Fracture mechanism of air percussive rotary bit matrix based on impact stress wave theory

2.3.1. Simplification and hypothesis for piston and bit

As shown in Fig. 2(a), the drill bit is consisting of airhole, spline, matrix, nozzle and so on. The piston- drill rock model can be expressed as Fig. 2(c). To facilitate the theoretical analysis, the piston and bit are simplified (Fig. 3). $L_1$ is the total length of the piston, mm; $L_{1i}$ is the length of each section of the piston, mm; $A_{1i}$ is the cross-sectional area of each section of the piston, mm; $L_2$ is the total length of the drill, mm; $L_{2i}$ is the length of each section of the bit, mm; and $A_{2i}$ is the cross-sectional area of each section of bit, mm². The detailed simplification and hypothesis are as follows:

(1) The geometric features of spline, nozzle, flute, tooth and airhole are ignored (Fig. 1);

(2) The material of the piston and bit are the same and isotropic;

(3) The principal impact stress and main reflection impact stress are considered. The aftermath is not considered.

Fig. 2Pistons-drill-rock three-dimensional solid model: a) bit, b) piston, c) pistons-drill-rock model and its inner structure

a)

b)

c)

Fig. 3Piston and bit

2.3.2. Maximum principal impact stress appears in each section of the bit when an impact stress wave propagates from the bit tail to the head

Since the cross-sections of the piston and drill are uneven, there are reflections and transmission phenomena in the dissemination of impact stress waves produced by the piston impacting the bit. The density, elastic modulus and wave speed of piston and bit are same since they are of the same material.

Eqs. (14-15) is used to calculate the coefficients of reflection and transmission at each section interface of the piston and bit. The expressions are given in Tables 1 and 2.

The shock waves propagating in the $+x$ and $-x$ directions in the piston are respectively denoted as $F_1$ and $G_1$. Before impact, the speed of the piston is $v_1$ and the stress of the piston is zero. From Eqs. (3-4) and (5-6), it follows that:

where ${\sigma }_{F11}$ is the stress of the stress wave propagating within the piston section $L_{11}$ in the $-x$ direction, MPa; ${\sigma }_{G11}$ is the stress of the stress wave propagating within section $L_{11}$ of the piston in the $+x$ direction, MPa; $F_{11}\left(x-C_{1}t\right)$ is the displacement of the stress wave propagating within the piston section $L_{11}$ in the $-x$ direction at speed $C_1$, m; $G_{11}\left(x+C_{1}t\right)$ is the displacement of the stress wave propagating within the piston section $L_{11}$ in the $+x$ direction at speed $C_1$, $m$; $v_{F11}$ is the speed of the stress wave propagating within the piston section $L_{11}$ in the $-x$ direction, m/s; $v_{G11}$ is the speed of the stress wave propagating within the piston section $L_{11}$ in the $+x$ direction, m/s; and $v_1$ is speed of the stress wave 1, m/s.

The solution can be obtained as:

where ${\sigma }_1$ is stress of the stress wave 1, MPa.

When the piston impacts the bit, the largest stress acting on the bit tail section $L_{21}$ is:

where ${\sigma }_{T21}$ is the stress of a wave transmitted from interface $A_{15}{A}_{21}$ into section $A_{21}$, MPa; $T_{21}$ is the coefficient of transmission from interface $A_{15}{A}_{21}$ into section $A_{21}$(Table 2); $A_{21}$ is the cross-sectional area of $L_{21}$; $A_{15}$ is the cross-sectional area of $L_{15}$; and $A_{22}$ is the cross-sectional area of $L_{22}$.

Since there are section mutations (sudden change between two different section areas) between the section of length $L_{21}$ and section of length $L_{22}$ when the impact stress wave arriving from the bit tail propagates to interface $A_{21}{A}_{22}$, there is a separate propagation of right-traveling impact stress waves in section $L_{22}$ and left-traveling impact stress waves in section $L_{21}$; i.e., there are transmitted waves and reflected waves at interface $A_{21}{A}_{22}$.

