Introduction

Mechanical excavation methods, represented by tunnel boring machines (TBMs), have been widely used in long-distance, large-diameter tunnel construction projects due to their safety and efficiency1,2. Efficient excavation of machines requires the appropriate setting of TBM specifications and operating parameters according to the geological conditions, while evaluating the duration and cost of the project requires accurate prediction of the cutting performance of the cutters under the specified rock conditions.

Predicting the cutting force or performance of the disc cutter is a complex task due to the combined influence of various factors, such as rock mass conditions, cutter geometry, and cutting parameters. Full-scale laboratory tests utilizing rocks from the excavation site effectively reduce the size and boundary effects, allowing test results to be used directly to guide the design and operation of TBMs. Therefore, full-scale laboratory cutting tests are generally considered the most reliable and accurate method to study the cutting performance of disc cutters3,4,5,6,7,8. Data such as normal force, rolling force, and penetration under different test parameters are measured during the cutting tests, and the rock debris from the cut rock surface is collected and weighed. By analyzing the test results, parameters such as cutting coefficient (CC)6,9, normalized resultant force (NRF)8,10, boreability indicated (BI)10, and specific energy (SE)11,12 were proposed to identify the optimal cutting parameters and evaluated the cutting performance. A series of prediction models were proposed based on different rock cutting mechanisms, among which the CSM model was widely used1,13. Although full-scale rock cutting tests exhibit satisfactory results in predicting the performance of TBMs, the difficulty of mining and transporting large rock blocks and the high cost of equipment construction have constrained the utilization of the method14,15,16.

Small-scale rock cutting tests are more feasible to conduct due to the use of smaller cutters and rock samples. Despite the fact that the scale effect cannot be eliminated, small-scale rock cutting tests have been performed to study the rock cutting performance due to the advantages of low cost and ease to conduct. A series of small-scale machines were built to conduct linear rock cutting tests, such as PLCM14,17, ILCM18,19, and small devices that can be utilized for tests with conventional hydraulic equipment20,21. The results obtained from the small-scale linear rock cutting test are very similar to those obtained from the full-scale test, proving that the small-scale test can be used to guide the design and selection of TBMs. In contrast to the small-scale linear test, Macias et al.22 developed the Rolling Indentation Abrasion Test method (RIAT) to evaluate the cutter life. Based on the idea of RIAT, Zhao et al.23 and Yang et al.24 investigated the effects of joint spacing and joint angle on the cutting performance. The test conducted based on the RIAT method cannot change the cutting spacing, and the applied thrust is insufficient. Duan et al.25 developed reduced-scale cutting equipment with adjustable cutting space to reveal the influence of the cutter profile on rock fragmentation. The small-scale test was also used to investigate the effect of assisted rock cutting methods, such as free face and microwave radiation, on the performance of disc cutters26,27. A rotary-percussive penetration apparatus was designed to study the effect of supplemented percussive load on the cutting process of multiple cutters28.

In prior studies, small-scale tests can be used to investigate the cutting performance of disc cutters under various conditions. However, the feasibility of using small-scale tests to study and predict the cutting performance of disc cutters remains to be further investigated due to issues such as insufficient stiffness and loading capacity of the test equipment, the simplification of rotary cutting motion, the limitations of the test parameter adjustment, and the incompleteness of the performance parameter acquisition. In order to validate the feasibility of the small-scale rock cutting test, this paper introduces a small-scale rotary cutting machine with high stiffness and load capacity. Rotary rock cutting tests are performed on three different rocks used in the full-scale tests, and the cutting performance is analyzed. Based on the test results, a scaled prediction model for predicting full-scale cutting force is proposed. The predictive capability of the proposed model is validated using 72 sets of full-scale test data.

