Weibull statistics effective area and volume in the ball-on-ring testing method

• First treatment of the effective volume of the ball-on-ring specimen.
• First analytical derivation of effective volume for a multiaxial stressed specimen.
• Handling of multiple terms in effective volume integral with multinomial theorem.
• Analytical solution found to match FEM for a range of geometric parameters.
• Analytical solution compares well with other multiaxial failure theorems.

The ball-on-ring method is together with other biaxial bending methods often used for measuring the strength of plates of brittle materials, because machining defects are remote from the high stresses causing the failure of the specimens. In order to scale the measured Weibull strength to geometries relevant for the application of the material, the effective area or volume for the test specimen must be evaluated. In this work analytical expressions for the effective area and volume of the ball-on-ring test specimen is derived. In the derivation the multiaxial stress field has been accounted for by use of the Weibull theory, and the multinomial theorem has been used to handle the integration of multiple terms raised to the power of the Weibull modulus. The analytical solution is verified with a high number of finite element models for various geometric parameters. The finite element model was also used to study the difference from other multiaxial failure theories, i.e. the Batdorf theories.