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

(215)

${\sigma }_{rz}=2\mu {\epsilon }_{rz}=\mu \left\{2\frac{{\partial }^2\mathrm{\Phi }}{\partial r\partial z}+\frac{\partial }{\partial r}\left(\frac{1}{r}\frac{\partial (rH)}{\partial r}\right)-\frac{{\partial }^{2}H}{{\partial }^{2}z}\right\}$ (216)

(216)

6.5.5.4 Calculation of Stresses in Terms of the Unknowns A₁,A₂,B₁,B₂

Recall the identities of Eq. (90), i.e.,

$\begin{array}{l}(\lambda +2\mu ){\xi }^2+\lambda {\zeta }_P^2=\mu ({\xi }^2+{\zeta }_S^2-2{\zeta }_P^2)\hfill \\ \lambda {\xi }^2+(\lambda +2\mu ){\zeta }_P^2=\mu ({\zeta }_S^2-{\xi }^2)\hfill \end{array}$ (217)

(217)

Substitution of Eqs. (201) and (208)–(211) into Eq. (197) and use of Eq. (217) yields

$\begin{array}{l}{\sigma }_{rr}=\lambda \mathrm{\Delta }+2\mu {\epsilon }_{rr}=-\lambda ({\xi }^2+{\zeta }_P^2)f{J}_0-2\mu (\xi f+h\prime )\left(\xi J_0-\frac{J_1}{r}\right)\hfill \\ =-\lambda ({\xi }^2+{\zeta }_P^2)f{J}_0-2\mu \xi f\left(\xi J_0-\frac{J_1}{r}\right)-2\mu h\prime \left(\xi J_0-\frac{J_1}{r}\right)\hfill \\ =-\lambda ({\xi }^2+{\zeta }_P^2)f{J}_0-2\mu \xi f\xi J_0+2\mu \xi f\frac{J_1}{}\hfill \end{array}$