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Subcritical Fracturing of Sandstone Characterized by the Acoustic Emission Energy

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

Introduction

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 (\({\text{d}}a/{\text{d}}N=C{\left( {\Delta {K_{\text{I}}}} \right)^m}\)

Keywords

Subcritical fracture General law Fracture process zone AE energy analysis 

List of Symbols

a

Effective crack length

\(N\)

Number of loading cycles

\(C\)

Paris constant

\(m\)

Paris exponent

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

Amplitude of the stress intensity factor in a load cycle

\(E\)

Elastic modulus

\(\nu\)

Poisson’s ratio

\(r\), \(b\)

Specimen radius and thickness

\({a_0}\)

Specimen notch length

\(s\)

Specimen span length

\({r_{\text{p}}}\)

Pore radii

\({\text{UCS}}\)

Uniaxial compressive strength

\({T_0}\)

Tensile strength

MFL

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

\({C_{\text{m}}}\)

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

\(R\)

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

\({E^{{\text{ae}}}}\)

AE energy

\(E_{i}^{{{\text{ae}}}}\)

AE energy of a waveform

\(d\)

Distance between the AE source and a sensor

\(n\)

Number of synchronous waveforms

\({a_{\text{e}}}\)

Length of the effective crack

\({K_{{\text{eff}}}}\)

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

\(\sigma\)

Closing cohesive stress

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

External tensile stress of the cracked plate

\(\delta\)

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

\(l\)

Length of the developing FPZ

\(L\)

Length of the fully developed FPZ

\(x\)

Coordinate along the FPZ growth path

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

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

\({G_{\text{D}}}\)

Accumulated dissipated energy of the FPZ

\({G_{\text{f}}}\)

Dissipated energy per unit length of the FPZ

\({C_{\text{G}}}\)

Combined modulus of material constants

\({U_{\text{w}}}\)

Work done by the external forces

\({U_{\text{c}}}\)

Stored strain energy

\(A\)

Cracked plate area

\({A_{\text{e}}}\)

Elastic area of cracked plate

\({E_{\text{p}}}\)

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

\({E_{\text{e}}}\)

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

\({B^*}\)

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

\({A^*}\)

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

Notes

Acknowledgements

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).

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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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