The construction of deep mine shafts in Western China’s thick, unconsolidated, and water-bearing strata presents significant engineering challenges. The shaft sinking by drilling method has become a preferred technique due to its high degree of mechanization, enhanced safety by eliminating the need for personnel underground, and successful application in such complex geology. This method relies heavily on the efficient performance of downhole tools, particularly the gear milling roller cutters mounted on the drill bit’s cutterhead.
However, field practices in Western mining areas, especially within the Jurassic system, have revealed persistent issues of severe wear and suboptimal rock-breaking efficiency of these roller cutters. The Jurassic weakly cemented rocks exhibit characteristics of mud disintegration and sandification upon contact with water, further complicating the drilling process. Existing models for predicting cutter forces, developed primarily for PDC bits, disc cutters, or button-type insert cutters, are not directly applicable due to fundamental differences in cutter geometry, rock failure mechanism, and the specific mechanical properties of these weak rocks. Consequently, there is a pressing need to develop a dedicated mechanical model to accurately predict the rock-breaking forces for the gear milling roller cutters used in this context. This study aims to establish such a calculation model, validate it through experimentation, and analyze the influence of key operational and geometric parameters.
Mechanical Interaction and Rock-Breaking Modes
The interaction between a gear milling cutter tooth and the rock is complex. Analysis of the tooth’s motion trajectory reveals that for typical single-revolution feed distances (penetration) of 1 mm or less, the horizontal cutting displacement of the tooth tip is negligible compared to the cutter’s dimensions. Therefore, the tooth’s rock penetration process can be reasonably approximated as a vertical indentation. Furthermore, under conditions of pure rolling, only one row of teeth is in active contact with the rock formation at any given instant during a full revolution of the cutter.
Based on these observations, we establish the following assumptions for model development: 1) The gear milling roller cutter undergoes pure rolling without slippage. 2) The influence of lateral forces on rock-breaking efficiency is not considered in this model, focusing solely on the vertical (normal) force and the rolling (tangential) force.
Single-Tooth Rock-Breaking Mechanism
The rock failure beneath a single gear milling tooth can be partitioned into distinct zones, as illustrated in spatial and planar models. These zones are:
- Crushing Zone: The rock directly beneath the tooth tip fails under compression once the load exceeds the rock’s unconfined compressive strength.
- Frontal Shear Zone: Rock ahead and behind the tooth along the direction of the cutter ring fails primarily in shear.
- Lateral Shear Zone: Rock on both sides of the tooth fails due to shear induced by the tooth’s lateral挤压.
- Tensile Zone: At the corners of the crushed pit, rock experiences tensile stress due to the interaction between the frontal and lateral shear zones, leading to tensile failure. The projection of this zone on the free surface approximates quarter-circles.
Experimental observations confirm the formation of a dense core within the rock beneath the tooth, comprising bottom, frontal, and lateral dense nuclei, which act as stress concentrators and energy transfer media, leading to crack propagation.
Rock-Breaking Modes for Cutter Rows and Rings
The sequential action of teeth during cutter rotation leads to two primary rock-breaking modes, defined by whether cracks from adjacent failure zones coalesce.
For Adjacent Teeth in the Same Row (Single-Row Mode):
- Non-Cooperative Rock-Breaking: At lower penetrations, lateral cracks from adjacent teeth do not intersect. A rock ridge remains between the individual crushed pits formed by each tooth.
- Cooperative Rock-Breaking: At higher penetrations, lateral cracks from adjacent teeth intersect and propagate to the free surface. The rock between teeth fails in tension, forming rock chips. The tensile region between the lateral shear zones is termed the cooperative tensile zone.
For Adjacent Rows of Teeth (Single-Ring Mode):
- Non-Progressive Rock-Breaking: Cracks from the crushed pits of teeth in front and rear rows do not connect. Rock ridges remain between rows.
- Progressive Rock-Breaking: Cracks from front and rear row teeth intersect. The intervening rock ridge fails under tension, forming larger fragments. This tensile region is the progressive tensile zone.
Thus, the overall gear milling process can involve a combination of cooperative breaking within a row and progressive breaking between rows, depending on the operating conditions.
Development of the Rock-Breaking Force Calculation Model
Vertical Force for a Single Tooth
The total vertical force ($F_n$) on a single tooth is the sum of components required to overcome resistance in each failure zone. We denote the rock’s unconfined compressive strength as $\sigma_c$, tensile strength as $\sigma_t$, and define geometric parameters: tooth edge width $T$, penetration $h$, half tooth wedge angle $\theta$, tooth side relief angle $\alpha$, and effective cutting edge length for the i-th tooth $a_i$.
