Skip to main content
SpringerLink
Log in

Post-Peak Stress–Strain Curves of Brittle Hard Rocks Under Different Loading Environment System Stiffness

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

To systematically study the influence of loading environment system stiffness (LESS) on post-peak stress–strain curves and failure modes of brittle hard rocks, marble and granite specimens are tested under uniaxial compression under axial-strain-controlled loading using a brittle hard rock testing system named Stiffman with variable LESS. The test results show that the post-peak descending slopes of the stress–strain curves under low LESS are steep, and with the increase of the LESS, the slopes are gentle. In addition, as LESS increases, AE events in the post-peak deformation stage gradually transit from a concentrated distribution near the peak strength to a dispersed distribution in the whole post-peak deformation stage, and the maximum AE rate and the maximum cumulative AE count decrease approximately linearly with the increase of LESS. The number of splitting macrocracks and the degree of damage decrease as LESS increases, indicating that the post-peak failure process is more stable when LESS is high. The mechanism that leads to different post-peak descending slopes of stress–strain curves under different LESS is analyzed by comparing the variations of axial stress, axial and lateral strains, rates of axial and lateral strains over time in the loading process, and analyzing the relations between the maximum axial stress drop rate, the average axial strain rate amplitude, the maximum lateral strain rate and LESS. Fitting formulas relating LESS to the post-peak deformation modulus are established. Finally, pillar stability is discussed based on energy absorption and release considering the post-peak slopes of the pillar and surrounding rock. Considering the influence of LESS on the post-peak stress–strain curves of rocks, it is advisable to obtain strength and deformation properties of brittle rocks in line with the field LESS so as to provide more accurate parameters for rock engineering design and analysis.

Highlights

  • Influence of loading environment system stiffness (LESS) on post-peak deformation of rock is studied experimentally.

  • LESS affects the amount of loading/unloading adjustment by the servo-control system.

  • Rock dilation is reduced with the increase of LESS, which will result in smaller unloading adjustments.

  • As LESS increases, the post-peak stress–strain curve becomes flatter.

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
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  • Bažant ZP (1976) Instability, ductility, and size effect in strain-softening concrete. J Eng Mech Division 102:331–344. https://doi.org/10.1061/JMCEA3.0002111

    Article  Google Scholar 

  • Bieniawski Z (1967) Mechanism of brittle fracture of rock, Parts I, II and III. Int J Rock Mech Mining Sci 4:395–430. https://doi.org/10.1016/0148-9062(67)90030-7

    Article  Google Scholar 

  • Bieniawski Z, Denkhaus H, Vogler U (1969) Failure of fractured rock. Int J Rock Mech Min Sci 6:323–341

    Article  Google Scholar 

  • Bieniawski Z (1966) Mechanism of rock fracture in compression. Rock Mechanics Division, National Mechanical Engineering Research Institute, Council for Scientific and Industrial Research, South Africa

  • Brady B, Brown E (1993) Rock mechanics for underground mining. Springer, Netherlands

    Google Scholar 

  • Cai M, Kaiser P (2018) Rockburst support reference book–Volume I: rockburst phenomenon and support characteristics. Laurentian University, Sudbury

    Google Scholar 

  • Cai M, Kaiser P, Uno H, Tasaka Y, Minami M (2004) Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system. Int J Rock Mech Mining Sci 41:3–19. https://doi.org/10.1016/S1365-1609(03)00025-X

    Article  Google Scholar 

  • Cai M, Kaiser P, Tasaka Y, Minami M (2007) Determination of residual strength parameters of jointed rock masses using the GSI system. Int J Rock Mech Mining Sci 44:247–265. https://doi.org/10.1016/j.ijrmms.2006.07.005

    Article  Google Scholar 

  • Cai M, Hou P, Zhang X, Feng X (2021) Post-peak stress–strain curves of brittle hard rocks under axial-strain-controlled loading. Int J Rock Mech Mining Sci 147:10421. https://doi.org/10.1016/j.ijrmms.2021.104921

