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Rock Mechanics and Rock Engineering

, Volume 52, Issue 11, pp 4257–4272 | Cite as

A Novel Application of Strain Energy for Fracturing Process Analysis of Hard Rock Under True Triaxial Compression

  • Yan Zhang
  • Xia-Ting FengEmail author
  • Xiwei Zhang
  • Zhaofeng Wang
  • Mostafa Sharifzadeh
  • Chengxiang Yang
Original Paper
  • 282 Downloads

Abstract

Energy principles, which can favorably explain the complete rock failure process, are one of the most reliable analysis approaches in rock mechanics and engineering. In this study, a strain energy approach under true triaxial compression (TTC) is proposed. On this basis, the energy evolution characteristics and variations of different failure behavior types (Class I, Class II and ductile failure) under TTC are analyzed. The variations of the strain energy characteristics of Beishan granite with σ2 and σ3 under TTC are studied. The results indicate that the total strain energy U and the elastic strain energy \(U^{e}\) of Beishan granite increase with the increasing σ2 or σ3. The dissipated strain energy \(U^{d}\) rapidly increases when the value of ε1/ε1peak is approximately 0.6–0.8. The influence of σ3 on the rock failure mode and energy evolution characteristics is greater than that of σ2. In highly brittle rocks, the tensile cracking of the rock microstructure is dominant, and the rock has a high strain energy storage capacity and a low strain energy dissipation capacity. The cumulative acoustic emission (AE) count rate curve shows the same trend as the total dissipated strain energy \(U^{d}\) curve. The research results show that the proposed strain energy analysis method for TTC can explain the macroscopic failure behaviors, microscopic failure mechanism and AE characteristics of Beishan granite under TTC, thereby providing new ideas and methods for investigating the behaviors of deep underground hard rock.

Keywords

Strain energy Energy evolution True triaxial compression Fracturing process Hard rock 

List of Symbols

\(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\)

Maximum, intermediate, and minimum principal stresses

\(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\)

Maximum, intermediate, and minimum principal strains

\(U\), \(U^{e}\), \(U^{d}\)

Total strain energy, elastic strain energy, and dissipated strain energy per unit volume of rock

\(A\), \(B\) and \(C\)

\(\sigma_{3}\), \(\sigma_{2}\) loading phase end points and \(\sigma_{1}\) peak point

\(\sigma_{A}\), \(\sigma_{B}\) and \(\sigma_{C}\)

Stresses corresponding to points A, B and C

\(\varepsilon_{A}^{t}\) and \(\varepsilon_{B}^{t}\)

Strains corresponding to points A and B

\(\varepsilon_{1}^{{t{\text{c}}}}\), \(\varepsilon_{2}^{{t{\text{c}}}}\) and \(\varepsilon_{3}^{{t{\text{c}}}}\)

Peak strains corresponding to \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\)

\(n^{{}}\) and \(i^{{}}\)

The numbers of small trapezoidal segments and segmentation points at any specific time t

\(\varepsilon_{1}^{t}\), \(\varepsilon_{2}^{t}\) and \(\varepsilon_{3}^{t}\)

Strains corresponding to \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) at any specific time t

\(\varepsilon_{1}^{{{\text{e}}t}}\), \(\varepsilon_{2}^{et}\) and \(\varepsilon_{3}^{et}\)

Elastic strains corresponding to \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) at any specific time t

\(\varepsilon_{2}^{{e_{1} }}\) and \(\varepsilon_{2}^{{e_{2} }}\)

Elastic strains corresponding to \(\sigma_{2}\) at point B and from point B to point C

\(\varepsilon_{3}^{{e_{1} }}\), \(\varepsilon_{3}^{{e_{2} }}\) and \(\varepsilon_{3}^{{e_{2}^{'} }}\)

Elastic strains corresponding to \(\sigma_{3}\) at point A, from point A to point B and from point A to point C

\(\theta\)

Failure angle between the failure plane and the \(\sigma_{1}\) loading surface

Notes

Acknowledgements

We sincerely acknowledge the financial support from the National Natural Science Foundation of China (Grant nos. 51621006, 51579043 and 51709043). The authors are grateful to Mr. Yangyi Zhou, Mr. Gaolei Song, Mr. Rui Kong, Mr. Jun Zhao, Mr. Qiang Han, Mr. Hong Xu and Mr. Yuemao Zhao at Northeastern University, China and Mr. Yaohui Gao and Mr. Zhi Zheng at the Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, for their valuable academic discussions and generous assistance with the laboratory tests. The authors would also like to thank the journal editor and anonymous reviewers for their valuable suggestions.

