Introduction

Normally, impact drilling is one of the main processes for breaking hard rock1. The force-penetration characteristics and energy transfer characteristics during impact drilling are the important theoretical basis for the structural optimization design and practical application of pneumatic impactor, as well as the important basis for the configuration and the prediction of drilling velocity. Scholars at home and abroad have carried out relevant research on the drilling performance analysis and dynamic simulation of pneumatic impactor2,3. B. Lundberg4 (1973) analyzed the influence factors of impactor energy transfer on impact efficiency (impact energy) in the process of rock breaking by impactor. The transfer of energy carried by stress waves during the impact period was studied, and the relationship between energy transfer efficiency and bit incidence angle was expounded. An example was used to show the linearization approximation effect of impact load nonlinear force-displacement. WANG Jianhua5 et al. (2014) analyzed the correlation between the penetration characteristics of drilling and rock force and obtained the correlation model to study the energy transfer efficiency during impact drilling. However, in the process of drilling, the distribution of viscoelastic force-penetration curve is affected by the damping of drilling velocity, and the corresponding distribution of force-penetration for viscoelastic and elastic rocks is different, which can attribute to the consumption of viscous energy and low efficiency of rocks. WANG Shenglin6 et al. (2022) used Abaqus software to build the numerical simulation model of granite drilling by pneumatic drill hammer. The drilling efficiency and drilling capability under the technological parameters in the process of drilling were analyzed. However, when selecting the drilling technological parameters, only the impact velocity and rotary speed are considered on the rock breaking efficiency of pneumatic drill hammer. Aiming at the fact that traditional PDC bit could not meet the rock breaking requirements of complex rock strata, WU Zebing7 et al. (2020) established the numerical analysis system of drilling based on D-P rock dynamic constitutive model by Abaqus software, and analyzed the characteristics of drilling process. However, in order to simplify the three-dimensional rock-breaking model, only the teeth of the hybrid bit head were retained, and the contact between other head body and rock in the drilling process was not considered, resulting in the numerical analysis error of drilling force. WAN Lirong8 et al. (2017) used ANSYS LS-DYNA research on the characteristics of rock sawing with linear cutter, and obtained the force curve of cutter. BI Yongsheng9 et al. (2022) simulated the geotechnical drilling process with ABAQUS software and analyzed the drilling data of different drilling parameters, so as to reveal the correlation between drilling torque and axial pressure under certain conditions. YANG Zhiying10 (2019) selected sandstone and red brick with hardness close to lunar rock to establish HPBT simulation model of HJC constitutive model of drilling tools by ANSYS LS-DYNA software. Based on this model, the impact model of bit head-drill rod-lunar rock was established, and the trends of crushing efficiency under bit rotary speed, impact velocity and different loads were investigated. XIE Shoudong11 et al. (2022) used the separated Hopkinson pressure bar (SHPB) technology to conduct various drilling tests on granite, cyan red sandstone and other materials, then analyzed the energy consumption trends of different rock breaking methods. The dynamic behavior of rock was studied to obtain the relevant drilling curve where the influence of drilling pressure and rotary speed on drilling velocity was researched. SHI Changshuai12 et al. (2017) studied the drilling process of bit head and rock under random load and impact conditions, and proposed a model to simulate the contact impact between drill rod and formation, which revealed that contact impact is the main factor resulting in premature failure of drill rod. BO Kun13 et al. (2022) studied the mechanical structure and working principle of DTH pneumatic drill whose impact dynamics was numerically simulated by ANSYS software, and the energy transfer characteristics during impact drilling was analyzed. In summary, there is relatively little research on the impact of a certain inclination angle of the rock surface on the distribution of drilling forces. It is of practical significance for the mechanical optimization design of impactor system to establish the drilling dynamics model of pneumatic impactor, conduct numerical analysis on the impact drilling system composed of bit head (including column teeth), impact piston and rock, and study the effects of different bit incidence angle on the impact drilling force and the law of rock breaking, provide basic data for fuzzy control14 of impactor.