According to Eqs. (7-8), the stress ${\sigma }_{r21}$ reflected into $L_{21}$ at interface $A_{21}{A}_{22}$ is:

where $F_{21}$ is the reflection coefficient of the impact stress wave at interface $A_{21}{A}_{22}$ (Table 1). ${\sigma }_{r21}$ is stress of a reflected wave from interface $A_{21}{A}_{22}$ into section $A_{21}$, MPa.

Because $A_{22}<A_{21}$, it follows that $F_{21}<0$, which shows that the reflection wave reflected into the bit tail $L_{21}$ at interface $A_{21}{A}_{22}$ is a tensile wave; i.e., tensile stress will appear in section $L_{21}$ of the bit tail.

According to the principle of wave superposition, the total stress in section $L_{21}$ of the bit tail is:

where ${\sigma }_{total21}$ is the total stress in section $L_{21}$ of the bit tail, MPa.

Therefore, the greatest impact stress ${\sigma }_{max21}$ in section $L_{21}$ is ${\sigma }_{T21}$_.

When the impact stress wave arriving from section $L_{21}$ transmits into section $L_{22}$ at interface $A_{21}{A}_{22}$, the impact stress in section $L_{22}$ is:

where ${\sigma }_{T22}$ is the stress of a wave transmitted from interface $A_{21}{A}_{22}$ into section $A_{22}$, MPa, and $T_{23}$ is the coefficient of transmission from interface $A_{21}{A}_{22}$ into section $A_{22}$.

Similarly, since there are section mutations in the $L_{22}{L}_{23}$ section, when the impact stress wave arriving from section $L_{22}$ propagates at interface $A_{22}{A}_{23}$, there is the transmission and reflection of waves.

According to Eqs. (7-8), the reflecting stress ${\sigma }_{r22}$ entering $L_{22}$ at interface $A_{22}{A}_{23}$ is:

where ${\sigma }_{r22}$ is the stress of a wave reflected from interface $A_{22}{A}_{23}$ into section $A_{22}$, MPa, and $F_{22}$ is the reflection coefficient of the impact stress wave at interface $A_{22}{A}_{23}$ (Table 1).

Since $A_{23}>A_{22}$, in cross-sectional area, it follows that$F_{22}>0$, which shows that the reflection wave entering section $L_{22}$ at interface $A_{22}{A}_{23}$ is a compressional wave.

According to the principle of wave superposition, the maximum stress in section $L_{22}$ of the bit tail is:

where ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}22}$ is the maximum stress in section $L_{22}$ of the bit, MPa; ${\sigma }_{r22}$ is the stress of a wave reflected from interface $A_{22}{A}_{23}$ into section $A_{22}$, MPa.

Similarly, stresses in section $L_{23}$ are:

where ${\sigma }_{T23}$ is the stress of a wave transmitted from interface $A_{22}{A}_{23}$ into section $A_{23}$, MPa; $T_{25}$ is the coefficient of transmission from interface $A_{22}{A}_{23}$ into section $A_{23}$; ${\sigma }_{r23}$ is the stress of a wave reflected from interface $A_{23}{A}_{24}$ into section $A_{23}$, MPa; $F_{23}$ is the coefficient of reflection from interface $A_{23}{A}_{24}$ into section $A_{23}$; ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}23}$ is the maximum stress in section $L_{23}$ of the bit, MPa; and $T_{25}$ is the coefficient of transmission from interface $A_{22}{A}_{23}$ into section $A_{23}$.

Stresses in section $L_{24}$ are:

where ${\sigma }_{T24}$ is the stress of a wave transmitted from interface $A_{23}{A}_{24}$ into section $A_{24}$, MPa; $T_{27}$ is the coefficient of transmission from interface $A_{23}{A}_{24}$ into section $A_{24}$; ${\sigma }_{r24}$ is the stress of a wave reflected from interface $A_3{A}_{24}$ into section $A_{24}$, MPa; ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}24}$ is the maximum stress in section $L_{24}$ of the bit, MPa; and $F_3$ is the coefficient of reflection from the interface between section $A_{24}$ and the tooth into section $A_{24}$.

