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

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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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\(\beta\) :

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)


Velocity anisotropy index (%)


Maximum absolute relative error


Average absolute relative error


Standard error


  1. Aliabadian Z, Zhao GF, Russell AR (2017) An analytical study of failure of transversely isotropic rock discs subjected to various diametrical loading configurations. Proc Eng 191:1194–1202. https://doi.org/10.1016/j.proeng.2017.05.295

  2. Amadei B (1996) Importance of anisotropy when estimating and measuring in situ stresses in rock. Int J Rock Mech Min Sci Geomech Abst 33:293–325. https://doi.org/10.1016/0148-9062(95)00062-3

  3. Amadei B (2012) Rock anisotropy and the theory of stress measurements, vol 2. Springer Science & Business Media, Berlin

  4. ASTM (2016) Standard test method for splitting tensile strength of intact rock core specimens. ASTM International, West Conshohocken

  5. Bahaaddini M, Serati M, Masoumi H, Rahimi E (2019) Numerical assessment of rupture mechanisms in Brazilian test of brittle materials. Int J Solids Struct 180–181:1–12. https://doi.org/10.1016/j.ijsolstr.2019.07.004

  6. Barla G, Innaurato N (1973) Indirect tensile testing of anisotropic rocks. Rock mechanics 5:215–230. https://doi.org/10.1007/bf01301795

  7. Barron K (1971) Brittle fracture initiation in and ultimate failure of rocks: part II—anisotropic rocks: theory. Int J Rock Mech Min Sci Geomech Abst 8:553–563

  8. Bažant ZP (1984) Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 110:518–535

  9. Bažant ZP (1997) Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection. Int J Fract 83:41. https://doi.org/10.1023/a:1007335506684

  10. Cai M, Kaiser PK (2004) Numerical simulation of the Brazilian test and the tensile strength of anisotropic rocks and rocks with pre-existing cracks. Int J Rock Mech Min Sci 41:478–483. https://doi.org/10.1016/j.ijrmms.2004.03.086

  11. Carpinteri A, Chiaia B, Ferro G (1995) Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Mater Struct 28:311. https://doi.org/10.1007/bf02473145

  12. Chen C-S, Hsu S (2001) Measurement of indirect tensile strength of anisotropic rocks by the ring test. Rock Mech Rock Eng 34:293–321

  13. Chen C-S, Pan E, Amadei B (1998) Determination of deformability and tensile strength of anisotropic rock using Brazilian tests. Int J Rock Mech Min Sci 35:43–61. https://doi.org/10.1016/S0148-9062(97)00329-X

  14. Cho J-W, Kim H, Jeon S, Min K-B (2012) Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist. Int J Rock Mech Min Sci 50:158–169. https://doi.org/10.1016/j.ijrmms.2011.12.004

  15. Chou YC, Chen CS (2008) Determining elastic constants of transversely isotropic rocks using Brazilian test and iterative procedure. Int J Numer Anal Meth Geomech 32:219–234

  16. Claesson J, Bohloli B (2002) Brazilian test: stress field and tensile strength of anisotropic rocks using an analytical solution. Int J Rock Mech Min Sci 39:991–1004. https://doi.org/10.1016/S1365-1609(02)00099-0

  17. Dan DQ, Konietzky H, Herbst M (2013) Brazilian tensile strength tests on some anisotropic rocks. Int J Rock Mech Min Sci 58:1–7. https://doi.org/10.1016/j.ijrmms.2012.08.010

  18. Darlington WJ, Ranjith PG, Choi SK (2011) The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials. Rock Mech Rock Eng 44:513. https://doi.org/10.1007/s00603-011-0161-6

  19. Exadaktylos GE, Kaklis KN (2001) Applications of an explicit solution for the transversely isotropic circular disc compressed diametrically. Int J Rock Mech Min Sci 38:227–243. https://doi.org/10.1016/S1365-1609(00)00072-1

  20. Fairhurst C (1971) Fundamental considerations relating to the strength of rock. In: Colloquium on rock fracture, Ruhr University, Bochum, Germany, Veröff. Inst. Bodenmechanik und Felsmechanik (Karlsruhe). 1971. Citeseer, pp 1–56

