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Micromechanics of Fracture Propagation During Multistage Stress Relaxation and Creep in Brittle Rocks

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Time-dependent rock deformation caused by the initiation and growth of fractures leads to the weakening of the rock mass. Understanding the fracturing mechanisms involved in the time-dependent behavior in brittle rocks is very important and to achieve this goal, a systematic series of three types of experiments was performed on double-flawed prismatic Barre granite specimens under unconfined compression. The first series aimed to identify the failure mechanism in the short-term failure mode under monotonic loading, the, second series involved multistage relaxation (constant strain) experiments to analyze the damage at different strain levels, and the third series explored the fracture propagation under multistage creep (constant load) experiments. The spatial and temporal evolution of cracking mechanisms were evaluated using the acoustic emission (AE) and two-dimensional digital image correlation (2D-DIC) techniques to observe the whole crack growth process as well as the accumulated inelastic strain at the specified region of interest. Results suggest that in the case of multistage creep experiments, the time to failure was less compared to the multistage relaxation, when loaded above the crack damage threshold (CD) estimated from the monotonic testing. The frequency magnitude distribution of the AE events generated in the three loading conditions followed the Gutenberg Richter model. A relatively lower b-value was obtained for the creep experiments, indicative of high energy AE events and faster crack growth. In addition, the AE and DIC results also revealed high evolution of tensile cracks at-different stages of creep and relaxation compared to shear and mixed-mode cracks.


  • Fracturing mechanisms under multistage relaxation and creep experiments were identified.

  • AE and DIC results showed high evolution of tensile cracks as compared to shear and mixed-mode cracks.

  • Frequency-magnitude distribution illustrated a lower \(b\)-value in case of multistage creep as compared to multistage relaxation.

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Acoustic emissions


Barre granite


Crack initiation


Crack damage


Linear variable differential transformer


Digital image correlation


Uniaxial compressive strength


Region of interest


Scanning electron microscopy












  • Amitrano D (2003) Brittle-ductile transition and associated seismicity: experimental and numerical studies and relationship with the b value. J Geophys Res Solid Earth 108(B1)

  • Atkinson BK (1984) Subcritical crack growth in geological materials. J Geophys Res Solid Earth 89(B6):4077–4114

    Article  Google Scholar 

  • Atkinson BK, Rawlings RD (1981) Acoustic emission during stress corrosion cracking in rocks. Earthq Predict 4:605–616

    Google Scholar 

  • Aydan Ö, Akagi T, Kawamoto T (1996) The squeezing potential of rock around tunnels: theory and prediction with examples taken from Japan. Rock Mech Rock Eng 29(3):125–143

  • Baud P, Meredith PG (1997) Damage accumulation during triaxial creep of Darley Dale sandstone from pore volumometry and acoustic emission. Int J Rock Mech Mining Sci 34(3–4):24–e1

  • Bieniawski ZT (1967) Mechanism of brittle fracture of rock: part I—theory of the fracture process. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 4, pp 395–406

  • Brantut N, Heap MJ, Meredith PG, Baud P (2013) Time-dependent cracking and brittle creep in crustal rocks: a review. J Struct Geol 52:17–43

    Article  Google Scholar 

  • Cao P, Youdao W, Yixian W, Haiping Y, Bingxiang Y (2016) Study on nonlinear damage creep constitutive model for high-stress soft rock. Environ Earth Sci 75(10):1–8

  • Chugh YP, Nath R, Shankar S (1987) Time-dependent behavior of immediate weak floor strata from an Illinois coal mine. In Proceedings of the 6th International Conference on Ground Control in Mining. West Virginia University, Morgantown, WV, pp 204–218

  • Dai F, Xia K (2010) Loading rate dependence of tensile strength anisotropy of Barre granite. Pure Appl Geophys 167(11):1419–1432

    Article  Google Scholar 

  • Damjanac B, Fairhurst C (2010) Evidence for a long-term strength threshold in crystalline rock. Rock Mech Rock Eng 43(5):513–531

    Article  Google Scholar 

  • Das S, Scholz CH (1981) Theory of time-dependent rupture in the Earth. J Geophys Res Solid Earth 86(B7):6039–6051

    Article  Google Scholar 

  • Diederichs MS, Kaiser PK (1999) Tensile strength and abutment relaxation as failure control mechanisms in underground excavations. Int J Rock Mech Min Sci 36(1):69–96

