On the Effect of Lateral Confinement on Rock-Cutting Tool Interactions

Abstract

Previous investigations of rock-cutting tool interaction by normal indentation of rocks were mostly conducted at ambient pressure, a condition not representative of the lateral stresses at the excavation face. Despite recent advances in understanding rock-cutting tool interactions with lateral confinement, working out the details of the relationship between rock fracture due to indentation and confinement still requires experimental investigations and theoretical analyses. We conducted a series of normal indentation tests on samples of Gildehaus Bentheim sandstone at lateral confinement up to 20 MPa using a set-up that allowed us to monitor lateral sample deformation and acoustic-emission activity. Experimental observations show that lateral confinement has a significant effect on the load responses and suppresses sample dilation; furthermore, the propagation path of the macroscopic fracture deviates from the indentation direction as confinement increases. The peak indentation pressure increases with confinement and its occurrence is accompanied by significant dilation and acoustic emission activity indicating that it coincides with initiation of macroscopic tensile fracturing. A cavity-expansion-based theoretical model, that accounts for the increases in compressive strength and fracture toughness with lateral confinement, captures the trend of increasing indentation pressure with lateral confinement. In addition, the good agreement between theoretical predictions and experimental data indicates that lateral confinement promotes the growth of the damage zone under the indenter preceding potential macroscopic tensile failure. The model correlates thrust required to break rocks in-situ with rock strength parameters, tool shape parameters, and lateral confinement, thus providing a starting point for optimizing the design of cutting tools.

Highlights

• The set-up for normal indentation tests with lateral confinements allowed for monitoring the apparent lateral strain of specimens, providing information on the failure process during indentation

• Indentation pressure exhibits a peak in contrast to force at all confinements investigated and the peak value in indentation pressure is suitable for an indicator for fracture initiation

• The strengthening effect with confinement, evidenced by the increase in force that is required to break the rock, is associated with decreasing density and length of pre-existing flaws

• Theoretical models are proposed to integrate the influence of confinement on rock indentation, the predictions of which agree well with experimental data

• The strengthening effect can be accounted for by the increase in the compressive strength and in the fracture toughness as confinement increases, the former controls the damage-zone formation and the latter ultimate macroscopic failure

Appendices

Appendix 1: Results of Triaxial Compression Tests

Triaxial compression tests were conducted at confining pressures up to 150 MPa in a conventional triaxial apparatus. Testing results were processed following the procedure outlined in Duda and Renner (2013) (Fig. 

Fig. 17

Results of conventional triaxial compression tests at confining pressures up to 150 MPa, as indicated in the legend that applies to a–c. a Differential stress versus axial strain; b differential stress versus volumetric strain; c volumetric strain versus axial strain, and d the polynomial fitting of peak and yield strength (Δσ_st) for tests covering the confining pressures applied in indentation tests

17a–c). In particular, volumetric strain is obtained from the operation of a volumometer, and axial strain is calculated from the movement of the axial piston accounting for system stiffness. For the derivation of failure criteria, we focus on test results for confining pressures up to 20 MPa, i.e., the range applied in the indentation tests. The dilation angle is estimated to be about 30° and 15° for confining pressures of 10 and 20 MPa. The yield strength was determined as the one where volumetric strain is the largest, and increases from 104 MPa to around 140 MPa at confining pressures of 10 MPa and 20 MPa, respectively. The peak strength amounts to 131 MPa and 166 MPa at confining pressures of 10 MPa and 20 MPa, respectively. Yield and peak strength at ambient pressure are considered equal to uniaxial compressive strength. Peak and yield strength were fitted by second order polynomials ($\Delta \sigma_{{\rm max} } = - 0.165\sigma_{\text{c}}^2 + 8.95\sigma_{\text{c}} + \sigma_{{\text{uni}}}$ and $\Delta \sigma_{\text{y}} = - 0.0432\sigma_{\text{c}}^2 + 5.1317\sigma_{\text{c}} + \sigma_{{\text{uni}}}$, respectively, Fig. 17b). These fits were used for normalization of peak indentation pressure in Fig. 9.

Appendix 2: Numerical Simulations of the Stress Field Induced by Application of Lateral Confinement with the Used Experimental Set-Up

To investigate the stress distribution caused by application of the lateral confinement, we performed numerical simulations at different lateral confinements (0, 5, 10, 20, 40, and 50 MPa) using the commercial software MIDAS that is based on the finite element method. The applied lateral confinement is beyond actually applied in the experiment to investigate largest admissible lateral confinement. Specimen geometry (30 mm in diameter and 50 mm in height) and material parameters, such as Young’s Modulus, Poisson’s ratio, internal frictional angle and cohesion back-calculated based on uniaxial compressive strength according to Labuz and Zang (2012), are the same as used in the experiments and the rock is defined as an isotropic material obeying Mohr–Coulomb failure criterion. The mesh size is 1 mm globally and reduces to 0.5 mm at the top and bottom surfaces. The boundary and loading conditions are as defined in the experimental set-up. Specifically, in lateral direction zero-displacement boundary condition is applied to the bottom of specimen and to the ring area on top of surface and loading is applied to the circumferential direction (Fig. 

