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Closed-Form Solution for the Stresses in Brazilian Disc Tests Under Vertical Uniform Loads

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Abstract

Loading arcs, which create a definite contact range and have a radius equal to the test specimen, have been recently introduced into Brazilian disc tests, which are common indirect tensile strength tests. Inspired by the testing apparatus, a new kind of loading pattern acting on the disc is assumed: the vertical uniform load. This load can be resolved into radial and tangential surface forces on the loaded rims of the specimen. Under the assumptions of isotropic, linear-elastic, and homogenous disc materials, the explicit closed-form expressions for the full-field stresses in the disc are derived by the complex variable method. The analytical solutions satisfy the stress boundary condition, and the analytical results are in good agreement with the numerical simulations. Based on the closed-form solutions, it is analytically proven that the maximum tensile stress occurs at the disc center. Then, the suggested formula for the tensile strength, which is related to the contact range, the external load, and the size of specimen, is provided.

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Abbreviations

A, B, C, D :

The end points of contact boundary on disc in physical plane

A′, B′, C′, D′:

The points of A, B, C, D mapped in imagine plane

d, t :

The diameter and thickness of specimen

\(E,\mu\) :

Young’s modulus and Poisson’s ratio of the disc

\(f_{j}^{y}\) :

The vertical force applied along the y direction on the jth node

l :

The horizontal length of contact range on disc

L :

The boundary of disc in physical plane

P :

Externally applied load

\({P_{\text{max} }}\) :

Failure load

q :

The vertical loading density on disc

\({q_\rho },{q_{\rho \theta }}\) :

The surface forces along the radial and tangential directions on disc

R :

The radius of disc

\(r,\theta\) :

Polar radius and angle in physical plane (z plane)

\({x_j}\) :

The x coordinate of the jth node

z :

The complex variable, \(z=x+iy=r{{\text{e}}^{i\theta }}\)

\({X_n},{Y_n}\) :

The surface forces on unit area along x and y directions

\(\alpha\) :

The semi-angle of contact range on disc

\(\beta\) :

The angle in contact range, \(\beta \in [ - \alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha ]\)

\(\zeta\) :

The complex variable of unit disc in imagine plane, \(\zeta =\xi +i\eta =\rho {{\text{e}}^{i\theta }}\)

\(\gamma\) :

The boundary of unit circle in imagine plane

\(\rho\) :

The modulus of \(\zeta\)

\(\sigma\) :

The boundary point of unit circle in image plane, \(\sigma ={{\text{e}}^{i\theta }}\)

\({\sigma _\rho },{\sigma _\theta },{\tau _{\rho \theta }}\) :

Radial, tangential, and shear stress components of disc

\({\sigma _t}\) :

The tensile strength of disc suggested by ISRM and ASTM

\({\sigma _t}^{\prime }\) :

The suggested tensile strength of disc under the vertical uniform load

\(\varphi (z),\psi (z)\) :

The complex potentials in physical plane

\(\varphi (\zeta ),\psi (\zeta )\) :

The complex potentials in imagine plane

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Acknowledgements

The study is supported by the Natural Science Foundation of China (Grant numbers 11572126 and 51704117) and the Fundamental Research Funds for the Central Universities (Grant number NCEPU2016XS59).

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Authors and Affiliations

  1. Renewable Energy School, Institute of Hydroelectric and Geotechnical Engineering, North China Electric Power University, No. 2 Beinong Road, Beijing, 102206, China

    Aizhong Lu, Shaojie Wang & Hui Cai

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  1. Aizhong Lu
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  2. Shaojie Wang
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  3. Hui Cai
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Correspondence to Shaojie Wang.

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Lu, A., Wang, S. & Cai, H. Closed-Form Solution for the Stresses in Brazilian Disc Tests Under Vertical Uniform Loads. Rock Mech Rock Eng 51, 3489–3503 (2018). https://doi.org/10.1007/s00603-018-1511-4

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