2.3.3. Main reflection impact stress appears in each section when impact stress wave propagates from bit head to tail

When the main impact stress wave propagates to the bit head $L_{24}$, some of the stress wave is transmitted into a tooth, and the remainder is reflected into $L_{24}$. When the tooth impacts rock, part of the stress wave is reflected in the tooth and transmitted into $L_{24}$. Therefore, the main reflection stress in the bit head $L_{24}$ is:

where ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}r24}$ is the maximum stress of the main reflection stress in bit $L_{24}$, MPa; $T_{32}$ is the coefficient of transmission from interface $A_3{A}_{24}$ into section $L_{24}$; $T_{31}$ is the coefficient of transmission from interface $A_{24}{A}_3$ into the tooth; $F_4$ is the coefficient of reflection from the rock interface into the tooth; ${\sigma }_3$ is the stress of the superposition wave of stress waves 1 and 2, MPa; and ${\sigma }_{r3}$ is the stress of a reflected superposition wave of stress waves 1 and 2, MPa; $T_{32}$ is transmission coefficient from interface $A_3{A}_{24}$ into section $L_{24}$.

Similarly, the main reflection stress in section $L_{23}$ is:

where ${\sigma }_{maxr23}$ is the maximum stress of the main reflection stress in bit $L_{23}$, MPa, and $T_{28}$ is the coefficient of transmission from interface $A_{24}{A}_{23}$ into section $A_{23}$.

The main reflection stress in section $L_{22}$ is:

where ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}r22}$ is the maximum stress of the main reflection stress in bit $L_{22}$, MPa, and $T_{26}$ is the coefficient of transmission from interface $A_{23}{A}_{22}$ into section $A_{22}$.

The main reflection stress in section $L_{21}$ is:

where ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}r21}$ is the maximum stress of the main reflection stress in bit $L_{21}$, MPa. $T_{24}$ is transmission coefficient from interface $A_{22}{A}_{21}$ into section $A_{21}$.

2.4.1. Maximum principal impact stress

From Eqs. (20) and (23), the largest impact stress in section $L_{21}$ is

where the negative sign represents compressive stress.

According to Eqs. (26), the largest impact stress in section $L_{22}$ is:

According to Eqs. (27-29), the largest impact stress in section $L_{23}$ is:

According to Eqs. (30-32), the largest impact stress in section $L_{24}$ is:

In summary, a change in the size of the bit section provides a strong discontinuity surface for stress wave propagation, which affects both the amplitude of the stress wave (the amplitude increases as the cross-sectional area decreases) and the nature of the reflection wave (tensile stress or compressive stress). When the compression stress wave is reflected from a smaller section into a larger section, the reflection coefficient is positive, and the reflection stress wave is a compressive wave. In contrast, when the compression stress wave is reflected from a larger size section into a smaller section, the reflection coefficient is negative, and the reflection stress wave is a tensile wave. When the reflection stress wave is a tensile wave, tensile stress is generated in the bit and the bit cracks. Under the repeated action of tensile stress, fatigue cracks grow and extend, and finally, the drill fractures. Therefore, the situation that the reflection stress wave is a tensile stress which is studied in the following.

2.4.2. Main reflection impact stress

The main reflection impact stress propagating in the air hammer bit may be compressive stress or tensile stress, depending on the reflection coefficient $F_4$. According to the discussion in Section 2.3, if the surface of the bit tooth is a free end surface (i.e., poor chip removal leads to sticking or the bit works in cavity rock formations, resulting in poor contact between the tooth and rock and the rock non-breaking by the bit), then the reflection stress is tensile stress when $F_4$ is –1. When the bit tooth surface is fixed (i.e., the bit works normally), $F_4$ is 1 and the reflection stress is compressive stress. Since tensile stress is the main reason for a bit to crack, later in this work, only the situation is considered that reflection stress is tensile stress.

According to Eq. (33), the main reflection impact stress in section $L_{24}$ is:

According to Eq. (34), the main reflection impact stress in $L_{23}$ is:

According to Eq. (35), the main reflection impact stress in $L_{22}$ is:

According to Eq. (36), the main reflection impact stress in $L_{21}$ is:

Therefore, the total stress in each section of the bit is:

As seen from the above analysis, for a certain bit section, (1) there is only the effect of the main impact stress wave before the reflection wave arrives, and the section is pressed, and (2) the section is under the combined action of the main impact stress wave and main reflection stress wave after the arrival of the reflection wave. When the amplitude of the main impact stress wave is greater than that of the main reflection stress wave, the bit sections (sections $L_{21}$, $L_{22}$ and $L_{23}$) are pressed. In contrast, when the amplitude of the main impact stress wave is less than that of the main reflection stress wave, the bit sections are stretched (i.e., the total stress in section $L_{24}$ is always positive). Under the repeated effects of tensile stress, fatigue failure easily occurs, which is the root cause for the easy fracture failure of the bit matrix.