Fig. 18

Boundary conditions (no vertical displacement at bottom and at the ring area in the top surface) and loading conditions at circumferential directions

18).

The modeling reveals that the largest von Mises stress occurs at the ring area at the top specimen surface and the largest displacement in the middle of the side surface (Fig. 

Fig. 19

Distribution of a displacement, b von Mises stress, c maximum and d minimum principal stress distribution (compressive stress is negative for the latter two), exemplified at a lateral confinement of 20 MPa

19). Results at other confinement show a similar pattern. The axial loading with the ring limits the maximum confining pressure that can be applied without causing stress-induced failure. For the tested rock the largest lateral confinement that can be applied without causing stress-induced failure is around 29 MPa.

Appendix 3: Cavity Expansion for Brittle Samples with Finite Radial Extent

To address the lack of a theoretical model for interpretation of rock failure under indentation at confinement conditions, we developed a model by modifying the boundary conditions used for the cavity expansion solutions of Yang et al. (2022) for specimens of limited size. Constant lateral stress or zero lateral displacement condition, introduced by Cheng et al. (2022) is adopted in the model derivation. Different to Cheng et al. (2022), the model below also addresses brittle failure. The procedure for model derivations is similar to Yang et al. (2022) and only the key features are presented below.

3.1 Evolution of Damage Zone

With the boundary conditions of finite lateral stress, exploiting the stress field presented in Cheng et al. (2018) and Cheng and Yang (2019) as

$$\sigma_{\text{r}} = - h + \left( {h + \sigma_{\text{c}} } \right)f\left( m \right)\left( {\frac{{\xi_{\text{R}} }}{\xi }} \right)^\beta ,$$

(8)

and

$$\sigma_\theta = - h + \left( {h + \sigma_{\text{c}} } \right)\frac{f\left( m \right)}{{K_{\text{p}} }}\left( {\frac{{\xi_{\text{R}} }}{\xi }} \right)^\beta ,$$

(9)

and combining the Mohr–Coulomb failure criterion,

$$\Delta \sigma_{{\text{st}}} = \left( {K_{\text{p}} - 1} \right)\sigma_{\text{c}} + \sigma_{{\text{uni}}} ,$$

(10)

the equation describing the evolution of indentation pressure with damage zone size of Yang et al. (2022) becomes

$$\frac{p}{{\sigma_{{\text{uni}}} }} = \frac{1}{{K_{\text{p}} - 1}}\left( {\frac{{\Delta \sigma_{{\text{st}}} }}{{\sigma_{{\text{uni}}} }}f\left( m \right)\xi_{\text{R}}^\beta - 1} \right),$$

(11)

where the normalized damage zone size $\xi_{\text{R}}$ is given in incremental form as

$$\begin{aligned} & \left( {\frac{\tan \alpha }{2} + \delta } \right)\frac{{{\text{d}}\xi_{\text{R}} }}{{{\text{d}}\delta }} \\ & \quad = \frac{{\chi \xi_{\text{R}}^\beta - \left( {\chi + C_{\text{v}} (\xi_{\text{R}} )\frac{2kG}{{f\left( m \right)\left( {h + \sigma_{\text{c}} } \right)\beta }}} \right)\xi_{\text{R}}^{\frac{k}{{K_{\text{d}} }} + 1} + \frac{2kG}{{f\left( m \right)\left( {h + \sigma_{\text{c}} } \right)\beta }}\frac{{2^{3 - 2k} }}{{\pi^{2 - k} }}\tan \alpha }}{{\left( {\chi \left( {1 + C_{{\text{BD}}} m(\xi_{\text{R}} )f\left( m \right)} \right) + C_{\text{v}} (\xi_{\text{R}} )\frac{2kG}{{f\left( m \right)\left( {h + \sigma_{\text{c}} } \right)\beta }}} \right)\xi_{\text{R}}^{\frac{k}{{K_{\text{d}} }}} - \chi \left( {1 + C_{{\text{BD}}} m(\xi_{\text{R}} )f\left( m \right)} \right)\xi_{\text{R}}^{\beta - 1} }} \\ \end{aligned}$$

