Naturally Accepted Boundary Conditions for the Brazilian Disc Test and the Corresponding Stress Field

Abstract

The stress field developed in the Brazilian disc is determined assuming that the disc is under the influence of a combination of two load distributions, namely normal (radial)- and shear (frictional)-stresses, both of them acting along two finite symmetric arcs of the disc periphery. The nature of the two distributions as well as the extent of the loaded arcs are obtained from the solution of the respective contact problem according to which the disc and the jaw are considered as a system of two elastic plane bodies in contact pressed against each other. Emphasis is given to the stresses due to friction since the stress field due to the specific distribution of radial pressure is already known. It is concluded that the role of friction cannot be ignored, especially in the immediate vicinity of the contact rim. Moreover, for high surface roughness the overall transverse normal stress component in this region becomes of tensile nature indicating increased possibility of premature local cracking (taking into account the low tensile strength of the materials tested using the Brazilian disc test). On the other hand, the stress field at the disc’s center is totally insensitive to both the exact distribution of radial pressure and also to the presence or absence of friction. It is thus indicated that in case fracture in the immediate vicinity of the contact rim is by some means suppressed (reducing for example drastically the coefficient of friction) the results of the Brazilian disc test correspond, in a satisfactory manner, to those predicted by Hondros’ classic approach.

Appendices

Appendix 1

The boundary values of the displacement components for the disc (1) and the jaw (2) in the contact region, as obtained from the contact problem (Kourkoulis et al. 2012b, c) (the disc and the jaw occupy the lower (−) and the upper (+) half planes, respectively).

$$ u_{1}^{ - } \left( \tau \right) = - \frac{{\kappa_{1} - 1}}{{24R \mu_{1} K}}\left( {\tau \sqrt {\ell^{2} - \tau^{2} } + \ell^{2} {\text{Arc}}\sin \frac{\tau }{\ell }} \right),\quad v_{1}^{ - } \left( \tau \right) = \frac{{\kappa_{1} + 1}}{{24R\mu_{1} K}}\tau^{2} $$

(7.1)

$$ u_{2}^{ + } \left( \tau \right) = - \frac{{\kappa_{2} - 1}}{{24R\mu_{2} K}}\left( {\tau \sqrt {\ell^{2} - \tau^{2} } + \ell^{2} {\text{Arc}}\sin \frac{\tau }{\ell }} \right),\quad v_{2}^{ + } \left( \tau \right) = \frac{{\kappa_{2} + 1}}{{24R \mu_{2} K}}\tau^{2} $$

(7.2)

Appendix 2

The stress field on the isolated disc due to a parabolically varying distribution P(ϑ) of radial pressure along the actual loaded rim (Markides and Kourkoulis 2012).

