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Measurement of elastic properties in Brazilian disc test: solution derivation and numerical verification

  • Yanhui Han
  • Bitao Lai
  • Hui-Hai Liu
  • Hui Li
Original Article

Abstract

Mechanical properties of rocks/soils are key inputs in the design and analysis of surface structures (e.g., slope, foundation) and underground excavations (e.g., tunnel, cavern, borehole, and fracture) in earth engineering practice. Brazilian disc test has been extensively used to measure tensile strength for various rocks. In this study, we propose to measure Young’s modulus, Poisson’s ratio and tensile strength in one Brazilian test through making two additional measurements, i.e., two relative displacements using LVDT, or normal strains by pasting two strain gauges, which are perpendicular to each other, at the center of the disc’s side faces. The analytical solutions of the equivalent strains calculated from LVDT measurements or measured directly by strain gauges are derived from elasticity theory. The analytical solutions are verified with measurements in a numerical model. New formulae for calculating Young’s modulus and Poisson’s ratio from two measured strains are provided and their insensitivity to gauge size is illustrated.

Keywords

Brazilian disc Young’s modulus Poisson’s ratio Strain gauge Closed-form solution 

List of symbols

\(E\)

Young’s modulus of sample (Pa)

\(v\)

Poisson’s ratio of sample

\(P\)

Applied load in y-direction (N)

\(t\)

Thickness of Brazilian disc (m)

\(R\)

Radius of Brazilian disc (m)

\(x\)

x-coordinate of a point on side surface of Brazilian disc (m)

\(y\)

y-coordinate of a point on side surface of Brazilian disc (m)

\(\sigma_{xx}\)

x-normal stress component in 2D plane (Pa)

\(\sigma_{yy}\)

y-normal stress component in 2D plane (Pa)

\(\sigma_{xy}\)

Shear stress component in 2D plane (Pa)

\(\sigma_{x0}\)

x-normal stress component at disc center (Pa)

\(\sigma_{y0}\)

y-normal stress component at disc center (Pa)

\(\varepsilon_{xx}\)

x-normal strain component in 2D plane

\(\varepsilon_{yy}\)

y-normal strain component in 2D plane

\(\varepsilon_{x0}\)

x-normal strain component at disc center

\(\varepsilon_{y0}\)

y-normal strain component at disc center

\(u_{lx}\)

Relative displacement between two ends of the strain gauge in x-direction (m)

\(u_{ly}\)

Relative displacement between two ends of the strain gauge in y-direction (m)

\(\varepsilon_{lx}\)

Averaged normal strain between two ends of the strain gauge in x-direction

\(\varepsilon_{ly}\)

Averaged normal strain between two ends of the strain gauge in y-direction

C1–C4

Strain gauge and disc size related constants

\(A\)

Left end point of horizontal strain gauge

\(B\)

Right end point of horizontal strain gauge

\(C\)

Bottom end point of vertical strain gauge

\(D\)

Top end point of vertical strain gauge

\(x^{A}\)

x-coordinate of point A (m)

\(x^{B}\)

x-coordinate of point B (m)

\(x_{d}^{A}\)

x-displacement at point A (m)

\(x_{d}^{B}\)

x-displacement at point B (m)

\(y^{C}\)

y-coordinate of point C (m)

\(y^{D}\)

y-coordinate of point D (m)

\(y_{d}^{C}\)

y-displacement at point C

\(y_{d}^{D}\)

y-displacement at point D

Notes

Acknowledgements

Authors are very grateful to Aramco management for permission to publish this work.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Aramco Research CentersAramco Services CompanyHoustonUSA
  2. 2.China University of PetroleumBeijingChina

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