Abstract
Study of intact rock failure criteria is an important topic in rock mechanics. In this study, applicability of nine different intact rock failure criteria is investigated for intact coal strength data. PFC3D modeling was used to simulate the laboratory polyaxial tests for cubic intact coal blocks of side dimension 110 mm under different confining stress combinations. A modified grid search procedure is proposed and used to find the best-fitting parameter values and to calculate the coefficient of determination (R 2) values for each criterion. Detailed comparisons of the nine criteria are made using the following aspects: R 2 values, σ 1 − σ 2 plots for different σ 3, shapes on the deviatoric plane, linearity or nonlinearity on the meridian planes. Through the comparisons of R 2 values, σ 1 − σ 2 plots and meridian lines, the modified Wiebols–Cook and modified Lade criteria were found to fit the intact coal strength data best. The nine failure criteria are categorized into three types based on the appearances on the deviatoric plane.
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Abbreviations
- σ 1, σ 2, σ 3 :
-
Major, intermediate and minor principal stresses at failure, respectively
- σ 1′, σ 2′, σ 3′:
-
Major, intermediate and minor effective principal stresses at failure, respectively
- I 1, I 3 :
-
First and third invariant of stress tensor
- I 1′, I 3′:
-
Modified first and third invariant of stress tensor
- J 2 :
-
Second invariant of deviatoric stress tensor
- σ oct :
-
Octahedral normal stress or mean stress
- τ oct :
-
Octahedral shear stress
- σ m,2 :
-
(σ 1 + σ 3)/2
- σ ci :
-
Uniaxial compressive strength of the intact rock
- σ ti :
-
Uniaxial tensile strength of the intact rock
- σ bi :
-
Strength of the intact rock under triaxial extension stress state for σ 3 = 0
- τ :
-
Shear stress
- σ n :
-
Normal stress
- τ 13 :
-
Major principal shear stress
- τ 12 :
-
Intermediate principal shear stress or minor principal shear stress
- τ 23 :
-
Intermediate principal shear stress or minor principal shear stress
- σ 13 :
-
Normal stresses acting on the τ 13 plane
- σ 12 :
-
Normal stresses acting on the τ 12 plane
- σ 23 :
-
Normal stresses acting on the τ 23 plane
- q :
-
Material constant determined by a certain function of the coefficient of friction in the W–C criterion
- c 0 :
-
Cohesion of the intact rock
- φ :
-
Internal friction angle of the intact rock
- m b :
-
Hoek–Brown constant for the rock mass
- m i :
-
Hoek–Brown constant for the intact rock
- s, a :
-
Constants depending on the rock mass characteristics
- D :
-
Disturbance factor
- GSI :
-
Geological Strength Index
- α, k :
-
Material constants of the D–P criterion
- ρ c , ρ t :
-
Compressive and tensile meridian on the deviatoric planes
- κ :
-
Material constant depending on the density of the soil in Lade–Duncan criterion (1975)
- p a :
-
Atmospheric pressure
- m′, η 1 :
-
Material constants of the modified Lade criterion (1977)
- S :
-
Material constant related to the cohesion and internal friction angle of the rock in the modified Lade criterion (1999)
- η :
-
Material constant representing the internal friction of the rock
- A, B, C :
-
Parameters in the modified W–C criterion
- β :
-
Constant smaller than 1 in Mogi (1967) criterion
- C′:
-
Parameter related to c 0 and φ in the linear unified failure criterion
- δ :
-
Parameter related to φ in the linear unified failure criterion
- b :
-
Intermediate stress parameter in both linear and nonlinear unified failure criteria
- ω :
-
Ratio of σ ci to σ ti for the intact rock in the linear unified failure criterion
- θ :
-
Lode angle
- ρ, ξ :
-
Abscissa and ordinate of the plots on the meridian planes
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Acknowledgements
The research was funded by the National Institute for Occupational Safety and Health (NIOSH) of the Centers for Disease Control and Prevention (Contract No. 200-2011-39886) and the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology, Beijing (Contract No. SKLGDUEK1416).
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He, Pf., Kulatilake, P.H.S.W., Yang, Xx. et al. Detailed comparison of nine intact rock failure criteria using polyaxial intact coal strength data obtained through PFC3D simulations. Acta Geotech. 13, 419–445 (2018). https://doi.org/10.1007/s11440-017-0566-9
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DOI: https://doi.org/10.1007/s11440-017-0566-9