Methods

Description of the test machine and parameters

The Small-scale Rotary Cutting Machine (SRCM) for conducting rock cutting tests consists primarily of mechanical part, hydraulic part, electrical control part, and data acquisition systems, as illustrated in Fig. 1a. The mechanical part of the SRCM is shown in Fig. 1b, where the power is transmitted from the reducer-1 to the cutter head-4 via the torque sensor-2 and reducer-3. The longitudinal hydraulic cylinder-7 propels the rock chamber-6 upwards along the guide post to enable the contact of the assembly cutter frame-5 with the rock surface. Longitudinal hydraulic cylinders provide thrust up to 10 T, and two reducers provide torque up to 700 N·m for the cutter head. The cutter head can install two sets of assembly cutter frames. The installation radius can be adjusted from 44 to 116 mm. Two sets of hydraulic jacks (two in each direction) are horizontally arranged in the rock chamber to provide uniform clamping or confining stress for the rock sample. The assembly cutter frame comprises the connecting plate-8, pressure sensor-9, cutter frame-10, cutter spindle-11, copper bush-12, bearing-13, and small-scale disc cutter-14, as shown in Fig. 1c. The cutting forces in the test are collected at 40 Hz. The assembly cutter frame is calibrated on the standard universal testing machine to ensure accurate data collection. The geometry of the small-scale disc cutter is shown in Fig. 1d, which is based on the constant cross section disc cutter (CCS) with a radius of 216 mm and a tip width of 19 mm, reduced by the scale coefficient of 5.4. Due to the manufacturing errors, the actual measured tip width is 3.4 mm. Figure 1e illustrates the parameters involved in the rock fragmentation process by the disc cutter, where Fn is the normal force, Fr is the rolling force, Pcr is the contact stress, φ is the cutter-rock contact angle, β is the resultant force acting angle, s is the cutting spacing, p is the penetration depth, R is the radius of the cutter, T is the tip width, and α is the tip angle, respectively.

Fig. 1
Fig. 1
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Device and parameters for the small-scale rotary rock cutting test. (a) General view of the equipment, (b) components of the mechanical structure of the equipment, (c) components of assembled cutter carrier, (d) dimensions of the small-scale disc cutter, (e) parameters involved in the rock cutting test.

Preparation and mechanical tests of samples

The texture, mineral composition, and strength of rocks can significantly affect the cutting force. Reducing the introduced variables is significant for directly comparing the rock cutting performance between small-scale and full-scale tests. Therefore, the same sample types used in the full-scale tests were adopted in this study, including cement mortar samples, Chongqing sandstone, and Beishan granite. The dimension of the samples was set to 240 mm–240 mm-100 mm. Natural rock samples were obtained by cutting from intact rock blocks, and rock columns were drilled at adjacent locations. The mass mixing ratio of cement mortar was cement: fine sand: water = 2.71: 4.75: 1. The mixture was poured into the mold and cured under standard water conditions for 28 days to obtain cement mortar blocks and cylinders. All sample surfaces were polished. Before conducting the rock cutting tests, the basic mechanical properties of the samples were tested based on the recommended procedure of ISRM29. Uniaxial compressive strength (σc) was determined using samples with a diameter of 50 mm and a height of 100 mm, while Brazilian splitting strength (σt) was determined using samples with a diameter of 50 mm and a height of 50 mm. The mechanical properties of the samples are presented in Table 1.

Table 1 The mechanical properties of the samples.
Full size table

Process and results of the small-scale rotary rock cutting test

To simulate the operating state of the cutterhead, the constant load test mode is adopted in the small-scale test, where the thrust is constant, and the penetration is measured. The test process is shown in Fig. 2. In order to investigate the thrust force at which the cutting tool begins to effectively fragment rock, a series of pre-tests were conducted prior to the small-scale tests. It was found that when the thrust forces applied to cement mortar, Chongqing sandstone, and Beishan granite approached 7.5 kN, 7.5 kN, and 17.5 kN, respectively, the cutter began to break the rock effectively. Consequently, in the small-scale tests, the initial thrust forces for the three types of samples were designed to be 4 kN, 4 kN, and 16 kN, respectively, and were increased at intervals of 4 kN. Based on the designed thrust levels, the transition from ineffective rock breaking to effective rock breaking can be observed. Due to the structural limitations of the assembly cutter frame, the cumulative penetration depth is constrained to 10 mm. Except for the highest thrust level, at least three layers, including nine cuts, were conducted on different rock surfaces for each thrust level. Based on the scale coefficient of 5.4, the cutting space was determined to be 15 mm, which corresponds to 80 mm commonly used in the field3,6. The distance between the outside cutting groove and the side of the sample was set to 45 mm to prevent the edge effect, and the installation radii in the test were 45 mm, 60 mm, and 75 mm, respectively. The first cut layer on the intact rock is regarded as a precondition of the rock surface, aiming to create a damaged rock surface similar to the real tunnel face. After each layer of the cutting test was completed, the rock debris remaining on the rock surface was collected and weighed. Considering that the penetration depth obtained from the displacement sensor was affected by the stiffness of the machine frame, a depth dial indicator was used to measure eight points of the penetration depth along the circle trajectory, and the average value was calculated as the result of the penetration depth. The data corresponding to the precondition stage and the maximum installation radius were discarded because they were unstable and not facilitated by the adjacent cutting grooves. The cutting force fluctuated when breaking the rock. Thus, the mean value recorded by the pressure sensor was adopted instead of the ideal applied thrust. Eventually, 105 small-scale rotary rock cutting tests were conducted on three different types of samples, and the performance of the disc cutter was calculated and presented in Table 2. The parameters CC, NFR, BI, and SE can be calculated based on Eqs. (1)–(4).