    Article  Google Scholar 

  • Diederichs MS, Kaiser PK, Eberhardt E (2004) Damage initiation and propagation in hard rock during tunnelling and the influence of near-face stress rotation. Int J Rock Mech Min Sci 41(5):785–812

    Article  Google Scholar 

  • Eberhardt E, Stead D, Stimpson B, Read RS (1998) Changes in acoustic event properties with progressive fracture damage. Int J Rock Mech Min Sci 34(3–4):71-e1

    Google Scholar 

  • Gdoutos EE (2005) Fracture mechanics, vol 123. Springer, Dordrecht

    Google Scholar 

  • Ghazvinian E (2015) Fracture initiation and propagation in low porosity crystalline rocks: implications for excavation damage zone (EDZ) mechanics. Doctoral dissertation

  • Gutenberg B, Richter CF (1944) Frequency of earthquakes in California. Bull Seismol Soc Am 34(4):185–188

    Article  Google Scholar 

  • Heap MJ, Baud P, Meredith PG, Bell AF, Main IG (2009) Time‐dependent brittle creep in Darley Dale sandstone. J Geophys Res Solid Earth 114(B7)

  • Hedayat A, Pyrak-Nolte LJ, Bobet A (2014) Detection and quantification of slip along non-uniform frictional discontinuities using digital image correlation. Geotech Test J 37(5):786–799

    Article  Google Scholar 

  • Hudson JA, Harrison JP (2000) Engineering rock mechanics: an introduction to the principles. Elsevier, Amsterdam

    Google Scholar 

  • Innocente JC, Paraskevopoulou C, Diederichs MS (2021) Estimating the long-term strength and time-to-failure of brittle rocks from laboratory testing. Int J Rock Mech Min Sci 147:104900

    Article  Google Scholar 

  • Iqbal M, Mohanty B (2007) Experimental calibration of ISRM suggested fracture toughness measurement techniques in selected brittle rocks. Rock Mech Rock Eng 40(5):453

    Article  Google Scholar 

  • Kemeny J (2005) Time-dependent drift degradation due to the progressive failure of rock bridges along discontinuities. Int J Rock Mech Min Sci 42(1):35–46

    Article  Google Scholar 

  • Ko TY, Kemeny J (2007) Effect of confining stress and loading rate on fracture toughness of rocks. In: 1st Canada-US rock mechanics symposium. OnePetro

  • Kranz RL (1983) Microcracks in rocks: a review. Tectonophysics 100(1–3):449–480

    Article  Google Scholar 

  • Lajtai EZ (1974) Brittle fracture in compression. Int J Fract 10(4):525–536

    Article  Google Scholar 

  • Lennartz-Sassinek S, Main IG, Zaiser M, Graham CC (2014) Acceleration and localization of subcritical crack growth in a natural composite material. Phys Rev E 90(5):052401

    Article  Google Scholar 

  • Li Y, Xia C (2000) Time-dependent tests on intact rocks in uniaxial compression. Int J Rock Mech Min Sci 37(3):467–475

    Article  Google Scholar 

  • Li BQ, da Silva BG, Einstein H (2019) Laboratory hydraulic fracturing of granite: Acoustic emission observations and interpretation. Eng Fract Mech 209:200–220

    Article  Google Scholar 

  • Liu J, Yang H, Xiao Y, Zhou X (2018) Macro-mesoscopic fracture and strength character of pre-cracked granite under stress relaxation condition. Rock Mech Rock Eng 51(5):1401–1412

    Article  Google Scholar 

  • Main IG, Meredith PG, Jones C (1989) A reinterpretation of the precursory seismic b-value anomaly from fracture mechanics. Geophys J Int 96(1):131–138

    Article  Google Scholar 

  • Malan DF (2002) Manuel Rocha medal recipient simulating the time-dependent behaviour of excavations in hard rock. Rock Mech Rock Eng 35(4):225–254

    Article  Google Scholar 

  • Martin CD, Chandler NA (1994) The progressive fracture of Lac du Bonnet granite. In: International journal of rock mechanics and mining sciences and geomechanics abstracts, vol 31, no 6. Pergamon, pp 643–659

  • Matsushima S (1960) On the flow and fracture of igneous rocks. Bulletins-Disaster Prevention Research Institute, Kyoto University, vol 36, pp 1–9

  • Meredith PG, Atkinson BK (1983) Stress corrosion and acoustic emission during tensile crack propagation in Whin Sill dolerite and other basic rocks. Geophys J Int 75(1):1–21