$$ \begin{gathered} \sigma_{{\,\,\begin{array}{*{20}c} {r\,r} \\ {\vartheta \vartheta } \\ \end{array} }}^{\,P(\vartheta )} = \frac{{P_{c} }}{{4\pi \sin^{2} \omega_{o} }}\left\{ {\left. {\begin{array}{*{20}c} {\frac{{\left( {R^{2} - r^{2} } \right)^{2} }}{{2r^{4} }}} \\ {\frac{{2r^{6} - R^{6} - r^{4} R^{2} }}{{2r^{4} R^{2} }}} \\ \end{array} } \right\rangle \cdot \sin 2\vartheta \cdot \ell n\frac{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left\langle {\begin{array}{*{20}c} \begin{gathered} \,\frac{{4\omega_{o} R^{2} \left( {R^{2} - 2r^{2} } \right)}}{{r^{4} }}\cos 2\vartheta + \left( {\frac{{r^{4} - R^{4} + 2r^{2} R^{2} }}{{r^{4} }}\cos 2\vartheta + 2\cos 2\omega_{o} } \right) \hfill \\ \hfill \\ \end{gathered} \\ { - \frac{{4\omega_{o} R^{4} }}{{r^{4} }}\cos 2\vartheta + \left( {\frac{{2r^{6} + R^{6} - r^{4} R^{2} }}{{r^{4} R^{2} }}\cos 2\vartheta + 2\cos 2\omega_{o} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right. \hfill \\ \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\left\langle {\begin{array}{*{20}c} \begin{gathered} \left. \begin{gathered} 2\pi - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} \hfill \\ \,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\, - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} \hfill \\ \end{gathered} \right\}\,\,\,\,region\,\,I \hfill \\ \hfill \\ \hfill \\ \end{gathered} \\ {the\,\,same\,\,\exp ression\,\,without\,\,2\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,region\,\,II} \\ \end{array} } \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \pm \left( {R^{2} - r^{2} } \right)\left[ {\left( {\frac{{r^{2} }}{{R^{2} }}\sin 4\vartheta + 2\cos 2\omega_{o} \sin 2\vartheta } \right)\left( {\frac{{ - R^{2} \cos 2\omega_{o} - r^{2} \cos 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + } \right.} \right. \hfill \\ \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. {\,\,\,.\frac{{R^{2} \cos 2\omega_{o} + r^{2} \cos 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right) - \left( {\frac{{r^{2} }}{{R^{2} }}\cos 4\vartheta + 2\cos 2\omega_{o} \cos 2\vartheta + \frac{{R^{2} }}{{r^{2} }}} \right) \cdot \hfill \\ \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. {\left. { \cdot \left( {\frac{{R^{2} \sin 2\omega_{o} - r^{2} \sin 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + \frac{{R^{2} \sin 2\omega_{o} + r^{2} \sin 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right)} \right] - 4\omega_{o} \cos 2\omega_{o} } \right\} \hfill \\ \end{gathered} $$

(8.1)

$$ \begin{gathered} \sigma_{\,\,\,r\vartheta }^{\,P(\vartheta )} = \frac{{P_{c} \left( {R^{2} - r^{2} } \right)}}{{4\pi \sin^{2} \omega_{o} }}\left[ {\frac{{r^{4} - R^{4} }}{{2r^{4} R^{2} }} \cdot \cos 2\vartheta \cdot \ell n\frac{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + } \right.\,\,\frac{{4\omega_{o} R^{2} }}{{r^{4} }}\sin 2\vartheta - \frac{{r^{4} + R^{4} }}{{r^{4} R^{2} }}\sin 2\vartheta \cdot \hfill \\ \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cdot \left\langle {\begin{array}{*{20}c} \begin{gathered} \left. \begin{gathered} 2\pi - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} \hfill \\ \,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\, - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} \hfill \\ \end{gathered} \right\}\,\,\,\,\,\,region\,\,I \hfill \\ \hfill \\ \end{gathered} \\ {\,the\,\,same\,\,\exp ression\,\,without\,\,2\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,region\,\,II} \\ \end{array} } \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{{r^{2} }}{{R^{2} }}\sin 4\vartheta + 2\cos 2\omega_{o} \sin 2\vartheta } \right)\left( \begin{gathered} \frac{{R^{2} \sin 2\omega_{o} - r^{2} \sin 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} \hfill \\ \hfill \\ \end{gathered} \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\left. { + \frac{{R^{2} \sin 2\omega_{o} + r^{2} \sin 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right) + \left( {\frac{{r^{2} }}{{R^{2} }}\cos 4\vartheta + 2\cos 2\omega_{o} \cos 2\vartheta + \frac{{R^{2} }}{{r^{2} }}} \right) \cdot \hfill \\ \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. { \cdot \left( {\frac{{ - R^{2} \cos 2\omega_{o} - r^{2} \cos 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + \frac{{R^{2} \cos 2\omega_{o} + r^{2} \cos 2\vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right)} \right] \hfill \\ \end{gathered} $$

(8.2)

Appendix 3

The u _ϑ -component of the displacement field in the isolated disc due to a parabolically varying distribution P(ϑ) of radial pressure along the actual loaded rim (Kourkoulis et al. 2012a).