1. Crushing Zone Force ($F_{n1}$):
$$F_{n1} = \sigma_c \cdot S_1 = \sigma_c (T + 2h \tan \theta)(2a_i \tan \alpha + 2h \tan \alpha)$$
This simplifies to:
$$F_{n1} = \sigma_c (T + 2h \tan \theta)(a_i + h) 2 \tan \alpha$$
2. Frontal Shear Zone Force ($F_{n2}$):
This force is derived from the contact stress $P$ on the tooth’s front face. Using the force equilibrium on a rock wedge in the shear zone and an energy-based strength criterion developed for Jurassic weak rocks, $F_{n2}$ is expressed as:
$$F_{n2} = 2 P h (a_i + h) \tan \alpha \tan \theta$$
Where $P$ is solved from the system involving shear stress $\tau$ and normal stress $\sigma$ on the failure plane:
$$
\begin{aligned}
P h \cos(\theta + \psi) &= \tau h / (\cos \theta \sin \psi) \\
P h \sin(\theta + \psi) &= \sigma h / (\cos \theta \sin \psi)
\end{aligned}
$$
The shear stress $\tau$ is given by the energy strength criterion:
$$\tau = \sqrt{ \sigma_c^2 + a E ( \sigma_3^2 – \sigma_c \sigma_3 ) – b E ( \sigma_3^2 – \sigma_c \sigma_3 )^{3/2} }$$
Here, $E$ is the elastic modulus, $\sigma_3$ is the confining pressure (approximated for the near-free surface condition), $\psi = \pi/2 – \varphi$ with $\varphi$ as the rock breakage angle (~130° for these rocks), and $a$, $b$ are fitting parameters from triaxial tests.
3. Lateral Shear Zone Force ($F_{n3}$):
By analogy with the frontal zone, but considering the side geometry:
$$F_{n3} = 2 P’ h (T + h) \tan \theta \tan \alpha$$
The contact stress $P’$ is derived similarly, using angle $\alpha$ instead of $\theta$ in the equilibrium equations.
4. Tensile Zone Force ($F_{n4}$):
$$F_{n4} = \sigma_t \cdot S_3 = \sigma_t \cdot (\pi h^2 / \tan^2 \psi)$$
5. Progressive Tensile Zone Force ($F_{n5}$): This component exists only in the progressive rock-breaking mode. If the row spacing $C$ is less than or equal to twice the lateral crack length $H$ plus the tooth width $T$, the rock ridge between rows fails in tension.
$$F_{n5} = \sigma_t \cdot (a_i + 2h \tan \alpha) \cdot ( 2\pi r / m – T )$$
Where $r$ is the average cutter radius and $m$ is the number of tooth rows on the cutter.
The total vertical force for a single tooth is therefore:
$$F_n = F_{n1} + F_{n2} + F_{n3} + F_{n4} \quad \text{(Non-Progressive Mode, if } C > 2H+T \text{)}$$
$$F_n = F_{n1} + F_{n2} + F_{n3} + F_{n4} + F_{n5} \quad \text{(Progressive Mode, if } 0 \leq C \leq 2H+T \text{)}$$
Vertical Force for a Three-Tooth Cutter Row
A standard gear milling roller cutter has three teeth per row that contact the rock simultaneously. The total vertical force for the row ($F_N$) depends on whether the teeth are in cooperative or non-cooperative mode.
Non-Cooperative Mode ($2H < b_j$, where $b_j$ is tooth spacing):
The force is simply three times the single-tooth force (for non-progressive mode at the row level), but summed for each of the three teeth with their respective edge lengths $a_1, a_2, a_3$. Let $L_1 = a_1 + a_2 + a_3$.
$$
F_N = \sum_{i=1}^{3} F_n^{(i)} \quad \text{(Specific form derived by summing Eqns for each tooth)}
$$
Cooperative Mode ($0 < 2H \leq b_j$):
An additional force component ($F_N^t$) is required to fail the cooperative tensile zone between teeth. The projected area of this zone is a rectangle with total length $L_2 = b_1 + b_2$.
$$F_N^t = \sigma_t \cdot S_5 = \sigma_t \cdot (T + 2h \tan \theta) L_2$$
The total row force is then:
$$F_N = \sum_{i=1}^{3} F_n^{(i)} + F_N^t$$
Rolling Force Model
The rolling force $F_R$ is primarily related to the energy required for the teeth to crush and displace rock in the direction of motion. Ignoring friction, it can be modeled as the product of the projected contact area in the rolling direction and the rock’s compressive strength, scaled by empirical coefficients.
$$F_R = \eta \cdot \xi \cdot k_d \cdot \sigma_c \cdot 3h (L_1 + 3h \tan \alpha)$$
Where:
- $\eta$: Rotational speed influence coefficient.