Dynamics equation and finite element model of rock drilling

Nonlinear contact dynamics equation of rock drilling by pneumatic impactor

Based on the mechanical properties of drill rod at the contact surface between piston and rock, the boundary conditions for the impact drilling of pneumatic impactor are established1. The process of bit head drilling rock is a nonlinear problem; The geometric nonlinear problem is caused by instantaneous large displacement; The contact problem is aroused by the varying contact surface during the process of bit head drilling rock. According to Darren Bell Principle and finite element method, the contact dynamics equation between rock and bit head is established1,2,6 is:

$$\left\{ \begin{gathered} F_{{\text{B}}}^{{\text{t}}}=M_{{\text{B}}}^{{}} \cdot \mathop {U_{{\text{B}}}^{{\text{t}}}}\limits^{{\cdot \cdot }} +C_{{\text{B}}}^{{}} \cdot \mathop {U_{{\text{B}}}^{{\text{t}}}}\limits^{\cdot } +K_{{\text{B}}}^{{}} \cdot U_{{\text{B}}}^{{\text{t}}}+f_{{\text{B}}}^{{\text{t}}} \hfill \\ F_{{\text{R}}}^{{\text{t}}}=M_{{\text{R}}}^{{}} \cdot \mathop {U_{{\text{R}}}^{{\text{t}}}}\limits^{{\cdot \cdot }} +C_{{\text{R}}}^{{}} \cdot \mathop {U_{{\text{R}}}^{{\text{t}}}}\limits^{\cdot } +K_{{\text{R}}}^{{}} \cdot U_{{\text{R}}}^{{\text{t}}}+f_{{\text{R}}}^{{\text{t}}} \hfill \\ \end{gathered} \right.$$
(1)

Where: \(F_{{\text{B}}}^{{\text{t}}}\)and \(F_{{\text{R}}}^{{\text{t}}}\)are the vector of the contact force on bit and rock, respectively. \(f_{{\text{B}}}^{{\text{t}}}\) and \(f_{{\text{R}}}^{{\text{t}}}\)are the drilling pressure on bit and rock, respectively; \(M_{{\text{B}}}^{{}}\) and \(M_{{\text{R}}}^{{}}\)are the mass matrix of bit head and rock, respectively. \(C_{{\text{B}}}^{{}}\) and \(C_{{\text{R}}}^{{}}\)are the damping matrix of bit head and rock, respectively. \(K_{{\text{B}}}^{{}}\) and \(K_{{\text{R}}}^{{}}\)are the stiffness matrix of bit head and rock, respectively. \(\mathop {U_{{\text{B}}}^{{\text{t}}}}\limits^{{\cdot \cdot }}\), \(\mathop {U_{{\text{B}}}^{{\text{t}}}}\limits^{\cdot }\) and \(\mathop {U_{{\text{B}}}^{{\text{t}}}}\limits^{{}}\)are the acceleration, velocity and displacement of bit head; \(\mathop {U_{{\text{R}}}^{{\text{t}}}}\limits^{{\cdot \cdot }}\), \(\mathop {U_{{\text{R}}}^{{\text{t}}}}\limits^{\cdot }\) and \(\mathop {U_{{\text{R}}}^{{\text{t}}}}\limits^{{}}\)are the acceleration, velocity and displacement of rock.

Three-dimensional numerical analysis model of pneumatic impactor

Taking Type 80 pneumatic impactor as an example, its mechanical structure is simplified according to the model composition required for the analysis of problem. Then, the parameters like incidence angle are defined. Finally, it is determined that the three-dimensional numerical analysis model of pneumatic impactor only includes three parts: piston, bit head and rock, as shown in Fig. 1.

Fig. 1
Fig. 1
Full size image

Schematic diagram of bit head (column tooth) -impact piston-rock (with initial hole) system of Type 80 pneumatic impactor. 1-bit head; 2-column teeth H5 (six-φ14 mm*22 mm, evenly distributed); 3-column teeth H1 ~ H4 (four-φ13*19); α-drill bit teeth distribution angle; γ-transition angle of rock initial hole modeling (21.48°); Dhole-the top diameter of rock initial hole; β-deviation angle between rock surface and drilling center (in ZOY plane); δ-deviation angle between rock surface and drilling center (in ZOX plane); β and δ are collectively referred to drilling incidence angle.