$$CC = F_{{\text{n}}} /F_{{\text{r}}}$$
(1)
$$NRF = \arctan (F_{{\text{n}}} /F_{{\text{r}}} )/\arccos ((R - P)/R)$$
(2)
$$BI = F_{{\text{n}}} /p$$
(3)
$$SE = 2\pi F_{{\text{r}}} \sum\limits_{1}^{n} {r_{i} /V_{{{\text{rock}}}} }$$
(4)

Where ri is the installation radius, n is the number of cuts conducted at a certain thrust level, Vrock is the volume of rock fragmentation.

Fig. 2
Fig. 2
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The process of the small-scale rock cutting test.

Table 2 Results of the small-scale rotary rock cutting test.
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Results and discussion

The rock cutting force

Obtaining the cutting force of a single cutter contributes to estimating the thrust and torque of the cutterhead. The relationship between the average cutting force and penetration depth is plotted in Fig. 3. Similar to the full-scale test results, Fn and Fr obtained in the small-scale test showed good power function and linear function relationships with p, respectively. The exponent value of the Fn–p curve remains relatively stable, while the curve coefficient decreases as the rock strength increases, indicating that rock strength does not change the type of the curve but affects the coefficient9,30,31. The rock cutting process involves energy accumulation and dissipation. As the cutter gradually penetrates the rock, cracks initiate and propagate, making it more susceptible to failure under subsequent loading, accounting for the decreased growth rate of normal force.

Fig. 3
Fig. 3
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The relationship between cutting force and penetration depth. (a) Fn-p, (b) Fr-p.

The cutting coefficient and normalized resultant force

The cutting coefficient (CC) is used to evaluate the predicted torque of the cutterhead under a given TBM thrust and can be calculated using Eq. (1). The normalized resultant force (NRF) indicates the degrees of the indentation and damaging effects during the rock crushing failure stage and can be calculated using Eq. (2). As shown in Fig. 4a, the relationship between CC and p can be fitted using a positive linear function, indicating that Fr increases faster as p increases compared to Fn. In Fig. 4b, NRF exhibits a trend of initially decreasing and then stabilizing as p increases. Fr consists of the force to break the rock and the starting torque. The cutter grinds against the rock at low thrust levels, and Fr is mainly used to overcome the startup torque rather than to break the rock. As thrust increases, the cutter penetrates and fractures the rocks, causing the rolling force to be utilized for rock fragmentation, subsequently reducing and stabilizing the value of NRF. In Fig. 4, the overlapping curves corresponding to different samples indicate that strength parameters do not significantly impact CC and NRF, especially when p is high. Numerous studies have shown that CC increases linearly and will not exceed 25% during the rock cutting process, while NRF initially decreases and then stabilizes around 0.54,8,9. Since the copper bush instead of bearings is used between the cutter and the spindle, resulting in the increase of starting torque, the CC and NRF values obtained in this study are higher than the empirical values. Based on the kinetic friction coefficient (0.1) between copper bushes and steel, the effect of the starting torque on Fr is eliminated. The modified CC’ and NRF’ are represented by dashed lines. The results show that CC’ is below 25%, and NRF’ decreases from 0.753 to 0.493, which aligns with the empirical values.

Fig. 4
Fig. 4
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The relationship between cutting coefficient, normalized resultant force and penetration depth. (a) CC-p, (b) NRF-P.