(12)

with

$$f\left( m \right) = \frac{{K_{\text{p}} \left( {k + 1} \right)}}{{K_{\text{p}} + k\left[ {1 + C_{{\text{BD}}} m\left( {1 - K_{\text{p}} } \right)} \right]}} \ge 1$$

(13)

and

$$m = \left( \frac{R}{B} \right)^{k + 1} = \left[ {\frac{a_0 }{B}\left( {1 + \frac{2\delta }{{\tan \alpha }}} \right)\xi_{\text{R}} } \right]^{k + 1} ,$$

(14)

parameters $C_{{\text{BD}}}$ (the subscript BD refers to boundary) and $C_{\text{v}}$ (the subscript v refers to velocity) depending on how the confinement is applied,

$$C_{{\text{BD}}} = \left\{ {\begin{array}{ll} {C_{{\text{CS}}} = - 1} &\quad {\text{for Constant Stress (CS) condition}} \\ {C_{{\text{ZD}}} = \frac{{1 + \mu \left( {k - 1} \right)}}{{k\left( {1 - 2\mu } \right)}}} &\quad {\text{for Zero Displacement (ZD) condition}}\end{array} } \right.,$$

(15)

$$C_{\text{v}} = \left\{ {\begin{array}{ll} {\frac{{\left( {k + 1} \right)\gamma }}{{\left( {k + 1} \right)\left( {1 + C_{{\text{CS}}} m} \right) + \gamma k\left( {1 - \frac{m}{{C_{{\text{ZD}}} k}}} \right)}}} &\quad {\text{for Constant Stress (CS) condition}} \\ {\frac{{\left( {k + 1} \right)\gamma }}{{\left( {k + 1} \right)\left( {1 + C_{{\text{ZD}}} m} \right) + \gamma k\left( {1 - \frac{{C_{{\text{CS}}} m}}{k}} \right)}}} &\quad {\text{for Zero Displacement (ZD) condition}} \end{array} } \right.,$$

(16)

and

$$\gamma = \left( {k + 1} \right)\frac{{\left( {h + \sigma_{\text{c}} } \right)}}{2kG}\left( {f\left( m \right) - 1} \right)$$

(17)

with the other parameters in (11) and (12), including $\chi = {{\left( {k\lambda K_{\text{p}} } \right)} / {\left[ {k\left( {K_{\text{p}} + K_{\text{d}} } \right) + \left( {1 - k} \right)K_{\text{p}} K_{\text{d}} } \right]}}$,$h = {{\sigma_{{\text{uni}}} } / {\left( {K_{\text{p}} - 1} \right)}}$, $K_{\text{d}} = {{\left( {1 + \sin \psi } \right)} / {\left( {1 - \sin \psi } \right)}}$ and $K_{\text{p}} = {{\left( {1 + \sin \phi } \right)} / {\left( {1 - \sin \phi } \right)}}$ the same as in Yang et al. (2022). The initial value problem (12) with the initial condition given by

$$\xi_{\text{R}} = 1\quad{\text{at }}\delta = 0,$$

(18)

can be solved using, e.g., the fourth Runge–Kutta method (Yang and Russell 2015).

3.2 Brittle Failure

The model addresses failure initiation and propagation based on the Mohr–Coulomb constitutive model. As outlined in Yang et al. (2022), the existence of a crack size $c_{{\rm max} }$, at which stress intensity is the largest for a certain indentation pressure/load, is central in the model. Exploiting the analytical relations for this extreme feature one can associate crack growth with load/indentation pressure. When the stress intensity of a pre-existing flaw with $c_{{\text{ini}}} < c_{{\rm max} }$ situated on the damage-zone boundary reaches the critical stress intensity, i.e., $K_{\text{I}} (c_{{\text{ini}}} ) = K_{{\text{IC}}}^{\,} \left( {\sigma_{\text{c}} } \right)$, fracture initiation occurs, presumably around $p_{{\rm max} }$, as indicated by acoustic-emission technique (Yang et al. 2022) and the lateral strain records presented here. The growth of the pre-existing flaw is associated with decreasing indentation pressure until the initial flaw reaches the length $c_{{\rm max} }$ at which a characteristic indentation pressure $p^*$ is reached.