The 3D structure of simplified model is modeled by AUTODESK INVENTOR 2023 3D design software, and then imported to ANSYS 19.2 LS-DYNA for grid division. The piston contains 32,960 units, the bit head contains 213,123 units, and the rock contains 451,328 units, totalling 697,411 units. The numerical model of system obtained is shown in Fig. 2.

Fig. 2
Fig. 2
Full size image

Simplified 3D model of bit head(column tooth)-impact piston-rock(with initial hole) 1-ANSYS 19.2 available from: https://www.ansys.com/zh-cn/products/fluids/ansys-fluent; 2-AUTODESK INVENTOR 2023 avaliable from: https://www.autodesk.com.cn/products/inventor/overview?term=1-YEAR&tab=subscription. (a) Simplified diagram of Mechanical structure. (b) Simplified diagram of 3D finite element.

Table 1 Rock material characteristic parameters of HJC criterion.
Full size table

Three-dimensional numerical analysis model of pneumatic impactor

In order to facilitate analysis and calculation, relevant simplification and assumptions should be made for the finite element model6,7. Since the material stiffness of piston2 and piston is much higher than that of rock, piston and bit head (including column teeth) are considered as continuous uniform materials and isotropy elastic material model. At the same time, the wear of bit head and column teeth are ignored, both simplified as rigid body model. The rock materials15,16 are assumed continuous uniform and isotropic elastoplastic model. The HJC criterion which can simulate large deformation, high strain rate and high-pressure effects are adopted. Besides, the damage model of rock materials is defined by keyword *Mat_Add_Erosion. Failure units of rock will be removed by the program, so as to simulate the release of small rock fragments. The rock characteristic parameters of HJC criteria are shown in Table 1. At the same time, the effect of temperature and the compressed air through hole between bit and piston is ignored. In order to maintain the continuous contact between bit column tooth and rock, as well as overcoming the rebound of bit under impact drilling, a certain drilling axial pressure17 Fr should be applied to bit, and the drilling axial pressure is 9 kN.

Process analysis of bit head impacting rock

The model with the impact velocity vp of 7.37 ms− 1, the drilling axial pressure Fr of 9 kN, the rotary speed nr of 15 rpm, the impact frequency fp of 14 Hz, and the drill bit distribution angle α of 0° are taken for analysis. The stress distribution of rock drilling without and with initial holes is compared.

Drilling process of rock without initial hole

Stress distribution states of rocks at different times were extracted, as shown in Fig. 3. When the simulation running time is 0.11897 ms, the column teeth H2 and H3 in bit head (see Fig. 1) first make initial contact with rock, at which time the maximum VM equivalent stress is 323.8 MPa, as shown in Fig. 3(a). When the simulation running time is 0.4759 ms, the column teeth H2 and H3 in bit head will continuously deepen and expand the drilled holes on rock. Meanwhile, the column teeth numbered H1 and H4 also contact with the rock surface, with the maximum VM equivalent stress of 582.1 MPa, as shown in Fig. 3(b). When the simulation running time is 0.71398ms, the column teeth numbered H1 ~ H4 in bit head will continuously deepen and expand the drilled holes on rock, as shown in Fig. 3(c). When the simulation running time is 1.071 ms, large holes are drilled on rock surface under the continuous impact of column teeth, with the maximum VM equivalent stress of 672.4 MPa, as shown in Fig. 3(d). With continuous contact between bit head and rock surface, the failure area of rock surface will carry on expanding, as shown in Fig. 3(e) and (f). Thereafter, the piston will return to the initial impact state under the action of external forces and move downward again. At the same time, the bit head will do circumferential rotation under the action of rotating mechanism, and the whole impact system will produce continuous impact on rock, drilling out more rock fragments. At this stage, the column teeth H5 which are uniformly distributed on the side of bit head do not contact with rock. The column teeth H1-H4 are the main roles to drill rock surface.

