Subcritical Fracturing of Sandstone Characterized by the Acoustic Emission Energy

  • Yuekun Xing
  • Guangqing ZhangEmail author
  • Bin Wan
  • Hui Zhao
Technical Note


Subcritical fracture growth in rock may be attributable to several competing mechanisms, including cyclic loading. Experimental investigations (Tao and Mo 1990; Bagde and Petroš 2005; Xiao et al. 2010) have demonstrated the progressive weakening of rock due to cyclic loading. Among the techniques employed to interpret the extension of subcritical fractures under cyclic loading, the Paris law (Paris et al. 1961; Paris and Erdogan 1963) is probably the most popular method. The Paris law (\({\text{d}}a/{\text{d}}N=C{\left( {\Delta {K_{\text{I}}}} \right)^m}\)


Subcritical fracture General law Fracture process zone AE energy analysis 

List of Symbols


Effective crack length


Number of loading cycles


Paris constant


Paris exponent

\(\Delta {K_{\text{I}}}\)

Amplitude of the stress intensity factor in a load cycle


Elastic modulus


Poisson’s ratio

\(r\), \(b\)

Specimen radius and thickness


Specimen notch length


Specimen span length


Pore radii


Uniaxial compressive strength


Tensile strength


Specimen mean failure load

\({P_{\hbox{min} }}\), \({P_{\hbox{max} }}\)

Minimum and maximum values of the cyclic load

\(\Delta P\)

Load amplitude

CMOD, \({\delta _{\text{m}}}\)

Crack mouth opening displacement

\(\Delta {\delta _{\text{m}}}\)

CMOD amplitude in a load cycle


Compliance of the \(P\)\({\delta _{\text{m}}}\) curve (\({C_{\text{m}}}=\Delta {\delta _{\text{m}}}/\Delta P\))


Stress ratio (\({P_{\hbox{min} }}/{P_{\hbox{max} }}\))


AE energy


AE energy of a waveform


Distance between the AE source and a sensor


Number of synchronous waveforms


Length of the effective crack


Stress intensity factor obtained from \({a_{\text{e}}}\)


Closing cohesive stress

\({\sigma _{\text{e}}}\)

External tensile stress of the cracked plate


Simplified crack opening displacement

\({\delta _{\text{c}}}\)

Crack tip opening displacement

\({\delta _{\text{cmax}}}\)

Critical value of the crack tip opening displacement

\({\sigma _{\text{t}}}\)

Tensile stress of the cohesive crack model


Length of the developing FPZ


Length of the fully developed FPZ


Coordinate along the FPZ growth path

\({b_1}\), \({b_2}\), \({b_3}\)

Slopes of \({\delta _{\text{c}}} - x\), \(\delta - x\) and \(\sigma - \delta\)


Accumulated dissipated energy of the FPZ


Dissipated energy per unit length of the FPZ


Combined modulus of material constants


Work done by the external forces


Stored strain energy


Cracked plate area


Elastic area of cracked plate


Effective elastic modulus obtained from the global responses (including both the elastic zone and the inelastic zone)


Elastic modulus of the linear elastic area

\({l_{{\text{min}}}}\), \({l_{{\text{max}}}}\)

FPZ length corresponding to the peak and valley values of the load in an arbitrary loading loop

\(\Delta l\)

FPZ growth length in an arbitrary loading loop

\(\Delta \sigma\)

Stress amplitude

\(\bar {\sigma }\)

Mean level of the stress amplitude


Experimentally fitted constant (\({E^{{\text{ae}}}}\) vs \(l\))


Experimentally fitted constant (\({l^{ - 1}}({\text{d}}{E^{{\text{ae}}}}/{\text{d}}N\)) versus (\({\text{d}}l/{\text{d}}N\))



This study was supported by the National Natural Science Foundation of China (General Program 51774299) and the Youth Innovation Team Program of China University of Petroleum-Beijing (C21601).