According to the analytical expression for the stress-intensity factor at ambient pressure in Yang et al. (2022), the $c_{{\rm max} }$ is associated with a minimum in indentation pressure. Yet, the minimum indentation pressure may not occur due to the growing fracture’s interaction with the free sample surfaces. Thus, we approximated $c_{{\rm max} }$ with $p^*$. As confinement increases, indentation pressure exhibits extended plateaus around the peak values, indicating that the penetration-depth relation $d(c_{{\text{ini}}} ) \simeq d(c_{{\rm max} } )$, previously assumed for tests at ambient pressure (Yang et al. 2022), no longer holds. As the significant decrease in indentation pressure after the peak force occurs over a small range in penetration compared to the penetration associated with reaching the peak force, we approximate $d(p^* ) \simeq d(c_{{\rm max} } )$, $p^*$ coinciding with peak force when present, at which also a significant drop in indentation pressure occurs. Combined with the approximation of a constant damage zone size as a consequence of the reduction in indentation pressure (the damage zone will exhibit limited growth during fracture growth), we arrive at $\xi_{\text{R}} (p^* ) \simeq \xi_{\text{R}} (c_{{\rm max} } )$. The subsequent sample’s fate is controlled by the macroscopic crack’s interaction with the samples surface, and thus a phase with a modestly increasing load follows whose extent depends on sample size (Yang et al. 2022). Therefore, with the boundary conditions of finite lateral stress, Eq. (51) of Yang et al. (2022) becomes

$$\frac{p^* }{{\sigma_{{\text{uni}}} }} = \frac{1}{{K_{\text{p}} - 1}}\left\{ {\frac{{\Delta \sigma_{{\text{st}}} }}{{\sigma_{{\text{uni}}} }}\frac{f\left( m \right)}{{B_{\text{g}}^{2\beta } }}\left[ {\frac{{\left( {\frac{{K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} } \right)}}{{\sigma_{{\text{uni}}} }}} \right)^2 }}{{a_0 + \frac{d^* }{{\tan \alpha }}}}} \right]^\beta - 1} \right\},$$

(19)

where

$$B_{\text{g}} = \underbrace {\frac{8}{{3\sqrt {3} \pi }}}_{0.4901}\sqrt {{C_{\text{b}} }} \frac{{\left( {1 - kC_{{\text{BD}}} m} \right) - \frac{{\sigma_{\text{c}} }}{{\sigma_{{\text{uni}}} }}\left( {k + 1} \right)}}{{k\left( {1 + C_{{\text{BD}}} m} \right) + K_{\text{p}} \left( {1 - kC_{{\text{BD}}} m} \right)}},$$

(20)

and

$$C_{\text{b}} = \left[ {\frac{{\sigma_{{\text{uni}}} }}{{\Delta \sigma_{{\text{st}}} }} - \frac{1}{{K_{\text{p}} }}f\left( m \right)} \right]\frac{k}{k + 1}\frac{{1 + C_{{\text{BD}}} m}}{f\left( m \right) - 1}.$$

(21)

The approximation underlying Eq. (19), necessary for a closed form solution, suggest that it results in underestimations of the penetration.

To account for the pressure dependency of $K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} } \right)$ as observed in Yang et al. (2021) for the tested rock, three process-zone models are considered that relate the $K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} } \right)$ at elevated confinement to $K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} = 0} \right)$ through confining pressure $\sigma_c$ and a fitting parameter $\sigma^*$, previously identified as intrinsic tensile strength (Sato and Hashida 2006) but possibly relates to crack-closure pressure study (Yang et al. 2021), including

$$\frac{{K_{{\text{IC}}}^{\,} \left( {\sigma_{\text{c}} } \right)}}{{K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} = 0} \right)}} = \sqrt {{1 + 2\frac{{\sigma_{\text{c}} }}{\sigma^* }}}\ ({\text{Sato and Hashida }}2006),$$

(22)

$$\frac{{K_{{\text{IC}}}^{\,} \left( {\sigma_{\text{c}} } \right)}}{{K_{{\text{IC}}}^{\,} \left( {\sigma_{\text{c}} = 0} \right)}} = 1 + \frac{{\sigma_{\text{c}} }}{\sigma^* }\ ({\text{M\"uller }}1986),$$

(23)

and

$$\frac{{K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} } \right)}}{{K_{{\text{IC}}}\, \left( {\sigma_{\text{c}} = 0} \right)}} = \frac{1}{{1 - \sqrt {{{{8\alpha_0 } / \pi }}} \frac{{\sigma_{\text{c}} }}{{\sigma_{\text{c}} + \sigma^* }}}} \, \left( {\text{Zhao and Roegiers }}1993 \right),$$

(24)

in which $\alpha_0$ assumes a value from ${1 / {\left( {2\pi } \right)}}$ to ${\pi / 8}$. The model of Sato and Hashida (2006) captures the relationship between toughness and confinement observed in Yang et al. (2022) for the tested rock (Fig. 

Fig. 20

Comparison of experimental data in Yang et al. (2022) for Gildehaus Bentheim sandstone with three model predictions. In the calculations based on (22)–(24), we used $\sigma^*$ = 0.7 MPa

20).