- $\xi$: Conversion coefficient (0.8 for rough rock surfaces).
- $k_d$: Rolling rock-breaking coefficient (0.5 for soft/weak rock).
Incorporating Rotational Speed Effect
Rotational speed significantly impacts the dynamic rock-breaking process. An influence coefficient $\eta$ is introduced, determined experimentally as a function of speed $n$ (rpm):
$$\eta = \frac{n}{7.85}$$
Thus, the final models for vertical force (in cooperative mode) and rolling force become:
$$F_N = \frac{n}{7.85} \left[ \sum_{i=1}^{3} F_n^{(i)} + \sigma_t (T + 2h \tan \theta) L_2 \right]$$
$$F_R = \frac{n}{7.85} \cdot \xi \cdot k_d \cdot \sigma_c \cdot 3h (L_1 + 3h \tan \alpha)$$
Similar equations apply for non-cooperative and non-progressive modes.
Model Validation and Critical Penetration
Defining Rock-Breaking Modes via Numerical Simulation
Finite-Discrete Element Method (FDEM) simulations were conducted for a three-tooth row indenting into a rock model. Lateral crack length $H$ was measured for varying penetrations $h$. A strong linear correlation was found:
$$H = 5.228h – 0.487 \quad (R^2 = 0.992)$$
The critical penetration for mode transition is determined by setting the geometric condition. For a 12-inch 3° Type I gear milling cutter ($b_j \approx 31$ mm, $C \approx 58.74$ mm for 17 rows):
- Cooperative Rock-Breaking: Occurs when $2H \geq b_j$. Solving gives $h \geq 3.06$ mm.
- Progressive Rock-Breaking: Occurs when $2H + T \geq C$. Solving gives $h \geq 5.33$ mm.
These values provide the criteria for selecting the correct force calculation formula based on operational penetration.
Experimental Verification
Rotary rock-breaking tests were performed on a large-scale platform using a 12-inch 3° Type I gear milling cutter and simulated Jurassic rock-like material. Tests varied penetration (0.5 to 4.0 mm) and rotational speed (5.24 to 13.09 rpm).
The table below compares the theoretical and experimental average vertical and rolling forces for a speed of 7.85 rpm, showing good agreement.
| Penetration, h (mm) | Theoretical Vertical Force, FN (kN) | Experimental Vertical Force (kN) | Relative Error (%) | Theoretical Rolling Force, FR (kN) | Experimental Rolling Force (kN) | Relative Error (%) |
|---|---|---|---|---|---|---|
| 0.5 | 18.2 | 19.8 | 8.1 | 0.75 | 0.70 | 6.1 |
| 1.0 | 24.5 | 23.6 | 3.6 | 1.65 | 1.84 | 10.2 |
| 1.5 | 31.1 | 29.6 | 5.0 | 2.70 | 2.61 | 3.1 |
| 2.0 | 38.0 | 36.4 | 4.3 | 3.89 | 3.77 | 3.1 |
| 2.5 | 45.1 | 42.8 | 5.3 | 5.21 | 4.91 | 5.8 |
| 3.0 | 52.6 | 51.1 | 2.8 | 6.68 | 6.40 | 4.6 |
| 3.5 | 60.4* | 57.9* | 4.2 | 8.29 | 8.02 | 3.4 |
| 4.0 | 68.5* | 65.5* | 4.6 | 10.04 | 9.64 | 3.9 |
*Indicates cooperative rock-breaking mode (h ≥ 3.06 mm). The model transitions accordingly.
The average relative errors for vertical and rolling forces across all tests were below 8.2% and 10.3%, respectively, validating the accuracy of the proposed gear milling cutter force model.
Analysis of Influencing Factors
The validated model allows for a systematic analysis of how key parameters affect rock-breaking forces in gear milling operations.