Fig. 3
Fig. 3
Full size image

Stress nephogram of the impact process between rock and bit head.

Drilling process of rock with initial holes

In addition to the above modelling parameters and initial solving conditions, a model with β and δ of 0°, γ of 21.48°, and Dhole of φ95 mm was taken for analysis, and the stress distribution diagrams of rock under different time were extracted. When the simulation running time is 1.5 ms, the maximum VM equivalent stress is 386.32 MPa, and the column teeth H1 and H4 in bit head first make initial contact with rock, as shown in Fig. 4(a). When the simulation running time is 1.63 ms, the maximum drilling force is 133.970 kN. When the simulation running time is 1.8 ms, the maximum VM equivalent stress is 589.77 MPa, all the column teeth are in contact with the rock, and the rock unit has been damaged, that is, the bit head has broken the rock, as shown in Fig. 4(b). With the continuous contact between bit head and rock surface, the rock surface will continue to be impacted and circumferentially sheared by column teeth, the damage area will continue to expand, and the rock will keep being broken, as shown in Fig. 4(c) and (d), until the arrival of the next impact cycle, the continuous rock drilling will eventually form rock holes.

Fig. 4
Fig. 4
Full size image

Stress nephogram of the contact process between rock and bit.

Analysis and discussion of the numerical results of pneumatic impactor

Maximum impact force of rock without initial hole under different incidence angles

The impact models with flat rock surfaces (i.e., without initial holes) in which only incidence angle δ is considered, only incidence angle β is considered and both β and δ are considered (β and δ are the same value), are established and solved, respectively. Maximum drilling force under different incidence angles is shown in Fig. 5.

Fig. 5
Fig. 5
Full size image

Maximum impact force of rock without initial hole under different incidence angles.

When incidence Angle is -1°~1°, the maximum impact force Ft1max between drill bit head and rock of the above three models are distributed within the range of 64.777 kN ~ 201.691 kN, 69.117 kN ~ 218.459 kN and 53.226 kN ~ 206.167 kN, respectively. The corresponding ratio of maximum to minimum impact force is 3.114, 3.161 and 3.873, respectively.

The nonlinear function fitting of the maximum impact force is conducted by Origin software. Then, the maximum impact force Ft1max when only δ conditions is considered, is: Ft1max=\({\text{35}}{\text{.539}}+\frac{{{\text{146}}.{\text{15766}}}}{{4 \times {{(\delta +0.01162)}^2}+0.89172}}\)(kN), with relative error of -7.17 %~3.93 %. The maximum drilling force Ft1max when only β conditions is considered, is: Ft1max=\({\text{32}}{\text{.393}}+\frac{{181.23092}}{{4 \times {{(\beta - 0.12518)}^2}+0.99958}}\)(kN), with relative error of 10.19 %~6.67 %. The maximum drilling force Ft1max when both β and δ conditions are considered, is Ft1max=\({\text{44}}{\text{.495}}+\frac{{{\text{58}}.{\text{87513}}}}{{4 \times {{(\beta - 0.1)}^2}+0.36435}}\)(kN) or Ft1max=\({\text{44}}{\text{.495}}+\frac{{{\text{58}}.{\text{87513}}}}{{4 \times {{(\delta - 0.1)}^2}+0.36435}}\)(kN), with relative error of -11.72%~15.58%. When both rock surface and drilling center are offset in ZOY plane, the maximum impact force of the contact surface between the spherical top surface of column teeth H2, H3 and rock surface will be 218.4589 kN, which is 8.31 % higher than 201.691 kN without deviation. It means that greater impact force will be generated on column teeth, thus weakening the strength of column teeth. The root of column tooth is more prone to fatigue fracture.

Maximum impact force of rock with initial hole when both β and δ are considered (β and δ are the same value)

The top diameter of rock initial hole Dhole is 95 mm, the transition angle γ is 21.48°, both β and δ are considered, the model is analyzed, as shown in Fig. 6.