  1. Anderson TL (2005) Fracture mechanics: fundamentals and applications. CRC Press, Boca RatonCrossRefGoogle Scholar
  2. Bagde MN, Petroš V (2005) Fatigue properties of intact sandstone samples subjected to dynamic uniaxial cyclical loading. Int J Rock Mech Min Sci 42(2):237–250. CrossRefGoogle Scholar
  3. Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129. CrossRefGoogle Scholar
  4. Bažant ZP (2002) Concrete fracture models: testing and practice. Eng Fract Mech 60(2):165–205. CrossRefGoogle Scholar
  5. Bažant ZP, Schell WF (1993) Fatigue fracture of high-strength concrete and size effect. ACI Mater J 90(5):472–478Google Scholar
  6. Bažant ZP, Xu K (1991) Size effect in fatigue fracture of concrete. ACI Mater J 88(4):390–399Google Scholar
  7. Bieniawski ZT, Bernede MJ (1979) Suggested methods for determining the uniaxial compressive strength and deformability of rock materials: part 1. Suggested method for determining deformability of rock materials in uniaxial compression. Int J Rock Mech Min Sci Geomech Abstr 16(2):138–140. CrossRefGoogle Scholar
  8. Ding X, Zhang GQ, Zhao B, Wang Y (2017) Unexpected viscoelastic deformation of tight sandstone: insights and predictions from the fractional Maxwell model. Sci Rep 7(1):11336. CrossRefGoogle Scholar
  9. Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–104. CrossRefGoogle Scholar
  10. Erarslan N (2016) Microstructural investigation of subcritical crack propagation and fracture process zone (FPZ) by the reduction of rock fracture toughness under cyclic loading. Eng Geol 208:181–190. CrossRefGoogle Scholar
  11. Hillerborg A, Modéer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 6:773–781. CrossRefGoogle Scholar
  12. Kim K, Mubeen A (1981) Relationship between differential stress intensity factor and crack growth rate in cyclic tension in Westerly granite. In: ASTM International (ed) Fracture mechanics for ceramics, rocks, and concrete. Testing materials, ASTM International, Philadelphia, pp 157–168CrossRefGoogle Scholar
  13. Kolluru SV, O’Neil EF, Popovics JS, Shah SP (2000) Crack propagation in flexural fatigue of concrete. J Eng Mech 126:891–898. CrossRefGoogle Scholar
  14. Labuz JF, Shah SP, Dowding CH (1985) Experimental analysis of crack propagation in granite. Int J Rock Mech Min Sci Geomech Abstr 22:85–98. CrossRefGoogle Scholar
  15. Labuz JF, Shah SP, Dowding CH (1987) The fracture process zone in granite: evidence and effect. Int J Rock Mech Min Sci Geomech Abstr 24(4):235–246. CrossRefGoogle Scholar
  16. Landis EN, Baillon L (2002) Experiments to relate acoustic emission energy to fracture energy of concrete. J Eng Mech 128(6):698–702. CrossRefGoogle Scholar
  17. Le JL, Bažant ZP (2011) Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: II. Fatigue crack growth, lifetime and scaling. J Mech Phys Solids 59(7):1322–1337. CrossRefGoogle Scholar
  18. Le JL, Manning J, Labuz JF (2014) Scaling of fatigue crack growth in rock. Int J Rock Mech Min Sci 72:71–79. CrossRefGoogle Scholar
  19. Li G, Moelle KHR, Lewis JA (1992) Fatigue crack growth in brittle sandstones. Int J Rock Mech Min Sci Geomech Abstr 29(5):469–477. CrossRefGoogle Scholar
  20. Lin Q, Labuz JF (2013) Fracture of sandstone characterized by digital image correlation. Int J Rock Mech Min Sci 60:235–245. CrossRefGoogle Scholar
  21. Lu G, Gordeliy E, Prioul R, Bunger A (2017) Modeling initiation and propagation of a hydraulic fracture under subcritical conditions. Comput Methods Appl Mech Eng 318:61–91. CrossRefGoogle Scholar
