Skip to main content
SpringerLink
Log in

Fracture Mechanism of the Formation with a Natural Flaw Under a Single-Tooth Cutting

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

The process of a single-tooth cutting formation is taken as the representative to analyze the fracture mechanism of the formation with natural flaws under the PDC drill bit. The modeling of the stress field distribution of the formation under a single tooth is proposed first. Subsequently, the modeling of the stress intensity factor K, T-stress and the initiation angle \(\theta \) of the flaw under two far-field compressive loads in two cases are presented based on the mode-II fracture. Finally, the results of theoretical calculations and simulations are given and discussed. Based on the analysis undertaken, it can be concluded that the confinement ratio \(\lambda \) applying on the flaw changes nonlinearly with the variation in \(\alpha +\varphi \). The relationship between the dimensionless ratio \(\beta _{{x}}/\beta _{{y}}\) of T-stress and the rotational polar angle \(\varphi \) is a parabola with an upward opening. When the direction of the cutting force is consistent with the flaw, the stress intensity factor K and the initiation angle \(\theta \) at the tip of the flaw reach the maximum. In addition, the angle between the cutting force and the flaw weakens the T-stress and inhibits the crack initiation and propagation of the flaw. The results of numerical simulations show good agreement with theoretical calculations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

F :

Cutting force (N)

\(\alpha \) :

Force angle of cutting tooth (\(^{\circ }\))

\(\varphi \) :

Angle between the force and the flaw, namely rotational polar angle (\(^{\circ }\))

\(\chi \) :

Stress function

x, y :

Horizontal and vertical distances between the cutting force and the flaw (m)

a :

Half-length of the flaw (m)

\(\theta \) :

Initiation angle of the flaw (\(^{\circ }\))

\(\beta \) :

Inclination angle of the flaw (\(^{\circ }\))

\(\sigma _{{x}}\), \(\sigma _{{y}}\) :

Far-field normal stresses applying on the flaw along the x- and y-directions (MPa)

\(K_{\mathrm {II}}\) :

Mode-II stress intensity factor at the tip of the flaw (\(\hbox {MPa}\,\hbox {mm}^{\mathrm {0.5}}\))

\(T_{{x}}\), \(T_{{y}}\) :

T-stresses along the x- and y-directions (MPa)

\(\mu \) :

Coulomb friction coefficient

\(\lambda \) :

Confinement ratio

\(r_{\mathrm {c}}\) :

Critical distance of crack initiation (m)

References

  1. Hoek E, Martin CD. Fracture initiation and propagation in intact rock: a review. J Rock Mech Geotech Eng. 2014;6(4):287–300.

    Article  Google Scholar 

  2. Wong LNY, Einstein HH. Crack coalescence in molded gypsum and Carrara marble: part 1. Macroscopic observations and interpretation. Rock Mech Rock Eng. 2009;42(2):475–511.

    Article  Google Scholar 

  3. Silva B, Gonçalves D, Einstein H. Modeling of crack initiation, propagation and coalescence in rocks. Int J Fract. 2013;182(2):167–86.

    Article  Google Scholar 

  4. Li H, Wong LNY. Influence of flaw inclination angle and loading condition on crack initiation and propagation. Int J Solids Struct. 2012;49(18):2482–99.

    Article  Google Scholar 

  5. Zhu XH, Tang LP, Tong H. Effects of high-frequency torsional impacts on rock drilling. Rock Mech Rock Eng. 2014;47(4):1345–54.

    Article  Google Scholar 

  6. Ma LF, Yari N, Wiercigroch M. Shear stress triggering brittle shear fracturing of rock-like materials. Int J Mech Sci. 2018;146–147:295–302.

    Article  Google Scholar 

  7. Li YW, Long M, Zuo LH, et al. Brittleness evaluation of coal based on statistical damage and energy evolution theory. J Pet Sci Eng. 2019;172:753–63.

    Article  Google Scholar 

  8. Jin XQ, Wang GY, Keer LM, et al. Modeling crack growth from pores under compressive loading with application to metallic glasses. J Alloys Compd. 2011;509S:S115–8.