2.4.3. Discussions

According to the above mentioned theory analysis and calculation, the propagation law and characteristics of the stress wave in the bit are obtained. The stress of the bit under different conditions can be easily calculated. The bit section mutation is the root cause for the shock of the impact wave and the change in the nature of the wave during propagation. Typically, the maximal amplitude of the stress wave appears in the smallest section $A_{22}$, and section $A_{24}$ is always subjected to tensile stress. However, as previously mentioned in Section 2.3.1, the effects of the principal impact stress and the main reflection impact stress are only considered in the propagation of stress wave. The aftermath is not considered. In addition, the key geometrical characteristics, such as air hole, spline, nozzle, flute, and tooth, are not considered.

Therefore, it is necessary to use the finite element simulation to explore the cause of the bit fracture. These stress amplitudes and the propagation law can be used as reference values of the result of finite element simulation.

3.1.1. Fundamental assumption

There are many factors to influence the FE simulation results. For the convenience of analysis and calculation, the factors which have a little effect on the process of FE simulation are ignored. The detailed assumption is shown in the following.

(1) The materials of the piston and bit and rock are isotropic and constant;

(2) The bit and piston are set to elastomer;

(3) The piston and bit are always coaxial during the impact process;

(4) The end face of piston contacts with the bit tail face completely;

(5) Air quality is ignored. The pressure of fluid column caused by drilling medium is not considered;

(7) The effects of temperature on the materials of piston, bit and rock are not considered.

(8) The rock elements are immediately deleted after failure.

3.1.2. Verification for FE model

Further, to verify the correctness of FE simulation setting, the coaxial collision of two elastic bars is analyzed by theoretical calculation and FE simulation. The materials and section areas of two bars are the same.

As shown in Fig. 4(a), from the view of stress wave theory, the stress amplitude of the bar B can be expressed as equation 49 according to the Eqs. (20) and (21) and Table 2. Let’s set the initial velocity of the bar A to 6, 7, 8, 9 m/s, and then, the stress amplitudes of the bar B under different conditions can be obtained by the Eq. (49):

where ${\sigma }_B$ is the stress amplitude in the bar B; ${\rho }_A$ is the density of bar A; $E_A$ is the elasticity modulus of bar A; $v$ is the initial velocity of bar A.

From the view of FE simulation, the above fundamental assumption is adopted. The material parameters are the same with the ones of theoretical analysis. Besides, the bar A and bar B are meshed with hexahedron elements as shown in Fig. 4(b).

Then, given the different initial velocities of bar A, the different wave stress in the rod B can be obtained using the FE simulation. Furthermore, take 8 elements at the different positions of the bar B. The upper and lower stress limits of the 8 elements are obtained, and the stress limits are expressed as the green curves in Fig. 5.

Fig. 4Coaxial collision model of two bars: a) model for theoretical analysis, b) model for FE analysis

Fig. 5Stress amplitude of bar B by FEA and theoretical calculation

The change trend of the stress wave of FEA is consistent with the stress calculated by equation 70 (the red solid curve in Fig. 5). The mean stress (the red dashed curve in Fig. 5) calculated by FEA is very close to the stress calculated by equation 70, and the average error of FEA compared with the theoretical calculation is less than 5.56 % (Table 5).

Therefore, the simulation model is reliable. The parameters setting of the piston-drill-rock model is same with the coaxial collision model of two elastic bars.

3.6.1. Propagation analysis of impact stress wave in percussive rotary bit

In order to better display the propagation process of an impact stress wave, the WOB is not considered in this section. At this point, the drilling pressure is set to 0 KN, and the velocity of piston is set to 6.5 m/s. Fig. 8 shows the propagation of the main impact stress wave in the drill body. A red color indicates maximum tensile stress (positive value). The analysis results are as follows.

