(215)

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

$\begin{array}{l}\left(\lambda +2\mu \right){\xi }^{2}+\lambda {\zeta }_{P}^{2}=\mu \left({\xi }^{2}+{\zeta }_{S}^{2}-2{\zeta }_{P}^{2}\right)\hfill \\ \lambda {\xi }^{2}+\left(\lambda +2\mu \right){\zeta }_{P}^{2}=\mu \left({\zeta }_{S}^{2}-{\xi }^{2}\right)\hfill \end{array}$ (217)

(217)

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

$\begin{array}{l}{\sigma }_{rr}=\lambda \mathrm{\Delta }+2\mu {\epsilon }_{rr}=-\lambda \left({\xi }^{2}+{\zeta }_{P}^{2}\right)f{J}_{0}-2\mu \left(\xi f+h\prime \right)\left(\xi {J}_{0}-\frac{{J}_{1}}{r}\right)\hfill \\ =-\lambda \left({\xi }^{2}+{\zeta }_{P}^{2}\right)f{J}_{0}-2\mu \xi f\left(\xi {J}_{0}-\frac{{J}_{1}}{r}\right)-2\mu h\prime \left(\xi {J}_{0}-\frac{{J}_{1}}{r}\right)\hfill \\ =-\lambda \left({\xi }^{2}+{\zeta }_{P}^{2}\right)f{J}_{0}-2\mu \xi f\xi {J}_{0}+2\mu \xi f\frac{{J}_{1}}{}\hfill \end{array}$

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