    Article  Google Scholar 

  • Meredith PG, Main IG, Jones C (1990) Temporal variations in seismicity during quasi-static and dynamic rock failure. Tectonophysics 175(1–3):249–268

    Article  Google Scholar 

  • Mishra B, Verma P (2015) Uniaxial and triaxial single and multistage creep tests on coal-measure shale rocks. Int J Coal Geol 137:55–65

    Article  Google Scholar 

  • Miura K, Okui Y, Horii H (2003) Micromechanics-based prediction of creep failure of hard rock for long-term safety of high-level radioactive waste disposal system. Mech Mater 35(3–6):587–601

    Article  Google Scholar 

  • Moore DE, Lockner DA (1995) The role of microcracking in shear-fracture propagation in granite. J Struct Geol 17(1):95–114

    Article  Google Scholar 

  • Moradian Z, Einstein HH, Ballivy G (2016) Detection of cracking levels in brittle rocks by parametric analysis of the acoustic emission signals. Rock Mech Rock Eng 49(3):785–800

    Article  Google Scholar 

  • Munoz H, Taheri A, Chanda EK (2016) Pre-peak and post-peak rock strain characteristics during uniaxial compression by 3D digital image correlation. Rock Mech Rock Eng 49(7):2541–2554

    Article  Google Scholar 

  • Munoz H, Taheri A (2018) Postpeak deformability parameters of localized and nonlocalized damage zones of rocks under cyclic loading. ASTM International

  • Nara Y, Takada M, Mori D, Owada H, Yoneda T, Kaneko K (2010) Subcritical crack growth and long-term strength in rock and cementitious material. Int J Fract 164(1):57–71

    Article  Google Scholar 

  • Nasseri M, Grasselli G, Mohanty B (2010) Fracture toughness and fracture roughness in anisotropic granitic rocks. Rock Mech Rock Eng 43(4):403–415

    Article  Google Scholar 

  • Nicksiar M, Martin CD (2012) Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mech Rock Eng 45(4):607–617

    Article  Google Scholar 

  • Nicksiar M, Martin CD (2013) Crack initiation stress in low porosity crystalline and sedimentary rocks. Eng Geol 154:64–76

    Article  Google Scholar 

  • Özsen H, Özkan İ, Sensögüt C (2014) Measurement and mathematical modelling of the creep behaviour of Tuzköy rock salt. Int J Rock Mech Min Sci (1997) 66:128–135

    Article  Google Scholar 

  • Paraskevopoulou C, Perras M, Diederichs M, Amann F, Löw S, Lam T, Jensen M (2017) The three stages of stress relaxation—observations for the time-dependent behavior of brittle rocks based on laboratory testing. Eng Geol 216:56–75

    Article  Google Scholar 

  • Paraskevopoulou C, Perras M, Diederichs M, Loew S, Lam T, Jensen M (2018) Time-dependent behaviour of brittle rocks based on static load laboratory tests. Geotech Geol Eng 36(1):337–376

    Article  Google Scholar 

  • Patel SM, Sondergeld CH, Rai CS (2017) Laboratory studies of hydraulic fracturing by cyclic injection. Int J Rock Mech Min Sci 95:8–15

    Article  Google Scholar 

  • Peng S, Podnieks ER (1972) Relaxation and the behavior of failed rock. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 9, no 6. Pergamon, pp 699–700

  • Peng SS (1973) Time-dependent aspects of rock behavior as measured by a servocontrolled hydraulic testing machine. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 10, no 3. Pergamon, pp 235–246

  • Pramthawee P, Jongpradist P, Sukkarak R (2017) Integration of creep into a modified hardening soil model for time-dependent analysis of a high rockfill dam. Comput Geotech 91:104–116

    Article  Google Scholar 

  • Qu H, Wu X, Huang P, Tang S, Wang R, Hu Y (2022) Acoustic emission and failure characteristics of shales with different brittleness under AWJ impingement. Rock Mech Rock Eng 1–16

  • Rao MVMS, Lakshmi KP (2005) Analysis of b-value and improved b-value of acoustic emissions accompanying rock fracture. Curr Sci 89:1577–1582

    Google Scholar 

  • Rydelek PA, Sacks IS (1989) Testing the completeness of earthquake catalogues and the hypothesis of self-similarity. Nature 337(6204):251–253