Fig. 6
Fig. 6
Full size image

Maximum impact force of rock with initial hole when both β and δ are considered (β and δ are the same value).

When the drilling incidence angle is between − 1° and 1°, the maximum impact force ranges from 74.017 kN to 133.970 kN and the ratio of maximum to minimum impact force is 1.810. According to the nonlinear function fitting of the maximum impact force, the maximum impact force Ft1max of rock with initial hole when both β and δ are considered (β and δ are the same value).

Ft1max=\({\text{72}}{\text{.001}}+\frac{{{\text{8}}0.0{\text{8755}}}}{{4*{{(\beta - 0.0{\text{3465}})}^2}+0.{\text{14696}}}}\)(kN) or

Ft1max=\({\text{72}}{\text{0.01}}+\frac{{{\text{8}}0.0{\text{8755}}}}{{4*{{(\gamma - 0.0{\text{3465}})}^2}+0.{\text{14696}}}}\)(kN). The maximum and minimum impact force after fitting is 125.290 kN and 73.827 kN, respectively, and the relative error of fit curve is -7.89 %~7.51 %.

Maximum impact force of rock with initial hole when both β and δ are considered (β and δ are different value)

The top diameter of rock initial hole Dhole is 95 mm, the transition angle γ is 21.48°, both β and δ are considered (β and δ are different value), the model is solved. The maximum force between piston and bit, as well as the maximum force between bit and rock is taken for analysis, as shown in Fig. 7.

Fig. 7
Fig. 7
Full size image

Maximum impact force of rock with initial hole when both β and δ are considered (β and δ are different value).

It can be seen that the maximum drilling force distribution curve under different β and δ is different. When γ are − 1°, -0.5°, 0°, 0.5°, and 1°, the maximum drilling force Ft1max is 87.23 kN when β is 0.75°, 93.921 kN when β is -0.25°, 133.970 kN when β is 0°, 124.325 kN when β is 0.1°, and 94.779 kN when β is -0.1°. When β and γ are − 0.25° and − 1°, respectively, the impact force has the minimum, which is 64.093 kN. The ratio of maximum to minimum impact force is 2.090. It can be seen that the distribution of bit column teeth will directly affect the drilling effect when the drilling incidence angle is not zero.

Limitations of the study and future research scope

The current research models consider the distribution of maximum and minimum drilling forces when considering the same values for the drilling angles of incidence β and δ, but do not account for the distribution of maximum and minimum drilling forces when considering different values for the drilling angles of incidence β and δ, which presents a limitation in the research. Moving forward, in addition to further investigating the aforementioned issues, the study will also focus on solving models under different strength parameters, damage constants, and pressure constants within the HJC criterion’s constitutive model. This will allow for a more comprehensive analysis of the impact of various factors on drilling forces.

Conclusions

  • The piston-bit head-rock impact system model of pneumatic impactor is established, the failure characteristics of rock materials are analyzed and solved numerically.

  • The stress breaking process of rock without and with initial holes are analyzed, so as to obtain the stress distribution of rock.

  • When modeling the rock without initial hole, the effects of only considering δ condition, only considering β conditions and both considering β and δ on the distribution of maximum impact force are enhanced successively, indicating that the maximum drilling force is greatly affected by the condition of considering two incidence angles simultaneously. The relative errors of nonlinear fitting are − 7.17 %~3.93 %, -10.19 %~6.67 % and − 11.72 %~15.58 %, respectively. In the case of rock with initial holes, considering the incidence angles of both directions, the distribution of drilling force is greatly affected, and the incidence angles of maximum drilling force are also inconsistent. Meanwhile, considering the same value of drilling incidence angle, the relative error distribution of fitting curve of maximum drilling force is within − 7.89 %~7.51 %.

  • The research results can guide the mechanical design and optimization of pneumatic impactor, and also restrict the angle of incidence of the drill bit during the drilling process, thereby avoiding the reduction in equipment service life caused by excessive changes in drilling forces.