Abstract
The brittle fracture behavior of rocks under mixed-mode loading is important in rock engineering. First, a new configuration called the notched deep beam (NDB) specimen was introduced for the fracture testing of rock materials under mixed-mode I/II loading, and a series of finite element analyses were performed to calibrate the dimensionless fracture parameters (i.e., Y I, Y II and \(T^{*}\)). The results showed that an NDB specimen subjected to three-point bending is able to generate pure mode I loading, pure mode II loading, and any mixed-mode loading in between. Then, several NDB specimens made of sandstone were used to investigate the brittle fracture behavior of rock under mixed-mode I/II loading. The fracture surfaces were theoretically described using a statistical method, and the results indicated that all the fracture surfaces generated under different mixed-mode loading were statistically identical; to some extent, these results experimentally showed that only tensile fracture occurs under mixed-mode I/II loading. The obtained fracture strengths were then analyzed using several brittle fracture criteria. The empirical criterion, maximum energy release rate criterion, generalized maximum tangential stress (GMTS) criterion, and improved R-criterion accurately predicted the fracture strength envelope of the sandstone. Finally, based on the concepts of point stress and mean stress, the micro-crack zones (MCZs) under different mixed-mode loading were theoretically estimated based on the MTS and GMTS criteria. The critical radius of MCZ in the crack propagation direction was not a constant for all mixed-mode loading conditions regardless of whether the T-stress was considered. This result suggests that the size of the core region used to predict the crack initiation direction and fracture strength based on the GMTS criterion should be chosen more carefully.
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Abbreviations
- a :
-
Crack length
- B :
-
Thickness of an NDB specimen
- C :
-
= 1/(2π) for the PS case, = 2/π for the MS case
- d :
-
Half-distance between the two bottom supports for an NDB specimen
- E :
-
Young’s modulus of elasticity
- f 1, f 2 :
-
Parameters related to the scale of the micro-crack zone
- G :
-
Energy release rate
- G c :
-
Critical energy release rate
- h 1, h 2, h 3, h 4 :
-
Heights corresponding to the four vertexes of a grid cell on the surface
- K I :
-
Mode I stress intensity factor
- K Ic :
-
Mode I fracture toughness
- K II :
-
Mode II stress intensity factor
- K IIc :
-
Mode II fracture toughness
- L :
-
Length of an NDB specimen
- P :
-
Applied load in an NDB three-point bending test
- P cr :
-
Peak load in an NDB three-point bending test
- r, ϑ :
-
Polar coordinates at the crack tip
- r 0 :
-
Critical radius of the core region in the crack propagation direction
- r c :
-
Radius of the core region
- R II,I :
-
Fracture toughness ratio
- r p :
-
Elastic–plastic boundary radius
- r pc :
-
Critical elastic–plastic boundary radius
- S :
-
Strain energy density factor
- S c :
-
Critical strain energy density factor
- T :
-
Nonsingular stress term
- \(T^{*}\) :
-
Nondimensional nonsingular stress term
- W :
-
Width of an NDB specimen
- Y I :
-
Nondimensional mode I stress intensity factor
- Y II :
-
Nondimensional mode II stress intensity factor
- α :
-
Crack inclination angle for an NDB specimen
- β I, β II :
-
Empirical parameters of the empirical criterion
- δ :
-
Scale of each grid cell
- η I, η II :
-
Parameters of the loading mode mixity
- κ :
-
= 3 − 4ν for the plane strain case, = (3 − ν)/(1 + ν) for the plane stress case
- ν :
-
Poisson’s ratio
- μ :
-
Shear modulus
- ξ :
-
= 2(1 − 2\(\nu^{*}\) )/3; \(\nu^{*}\) = ν for the plane strain case, \(\nu^{*}\) = 0 for the plane stress case
- σ rr , σ ϑϑ , σ rϑ :
-
Stress components in polar coordinates
- σ t :
-
Tensile strength of a given material
- σ xx , σ yy , σ xy :
-
Stress components in Cartesian coordinates
- σ ϑϑc :
-
Critical tangential stress
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Acknowledgements
The authors express their sincere gratitude to all the anonymous reviewers for their comments devoted to improving the quality of our paper. This paper was financially supported by the Major State Research Projects (2016YFC0600702) and Provincial Science and Technology Support Project of Sichuan Province (2015JY0280).
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Luo, Y., Ren, L., Xie, L.Z. et al. Fracture Behavior Investigation of a Typical Sandstone Under Mixed-Mode I/II Loading Using the Notched Deep Beam Bending Method. Rock Mech Rock Eng 50, 1987–2005 (2017). https://doi.org/10.1007/s00603-017-1227-x
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DOI: https://doi.org/10.1007/s00603-017-1227-x