relationship between the numerical and theoretical results and the relationship between the numerical and experimental results was verified by linear regression analysis. Accordingly, analysis of variance was carried out with a confidence level of 0.95. The cutting forces obtained from LSDYNA (3D) were taken as the independent variable, while the cutting forces obtained from the theoretical and experimental method were taken as the dependent variable. Since the
Regression results to predict cutter performance.
Variables  Regression equation  Correlation coefficient 



EvansDYNA (3D) 

0.69826  26.45539 

Roxborough DYNA (3D) 

0.81118  48.25786 

Goktan DYNA (3D) 

0.86833  73.54087 

LabDYNA (3D) 

0.76819  37.45178 

The linear correlation coefficient between the numerical study and Evans’ theory is lower than the coefficient from the theories of the other two researchers. The result occurs because the mechanical properties of compressive strength and tensile strength mainly lead to rock failure in Evans’ cutting theory; in contrast, the friction angle was taken into account in Roxborough’s and Goktan’s cutting theories, and only tensile strength determined rock failure. Moreover, the highest linear correlation coefficient was obtained between the numerical study and Goktan’s theory, in agreement with the studies performed by Biligin et al. [
The slopes of the fitting equation between the numerical and theoretical results are 0.513, 0.754, and 0.938. The regression slopes suggest that the cutting forces obtained with the numerical simulation are always greater than the values calculated by theoretical models, which is consistent with experimental results. The intercepts of 0.907, 1.452, and 1.88 were acceptable when compared with the numerical results. The numerical results underestimated the experimental results by a factor of 3.5 due to the instability of the experimental cutting forces. The cutting forces were significantly influenced by the joints, bedding planes, discontinuities, and hardness inclusion of rock specimens under the experimental conditions, but these factors were not considered in the numerical simulation. This assumption caused most of the difference between the cutting force obtained from numerical results and that obtained from experimental results [
The estimation of the specific energy is important for predicting the cutting efficiency, as explained in (
The results of the mean cutting forces and element separation at a cutting speed of 1 m/s.
Rock name  Mean cutting force (kN)  Number of separate elements  Specific energy (kWh/m^{3})  

Depth of cut (mm)  Depth of cut (mm)  Depth of cut (mm)  
3  6  9  12  3  6  9  12  3  6  9  12  
Sandstone1  1.2  2.3  3.8  5.6  5,002  11,616  20,697  32,908  5.33  4.4  4.08  3.78 
Sandstone2  1.0  1.8  2.5  3.6  4,831  10,547  15,973  24,342  4.60  3.79  3.48  3.29 
Sandstone3  0.9  1.6  2.6  3.8  4,578  10,589  17,676  27,963  4.37  3.36  3.27  3.02 
Limestone  1.2  2.1  3.6  5.2  5,431  11,873  21,740  33,849  4.91  3.93  3.68  3.41 
The relationship between the specific energy and rock mechanical properties: (a) the linear fitting between the uniaxial compressive strength and specific energy and (b) the exponential fitting between the cutting depth and specific energy.
The minimum value of the correlation coefficient is 0.917, indicating strong agreement between the uniaxial compressive strength and specific energy. An exponential relationship exists between the uniaxial compressive strength and specific energy, consistent with the experimental results [
Regression results to predict cutter performance.
Variables  Cutting parameter  Regression equation  Correlation coefficient 



SEUCS  Cutting depth of 3 mm 

0.91699  26.45539 

SEUCS  Cutting depth of 6 mm 

0.94529  48.25786 

SEUCS  Cutting depth of 9 mm 

0.96059  73.54087 

SEUCS  Cutting depth of 12 mm 

0.97069  37.45178 

SECD  Sandstone1 

0.98492  3861.400  0.01 
SECD  Sandstone2 

0.97680  2175.379  0.02 
SECD  Sandstone3 

0.99714  20362.05  0.005 
SECD  Limestone 

0.94301  830.8230  0.02 
SE: specific energy, UCS: uniaxial compressive strength, and CD: cutting depth.
In this research, the explicit finite element code LSDYNA (3D) was used to model the interaction between a conical pick and base rock. In the simulation, damaged material and erosion criteria were combined to investigate the behavior during rock cutting. The crack propagation and fragment formation were studied with varying cutting depth, speed of the conical pick, and mechanical properties of the rock. The main conclusions are as follows:
The morphology and number of fragments obtained in the cutting process varied significantly with varying cutting depth and mechanical properties of the base rock. However, the morphology of the rock varied nonsignificantly at a lower cutting depth, since only a few fragments were generated at the lower cutting depth.
The cutting forces during the cutting process also varied significantly with the cutting depth and rock properties. With decreasing cutting depth, compressive strength, and elastic modulus of the base rock, the frequency of the cutting force increased, but the peak cutting force decreased.
The mean peak cutting forces were verified by theoretical and experimental results. A strong linear correlation coefficient was achieved between the numerical and theoretical results.
The linear correlation coefficient obtained between the numerical and experimental studies was lower than that obtained between the numerical and theoretical studies, since there were many uncertain factors in the experimental process that influenced the cutting force.
The specific energy increased linearly with increasing uniaxial compressive strength and decreased exponentially with increasing cutting depth.
This paper successfully proves that the explicit FEM code LSDYNA (3D) can be used to simulate fragment separation.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by projects of the National Natural Science Foundation of China (Grant no. 51375282), the Natural Science Foundation of Shandong Province (Grant no. ZR2014EEM021), the science and technology development program of Shandong province (Grant no. 2014GGX103043), and the Qingdao postdoctoral researcher applied research project (Grant no. 2016120).