Table 2 Main stresses and Griffith function for the three loading geometries shown in Table 1.

CUR

\( \sigma_{1} = \sigma_{\theta } = \frac{2P}{\pi Dt}\frac{1}{\alpha }\left[ {A - B} \right] \)

\( f_{G} \left( {r;\alpha } \right) = \frac{1}{4\sin \alpha }\left( {\frac{{A^{2} }}{B}} \right) \)

Hondros (1959), Mellor and Hawkes (1971), Satoh (1986) and Hung and Ma (2003)

\( \sigma_{3} = \sigma_{r} = - \frac{2P}{\pi Dt}\frac{1}{\alpha }\left[ {A + B} \right] \)

\( \tau_{r\theta } = 0;m = {r \mathord{\left/ {\vphantom {r R}} \right. \kern-0pt} R} \)

\( A = \frac{{\left( {1 - m^{2} } \right)\sin 2\alpha }}{{1 - 2m^{2} \cos 2\alpha + m^{4} }};B = \arctan \left( {\frac{{1 + m^{2} }}{{1 - m^{2} }}\tan \alpha } \right) \)

CUP

\( \sigma_{1} = \sigma_{\theta } = \frac{P}{\pi Rt}\left[ {\frac{1}{2}A - C} \right] \)

\( f_{G} \left( {r;\alpha } \right) = \frac{{\left( {B - C} \right)^{2} }}{{8\left( {A - B - C} \right)}} \)

Satoh (1986), Wang et al. (2004), and Yoshiaki (1980)

\( \sigma_{3} = \sigma_{r} = \frac{P}{\pi Rt}\left[ {\frac{1}{2}A - B} \right] \)

\( \tau_{r\theta } = 0;m = {r \mathord{\left/ {\vphantom {r R}} \right. \kern-0pt} R} \)

\( A = \frac{1}{2}\frac{{\left( {2\alpha + \sin 2\alpha } \right)}}{\sin \alpha };B = \frac{4}{{1 - m^{2} }};C = \frac{{4\left( {1 + 3m^{2} } \right)\sin^{2} \alpha }}{{3\left( {1 - m^{2} } \right)^{3} }} \)

FUP

\( \sigma_{1} = \left. { - \sigma_{y} } \right|_{x = 0} = \frac{P}{\pi Rt\sin \alpha }\left( {\frac{{B_{1} }}{{A_{1} }} - C_{1} + \frac{{B_{3} }}{{A_{3} }} - C_{3} } \right) - \frac{P\cos \alpha }{\pi Rt} \)

\( f_{G} \left( {y;\alpha } \right) = \frac{{ - \left( {\frac{{B_{1} }}{{A_{1} }} + \frac{{B_{3} }}{{A_{3} }}} \right)^{2} }}{{4\sin \alpha \left( {C_{1} + C_{3} + \sin \alpha \cos \alpha } \right)}} \)

Huang et al. (2014)

\( \sigma_{3} = \left. { - \sigma_{x} } \right|_{x = 0} = \frac{ - P}{\pi Rt\sin \alpha }\left( {\frac{{B_{1} }}{{A_{1} }} + C_{1} + \frac{{B_{3} }}{{A_{3} }} + C_{3} } \right) - \frac{P\cos \alpha }{\pi Rt} \)

\( \tau_{xy} = \frac{P}{2\pi Rt\sin \alpha }\left( {A^{2} \left( {\frac{1}{{A_{1} }} - \frac{1}{{A_{2} }}} \right) - B^{2} \left( {\frac{1}{{A_{3} }} + \frac{1}{{A_{4} }}} \right)} \right) \)

\( A = R\cos \alpha + y;B = R\cos \alpha - y \)

\( C = x + R\sin \alpha ;D = x - R\sin \alpha \)

\( A_{1} = A^{2} + D^{2} ;B_{1} = - AD;C_{1} = \arctan \left( {{D \mathord{\left/ {\vphantom {D A}} \right. \kern-0pt} A}} \right) \)

\( A_{2} = A^{2} + C^{2} ;B_{2} = AC;C_{2} = \arctan \left( {{C \mathord{\left/ {\vphantom {C A}} \right. \kern-0pt} A}} \right) \)

\( A_{3} = B^{2} + D^{2} ;B_{3} = - BD;C_{3} = \arctan \left( {{D \mathord{\left/ {\vphantom {D B}} \right. \kern-0pt} B}} \right) \)

\( A_{4} = B^{2} + C^{2} ;B_{4} = BC;C_{4} = \arctan \left( {{C \mathord{\left/ {\vphantom {C B}} \right. \kern-0pt} B}} \right) \)