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


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.


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


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



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.


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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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