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Tensile strength and elastic modulus determined in the Brazilian test: Theory and experiment

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Abstract

The real distribution of pressure on the disc in the Brazilian test in this paper is considered to be nonuniform. The simplest full-field analytical solutions for the stress and the displacement fields in a Brazilian disc under nonuniform pressure are given by the integral method. The formulae for tensile strength and elastic modulus determined by the Brazilian test under nonuniform pressures are presented based on the Griffith criterion and the analytical solutions in this paper. The Brazilian splitting test is carried out with uncracked sandstone discs under plat platens and curved jaws, and the contact angle is measured by the indentation method. The results show that the effects of the load distributions are great on the stress and the displacement fields near the loaded area of the disc. Using the formula recommended by the American Society for Testing and Materials (ASTM) and the International Society for Rock Mechanics (ISRM), the tensile strength of rock materials is underestimated or overestimated when the contact angle is small or large. With the increase in the contact angle, the initiation position of the crack moves from the edge to the centre of the disc, and the maximum load for the failure of the disc also increases. The distribution form has a certain influence on the determination of the tensile strength and the elastic modulus. The influence of the distribution form on the determination of tensile strength could be ignored when the contact angle γ ≥ 15°. The experimental results show that the results of the tensile strength calculated with the formula recommended by the ASTM and the ISRM are smaller than those in this paper. It is suggested that the calculation formula for the tensile strength in the Brazilian test be modified using the formula in this paper when γ ≥ 13.84°.

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Abbreviations

a n, b n, c n (n = i, i − 1):

Coefficients

ASTM:

American Society for Testing and Materials

A ji, C ji (j = 1, 2, 3; i = 1, 2,…,n) :

Angle coefficients

B :

Thickness of disc

C 0 :

Coefficient

C 1, C 2 :

Integral constants

E :

Elastic modulus

f i :

Coefficient

F E :

Normalised elastic modulus

F G :

Normalised Griffith equivalent stress

F u, F v :

Normalised radial and hoop displacements

F θ, F rθ, F r :

Normalised hoop, tangential, and radial stresses

F σt :

Normalised tensile strength

ISRM:

International Society for Rock Mechanics

J 1 :

First-order Bessel function

k :

Slope of the linear section in the load–displacement curve

n :

Sum coefficient

P :

Compression load

P max :

Maximum compression load

q 0 :

Coefficient

r, θ :

Polar coordinates

R :

Radius of disc

u, v :

Radial and hoop displacements

w :

Maximum displacement

γ :

Contact angle

δ 1, δ 2 :

Relative errors

ΔP, Δw :

Variations of load and displacement

ε r, ε θ :

Radial and hoop strains

μ :

Poisson's ratio

ρ :

Normalised coordinate

σ :

Coefficient

σ G :

Griffith equivalent stress

σ t :

Tensile strength

σ 0 :

Maximum distribution intensity of pressure

σ 1, σ 3 :

First and third principal stresses

σ θ, τ rθ, σ r :

Hoop, tangential, and radial stresses

References

  1. Zhou YX, Xia K, Li XB, Li HB, Ma GW, Zhao J et al (2012) Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. Int J Rock Mech Min Sci 49:105–112

    Article  Google Scholar 

  2. Yin TB, Li XB, Cao WZ, Xia KW (2015) Effects of thermal treatment on tensile strength of laurentian granite using Brazilian test. Rock Mech Rock Eng 48:2213–2223

    Article  Google Scholar 

  3. Yu JH, Shang XC, Wu PF (2019) Influence of pressure distribution and friction on determining mechanical properties in the Brazilian test: theory and experiment. Int J Solids Struct 161:11–22

    Article  Google Scholar 

  4. Kourkoulis SK, Markides CF, Pasiou ED (2015) A combined analytic and experimental study of the displacement field in a circular ring. Meccanica 50:493–515

    Article  Google Scholar 

  5. Aliha MRM, Ebneabbasi P, Karimi HR, Nikbakht E (2021) A novel test device for the direct measurement of tensile strength of rock using ring shape sample. Int J Rock Mech Min Sci 139:12

    Article  Google Scholar 

  6. Aliha MRM, Heidan-Rarani M, Shokrieh MM, Ayatollahi MR (2012) Experimental determination of tensile strength and K-Ic of polymer concretes using semi-circular bend (SCB) specimens. Struct Eng Mech 43:823–833

    Article  Google Scholar 

  7. Carneiro FLLB A new method to determine the tensile strength of concrete.In: 5th meeting of the Brazilian association for technical rules1943. pp 126–129

  8. Akazawa T (1943) New test method for evaluating internal stress due to compression of concrete (the splitting tension test) (Part 1). J Jpn Soc Civil Eng 29:777–787

    Google Scholar 

  9. ASTM. D3967-95a Standard test method for splitting tensile strength of intact rock core specimens. West Conshohocken, PA1995

  10. Bieniawski ZT, Hawkes I (1978) Suggested methods for determining tensile strength of rock materials. Int J R Mech Min Sci Geomech Abstr 15:99–103

    Article  Google Scholar 

  11. Fairhurst C (1964) On the validity of the ‘Brazilian’ test for brittle materials. Int J Rock Mech Min Sci Geomech Abstr 1:535–546

    Article  Google Scholar 

  12. Erarslan N, Williams DJ (2012) Experimental, numerical and analytical studies on tensile strength of rocks. Int J Rock Mech Min Sci 49:21–30

    Article  Google Scholar 

  13. Wang J, Xie L, Xie H, Ren L, He B, Li C et al (2016) Effect of layer orientation on acoustic emission characteristics of anisotropic shale in Brazilian tests. J Nat Gas Sci Eng 36:1120–1129