    Article  Google Scholar 

  • Carpinteri A (1989) Cusp catastrophe interpretation of fracture instability. J Mech Phys Solids 37:567–582. https://doi.org/10.1016/0022-5096(89)90029-X

    Article  Google Scholar 

  • Carpinteri A, Corrado M, Lacidogna G (2013) Heterogeneous materials in compression: correlations between absorbed, released and acoustic emission energies. Eng Failure Anal 33:236–250. https://doi.org/10.1016/j.engfailanal.2013.05.016

    Article  Google Scholar 

  • Carpinteri A, Lacidogna G, Corrado M, Di Battista E (2016) Cracking and crackling in concrete-like materials: a dynamic energy balance. Eng Fracture Mech 155:130–144. https://doi.org/10.1016/j.engfracmech.2016.01.013

    Article  Google Scholar 

  • Cook N (1965) The failure of rock. Int J Rock Mech Mining Sci Geomech Abstracts 2:389–403. https://doi.org/10.1016/0148-9062(65)90004-5

    Article  Google Scholar 

  • Cook N, Hojem J (1966) A rigid 50-ton compression and tension testing machine. J S Afr Mech Eng 1:89–92

    Google Scholar 

  • Fairhurst C, Hudson J (1999a) Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci 36:279–289

    Article  Google Scholar 

  • Fairhurst C, Hudson J (1999b) Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. Int J Rock Mech Mining Sci 36:279–289. https://doi.org/10.1016/S0148-9062(99)00006-6

    Article  Google Scholar 

  • Feng X (2017) Rockburst: mechanisms, monitoring, warning, and mitigation. Butterworth-Heinemann, Oxford

    Google Scholar 

  • Gao F, Yang L (2021) Experimental and numerical investigation on the role of energy transition in strainbursts. Rock Mech Rock Eng. https://doi.org/10.1007/s00603-021-02550-8

    Article  Google Scholar 

  • Gao F, Kang H, Lou J, Li J, Wang X (2019) Evolution of local mine stiffness with mining process: insight from physical and numerical modeling. Rock Mech Rock Eng 52:3947–3958. https://doi.org/10.1007/s00603-019-01798-5

    Article  Google Scholar 

  • Gao F, Kaiser P, Stead D, Eberhardt E, Elmo D (2018) A numerical study on the effect of loading system stiffness on strainbursts. In: 52nd US Rock Mechanics/Geomechanics Symposium, Seattle, Washington, USA, 2018. OnePetro: 323–345

  • Hemami B, Fakhimi A (2014) Numerical simulation of rock-loading machine interaction. In: 48th US Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA, 2014. OnePetro: 111–123

  • Hojem J, Cook N, Heins C (1975) A stiff, two meganewton testing machine for measuring the work softening behaviour of brittle materials. J S Afr Mech Eng 25:250–270

    Google Scholar 

  • Hou P, Cai M, Zhang X, Feng X (2021) Post-peak stress-strain curves of brittle rocks under axial-and lateral-strain-controlled loadings. Rock Mech Rock Eng. https://doi.org/10.1007/s00603-021-02684

    Article  Google Scholar 

  • Hudson J (1974) Shape of the complete stress-strain curve for rock. Int J Rock Mech Mining Sci Geomech Abstr 11:A6

    Google Scholar 

  • Hudson J, Harrison J (2000) Engineering rock mechanics: part 1: an introduction to the principles. Elsevier, Pergamon

    Google Scholar 

  • Hudson J, Crouch S, Fairhurst C (1972) Soft, stiff and servo-controlled testing machines: a review with reference to rock failure. Eng Geol 6:155–189. https://doi.org/10.1016/0013-7952(72)90001-4

    Article  Google Scholar 

  • Joseph T (2002) Estimation of the post-failure stiffness of rock. Dissertation, University of Alberta