(1) In the process of breaking rock, the energy of the piston propagates from the bit tail to the head in the form of a stress wave. When the stress wave reaches the contact surface between a tooth and rock, some of the energy propagates to the rock (wave transmission), and the rest is reflected (wave reflection) back to the bit at the contact surface and travels from the bit head to the tail.

Fig. 8Propagation process of main impact stress wave in drill body

(2) The section mutations (a sudden change between two different section areas) of the bit generate stress wave vibration in the bit body. When an impact stress wave propagates to the bit body, there are many transmissions and reflections at the sectional mutation positions. These results in the stress wave vibration in the bit body and make the bit work under the effect of alternating stress, which is consistent with the theoretical analysis.

(3) The bit matrix bears large tensile stress (of red color), and other parts bear compressive stress, which is consistent with theoretical results. The maximum stress is focused on the bit nozzle interchanges. The stress concentration can more easily induce fatigue fracture under the repeated vibration effect of the stress wave.

3.6.2. Stress analysis of bit matrix

3.6.2.1. Stress analysis of bit matrix for same drill pressure and piston speed

In this section, the drill pressure is set to 10 KN, and the piston speed is set to 7 m/s. The other parameters are set as shown in the Section 3.5. To compare the stress conditions of each section of the bit matrix, an element is considered for each section of the external and internal parts of the bit. Fig. 9 shows the selected element for each section, and Fig. 10 shows the time histories of the stresses of the external and internal elements of the bit body. The following results are obtained from Figs. 9 and 10.

Fig. 9External and internal unit number of bit body: a) external unit, b) internal unit

Fig. 10Stress time history of external and internal unit of bit body: a) external unit, b) internal unit

a)

b)

(1) In Fig. 10(a), the stress peaks earliest for the element corresponding to the bit tail (element $D$), then the elements corresponding to the smaller diameter around the bit tail (element $C$) and the spline (element $B$), at finally the element corresponding to the bit matrix (element $A$). Because the impact stress wave and the maximum stress both propagate from the bit tail to the head, the result is consistent with the propagation characteristic of the impact stress wave in the bit body.

(2) In Fig. 10(a), the peak stress in each element decreases in the order $C>D>B>A$.

According to the theoretical analysis, when a stress wave propagates from a small section to a large section, the amplitude of the transmission stress wave is less than that of the incident stress wave (Fig. 3); i.e., the area ratios $A_{22}>A_{23}$ and $A_{23}>A_{24}$ imply the peak stresses $B<C$ and $A<B$. Therefore, $C>B>A$.

When a stress wave propagates from a large section to a small section, the amplitude of the transmission stress wave is greater than that of the incident wave stress; i.e., $A_{21}>A_{22}$ and $C>D$.

However, energy of the incident stress wave is lost during the propagation; therefore, $D>B$.

In summary, the order of the magnitude of peak stress in the different section elements is $C>D>B>A$, and the simulation result is thus consistent with the theoretical result.

(3) In Fig. 10(a) and 7(b), during the propagation of the impact stress wave, the stress in each element fluctuates frequently, and the fluctuation amplitudes of the different sections are not the same. This is because when the stress wave propagates in the bit body, the bit is under the long-term effect of alternating stress caused by the superposition of transmission and reflection stress waves. The repeated alternating stress will readily lead to fatigue failure of the bit.

(4) In Fig. 10(b), the peak stress and average stress are both largest at stomatal interchanges (Fig. 1(a)) of the bit matrix. A high stress concentration and long-term effect of alternating stress can readily lead to fatigue fracture at stomatal interchanges of the bit matrix. The analysis of bit failures in the field shows that fracturing of the bit body often occurs at the interchanges. Therefore, the simulation result is consistent with the failure of bits used in the field, demonstrating the correctness and rationality of the simulation.