    Article  Google Scholar 

  14. Hertz H (1882) Ueber die Berührung fester elastischer Körper. J für die Reine und Angewandte Mathematik 92:156–171

    MathSciNet  MATH  Google Scholar 

  15. Kourkoulis SK, Markides CF, Chatzistergos PE (2013) The standardized Brazilian disc test as a contact problem. Int J Rock Mech Min Sci 57:132–141

    Article  Google Scholar 

  16. Hong T, Saka N (1991) Finite element analysis of an elastic-plastic two-layer half-space: mormal contact. Wear 148:47–68

    Article  Google Scholar 

  17. Japaridze L (2015) Stress-deformed state of cylindrical specimens during indirect tensile strength testing. J Rock Mech Geotech Eng 7:509–518

    Article  Google Scholar 

  18. 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

    Article  Google Scholar 

  19. Kourkoulis SK, Markides CF, Chatzistergos PE (2012) The Brazilian disc under parabolically varying load: theoretical and experimental study of the displacement field. Int J Solids Struct 49:959–972

    Article  Google Scholar 

  20. Guerrero-Miguel DJ, Alvarez-Fernandez MI, Garcia-Fernandez CC, Gonzalez-Nicieza C, Menendez-Fernandez C (2019) Analytical and numerical stress field solutions in the Brazilian Test subjected to radial load distributions and their stress effects at the centre of the disk. J Eng Math 116:29–48

    Article  MathSciNet  Google Scholar 

  21. Markides CF, Pazis DN, Kourkoulis SK (2012) The Brazilian disc under non-uniform distribution of radial pressure and friction. Int J Rock Mech Min Sci 50:47–55

    Article  Google Scholar 

  22. Atkinson C, Smelser RE, Sanchez J (1982) Combined mode fracture via the cracked Brazilian disk test. Int J Fract 18:279–291

    Article  Google Scholar 

  23. Dong S, Wang Y, Xia Y (2004) Stress intensity factors for central cracked circular disk subjected to compression. Eng Fract Mech 71:1135–1148

    Article  Google Scholar 

  24. Gutierrez-Moizant R, Ramirez-Berasategui M, Santos-Cuadros S, Garcia-Fernandez CC (2020) A novel analytical solution for the Brazilian test with loading arcs. Math Probl Eng 2020:19

    Article  MathSciNet  Google Scholar 

  25. Ye JH, Wu FQ, Sun JZ (2009) Estimation of the tensile elastic modulus using Brazilian disc by applying diametrically opposed concentrated loads. Int J Rock Mech Min Sci 46:568–576

    Article  Google Scholar 

  26. Aliha MRM (2014) Indirect tensile test assessments for rock materials using 3-D disc-type specimens. Arab J Geosci 7:4757–4766

    Article  Google Scholar 

  27. Heidari-Rarani M, Aliha MRM, Shokrieh MM, Ayatollahi MR (2014) Mechanical durability of an optimized polymer concrete under various thermal cyclic loadings – An experimental study. Constr Build Mater 64:308–315

    Article  Google Scholar 

  28. Aliha MRM, Bahmani A, Akhondi S (2016) A novel test specimen for investigating the mixed mode I+III fracture toughness of hot mix asphalt composites: experimental and theoretical study. Int J Solids Struct 90:167–177

    Article  Google Scholar 

  29. Hua W, Dong SM, Fan Y, Pan X, Wang QY (2017) Investigation on the correlation of mode II fracture toughness of sandstone with tensile strength. Eng Fract Mech 184:249–258

    Article  Google Scholar 

  30. Pan X, Huang JZ, Gan ZQ, Hua W, Dong SM (2021) Investigation on mixed-mode II-III fracture of the sandstone by using eccentric cracked disk. Theoret Appl Fract Mech 115:10

    Article  Google Scholar 

  31. Yuan R, Shen B (2017) Numerical modelling of the contact condition of a Brazilian disk test and its influence on the tensile strength of rock. Int J Rock Mech Min Sci 93:54–65

    Article  Google Scholar 

  32. Garcia-Fernandez CC, Gonzalez-Nicieza C, Alvarez-Fernandez MI, Gutierrez-Moizant RA (2018) Analytical and experimental study of failure onset during a Brazilian test. Int J Rock Mech Min Sci 103:254–265

    Article  Google Scholar 

  33. Erarslan N, Liang ZZ, Williams DJ (2012) Experimental and numerical studies on determination of indirect tensile strength of rocks. Rock Mech Rock Eng 45:739–751

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11872042 and 12132019), Sichuan Science and Technology Program (No. 2021YJ0357), Xichang University Science and Technology Program (YBZ202109), and China Postdoctoral Science Foundation (2019M653395).

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Authors and Affiliations

  1. Key Laboratory of Deep Underground Science and Engineering, Ministry of Education, College of Architecture and Environment, Sichuan University, Chengdu, 610065, China

    Huaizi Tang, Jinyuan He, Zhiqiang Gan, Wen Hua & Shiming Dong

  2. Failure Mechanics and Engineering Disaster Prevention Key Laboratory of Sichuan Province, Sichuan University, Chengdu, 610065, China

    Huaizi Tang, Zhiqiang Gan, Wen Hua & Shiming Dong

  3. College of Mechanical and Electrical Engineering, Xichang University, Xichang, 615013, China

    Jianxiong Li

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Correspondence to Shiming Dong.

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Tang, H., He, J., Gan, Z. et al. Tensile strength and elastic modulus determined in the Brazilian test: Theory and experiment. Meccanica 57, 2533–2552 (2022). https://doi.org/10.1007/s11012-022-01574-w

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