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Failure Behaviour of a Sandstone Subjected to the Systematic Cyclic Loading: Insights from the Double-Criteria Damage-Controlled Test Method

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Abstract

The post-peak behaviour of rocks subjected to cyclic loading is very significant to appraise the long-term stability of underground excavations. However, an appropriate testing methodology is required to control the damage induced by the cyclic loading during the failure process. In this study, the post-failure behaviour of Gosford sandstone subjected to the systematic cyclic loading at different stress levels was investigated using the double-criteria damage-controlled testing methodology, and the complete stress–strain relations were captured successfully. The results showed that there exists a fatigue threshold stress in the range of 86–87.5% of the average monotonic strength in which when the cyclic loading stress is below this threshold, no failure occurred for a large number of cycles and in turn, the peak strength improved up to 8%. Also, the variation of the energy dissipation ratio, rock stiffness and acoustic emission hits for hardening tests showed that cyclic loading in the pre-peak regime creates no critical damage in the specimen, and a quasi-elastic behaviour dominates the damage evolution. The post-failure instability of such tests was similar to those obtained for monotonic tests. On the other hand, by exceeding the fatigue threshold stress, the brittleness of the specimens increased with an increase in the applied stress level, and class II behaviour prevailed over total post-peak behaviour. A loose-dense-loose behaviour with different extents was also observed in the post-peak regime of all fatigue cyclic loading tests. This was manifested then as a secondary inverted S-shaped damage behaviour by the variation of the cumulative irreversible axial and cumulative irreversible lateral strains with the post-peak cycle number. Furthermore, it was confirmed that the damage per cycle in the post-peak regime decreases exponentially with an increase in the applied stress level.

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Fig. 1
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modified from Shirani Faradonbeh et al. (2020)

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Modified from Guo et al. 2012), c, d the loose-dense-loose behaviour in the post-peak regime of specimen GS-23, e, f the evolution of cumulative irreversible strains in the post-peak regime for specimen GS-23, g the secondary inverted S-shaped damage behaviour in the post-peak regime for specimen GS-23 and h the number of cycles after failure point versus the applied stress level for the fatigue cyclic loading tests

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Abbreviations

\(E_{{{\text{tan}}}}\) :

Tangent Young’s modulus

v :

Poisson’s ratio

\(\sigma _{1}\) :

Major principal stress

\(\sigma _{a}\) :

Axial stress

\(\sigma _{i}\) :

Indicator stress

\(\sigma _{{a - {\text{peak}}}}\) :

Axial peak stress

\(\sigma _{m}\) :

Average monotonic strength

\(\sigma _{a} /\sigma _{m}\) :

Applied stress level

\(\sigma _{h} /\sigma _{m}\) :

Strength hardening ratio

\(\sigma _{{ci}} /\sigma _{{a - {\text{peak}}}}\) :

Crack initiation stress ratio

\(\sigma _{{cd}} /\sigma _{{a - {\text{peak}}}}\) :

Crack damage stress ratio

\(\varepsilon _{a}\) :

Axial strain

\(\varepsilon _{l}\) :

Lateral strain

\({\text{d}}\varepsilon _{l} /{\text{d}}t\) :

Lateral strain rate

\(\varepsilon _{{a - {\text{peak}}}}\) :

Axial strain at peak stress

\(\varepsilon _{{l - {\text{peak}}}}\) :

Lateral strain at peak stress

\(\varepsilon _{{v - {\text{peak}}}}\) :

Volumetric strain at peak stress

\(\varepsilon _{{a - f}}\) :

Axial strain at the final cycle

\(\varepsilon _{a}^{{{\text{irr}}}}\) :

Irreversible axial strain

\(\varepsilon _{l}^{{{\text{irr}}}}\) :

Irreversible lateral strain

\({{\Sigma }}\varepsilon _{a}^{{{\text{irr}}}}\) :

Cumulative irreversible axial strain

\({{\Sigma }}\varepsilon _{l}^{{{\text{irr}}}}\) :

Cumulative irreversible lateral strain

\(U_{e}\) :

Elastic energy at peak stress

\(U_{e}^{i}\) :

Elastic energy of cycle i

\(U_{d}^{i}\) :

Dissipated energy of cycle i

\(U_{{{\text{pre}}}}\) :

Pre-peak dissipated energy

\(U_{{{\text{post}}}}\) :

Post-peak dissipated energy

\(U_{t}\) :

Total fracture energy

\({\text{Amp}}.~\left( {\sigma _{a} } \right)\) :

Loading amplitude

\({\text{Amp}}.~\left( {\varepsilon _{l} } \right)\) :

Lateral strain amplitude

n :

Cycle number

\(N_{{{\text{total}}}}\) :

Total number of cycles

\(N_{{{\text{after}}}}\) :

Number of cycles after failure point

\(BI\) :

Brittleness index

D :

Damage variable

M :

Post-peak modulus

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Acknowledgements

The first author acknowledges the University of Adelaide for providing the research fund (Beacon of Enlightenment Ph.D. Scholarship) to conduct this study. The authors would like to thank the laboratory staff, in particular, Simon Golding and Dale Hodson, for their aids in conducting the tests.

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  1. School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia

    Roohollah Shirani Faradonbeh, Abbas Taheri & Murat Karakus

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  1. Roohollah Shirani Faradonbeh
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  2. Abbas Taheri
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Correspondence to Roohollah Shirani Faradonbeh.

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Shirani Faradonbeh, R., Taheri, A. & Karakus, M. Failure Behaviour of a Sandstone Subjected to the Systematic Cyclic Loading: Insights from the Double-Criteria Damage-Controlled Test Method. Rock Mech Rock Eng 54, 5555–5575 (2021). https://doi.org/10.1007/s00603-021-02553-5

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