  • Kaiser P, Cai M (2012) Design of rock support system under rockburst condition. J Rock Mech Geotech Eng 4:215–227. https://doi.org/10.3724/SP.J.1235.2012.00215

    Article  Google Scholar 

  • Kaiser P, Tang C (1998) Numerical simulation of damage accumulation and seismic energy release during brittle rock failure—part II: rib pillar collapse. Int J Rock Mech Mining Sci 35:123–134. https://doi.org/10.1016/S0148-9062(97)00010-7

    Article  Google Scholar 

  • Karl R (1992) Analysis of stable and unstable pillar failure using a local mine stiffness method. In: Proceedings of the Workshop on Coal Pillar Mechanics and Design, 1992. US Department of the Interior, Bureau of Mines: 128–139

  • Kawakata H, Cho A, Kiyama T, Yanagidani T, Kusunose K, Shimada M (1999) Three-dimensional observations of faulting process in Westerly granite under uniaxial and triaxial conditions by X-ray CT scan. Tectonophysics 313:293–305

    Article  Google Scholar 

  • Keedy D, Volungis R, Kawai H (1960) The use of an instron testing machine for the determination of stress and strain-optical coefficients. Universtity of Massachusetts Amherst, Amherst

    Google Scholar 

  • Kias E, Ozbay U (2013) Modeling unstable failure of coal pillars in underground mining using the discrete element method. In: 47th US Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA, 2013. OnePetro: 178–194

  • Li Y, Li K, Feng X, Cai M (2018) Development and evaluation of artificial expandable pillars for hard rock mining. Int J Rock Mech Mining Sci 110:68–75. https://doi.org/10.1016/j.ijrmms.2018.07.014

    Article  Google Scholar 

  • Manouchehrian A, Cai M (2015) Simulation of unstable rock failure under unloading conditions. Can Geotech J 53:22–34. https://doi.org/10.1139/cgj-2015-0126

    Article  Google Scholar 

  • Paulding B (1965) Crack growth during brittle fracture in compression. Dissertation, Massachusetts Institute of Technology

  • Pen Y, Barron K (1994) The role of local mine stiffness in pillar bump prediction. In: 1st North American Rock Mechanics Symposium, Austin, Texas, 1994. OnePetro: 59–71

  • du Plessis M, Malan D (2018) The effect of rock mass stiffness on crush pillar behaviour. In: Litvinenko V (ed) EUROCK2018: Geomechanics and Geodynamics of Rock Masses. 1st Edition edn. CRC Press, Netherlands: 871–876

  • Rist M, Sammonds P, Murrell S (1991) Strain rate control during deformation of ice: an assessment of the performance of a new servo-controlled triaxial testing system. Cold Regions Sci Technol 19:189–200. https://doi.org/10.1016/0165-232X(91)90008-5

    Article  Google Scholar 

  • Rummel F, Fairhurst C (1970) Determination of the post-failure behavior of brittle rock using a servo-controlled testing machine. Rock Mech 2:189–204

    Article  Google Scholar 

  • Salamon M (1970) Stability, instability and design of pillar workings. Int J Rock Mech Mining Sci Geomech Abstr 7:613–631. https://doi.org/10.1016/0148-9062(70)90022-7

    Article  Google Scholar 

  • Sinha N, Frederking R (1979) Effect of test system stiffness on strength of ice. National Research Council, Division of Building Research, Canada

  • Stavrogin A, Tarasov B (2001) Experimental physics and rock mechanics. CRC Press, Florida

    Google Scholar 

  • Stavrogin A, Tarasov B, Shirkes O, Pevzner E (1981) Strength and deformation of rocks before and after the breakdown point. Sov Min Sci 17:487–493

    Article  Google Scholar 

  • Tarasov B, Potvin Y (2013) Universal criteria for rock brittleness estimation under triaxial compression. Int J Rock Mech Mining Sci 59:57–69. https://doi.org/10.1016/j.ijrmms.2012.12.011

