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Size Effects in a Transversely Isotropic Rock Under Brazilian Tests: Laboratory Testing

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Abstract

A transversely isotropic rock, slate, was utilized to investigate the size effect and anisotropy on its deformation, tensile strength, and failure mechanism. A series of Brazilian tests were conducted on slate samples of six different sizes from 25 to 100 mm in diameter at seven different loading-foliation angles from 0° to 90°. The results indicate that the Young’s modulus in the plane of transverse isotropy increases, while the Young’s modulus and shear modulus perpendicular to the plane of transverse isotropy decrease with specimen size. The tensile strength of the slate increases with increasing loading-foliation angle, the variation of which is well captured by the Nova–Zaninetti criterion. Furthermore, the tensile strength of the slate increases with specimen size at loading-foliation angles from 0° to 45°, while it increases first and then decreases with specimen size at loading-foliation angles from 60° to 90°. A unified size-effect relation including two equations is proposed and verified against the experimental data on slate. The size-effect relation reveals the relationship among the tensile strength, specimen size, and loading-foliation angle for the transversely isotropic rock. Finally, the slate samples exhibit an increased brittle failure with specimen size, which is consistent with the observations in various isotropic rocks. It is also found that the specimen size, loading-foliation angle, and loading configuration together control the failure mechanism of transversely isotropic rocks in the Brazilian test.

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Abbreviations

\(\beta\) :

Angle between the loading direction and the transversely isotropic plane (°)

\(\sigma_{x} ,\;\sigma_{y} \;{\text{and}}\;\tau_{xy}\) :

Stresses in global coordinate system (MPa)

\(\varepsilon_{x} ,\;\varepsilon_{y} ,\;{\text{and}}\;\gamma_{xy}\) :

Strains in global coordinate system

\(q_{xx} ,\;q_{yy} ,\;{\text{and}}\;q_{xy}\) :

Stress concentration factors

\(a_{ij}\) :

Compliance matrix

\(E\;{\text{and}}\;E^{\prime}\) :

Elastic moduli parallel to and perpendicular to the plane of transverse isotropy, respectively (GPa)

\(\nu \;{\text{and}}\;\nu^{\prime}\) :

Poisson’s ratios parallel to and perpendicular to the plane of transverse isotropy, respectively

\(G^{\prime}\) :

Shear modulus normal to the transversely isotropic plane (MPa)

\(T(\beta )\) :

Tensile strength of a specimen at \(\beta\) (MPa)

\(T_{\text{m}}\) and \(T_{\text{b}}\) :

Tensile strength of rock matrix and weak plane, respectively (MPa)

\(\sigma_{\text{t}}\) :

Tensile strength (MPa)

\(d\) :

Sample diameter (mm)

\(k\) :

Positive constant

\(\sigma_{{{\text{t}}50}}\) :

Tensile strength obtained from a specimen of 50 mm in diameter (MPa)

\(B\;{\text{and}}\;\lambda\) :

Dimensionless material constants

\(f_{\text{t}} \;{\text{and}}\;\sigma_{0}\) :

Strength of a specimen with an infinitesimal size (MPa)

\(f_{\text{c}}\) :

Strength of a specimen with an infinite size (MPa)

\(d_{0}\) :

Maximum aggregate size (mm)

\(l\) :

Material constant (mm)

\(d_{\text{f}}\) :

Fractal dimension

\(v_{\hbox{max} }\), \(v_{\hbox{min} }\) and \(v_{\text{mean}}\) :

Maximum, minimum, and average ultrasonic wave velocities, respectively (m/s)

\(T_{1} (d)\) and \(T_{2} (d)\) :

Tensile strength of a specimen with the diameter of \(d\) (MPa)

\(\sigma_{0}\) and \(\sigma_{\text{M}}\) :

Tensile strength when \(d \to 0\) and \(d \to \infty\), respectively (MPa)

\(\bar{\sigma }_{0}\) :

Tensile strength when \(d \to 0\) (MPa)

\(T_{1} (d,\beta )\) and \(T_{2} (d,\beta )\) :

Tensile strength of a specimen with d at β (MPa)

\(T_{{{\text{b}}0}}\) and \(T_{{{\text{m}}0}}\) :

Tensile strength of the weak plane and the rock matrix, respectively, when \(d \to 0\) (MPa)

\(T_{\text{bM}}\) and \(T_{\text{mM}}\) :

Tensile strength of the weak plane and the rock matrix, respectively, when \(d \to \infty\) (MPa)

\(d_{i}\) :

Specimen diameter at which the maximum tensile strength reaches (mm)

VA:

Velocity anisotropy index (%)

MARE:

Maximum absolute relative error

AARE:

Average absolute relative error

SE:

Standard error

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Acknowledgements

The authors would like to thank Dr. H. Masoumi for his precious suggestions and Mr. R. Leung for his assistance during the experiments. The work in this paper is financially supported by the Hong Kong Polytechnic University (account RUF4), National Natural Science Foundation of China (Grant No. 51778313), and Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone.

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Correspondence to Yungming Cheng.

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Li, K., Cheng, Y., Yin, Z. et al. Size Effects in a Transversely Isotropic Rock Under Brazilian Tests: Laboratory Testing. Rock Mech Rock Eng (2020). https://doi.org/10.1007/s00603-020-02058-7

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Keywords

  • Slate
  • Transversely isotropic rock
  • Brazilian test
  • Tensile strength
  • Size effect
  • Anisotropy