  21. Fu H, Zhang J, Huang Z, Shi Y, Chen W (2018) A statistical model for predicting the triaxial compressive strength of transversely isotropic rocks subjected to freeze–thaw cycling. Cold Reg Sci Technol 145:237–248. https://doi.org/10.1016/j.coldregions.2017.11.003

  22. Goodman RE (1989) Introduction to rock mechanics, vol 2. Wiley, New York

  23. Hobbs DW (1964) The tensile strength of rocks. Int J Rock Mech Min Sci Geomech Abst 1:385–396. https://doi.org/10.1016/0148-9062(64)90005-1

  24. Hobbs D (1967) Rock tensile strength and its relationship to a number of alternative measures of rock strength. Int J Rock Mech Min Sci Geomech Abst 4:115–127

  25. Hoek E, Brown ET (1980) Underground excavations in rock. Institution of Mining and Metallurgy, Spon Press, Hertford

  26. Hu S, Tan Y, Zhou H, Guo W, Hu D, Meng F, Liu Z (2017) Impact of bedding planes on mechanical properties of sandstone. Rock Mech Rock Eng 50:2243–2251. https://doi.org/10.1007/s00603-017-1239-6

  27. ISRM (1978) Suggested methods for determining tensile strength of rock materials. Int J Rock Mech Min Sci Geomech Abst 15:99–103. https://doi.org/10.1016/0148-9062(78)90003-7

  28. Khanlari G, Rafiei B, Abdilor Y (2015) An experimental investigation of the Brazilian tensile strength and failure patterns of laminated sandstones. Rock Mech Rock Eng 48:843–852

  29. Komurlu E, Kesimal A (2015) Evaluation of indirect tensile strength of rocks using different types of jaws. Rock Mech Rock Eng 48:1723–1730

  30. Kundu J, Mahanta B, Sarkar K, Singh TN (2018) The effect of lineation on anisotropy in dry and saturated himalayan schistose rock under Brazilian test conditions. Rock Mech Rock Eng 51:5–21. https://doi.org/10.1007/s00603-017-1300-5

  31. Lee Y-K, Pietruszczak S (2015) Tensile failure criterion for transversely isotropic rocks. Int J Rock Mech Min Sci 79:205–215. https://doi.org/10.1016/j.ijrmms.2015.08.019

  32. Li D, Wong LNY (2013) The Brazilian disc test for rock mechanics applications: review and new insights. Rock Mech Rock Eng 46:269–287. https://doi.org/10.1007/s00603-012-0257-7

  33. Li K, Cheng Y, Fan X (2018) Roles of model size and particle size distribution on macro-mechanical properties of Lac du Bonnet granite using flat-joint model. Comput Geotech 103:43–60. https://doi.org/10.1016/j.compgeo.2018.07.007

  34. Li K, Yin Z, Cheng Y, Cao P, Fan X, Meng J (2019) Size effect and anisotropy in a transversely isotropic rock under compressive conditions. International Journal of Rock Mechanics and Mining Sciences (submitted for publication)

  35. Liao JJ, Yang M-T, Hsieh H-Y (1997) Direct tensile behavior of a transversely isotropic rock. Int J Rock Mech Min Sci 34:837–849. https://doi.org/10.1016/S1365-1609(96)00065-4

  36. Lin H, Xiong W, Yan Q (2015) Three-dimensional effect of tensile strength in the standard Brazilian test considering contact length. Geotech Test J 39:137–143

  37. Loureiro-Pinto J (1979) Determination of the elastic constants of anisotropic bodies by diametral compression tests. In: 4th ISRM Congress, 1979. International Society for Rock Mechanics

  38. Ma T, Zhang QB, Chen P, Yang C, Zhao J (2017) Fracture pressure model for inclined wells in layered formations with anisotropic rock strengths. J Pet Sci Eng 149:393–408. https://doi.org/10.1016/j.petrol.2016.10.050

  39. Ma T, Peng N, Zhu Z, Zhang Q, Yang C, Zhao J (2018) Brazilian tensile strength of anisotropic rocks: review and new insights. Energies 11:304

  40. Markides CF, Kourkoulis SK (2012) The stress field in a standardized Brazilian disc: the influence of the loading type acting on the actual contact length. Rock Mech Rock Eng 45:145–158. https://doi.org/10.1007/s00603-011-0201-2