The rock boreability index

The rock boreability index (BI) is used to evaluate the cutting ability of disc cutters and can be calculated using Eq. (3). In Fig. 5, a decreasing power function can describe the relationship between BI and p. The strength of the rock do not influence the power and type of the curve, but the coefficients of the curve increase with increasing rock strength31. The mean value of the power (− 0.715) of the BI-p curve obtained from the small-scale tests is close to the empirical value from full-scale (− 0.73) and the field tests (− 0.75)10,32,33. In the BI-p curve, when p exceeds the critical value, BI begins to stabilize, indicating that further increasing p will not further improve the cutting ability of the cutter, and the disc cutter starts to break the rock efficiently. The critical penetration depth of disc cutters can be determined manually based on the inflection point of the BI-p curve, and the critical rock cutting force is the normal force corresponding to the critical penetration depth. The critical penetration depth obtained in the small-scale test falls within the range of 0.5–2 mm, close to the range of 0.5–1 mm/r obtained from field tests32. Based on the results above, it is recommended to maintain a penetration depth greater than 0.5 mm during the rock cutting process to ensure that the disc cutter has an acceptable rock cutting capability rather than engaging in grinding against the rock. Based on the BI-p curves, the critical rock cutting force corresponding to the three types of samples are 6.59 kN, 7.01 kN, and 15.77 kN, respectively. These values are close to the pre-test results, validating the rationality of the thrust setting in the test.

Fig. 5
Fig. 5
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The relationship of boreability index and penetration depth.

The sample surface morphology

Figure 6a presents the surface morphology of the tested samples under different thrust levels. When a relatively low thrust level (4 kN, 4 kN, 16 kN) is applied, even after three layers of cutting, effective rock fragmentation cannot be achieved, and few shallow indentations can be observed on the rock surface (circle in blue line). As the thrust increased (8 kN, 8 kN, 20 kN), the disc cutter starts to break the rock, but long rock ridges (circle in red dotted line) preventing rock fragmentation can still be found between adjacent cutting grooves. When the thrust approaches the critical rock cutting force (12 kN, 12 kN, 24 kN), the rock between adjacent cutting grooves is completely removed. Each cut generates significant rock debris and achieves a deep penetration depth at the maximum thrust level (16 kN, 16 kN, 28 kN). At a thrust force of 16 kN, the penetrating cracks (circle in yellow dotted line) can be observed in Fig. 6b. The cracks start from the outer cutting groove, point to the lateral free surface of the rock, and continue to propagate along the depth direction to the bottom of the rock. Penetrating cracks in soft sandstone indicate that the TBM penetration rate should be controlled when facing soft rock conditions to avoid security incidents such as instability and collapse of the tunnel face.

Fig. 6
Fig. 6
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The surface morphology of samples and rock cutting mechanism. (a) typical rock surface morphology, (b) the penetrative cracks.

The specific energy

The specific energy (SE) is the energy required to break a unit volume of rock during TBM tunneling, which is used to describe the cutting efficiency and can be calculated by Eq. (3)12. The relationship between SE and p is depicted in Fig. 7, and the relationship can be described using a quadratic function. It is generally accepted that with the increase of s/p, the synergetic cutting effect between adjacent cutters diminishes, and the pressure-induced lateral cracks fail to intersect with each other, leading to the formation of unfavorable ridges between adjacent trajectories and thus reducing the cutting efficiency. An excessively small s/p can also reduce the cutting efficiency, as the rock between adjacent cutters is excessively fragmented. The ratio of cutting spacing to penetration depth corresponding to the minimum value of SE is defined as the optimum cutting parameter (s/popt). Based on the fitted SE-p curves, p corresponding to the minimum values of SE for cement mortar, Chongqing sandstone, and Beishan granite are 3.31, 3.53, and 3.25, respectively. In the small-scale test, with a cutting spacing of 15 mm, s/popt for the three types of samples are calculated to be 4.53, 4.25, and 4.62, respectively. The results indicate that the rock type has no significant influence on the value of s/popt. The average value of s/popt in this study is calculated to be 4.47. The field and full-scale laboratory tests have demonstrated that the values of s/popt for different types of rocks generally approach 206,34. A result of 24.14, close to the empirical value, can be obtained by multiplying the average s/popt obtained in this study by the scale coefficient 5.4. Comakli et al.14 reported s/popt ranging from 4 to 6.7 when performing small-scale rock cutting tests on the PLCM, and the corresponding scale coefficient is 2.98. A range of 11.92 to 19.97 can be obtained by multiplying s/popt by the scale coefficient of 2.98, which is also close to the empirical value. The results indicate that the s/popt in full-scale tests can be predicted based on the scale coefficient and the results of small-scale tests, validating the feasibility of using small-scale tests to guide the layout of the cutter head.