  22. Otsuka K, Date H (2000) Fracture process zone in concrete tension specimen. Eng Fract Mech 65(2):111–131. CrossRefGoogle Scholar
  23. Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng 85(4):528–533CrossRefGoogle Scholar
  24. Paris PC, Gomez MP, Anderson WE (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14Google Scholar
  25. Perdikaris PC, Calomino AM, Chudnovsky A (1986) Effect of fatigue on fracture toughness of concrete. J Eng Mech 112(8):776–791. CrossRefGoogle Scholar
  26. Rice JR (1967) Mechanics of crack tip deformation and extension by fatigue. In: Fatigue crack propagation fatigue crack propagation. Special Technical Publication 415, American Society for Testing and Materials, Philadelphia, pp 247–309Google Scholar
  27. Sagar RV, Prasad BKR (2009) AE energy release during the fracture of HSC beams. Mag Concrete Res 61(6):419–435. CrossRefGoogle Scholar
  28. Scholz CH, Koczynski TA (1979) Dilatancy anisotropy and the response of rock to large cyclic loads. J Geophys Res Solid Earth 84:5525–5534. CrossRefGoogle Scholar
  29. Swartz SE, Go CG (1984) Validity of compliance calibration to cracked concrete beams in bending. Exp Mech 24(2):129–134. CrossRefGoogle Scholar
  30. Tao Z, Mo H (1990) An experimental study and analysis of the behaviour of rock under cyclic loading. Int J Rock Mech Min Sci Geomech Abstr 27(1):51–56. CrossRefGoogle Scholar
  31. Weertman J (1966) Rate of growth of fatigue cracks calculated from the theory of infinitesimal dislocations distributed on a plane. Int J Fract Mech 2(2):460–467. CrossRefGoogle Scholar
  32. Wu BS, Zhang GQ, Zhang X, Jeffrey RG, Kear J, Zhao T (2017) Semi-analytical model for a geothermal system considering the effect of areal flow between dipole wells on heat extraction. Energy 138:290–305. CrossRefGoogle Scholar
  33. Xiao JQ, Ding DX, Jiang FL, Xu G (2010) Fatigue damage variable and evolution of rock subjected to cyclic loading. Int J Rock Mech Min Sci 47(3):461–468. CrossRefGoogle Scholar
  34. Xing YK, Zhang GQ, Lin Q, Bu XQ, Da YP, Qi Y (2017) Subcritical fracture process of sandstone with AE energy analysis. In: Proceedings of the 51st U.S. rock mechanics/geomechanics symposium. American Rock Mechanics AssociationGoogle Scholar
  35. Zhang GQ, Xing YK, Wang LL (2018) Comprehensive sandstone fracturing characterization: Integration of fiber Bragg grating, digital imaging correlation and acoustic emission measurements. Eng Geol 246:45–56. CrossRefGoogle Scholar
  36. Zhao B, Zhang GQ, Zhao PY, Wang LL, Lin Y, Lv YJ (2017) Experimental study of mechanics and seepage characteristics of sandstones after liquid-nitrogen stimulation. J Nat Gas Sci Eng 47:11–21. CrossRefGoogle Scholar
  37. Zhou ZL, Zhang GQ, Dong HR, Liu ZB, Nie YX (2017) Creating a network of hydraulic fractures by cyclic pumping. Int J Rock Mech Min Sci 97:52–63. CrossRefGoogle Scholar
  38. Zhou DW, Zhang GQ, Zhao PY, Wang YY, Xu SF (2018) Effects of post-instability induced by supercritical CO2, phase change on fracture dynamic propagation. J Pet Sci Eng 162:358–366. CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Yuekun Xing
    • 1
  • Guangqing Zhang
    • 2
    Email author
  • Bin Wan
    • 1
  • Hui Zhao
    • 1
  1. 1.Department of Engineering Mechanics, College of Petroleum EngineeringChina University of PetroleumBeijingPeople’s Republic of China
  2. 2.State Key Laboratory of Petroleum Resources and ProspectingChina University of PetroleumBeijingPeople’s Republic of China

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