References

  1. Abu Al-Rub RK, Voyiadjis GZ (2003) On the coupling of anisotropic damage and plasticity models for ductile materials. Int J Solids Struct 40(2):2611–2643CrossRefGoogle Scholar
  2. Ai C, Zhang J, Li YW, Zeng J, Yang XL, Wang JG (2016) Estimation criteria for rock brittleness based on energy analysis during the rupturing process. Rock Mech Rock Eng 49(12):4681–4698CrossRefGoogle Scholar
  3. Chen ZQ, He C, Wu D, Xu GW, Yang WB (2017) Fracture evolution and energy mechanism of deep-buried carbonaceous slate. Acta Geotech 12:1243–1260CrossRefGoogle Scholar
  4. Dong XJ, Karrech A, Basarir H, Elchalakani M, Seibi A (2018) Energy dissipation and storage in underground mining operations. Rock Mech Rock Eng 52:229–245CrossRefGoogle Scholar
  5. Feng XT, Hudson JA (2011) Rock engineering design. CRC, OxfordCrossRefGoogle Scholar
  6. Feng XT, Zhang XW, Kong R, Wang G (2016) A novel Mogi type true triaxial testing apparatus and its use to obtain complete stress-strain curves of hard rocks. Rock Mech Rock Eng 49(5):1649–1662CrossRefGoogle Scholar
  7. Feng XT, Zhang XW, Yang CX, Kong R, Liu XY, Peng S (2017) Evaluation and reduction of the end friction effect in true triaxial tests on hard rocks. Int J Rock Mech Min Sci 97:144–148CrossRefGoogle Scholar
  8. Gao YH, Feng XT, Zhang XW, Feng GL, Jiang Q, Qiu SL (2018) Characteristic Stress Levels and brittle fracturing of hard rocks subjected to true triaxial compression with low minimum principal stress. Rock Mech Rock Eng 51:3681–3697CrossRefGoogle Scholar
  9. Gaziev E (2001) Rupture energy evaluation for brittle materials. Int J Solids Struct 38(42):7681–7690CrossRefGoogle Scholar
  10. Hua AZ, You MQ (2013) Rock failure due to energy release during unloading and application to underground rock burst control. Tunn Undergr Sp Technol 23(8):1572–1578Google Scholar
  11. Huang D, Li YR (2014) Conversion of strain energy in triaxial unloading tests on marble. Int J Rock Mech Min Sci 66:160–168CrossRefGoogle Scholar
  12. Jaeger JC, Cook NGW, Zimmerman RW (2007) Fundamentals of rock mechanics, 4th edn. Wiley-Blackwell, OxfordGoogle Scholar
  13. Kanaya T, Hirth G (2018) Brittle to semibrittle transition in quartz sandstone: energetics. J Geophys Res Solid Earth 123:84–106CrossRefGoogle Scholar
  14. Kong R, Feng XT, Zhang XW, Yang CX (2018) Study on crack initiation and damage stress in sandstone under true triaxial compression. Int J Rock Mech Min Sci 106:117–123CrossRefGoogle Scholar
  15. Li QM (2001) Strain energy density failure criterion. Int J Solids Struct 38:6997–7013CrossRefGoogle Scholar
  16. Li DY, Sun Z, Xie T, Li XB, Ranjith PG (2017) Energy evolution characteristics of hard rock during triaxial failure with different loading and unloading paths. Eng Geol 228:270–281CrossRefGoogle Scholar
  17. Meng QB, Zhang MW, Han LJ, Pu H, Nie TY (2016) Effects of acoustic emission and energy evolution of rock specimens under the uniaxial cyclic loading and unloading compression. Rock Mech Rock Eng 49:3873–3886CrossRefGoogle Scholar