1. Penetration (h)
Both vertical force ($F_N$) and rolling force ($F_R$) exhibit a strong positive linear correlation with penetration. For example, at 7.85 rpm, increasing $h$ from 0.5 mm to 4.0 mm causes a 139% increase in $F_N$ and a 710% increase in $F_R$. However, $F_R$ remains an order of magnitude smaller than $F_N$, confirming that the vertical force is the dominant component in the rock-breaking process for gear milling cutters.
2. Cutter Rotational Speed (n)
The forces increase linearly with speed due to the incorporated coefficient $\eta = n/7.85$. The rate of increase is itself dependent on penetration. At h=4 mm, the rate is highest: $\Delta F_N / \Delta n \approx 6.45$ kN/rpm and $\Delta F_R / \Delta n \approx 0.85$ kN/rpm. Higher speed increases the kinetic energy and dynamic impact effect of the teeth, which is captured by the model.
3. Cutter Geometry Parameters
Tooth Edge Length ($a_i$): The total effective cutting length $L_1$ directly influences the contact area. Forces show a positive linear relationship with $a_i$. A longer edge engages more rock, increasing both crushing and shearing resistance.
Row Spacing ($C$): This parameter determines the transition to progressive rock-breaking. The critical penetration $h’$ for progressive breaking is given by:
$$h’ = \frac{C – T + 0.974}{10.456}$$
Reducing $C$ (adding more rows) decreases $h’$, making it easier to achieve the more efficient progressive breaking mode at lower penetrations. The figure below illustrates how vertical force jumps at the critical $h’$ for different row spacings, while rolling force remains unaffected by this mode change.
Tooth Spacing ($b_j$): This controls the transition to cooperative breaking within a row. The critical penetration $h_c$ is:
$$h_c = \frac{b_j + 0.974}{10.456}$$
Reducing $b_j$ lowers $h_c$, promoting cooperative breaking. For typical gear milling cutter designs, a tooth spacing between 11 mm and 21 mm ensures the cooperative breaking mode is achievable within common operational feed rates (penetration less than 1.5 mm per revolution), thereby optimizing fragmentation efficiency.
| Parameter | Effect on Critical Penetration | Effect on Forces (FN, FR) | Design Implication for Gear Milling Cutters |
|---|---|---|---|
| Increase Penetration (h) | N/A (Independent variable) | Strong positive linear increase. | Primary control parameter for force and efficiency. |
| Increase Rotational Speed (n) | No direct effect on critical h. | Positive linear increase (via coefficient η). | Can increase production rate but also dynamic loads. |
| Increase Edge Length (ai, L1) | No direct effect. | Positive linear increase. | Longer edges increase force and wear area; optimal length balances efficiency and durability. |
| Decrease Row Spacing (C) | Decreases critical h for progressive breaking. | For h ≥ h’, FN increases significantly due to added tensile component. FR unchanged. | More rows can improve fragmentation efficiency at lower h, but may increase manufacturing complexity. |
| Decrease Tooth Spacing (bj) | Decreases critical h for cooperative breaking. | For h ≥ hc, FN increases due to cooperative tensile component. FR unchanged. | Closer tooth spacing promotes chip formation and efficient rock removal. Recommended range: 11-21 mm. |
Conclusion
This study establishes a comprehensive mechanical model for calculating the rock-breaking forces of gear milling roller cutters used in drilling shafts through Western China’s Jurassic weakly cemented strata. The model successfully integrates the distinct rock failure zones (crushing, shear, tensile) and accounts for the two key efficiency-enhancing rock-breaking modes: cooperative breaking within a tooth row and progressive breaking between rows. The criteria for these modes are quantitatively defined via lateral crack length, which is linearly related to penetration ($H = 5.228h – 0.487$).
The model introduces a rotational speed influence coefficient ($\eta = n/7.85$) to capture dynamic effects. Validation through rotary cutting tests on simulated rock confirmed the model’s accuracy, with average errors below 10.3%. The analysis demonstrates that vertical force is the dominant component and is more sensitive to changes in penetration, speed, and tooth geometry than rolling force. Furthermore, strategic reduction of cutter geometric parameters like row spacing ($C$) and tooth spacing ($b_j$) can lower the critical penetration required to activate the more efficient cooperative and progressive breaking modes, providing clear guidance for the design optimization of gear milling cutters. This model serves as a valuable theoretical tool for predicting loads, optimizing operational parameters, and designing next-generation cutters for efficient shaft sinking in challenging weak rock formations.