  41. Masoumi H, Saydam S, Hagan PC (2015) Unified size-effect law for intact rock. Int J Geomech 16:04015059

  42. Masoumi H, Serati M, Williams DJ, Alehossein H (2017) Size dependency of intact rocks with high brittleness: a potential solution to eliminate secondary fractures in Brazilian test. In: 51st US Rock Mechanics/Geomechanics Symposium, 2017. American Rock Mechanics Association

  43. Masoumi H, Roshan H, Hedayat A, Hagan PC (2018) Scale-size dependency of intact rock under point-load and indirect tensile brazilian testing. Int J Geomech 18:04018006

  44. Mighani S, Sondergeld CH, Rai CS (2016) Observations of tensile fracturing of anisotropic rocks. SPE J 21:1289–2812301

  45. Nova R, Zaninetti A (1990) An investigation into the tensile behaviour of a schistose rock. In: International Journal of rock mechanics and mining sciences & geomechanics abstracts, 1990. vol 4. Elsevier, pp 231–242

  46. Quiñones J, Arzúa J, Alejano LR, García-Bastante F, Mas Ivars D, Walton G (2017) Analysis of size effects on the geomechanical parameters of intact granite samples under unconfined conditions. Acta Geotech. https://doi.org/10.1007/s11440-017-0531-7

  47. Rocco C, Guinea GV, Planas J, Elices M (1999) Size effect and boundary conditions in the Brazilian test: experimental verification. Mater Struct 32:210. https://doi.org/10.1007/bf02481517

  48. Saint Venant B (1863) Sur la distribution des élasticités autour de chaque point d’un solide ou d’un milieu de contexture quelconque, particulièrement lorsqu’il est amorphe sans être isotrope. J Math Pures Appl 8:257–430

  49. Serati M, Masoumi H, Williams DJ, Alehossein H (2017) Modified Brazilian test for indirect measurement of tensile strength of brittle materials. In: 51st US Rock Mechanics/Geomechanics Symposium, 2017. American Rock Mechanics Association,

  50. Sesetty V, Ghassemi A (2018) Effect of rock anisotropy on wellbore stresses and hydraulic fracture propagation. Int J Rock Mech Min Sci 112:369–384. https://doi.org/10.1016/j.ijrmms.2018.09.005

  51. Setiawan NB, Zimmerman RW (2018) Wellbore breakout prediction in transversely isotropic rocks using true-triaxial failure criteria. Int J Rock Mech Min Sci 112:313–322. https://doi.org/10.1016/j.ijrmms.2018.10.033

  52. Shang J, Hencher SR, West LJ (2016) Tensile strength of geological discontinuities including incipient bedding, rock joints and mineral veins. Rock Mech Rock Eng 49:4213–4225. https://doi.org/10.1007/s00603-016-1041-x

  53. Tavallali A, Vervoort A (2010) Failure of layered sandstone under brazilian test conditions: effect of micro-scale parameters on macro-scale behaviour. Rock Mech Rock Eng 43:641–653. https://doi.org/10.1007/s00603-010-0084-7

  54. Tavallali A, Vervoort A (2013) Behaviour of layered sandstone under Brazilian test conditions: layer orientation and shape effects. J Rock Mech Geotech Eng 5:366–377. https://doi.org/10.1016/j.jrmge.2013.01.004

  55. Togashi Y, Kikumoto M, Tani K (2017) An experimental method to determine the elastic properties of transversely isotropic rocks by a single triaxial test. Rock Mech Rock Eng 50:1–15. https://doi.org/10.1007/s00603-016-1095-9

  56. Tsidzi K (1997) Propagation characteristics of ultrasonic waves in foliated rocks. Bull Int Assoc Eng Geol 56:103–114

  57. Vervoort A et al (2014) Failure of transversely isotropic rock under Brazilian test conditions. Int J Rock Mech Min Sci 70:343–352. https://doi.org/10.1016/j.ijrmms.2014.04.006

  58. Walton G (2017) Scale effects observed in compression testing of stanstead granite including post-peak strength and dilatancy. Geotech Geol Eng. https://doi.org/10.1007/s10706-017-0377-7

  59. Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297

  60. Xu G, He C, Chen Z, Wu D (2018) Effects of the micro-structure and micro-parameters on the mechanical behaviour of transversely isotropic rock in Brazilian tests. Acta Geotech 13:887–910. https://doi.org/10.1007/s11440-018-0636-7

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