Structural Health Monitoring with Piezoelectric Wafer Active Sensors, 2nd Edition

(136)

Substitution of A₁, B₂ from Eq. (120) into Eq. (136) yields

$\begin{array}{l}{\sigma }_{xx}^A=-\mu [({\xi }^2+{\eta }_S^2-2{\eta }_P^2)2\xi {\eta }_S\sin {\eta }_{S}d\sin {\eta }_{P}y-2\xi {\eta }_S({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d\sin {\eta }_{S}y]\hfill \\ {\sigma }_{yy}^A=\mu [({\xi }^2-{\eta }_S^2)2\xi {\eta }_S\sin {\eta }_{S}d\sin {\eta }_{P}y+2\xi {\eta }_S({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d(-\sin {\eta }_{S}y)]\hfill \\ {\sigma }_{xy}^A=i\mu [2\xi {\eta }_{P}2\xi {\eta }_S\sin {\eta }_{S}d\cos {\eta }_{P}y+({\xi }^2-{\eta }_S^2)({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d\cos {\eta }_{S}y]\hfill \end{array}$ (137)

(137)

Upon expansion and rearrangement, Eq. (137) becomes

$\begin{array}{l}{\sigma }_{xx}^A=-2\mu \xi {\eta }_S[({\xi }^2+{\eta }_S^2-2{\eta }_P^2)\sin {\eta }_{S}d\sin {\eta }_{P}y-({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d\sin {\eta }_{S}y]\hfill \\ {\sigma }_{yy}^A=2\mu \xi {\eta }_S({\xi }^2-{\eta }_S^2)(\sin {\eta }_{S}d\sin {\eta }_{P}y-\sin {\eta }_{P}d\sin {\eta }_{S}y)\hfill \\ {\sigma }_{xy}^A=i\mu [4{\xi }^2{\eta }_P{\eta }_S\sin {\eta }_{S}d\cos {\eta }_{P}y+{({\xi }^2-{\eta }_S^2)}^2\sin {\eta }_{P}d\cos {\eta }_{S}y]\hfill \end{array}$ (138)

(138)

Showing explicitly the $x,t$ dependency through the common factor ...