$$ \begin{gathered} u_{\vartheta } = \frac{{P_{c} }}{{8\pi \mu \sin^{2} \omega_{o} }}\left\{ { - \left[ {\left( {\kappa + 1} \right)\cos 2\omega_{o} r + \left[ {\frac{{\left( {\kappa + 3} \right)r^{3} }}{{6R^{2} }} - \frac{{\left( {\kappa - 1} \right)R^{2} }}{2r} - \frac{{3r^{4} + R^{4} }}{{3r^{3} }}} \right]\cos 2\vartheta } \right] \cdot } \right. \hfill \\ \,\,\,\,\,\,\,\,\,\, \cdot \ell n\sqrt {\frac{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} + \left[ {\frac{{\left( {\kappa + 3} \right)r^{3} }}{{6R^{2} }} + \frac{{\left( {\kappa - 1} \right)R^{2} }}{2r} - \frac{{3r^{4} - R^{4} }}{{3r^{3} }}} \right]\sin 2\vartheta \cdot \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, \cdot \left\langle {\begin{array}{*{20}c} \begin{gathered} \left. \begin{gathered} 2\pi - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} \hfill \\ \,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\, - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} \hfill \\ \end{gathered} \right\}\,\,\,\,\,\,\,\,\,\,region\,\,I \hfill \\ \hfill \\ \end{gathered} \\ {\,\,the\,\,same\,\,\exp ression\,\,without\,\,2\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,region\,\,II} \\ \end{array} } \right.\,\,\,\,\,\,\,\,\,\, \hfill \\ \,\,\,\,\,\, + \frac{{4\left( {\kappa + 1} \right)R}}{3}\left( {\sin^{3} \omega_{o} \cos \vartheta - \cos^{3} \omega_{o} \sin \vartheta } \right)\ell n\sqrt {\frac{{R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} + \vartheta } \right)}}{{R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} + \vartheta } \right)}}} + \frac{{4\left( {\kappa - 1} \right)R}}{3} \cdot \hfill \\ \,\,\,\,\, \cdot \left( {\sin^{3} \omega_{o} \sin \vartheta + \cos^{3} \omega_{o} \cos \vartheta } \right) \cdot \left\langle {\begin{array}{*{20}c} \begin{gathered} \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }}\,\,\,\,\,\,\,\,\,region\,\,I \hfill \\ \hfill \\ \end{gathered} \\ {\,\,the\,\,same\,\,\exp ression\,\,plus\,\,\,\,\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,region\,\,II} \\ \end{array} } \right. \hfill \\ \,\,\,\, + \frac{{4\left( {\kappa + 1} \right)R}}{3}\left( {\sin^{3} \omega_{o} \cos \vartheta + \cos^{3} \omega_{o} \sin \vartheta } \right)\ell n\sqrt {\frac{{R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} - \vartheta } \right)}}{{R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} - \vartheta } \right)}}} + \frac{{4\left( {\kappa - 1} \right)R}}{3} \cdot \hfill \\ \,\,\, \cdot \left( {\sin^{3} \omega_{o} \sin \vartheta - \cos^{3} \omega_{o} \cos \vartheta } \right) \cdot \left\langle {\begin{array}{*{20}c} \begin{gathered} - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}\,\,\,\,\,\,region\,\,I \hfill \\ \hfill \\ \end{gathered} \\ {\,the\,\,same\,\,\exp ression\,\,plus\,\,\,\,\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,region\,\,II} \\ \end{array} } \right.\,\left| {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \right\rangle \,\,\,\,\, - \,2\left( {\kappa - 1 + \frac{{2R^{2} }}{{3r^{2} }}} \right)\frac{{\omega_{o} R^{2} \sin 2\vartheta }}{r}\left. {\begin{array}{*{20}c} {} \\ {} \\ {} \\ {} \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(9.1)

