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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- 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
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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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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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DOI: https://doi.org/10.1007/s11012-022-01574-w