To further analyze the force condition of the bit matrix, elements are taken separately from the external part to the internal part of the bit matrix, as shown in Fig. 11(a). Fig. 11(b) shows the stress time histories for these external and internal elements of the bit matrix. The following results are obtained from Figs. 11(a) and 11(b). The magnitudes of peak stress decrease in order from the internal element to the external element. The peak stress of the internal element (element No. H1573160) is 322 MPa, and that of the external element (element No. H1340693) is about 40 MPa, which shows that there is a high stress concentration at stomatal interchanges. Under the effect of high stress, local stresses will increase suddenly, the formation and extension of cracking will be exacerbated, and the fatigue life of the bit will decrease; thus, bit fracture will ultimately occur. Therefore, in a future design, it is important to improve the structure to decrease the stress concentration and probability of the fracture failure of the bit matrix.

Fig. 11External and internal unit of bit matrix and stress time history: a) external and internal unit of bit matrix, b) stress time history

a)

b)

3.6.2.2. Stress analysis of bit matrix for different drill pressures and same piston speed

In this section, the piston speed is set to 7 m/s, the drill pressure is set as follows. To facilitate the analysis, an element on the bit matrix is taken (Fig. 12(a)). Fig. 13 shows the stress time history of the bit matrix unit under different drilling pressures. Fig. 12(b) shows the relationship between the drilling pressure and peak stress in the element. It is seen that the stress in the drill matrix element increases with increasing drilling pressure. The peak stress is 66 MPa when the drilling pressure is 10 kN, 94 MPa when the drilling pressure is 30 kN, and 138 MPa when the drilling pressure is 50 kN.

Fig. 12Bit matrix unit and relationship between drilling pressure and unit peak stress of bit matrix: a) bit matrix unit, b) relationship between drilling pressure and unit peak stress

a)

b)

Fig. 13Stress time history of bit matrix unit under different drilling pressure: a) 10 KN, b) 30 KN, c) 50 KN

a)

b)

c)

In percussive rotary drilling, the action of drilling pressure ensures that the bit compresses the rock tightly to break the rock and prevent otherwise from a poor contact between the bit and the rock affecting the stress wave propagation under the action of an impact reaction force and reducing the efficiency of rock breaking. It is clear that, when a bit drills through hard rock formations, with an increase in drilling pressure, wear of the bit body and tooth is not only increased but also the matrix stress is increased. Additionally, there is a high stress concentration at stomatal interchanges, and local stresses will thus increase to exacerbate the formation and extension of the crack, leading to bit fracture.

Therefore, in the drilling process, the high breaking efficiency should not be pursued to increase the drilling pressure. The drilling pressure must be low enough to ensure sufficient service life of the bit but high enough to ensure adequate contact between the bit tooth and rock. Therefore, it is important that a proper drilling pressure be chosen according to the specific geological situation to prevent from fracture of the bit matrix.

3.6.2.3. Stress analysis of bit matrix for same drill pressure and different piston speeds

In this section, the drill pressure is set to 10 KN, and the piston speed is set as follows. Fig. 14 shows the stress time history of internal elements of the bit body (as indicated in Fig. 9) for different piston speeds. Fig. 15 shows the relationship between the piston speed and peak stress in an element of the bit matrix. The following results are obtained from the figures. With an increase in piston speed (impact energy), the magnitude of peak stress increases. Table 7 shows that the rate of increase in the bit matrix’s (element A) peak stress is the largest (25, 28, 34 MPa), revealing that an increase in piston speed leads to too much stress in the matrix; the high stress easily leads to fatigue failure of the bit and especially the matrix under the action of the repeated alternating stress.

Table 7Reflection coefficients of piston and bit

Piston speed (m/s)		6.5	7	7.5	8
Element stress (MPa)	A	297	322	350	384
	B	142	155	170	190
	C	219	234	260	287
	D	227	246	269	295
Increase in stress (MPa)	A		25	28	34
	B		13	15	20
	C		15	26	27
	D		19	23	26

Fig. 14Stress time history of bit body internal unit under different piston speed: a) 6.5 m/s, b) 7 m/s, c) 7.5 m/s, d) 8 m/s

a)

b)

c)

d)

Fig. 15Relationship between piston speed and unit peak stress of bit matrix

It is thus important in the drilling process to choose proper air pressure to get a proper piston speed. It is not wise to pursue a high drilling speed, because early fracture failure resulting from a large drilling force will reduce the drilling efficiency and increase drilling costs.