Appendix 4

The stress field on the isolated disc due to distribution T(ϑ) of frictional stresses.

$$ \begin{gathered} \sigma_{rr}^{T(\vartheta )} = \frac{5\pi Rnw}{{288\left( {10 + \pi^{2} } \right)K^{2} P_{dev} }}\left( {R^{2} - r^{2} } \right)\left\{ {\left[ { - \left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{r^{2} + 3R^{2} }}{{2r^{3} R}}\cos \vartheta + \frac{{5R^{6} - r^{6} }}{{2r^{5} R^{3} }}\cos 3\vartheta } \right]} \right. \cdot \hfill \\ \ell n\frac{{\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}}{{\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}} + \left[ {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{r^{2} - 3R^{2} }}{{r^{3} R}}\sin \vartheta + \frac{{5R^{6} + r^{6} }}{{r^{5} R^{3} }}\sin 3\vartheta } \right] \cdot \hfill \\ \left( {\tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right) - \hfill \\ 2\left[ {\frac{{r^{4} \left( {4\sin^{2} \omega_{o} - 3} \right) - R^{4} }}{{r^{4} }}\cos 2\vartheta - \frac{{r^{2} }}{{R^{2} }}\cos 4\vartheta + \left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{R^{2} }}{{r^{2} }}} \right] \cdot \hfill \\ \left[ {\frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] + \hfill \\ 2\sin 2\vartheta \left( {4\sin^{2} \omega_{o} - 3 - \frac{{2r^{2} \cos 2\vartheta }}{{R^{2} }} + \frac{{R^{4} }}{{r^{4} }}} \right)\left[ {\frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} + 2r^{2} \sin \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right. - \hfill \\ \left. {\frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} - 2r^{2} \sin \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] + \frac{{3\pi^{2} }}{{8\sin \omega_{o} }}\left[ {\left( { - \cos^{2} \omega_{o} \frac{{2R^{2} }}{{r^{4} }}\cos 2\vartheta + \frac{{4\sin^{2} \omega_{o} - 3}}{{2r^{2} }} + \frac{{r^{8} - 3R^{8} }}{{4r^{6} R^{4} }}\cos 4\vartheta } \right)} \right. \cdot \hfill \\ \ell n\frac{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} } \right]^{2} }}{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} } \right]\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} } \right]}} - \sin 2\vartheta \left( {\cos^{2} \omega_{o} \frac{{4R^{2} }}{{r^{4} }} + \frac{{r^{8} + 3R^{8} }}{{r^{6} R^{4} }}\cos 2\vartheta } \right) \cdot \hfill \\ \left[ {2\left( {\tan^{ - 1} \frac{R + r\sin \vartheta }{r\cos \vartheta } - \tan^{ - 1} \frac{R - r\sin \vartheta }{r\cos \vartheta }} \right) - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right. + \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \hfill \\ \left. {\tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right] + \left[ {\frac{{r^{4} \cos 5\vartheta }}{{2R^{4} }} + } \right.\frac{{4r^{6} \cos^{2} \omega_{o} + R^{6} }}{{2r^{4} R^{2} }}\cos 3\vartheta + \hfill \\ \left. {\frac{{2R^{2} \cos^{2} \omega_{o} - r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{r^{2} }}\cos \vartheta } \right]\left[ {\frac{{2\left( {R^{2} + r^{2} } \right)\cos \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }}} \right. - \frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} - \hfill \\ \left. {\frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right] - \left[ {\frac{{r^{4} \sin 5\vartheta }}{{2R^{4} }} + } \right.\frac{{4r^{6} \cos^{2} \omega_{o} - R^{6} }}{{2r^{4} R^{2} }}\sin 3\vartheta - \hfill \\ \left. {\frac{{2R^{2} \cos^{2} \omega_{o} + r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{r^{2} }}\sin \vartheta } \right]\left[ {\frac{{2\left( {R^{2} - r^{2} } \right)\sin \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }}} \right. + \frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta - 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + \hfill \\ \left. {\left. {\left. {\frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta + 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right]} \right] + \frac{{4\sin \omega_{o} }}{{r^{2} }}\left[ {\left( {4 + \frac{{3\pi^{2} }}{8}} \right)\frac{{R^{2} }}{{r^{2} }}\cos 2\vartheta - \sin^{2} \omega_{o} \left( {\frac{16}{3} + \frac{{3\pi^{2} }}{8}} \right) + \frac{{3\pi^{2} }}{8} + 4} \right]} \right\} \hfill \\ \end{gathered} $$