The aforementioned research demonstrates a strong consistency between small-scale test results and full-scale test results, which demonstrates the feasibility of using small-scale tests to guide the design of TBM and predict the excavation performance.

Fig. 7
Fig. 7
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The relationship of specific energy and penetration depth.

The scale prediction model

Basis for the scale prediction model

Predicting the cutting force of disc cutters under specified rock conditions is crucial for predicting the performance of TBMs, as it allows for the estimation of the thrust and torque of the cutterhead based on the force acting on the disc cutter6. The challenges in predicting cutting forces are that several factors, such as rock mechanical properties, cutter profile, and cutting parameters affect cutting forces. Small-scale tests effectively control the variables by employing the same types of samples involved in the full-scale tests and designing the cutting parameters and cutter profiles according to the scale coefficient. Based on the small-scale test results, a scaled prediction model for predicting the cutting force of a full-scale disc cutter is developed. Based on the idea of Rostami, this study assumes that indenters with similar shapes, when indenting into the same material to the same depth and causing failure of the material, generate identical contact stresses. Although the same types of rock are used in both small-scale and full-scale tests, differences in mining locations and weathering degrees lead to variations in rock strength. The full-scale test involves various cutting spacings. However, in the small-scale test, the cutting spacing is fixed at 15 mm, which corresponds only to an 80 mm cutting spacing in the full-scale test. The influence of the scale effect cannot be avoided in the small-scale test. Ultimately, correction coefficients involving rock strength, cutting spacing, and scale effect are introduced, and the prediction model for predicting the full-scale cutting force based on the results of the small-scale test is shown in Eq. (5).

$$F_{{\text{n}}} = P_{{{\text{cr}}}} S_{{{\text{area}}}} k_{{{\text{UCS}}}} k_{{\text{s}}} k_{{{\text{scale}}}}$$
(5)

Where Pcr is the contact stress, Sarea is the contact area, kUCS, ks, and kscale are rock strength, cutting space, and scale effect correction factors, respectively.

Derivation of the scale prediction model

The Pcr corresponding to different rocks can be calculated based on the small-scale test results and is used to predict the full-scale cutting force. As depicted in Fig. 1e, the projection of the contact area between the cutter and rock in the vertical direction can be simplified as a trapezoid, which can be calculated using Eq. (6).

$$S_{{{\text{area}}}} = \sqrt {2Rp - p^{2} } (T + p\tan (\alpha /2))$$
(6)

The contact stress corresponding to different rocks was obtained by dividing the cutting force measured in the small-scale tests by the contact area. As shown in Fig. 8, Pcr exhibits a good decreasing power function relationship with p. Similar to the Fn-p curve, the power of the Pcr-p curve is independent from the rock type, while the coefficient of the curve increases with the rock strength. The corresponding contact stresses for the three different types of rock are shown in Eqs. (7)-(9).

Fig. 8
Fig. 8
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The relationship between Pcr and p.

Fig. 9
Fig. 9
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The relationship between normalized Fn and s.

$${\text{Cement}}\;{\text{mortar}}:\;P_{{{\text{cr}}}} = 251.313p^{ - 0.23}$$
(7)
$${\text{Chongqing}}\;{\text{sandstone}}:\;P_{{{\text{cr}}}} = 264.036p^{ - 0.252}$$
(8)
$${\text{Beishan}}\;{\text{granite}}:\;P_{{{\text{cr}}}} = 594.071p^{ - 0.283}$$
(9)

The predictive capability of the scale prediction model can be demonstrated by using it to predictive the cutting force in the full-scale test. Although the same types of samples are used in both full-scale and small-scale tests, variations in the mining locations result in minor differences in strength. Therefore, the correction factor kUCS is introduced to account for minor strength variations within the same type of rock. It is generally accepted that the cutting force increases with the uniaxial compressive strength3,16,35. The normal force is considered to be linearly related to penetration depth in the Fn-p formula mentioned by Entacher and Schuller21. Based on the idea of Entacher, kUCS can be calculated by Eq. (10). It is worth noting that if the in-situ rock is available, there will be no variations in rock strength, resulting in the value of kUCS equal to 1.

$$k_{{{\text{UCS}}}} = \sigma_{{{\text{c}} - {\text{full}}}} /\sigma_{{{\text{c}} - {\text{small}}}}$$
(10)

where σc−full and σc−small are the uniaxial compressive strengths of the samples in the full-scale and small-scale tests, respectively. For example, when utilizing the scale prediction model to predict the full-scale cutting force for the sandstone mentioned by Wang et al., the value of σc−small is 79.06, whereas σc−full is 60.67.