    Article  Google Scholar 

  • Sabitova A, Yarushina VM, Stanchits S, Stukachev V, Khakimova L, Myasnikov A (2021) Experimental compaction and dilation of porous rocks during triaxial creep and stress relaxation. Rock Mech Rock Eng 54(11):5781–5805

  • Schmidtke RH, Lajtai EZ (1985) The long-term strength of Lac du Bonnet granite. Int J Rock Mech Mining Sci Geomech Abstr 22(6):461–465

  • Scholz CH (2015) On the stress dependence of the earthquake b value. Geophys Res Lett 42(5):1399–1402

    Article  Google Scholar 

  • Shirole D, Hedayat A, Walton G (2019) Experimental relationship between compressional wave attenuation and surface strains in brittle rock. J Geophys Res Solid Earth 124(6):5770–5793

    Article  Google Scholar 

  • Shirole D, Hedayat A, Walton G (2020a) Illumination of damage in intact rocks by ultrasonic transmission-reflection and digital image correlation. J Geophys Res Solid Earth 125(7):e2020JB019526

    Article  Google Scholar 

  • Shirole D, Walton G, Hedayat A (2020b) Experimental investigation of multi-scale strain-field heterogeneity in rocks. Int J Rock Mech Min Sci 127:104212

    Article  Google Scholar 

  • Singh DP (1975) A study of creep of rocks. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 12, no 9, pp 271–276

  • Sutton MA, Orteu JJ, Schreier H (2009) Image correlation for shape, motion, and deformation measurements: basic concepts, theory, and applications. Springer Science & Business Media, Berlin

    Google Scholar 

  • Tapponnier P, Brace WF (1976) Development of stress-induced microcracks in Westerly granite. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 13, no 4. Pergamon, pp 103–112

  • Tian H, Chen W, Yang D, Dai F (2016) Relaxation behavior of argillaceous sandstone under high confining pressure. Int J Rock Mech Min Sci 88:151–156

    Article  Google Scholar 

  • Wang R, Li L, Simon R (2019a) A model for describing and predicting the creep strain of rocks from the primary to the tertiary stage. Int J Rock Mech Min Sci 123:104087

    Article  Google Scholar 

  • Wang Z, Gu L, Shen M, Zhang F, Zhang G, Wang X (2019b) Shear stress relaxation behavior of rock discontinuities with different joint roughness coefficient and stress histories. J Struct Geol 126:272–285

    Article  Google Scholar 

  • Woessner J, Wiemer S (2005) Assessing the quality of earthquake catalogues: estimating the magnitude of completeness and its uncertainty. Bull Seismol Soc Am 95(2):684–698

    Article  Google Scholar 

  • Wyss M, Wiemer S (2000) Change in the probability for earthquakes in southern California due to the Landers magnitude 7.3 earthquake. Science 290(5495):1334–1338

    Article  Google Scholar 

  • Yang XJ (2017) New general fractional-order rheological models with kernels of Mittag-Leffler functions. Rom Rep Phys 69(4):118

    Google Scholar 

  • Yu H, Li Y, Liu H (2012) Comparative study of conventional mechanical, creep and stress relaxation properties of silty mudstone under triaxial compression. Chin J Rock Mech Eng 31(1):60–70

    Google Scholar 

  • Zafar S, Hedayat A, Moradian O (2020) Evaluation of crack initiation and damage in intact barre granite rocks using acoustic emission. Geotechnical earthquake engineering and special topics

  • Zafar S, Hedayat A, Moradian O (2022) Evolution of tensile and shear cracking in crystalline rocks under compression. Theor Appl Fract Mech 118:103254

    Article  Google Scholar 

  • Zhang Y, Zhang Z, Xue S, Wang R, Xiao M (2020) Stability analysis of a typical landslide mass in the Three Gorges Reservoir under varying reservoir water levels. Environ Earth Sci 79(1):1–14

    Article  Google Scholar 

  • Zhuang L, Jung SG, Diaz M, Kim KY, Hofmann H, Min KB, Yoon JS (2020) Laboratory true triaxial hydraulic fracturing of granite under six fluid injection schemes and grain-scale fracture observations. Rock Mech Rock Eng 53(10):4329–4344

    Article  Google Scholar 

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This research article is based upon the work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award Number DE‐SC0019117. This support is gratefully acknowledged.


The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Zafar, S., Hedayat, A. & Moradian, O. Micromechanics of Fracture Propagation During Multistage Stress Relaxation and Creep in Brittle Rocks. Rock Mech Rock Eng 55, 7611–7627 (2022).

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