(10.1)

$$ \begin{gathered} \sigma_{\vartheta \vartheta }^{T(\vartheta )} = \frac{5\pi Rnw}{{288\left( {10 + \pi^{2} } \right)K^{2} P_{dev} }}\left\{ {\left[ {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{3\left( {r^{4} + R^{4} } \right) + 2r^{2} R^{2} }}{{2r^{3} R}}\cos \vartheta - \frac{{5\left( {r^{8} + R^{8} } \right) - r^{2} R^{2} \left( {r^{4} + R^{4} } \right)}}{{2r^{5} R^{3} }}\cos 3\vartheta } \right]} \right. \cdot \hfill \\ \ell n\frac{{\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}}{{\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}} - \left[ {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{3\left( {r^{4} - R^{4} } \right)}}{{r^{3} R}}\sin \vartheta } \right. + \hfill \\ \left. {\frac{{5\left( {R^{8} - r^{8} } \right) - r^{2} R^{2} \left( {R^{4} - r^{4} } \right)}}{{r^{5} R^{3} }}\sin 3\vartheta } \right]\left( {\tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + } \right.\tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \hfill \\ \left. {\tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right) + 2\left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} \left( {4\sin^{2} \omega_{o} - 3} \right) - R^{4} }}{{r^{4} }}\cos 2\vartheta - \frac{{r^{2} }}{{R^{2} }}\cos 4\vartheta + \left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{R^{2} }}{{r^{2} }}} \right] \cdot \hfill \\ \left[ {\frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] - \hfill \\ 2\left( {R^{2} - r^{2} } \right)\sin 2\vartheta \left( {4\sin^{2} \omega_{o} - 3 - \frac{{2r^{2} \cos 2\vartheta }}{{R^{2} }} + \frac{{R^{4} }}{{r^{4} }}} \right)\left[ {\frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} + 2r^{2} \sin \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right. - \hfill \\ \left. {\frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} - 2r^{2} \sin \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] + \frac{{3\pi^{2} }}{{8\sin \omega_{o} }}\left[ {\left[ {\cos^{2} \omega_{o} \frac{{4\left( {r^{6} + R^{6} } \right)}}{{2r^{4} R^{2} }}\cos 2\vartheta - \left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{r^{2} + R^{2} }}{{2r^{2} }}} \right. + } \right. \hfill \\ \left. {\frac{{3\left( {r^{10} + R^{10} } \right) - r^{2} R^{2} \left( {r^{6} + R^{6} } \right)}}{{4r^{6} R^{4} }}\cos 4\vartheta } \right]\ell n\frac{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} } \right]^{2} }}{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} } \right]\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} } \right]}} - \hfill \\ \sin 2\vartheta \left[ {\cos^{2} \omega_{o} \frac{{4\left( {r^{6} - R^{6} } \right)}}{{r^{4} R^{2} }} + \frac{{3\left( {r^{10} - R^{10} } \right) - r^{2} R^{2} \left( {r^{6} - R^{6} } \right)}}{{r^{6} R^{4} }}\cos 2\vartheta } \right]\left[ {2\left( {\tan^{ - 1} \frac{R + r\sin \vartheta }{r\cos \vartheta } - \tan^{ - 1} \frac{R - r\sin \vartheta }{r\cos \vartheta }} \right) - } \right. \hfill \\ \left. {\tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right] - \hfill \\ \left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} \cos 5\vartheta }}{{2R^{4} }} + \frac{{4r^{6} \cos^{2} \omega_{o} + R^{6} }}{{2r^{4} R^{2} }}\cos 3\vartheta + \frac{{2R^{2} \cos^{2} \omega_{o} - r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{r^{2} }}\cos \vartheta } \right] \cdot \hfill \\ \left[ {\frac{{2\left( {R^{2} + r^{2} } \right)\cos \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }} - \frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} - \frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right] + \hfill \\ \left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} \sin 5\vartheta }}{{2R^{4} }} + \frac{{4r^{6} \cos^{2} \omega_{o} - R^{6} }}{{2r^{4} R^{2} }}\sin 3\vartheta - \frac{{2R^{2} \cos^{2} \omega_{o} + r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{r^{2} }}\sin \vartheta } \right] \cdot \hfill \\ \left. {\left[ {\frac{{2\left( {R^{2} - r^{2} } \right)\sin \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta - 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta + 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right]} \right] - \hfill \\ \left. {\frac{{4\sin \omega_{o} }}{{r^{2} }}\left[ {\left( {4 + \frac{{3\pi^{2} }}{8}} \right)\frac{{r^{6} + R^{6} }}{{r^{2} R^{2} }}\cos 2\vartheta - \left( {r^{2} + R^{2} } \right)\left[ {\sin^{2} \omega_{o} \left( {\frac{16}{3} + \frac{{3\pi^{2} }}{8}} \right) - \frac{{3\pi^{2} }}{8} - 4} \right]} \right]} \right\} \hfill \\ \end{gathered} $$