The parameter s affects the synergistic cutting effect of adjacent cutters, thereby influencing the cutting force. The cutter spacing correction factor ks is introduced to incorporate the influence of the cutting spacing on the cutting force into the model. A series of full-scale test data are collected to investigate the influence of cutting spacing on the cutting force, among which only the data from tests using 17-inch disc cutters and including a cutter spacing of 80 mm are retained4,6,16,36,37. The data were grouped based on rock type and penetration depth, and each group was normalized separately. In each group of data, the normal force is divided by the normal force corresponding to a cutting space of 80 mm to obtain a coefficient, which was defined as the normalized normal force. The results of the normalized normal force are shown in Fig. 9. The normalized normal force of the disc cutter exhibits a linear increasing relationship with the cutting space. Based on the fitting results, ks is introduced to describe the influence of cutting space on the normal force, which can be calculated using Eq. (11).

$$k_{s} = 0.00527s + 0.57252 \, (20\;{\text{mm}} \le s \le 120\;{\text{mm}})$$
(11)

Numerous studies on the scale effect of rocks have demonstrated that the strength of rocks decreases with increasing rock volume in uniaxial compressive tests due to the inclusion of more microscopic defects in larger rock volumes38. Weibull39 suggested that the damage probability of rock materials is a function of their volume and proposed a formula, as shown in Eq. (12), to describe the effect of scale effect on rock strength. Under the same penetration depth, the volume ratio of rock penetrated by small-scale and full-scale cutters is equal to the ratio of the normal contact area. Thus, the scale correction factor can be derived from Eq. (13).

$$\sigma_{{\text{c - full}}} /\sigma_{{\text{c - small}}} = (V_{{{\text{small}}}} /V_{{{\text{full}}}} )^{1/m} = k_{{{\text{scale}}}}$$
(12)
$$k_{scale} = [\sqrt {80p - p^{2} } (3.4 + p\tan (\alpha /2))/(\sqrt {2Rp - p^{2} } (T + p\tan (\alpha /2)))]^{1/m}$$
(13)

where Vfull and Vsmall are different volumes of rocks, and m can be taken as 10 for homogeneous rocks.

Substituting Eqs. (6)-(13) into Eq. (5), the corresponding normal force prediction model can be obtained as shown in Eq. (14). The resultant force acting angle can be used to calculate the rolling force based on the normal force, which is usually assumed to be half of the cutter-rock contact angle1. The rolling force for the corresponding sample can be calculated using Eq. (15).

$$\begin{aligned} F_{{{\text{n}} - {\text{sf}}}} = & P_{{{\text{cr}}}} \sqrt {2Rp - p^{2} } (T + p\tan (\alpha /2))(\sigma_{{{\text{c}} - {\text{full}}}} /\sigma_{{{\text{c}} - {\text{small}}}} )(0.00527s + 0.57252) \\ \, & \left[ {\sqrt {80p - p^{2} } (3.4 + p\tan (\alpha /2))/(\sqrt {2Rp - p^{2} } (T + p\tan (\alpha /2)))} \right]^{1/m} \\ \end{aligned}$$
(14)
$$F_{{{\text{r}} - {\text{sf}}}} = F_{{\text{n}}} \tan (\beta ) = F_{{\text{n}}} \tan (\varphi /2) = F_{{\text{n}}} \tan (0.5\arccos ((R - p)/R))$$
(15)

where Fn−sf and Fr−sf are the predicted full-scale normal and rolling forces, respectively. Pcr represents the cutter-rock contact stress, Pcr = 251.313p−0.23 for cement mortar, Pcr = 264.036p−0.252 for Chongqing sandstone, and Pcr = 594.071p−0.283 for Beishan granite, respectively.