  18. Ning JG, Wang J, Jiang JQ, Hu SC, Jiang LS, Lu XS (2018) Estimation of crack initiation and propagation thresholds of confined brittle coal specimens based on energy dissipation theory. Rock Mech Rock Eng 51:119–134CrossRefGoogle Scholar
  19. Peng RD, Ju Y, Wang JG, Xie HP, Gao F, Mao LT (2015) Energy dissipation and release during coal failure under conventional triaxial compression. Rock Mech Rock Eng 48(2):509–526CrossRefGoogle Scholar
  20. Solecki R, Conant RJ (2003) Advanced mechanics of materials. Oxford University Press, LondonGoogle Scholar
  21. Su GS, Feng XT, Jiang Q, Chen GQ (2006) Study on new index of local energy release rate for stability analysis and optimal design of underground rock mass engineering with high geostress. Chin J Rock Mech Eng 25(12):2453–2460 (in Chinese) Google Scholar
  22. Tarasov BG, Potvin Y (2013) Universal criteria for rock brittleness estimation under triaxial compression. Int J Rock Mech Min Sci 59(4):57–69CrossRefGoogle Scholar
  23. Tarasov BG, Randolph MF (2011) Superbrittleness of rocks and earthquake activity. Int J Rock Mech Min Sci 48:888–898CrossRefGoogle Scholar
  24. Tarasov BG, Stacey TR (2017) Features of the energy balance and fragmentation mechanisms at spontaneous failure of Class I and Class II rocks. Rock Mech Rock Eng 50(10):2563–2584CrossRefGoogle Scholar
  25. Wang JA, Park HD (2001) Comprehensive prediction of rockburst based on analysis of strain energy in rocks. Tunn Undergr Sp Technol 16(1):49–57CrossRefGoogle Scholar
  26. Xie HP, Ju Y, Li LY (2005) Criteria for strength and structural failure of rocks based on energy dissipation and energy release principles. Chin J Rock Mech Eng 24(17):3003–3010 (in Chinese) Google Scholar
  27. Xie HP, Li LY, Peng RD, Ju Y (2009) Energy analysis and criteria for structural failure of rocks. J Rock Mech Geotech Eng 1:11–20CrossRefGoogle Scholar
  28. Zhang Y, Feng XT, Yang CX, Zhang XW, Sharifzadeh M, Wang ZF (2019) Fracturing evolution analysis of Beishan granite under true triaxial compression based on acoustic emission and strain energy. Int J Rock Mech Min Sci 117:150–161CrossRefGoogle Scholar
  29. Zhao XG, Cai M, Wang J, Ma LK (2013) Damage stress and acoustic emission characteristics of the Beishan granite. Int J Rock Mech Min Sci 64:258–269CrossRefGoogle Scholar
  30. Zhao J, Feng XT, Zhang XW, Zhang Y, Zhou YY, Yang CX (2018) Brittle-ductile transition and failure mechanism of Jinping marble under true triaxial compression. Eng Geol 232:160–170CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  • Yan Zhang
    • 1
  • Xia-Ting Feng
    • 1
    Email author
  • Xiwei Zhang
    • 1
  • Zhaofeng Wang
    • 2
  • Mostafa Sharifzadeh
    • 3
  • Chengxiang Yang
    • 1
  1. 1.Key Laboratory of Ministry of Education on Safe Mining of Deep Metal MinesNortheastern UniversityShenyangChina
  2. 2.State Key Laboratory of Geomechanics and Geotechnical EngineeringInstitute of Rock and Soil Mechanics, Chinese Academy of SciencesWuhanChina
  3. 3.Department of Mining Engineering, Western Australian School of Mines: Mineral, Energy and Chemical Engineering (WASM:MECE)Curtin UniversityKalgoorlieAustralia

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