(10.2)

$$ \begin{gathered} \sigma_{r\vartheta }^{T(\vartheta )} = \frac{5\pi Rnw}{{288\left( {10 + \pi^{2} } \right)K^{2} P_{dev} }}\left\{ {\left[ {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{r^{4} + 3R^{4} }}{{r^{3} R}}\cos \vartheta - \frac{{r^{6} \left( {3r^{2} - R^{2} } \right) - R^{6} \left( {3r^{2} - 5R^{2} } \right)}}{{r^{5} R^{3} }}\cos 3\vartheta } \right]} \right. \cdot \hfill \\ \left( {\tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right) - \hfill \\ \left[ {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{\left( {R^{2} - r^{2} } \right)\left( {r^{2} + 3R^{2} } \right)}}{{2r^{3} R}}\sin \vartheta + \frac{{r^{6} \left( {3r^{2} - R^{2} } \right) + R^{6} \left( {3r^{2} - 5R^{2} } \right)}}{{2r^{5} R^{3} }}\sin 3\vartheta } \right] \cdot \hfill \\ \ln \frac{{\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} - 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}}{{\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right]\left[ {R^{2} + r^{2} + 2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right]}} + 2\left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} \left( {4\sin^{2} \omega_{o} - 3} \right) - R^{4} }}{{r^{4} }}\cos 2\vartheta } \right. + \hfill \\ \left. {\left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{R^{2} }}{{r^{2} }} - \frac{{r^{2} }}{{R^{2} }}\cos 4\vartheta } \right]\left[ {\frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} + 2r^{2} \sin \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }} - \frac{{\left( {R^{2} + r^{2} } \right)\cos \omega_{o} - 2r^{2} \sin \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] + \hfill \\ 2\left( {R^{2} - r^{2} } \right)\sin 2\vartheta \left[ {\frac{{r^{4} \left( {4\sin^{2} \omega_{o} - 3} \right) + R^{4} }}{{r^{4} }} - \frac{{2r^{2} }}{{R^{2} }}\cos 2\vartheta } \right]\left[ {\frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right. + \hfill \\ \left. {\frac{{\left( {R^{2} + r^{2} } \right)\sin \omega_{o} - 2r^{2} \cos \vartheta \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }}} \right] - \frac{{3\pi^{2} }}{{4\sin \omega_{o} }}\left[ {\left[ {\cos^{2} \omega_{o} \frac{{R^{4} \left( {r^{2} - 2R^{2} } \right) - r^{6} }}{{r^{4} R^{2} }}\cos 2\vartheta + \left( {4\sin^{2} \omega_{o} - 3} \right)\frac{{R^{2} }}{{2r^{2} }}} \right.