The scaling prediction model establishes a relationship between the small-scale and full-scale cutting forces. For different rock types, the corresponding contact stresses can be obtained through small-scale tests, and the prediction model of full-scale cutting force can be established. The model is proposed to predict the cutting force during effective rock fragmentation by the disc cutter, thereby the penetration should not be less than 0.5 mm.

Validation of the scale prediction model

In order to validate the predictive capability of the proposed model, 72 sets of full-scale test data are collected, which are conducted on cement mortar, Chongqing sandstone, and Beishan granite8,11,31,37,40,41,42. The predictive ability of the proposed model and the CSM model for full-scale test results were compared. The CSM model is presented in Eqs. (16)-(17).

$$F_{{{\text{n}} - {\text{csm}}}} = C\sqrt[3]{{\frac{{s\sigma_{c}^{2} \sigma_{t} }}{{\varphi \sqrt {RT} }}}}\frac{TR\varphi }{{1 + \psi }}\cos (\varphi /2)$$
(16)
$$F_{{{\text{r}} - {\text{csm}}}} = C\sqrt[3]{{\frac{{s\sigma_{c}^{2} \sigma_{t} }}{{\varphi \sqrt {RT} }}}}\frac{TR\varphi }{{1 + \psi }}\sin (\varphi /2)$$
(17)

where the fitting coefficient (C) is generally taken as 2.12, and contact pressure distribution constant (ψ) ranges from − 0.2 to 0.2, which can be calculated using ψ = − 0.0229T + 0.371431.

As shown in Fig. 10, the experimental results and predicted values are plotted in the same graph, and a straight line passing through the origin is used to fit the prediction results of the scale prediction model and CSM model. The slopes of the fitting curves for Fn and Fr, as calculated by the proposed model, are 0.977 and 1.004, respectively, whereas the corresponding slopes for the CSM model are 0.682 and 0.714, respectively. The fitting results conclusively demonstrate that the scaling prediction model, proposed based on the small-scale test results, predicts full-scale rock cutting force more accurately than the CSM model. The CSM model tends to underestimate the cutting force for rocks with medium strength, potentially due to insufficient frame stiffness8. In the small-scale tests, the penetration depth is directly measured using a depth dial indicator, uninfluenced by frame stiffness, leading to the predictions from the proposed model being closer to the test results.

Fig. 10
Fig. 10
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The relationship between predicted and test results. (a) Fn, (b) Fr.

The research results in this paper indicate that the full-scale cutting performance can be predicted using small-scaled tests by assuming the same types of rock with similar strength exhibit identical cutter-rock contact stresses when cut and damaged by the cutter. The small-scale tests can be used instead of the full-scale tests to study and predict the cutting performance of the disc cutter and guide the design and construction of TBMs.

Conclusion

In this paper, a device (SRCM) is introduced that is capable of studying rock cutting performance and reproducing the rotary cutting motion of disc cutters. The results of small-scale rotary rock cutting tests indicate that by designing the cutter profile based on the scale coefficient and using the same types of samples, the small-scale tests can achieve similar cutting performance as the full-scale test. As the thrust and penetration depth increase, the sample is gradually damaged, reducing the difficulty of the penetration into the rock, which in turn slows down the growth rate of the normal force. At low penetration depths, the influence of startup torque on rolling force is significant, resulting in an overestimation of the cutting coefficient and normalized resultant force. After removing the interference of startup torque, the cutting coefficient is less than 25%, and the normalized resultant force approaches 0.5, which is consistent with empirical values. The inflection point of the boreability index and penetration curve indicates that the critical penetration depth for effective rock fragmentation ranges from 0.5 to 2 mm. When the penetration depth falls below 0.5 mm, the cutter only grinds on the rock surface. The lowest point of the specific energy and penetration curve indicates that the optimal cutting parameter in the small-scale test is 4.47. By multiplying 4.47 with the scale coefficient 5.4, a result that is highly consistent with empirical values can be obtained. Based on the small-scale test result, the contact stress corresponding to the failure of different samples is calculated, and then the scale prediction model is proposed. The predictive capability of the scale prediction model and CSM model are evaluated using 72 sets of full-scale test data, and the results indicated that predicted values calculated by the proposed model are closer to the test results than those of the CSM model. The aforementioned results indicate that small-scale tests can serve as a more economical and convenient approach to guide the design and construction of TBMs.