    Article  Google Scholar 

  9. Gratchev I, Dong HK, Chong KY. Strength of rock-like specimens with preexisting cracks of different length and width. Rock Mech Rock Eng. 2016;49(9):4491–6.

    Article  Google Scholar 

  10. Zhou YN, Lin JS. On the critical failure mode transition depth for rock cutting. Int J Rock Mech Min Sci. 2013;62:131–7.

    Article  Google Scholar 

  11. Liu WJ, Zhu XH, Jing J. The analysis of ductile-brittle failure mode transition in rock cutting. J Pet Sci Eng. 2018;163:311–9.

    Article  Google Scholar 

  12. Zhou XP, Bi J, Qian QH. Numerical simulation of crack growth and coalescence in rock-like materials containing multiple pre-existing flaws. Rock Mech Rock Eng. 2015;48:1097–114.

    Article  Google Scholar 

  13. Wong LNY, Zhang XP. Size effects on cracking behavior of flaw-containing specimens under compressive loading. Rock Mech Rock Eng. 2014;47:1921–30.

    Article  Google Scholar 

  14. Eftekhari M, Baghbanan A, Hashemolhosseini H. Crack propagation in rock specimen under compressive loading using extended finite element method. Arab J Geosci. 2016;9:145.

    Article  Google Scholar 

  15. Shi F. Study on the cracking process of rock using the extended finite element method. Doctoral Thesis. China University of Mining and Technology; 2015.

  16. Cao P, Liu TY, Pu CZ, et al. Crack propagation and coalescence of brittle rock-like specimens with pre-existing cracks in compression. Eng Geol. 2015;187:113–21.

    Article  Google Scholar 

  17. Zhou XP, Cheng H, Feng YF. An experimental study of crack coalescence behaviour in rock-like materials containing multiple flaws under uniaxial compression. Rock Mech Rock Eng. 2014;47:1961–86.

    Article  Google Scholar 

  18. Park CH, Bobet A. Crack initiation, propagation and coalescence from frictional flaws in uniaxial compression. Eng Fract Mech. 2010;77:2727–48.

    Article  Google Scholar 

  19. Willemse EJM, Pollard DD. On the orientation and patterns of wing cracks and solution surfaces at the tips of a sliding flaw or fault. J Geophys Res. 1998;103:B2427–38.

    Article  Google Scholar 

  20. Mutlu O, Pollard DD. On the patterns of wing cracks along an outcrop scale flaw: a numerical modeling approach using complementarity. J Geophys Res. 2008;113:B06403.

    Article  Google Scholar 

  21. Landau LD, Pitaevskii LP, Kosevich AM, et al. Theory of Elasticity. Theoretical Physics, vol. 7. 3rd ed. Amsterdam: Elsevier; 2012.

    Google Scholar 

  22. Li XF, Lee KY, Tang GJ. Kink angle and fracture load for an angled crack subjected to far-field compressive loading. Eng Fract Mech. 2012;82:172–84.

    Article  Google Scholar 

Download references

Acknowledgements

The support of National Natural Science Foundation of China (No. 51704074) and Youth Science Foundation of Heilongjiang Province (No. QC2018049) is gratefully acknowledged. The work is also supported by Talent Cultivation Foundation (Nos. SCXHB201703; ts26180119; td26180141) and Youth Science Foundation (No. 2019QNL-07) of Northeast Petroleum University.

Author information

Authors and Affiliations

  1. Institute of Petroleum Engineering, Northeast Petroleum University, Daqing, 163318, Heilongjiang, China

    Siqi Li, Wei Li, Tie Yan, Huan Zhao & Xin Ling

  2. The Fifth Oil Production Plant of Daqing Oilfield, Daqing, 163513, Heilongjiang, China

    Fuqing Bi

Authors
  1. Siqi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tie Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fuqing Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Huan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Xin Ling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Li, W., Yan, T. et al. Fracture Mechanism of the Formation with a Natural Flaw Under a Single-Tooth Cutting. Acta Mech. Solida Sin. 33, 719–730 (2020). https://doi.org/10.1007/s10338-020-00167-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10338-020-00167-y

Keywords

Search

Navigation