} \right. + \hfill \\ \left. {\frac{{r^{8} \left( {R^{2} - 2r^{2} } \right) + R^{8} \left( {2r^{2} - 3R^{2} } \right)}}{{4r^{6} R^{4} }}\cos 4\vartheta } \right]\left[ {2\left( {\tan^{ - 1} \frac{R + r\sin \vartheta }{r\cos \vartheta } - \tan^{ - 1} \frac{R - r\sin \vartheta }{r\cos \vartheta }} \right) - \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right. + \hfill \\ \tan^{ - 1} \frac{{R\cos \omega_{o} - r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} - \left. {\tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} + r\cos \vartheta }} + \tan^{ - 1} \frac{{R\cos \omega_{o} + r\sin \vartheta }}{{R\sin \omega_{o} - r\cos \vartheta }}} \right] - \sin 2\vartheta \left[ {\cos^{2} \omega_{o} \frac{{R^{6} - r^{6} + R^{4} \left( {R^{2} - r^{2} } \right)}}{{2r^{4} R^{2} }}} \right. + \hfill \\ \left. {\frac{{r^{8} \left( {R^{2} - 2r^{2} } \right) + R^{8} \left( {3R^{2} - 2r^{2} } \right)}}{{4r^{6} R^{4} }}\cos 2\vartheta } \right]\ln \frac{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} } \right]^{2} }}{{\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} } \right]\left[ {\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} } \right]}} - \hfill \\ \left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} }}{{4R^{4} }}\cos 5\vartheta + \frac{{4r^{6} \cos^{2} \omega_{o} + R^{6} }}{{4r^{4} R^{2} }}\cos 3\vartheta + \frac{{2R^{2} \cos^{2} \omega_{o} - r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{2r^{2} }}\cos \vartheta } \right] \cdot \hfill \\ \left[ {\frac{{2\left( {R^{2} - r^{2} } \right)\sin \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta - 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} + \frac{{\left( {R^{2} + r^{2} } \right)\sin \vartheta + 2R^{2} \cos \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right] - \hfill \\ \left( {R^{2} - r^{2} } \right)\left[ {\frac{{r^{4} }}{{4R^{4} }}\sin 5\vartheta + \frac{{4r^{6} \cos^{2} \omega_{o} - R^{6} }}{{4r^{4} R^{2} }}\sin 3\vartheta - \frac{{2R^{2} \cos^{2} \omega_{o} + r^{2} \left( {4\sin^{2} \omega_{o} - 3} \right)}}{{2r^{2} }}\sin \vartheta } \right] \cdot \hfill \\ \left. {\left[ {\frac{{2\left( {R^{2} + r^{2} } \right)\cos \vartheta }}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \vartheta } \right)^{2} }} - \frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} + \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} + \vartheta } \right)} \right)^{2} }} - \frac{{\left( {R^{2} + r^{2} } \right)\cos \vartheta - 2R^{2} \sin \omega_{o} \sin \left( {\omega_{o} - \vartheta } \right)}}{{\left( {R^{2} + r^{2} } \right)^{2} - \left( {2rR\sin \left( {\omega_{o} - \vartheta } \right)} \right)^{2} }}} \right]} \right] - \hfill \\ \sin \omega_{o} \left. {\left( {8 + \frac{{3\pi^{2} }}{4}} \right)\frac{{r^{6} - R^{6} - R^{4} \left( {R^{2} - r^{2} } \right)}}{{r^{4} R^{2} }}\sin 2\vartheta } \right\} \hfill \\ \end{gathered} $$

(10.3)