    Article  Google Scholar 

  • Van Mier J, Shah S, Arnaud M, Balayssac J, Bascoul A, Choi S, Dasenbrock D, Ferrara G, French C, Gobbi M (1997) Strain-softening of concrete in uniaxial compression. Mater Struct 30:195–209. https://doi.org/10.1007/BF02486177

    Article  Google Scholar 

  • Wang S, Lam K, Au S, Tang C, Zhu W, Yang T (2006) Analytical and numerical study on the pillar rockbursts mechanism. Rock Mech Rock Eng 39:445–467. https://doi.org/10.1007/s00603-005-0075-2

    Article  Google Scholar 

  • Wawersik W, Fairhurst C (1970) A study of brittle rock fracture in laboratory compression experiments. Int J Rock Mech Mining Sci Geomech Abstr 5:561–575

    Article  Google Scholar 

  • Wawersik W (1968) Detailed analysis of rock failure in laboratory compression tests. Doctoral thesis University of Minnesota

  • Whitney C (1943) Discussion on VP Jensen’s paper. ACI Mater Journal 39:584

    Google Scholar 

  • Xu Y, Cai M (2017) Influence of loading system stiffness on post-peak stress–strain curve of stable rock failures. Rock Mech Rock Eng 50:2255–2275. https://doi.org/10.1007/s00603-017-1231-1

    Article  Google Scholar 

  • Xu Y, Cai M (2018) Influence of strain energy released from a test machine on rock failure process. Can Geotech J 55:777–791. https://doi.org/10.1139/cgj-2017-0256

    Article  Google Scholar 

  • Xu T, Tang CA, Zhang Z, Zhang YB (2003) Theoretical, experimental and numerical studies on deformation and failure of brittle rock in uniaxial compression. J Northeast Univ 24:3164–3169

    Google Scholar 

  • Xu S, Hou P, Cai M, Li Y (2019) An experiment study on a novel self-swelling anchorage bolt. Rock Mech Rock Eng 52:4855–4862. https://doi.org/10.1007/s00603-019-01854-0

    Article  Google Scholar 

  • Xu S, Yang Z, Cai M, Hou P (2021) An experimental study on the anchoring characteristics of an innovative self-swelling Split-set. Tunn Undergr Space Technol 112:13919. https://doi.org/10.1016/j.tust.2021.103919

    Article  Google Scholar 

  • Zhang X, Yang C, Zhang J, Ren J (2013) Experimental study of mechanical behaviour of deep gneiss in Hongtoushan copper mine. Chin J Rock Mech Eng Geol 32:3228–3237

    Google Scholar 

  • Zhao X, Cai M, Wang J, Ma L (2013) Damage stress and acoustic emission characteristics of the Beishan granite. Int J Rock Mech Mining Sci. https://doi.org/10.1016/j.ijrmms.2013.09.003

    Article  Google Scholar 

Download references

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China under Grant No. 51974061, 42177169 and 52074062, the 111 project under grant no. B17009, and the Fundamental Research Funds for the Central Universities (N2001003, N2001001).

Author information

Authors and Affiliations

  1. Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, 110819, People’s Republic of China

    P. Y. Hou & M. Cai

  2. Bharti School of Engineering and Computer Science, Laurentian University, Sudbury, ON, Canada

    M. Cai

  3. MIRARCO, Laurentian University, Sudbury, ON, Canada

    M. Cai

Authors
  1. P. Y. Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Cai.

Ethics declarations

Conflict of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Table 4.

Table 4 Mechanical properties of marble and granite under different loading environment system stiffness (LESS)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, P.Y., Cai, M. Post-Peak Stress–Strain Curves of Brittle Hard Rocks Under Different Loading Environment System Stiffness. Rock Mech Rock Eng 55, 3837–3857 (2022). https://doi.org/10.1007/s00603-022-02839-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-022-02839-2

Keywords

Search

Navigation