Mechanical Effect of Electrode Plating

Abstract

We are here interested in determining the effect of the elastic stiffness and inertia of the electrodes shown in Fig. 22 on the truncated form of the two-dimensional equations of motion we have obtained previously in Chapter 13. The equations for the unplated crystal (13.51) and (13.52) are reproduced here

$$T_{aj,a}^{(0)} + F_j^{(0)} = 2\varrho h\ddot u_j^{\left( 0 \right)},\quad \quad T_{ab,a}^{(1)} - T_{2b}^{(0)} + F_b^{(1)} = \frac{2}{3}\varrho {h^3}\ddot u_b^{\left( 1 \right)},$$

((14.1))

$$[itex]D_{a,a}^{(0)} + D_2^{(1)} = 0,\quad \quad D_{a,a}^{(1)} + 2D_2^{(2)} =0,[/itex]$$

((14.2))

and the constitutive equations (13.49) and (13.50) are also reproduced here as

$$[itex]\begin{array}{l} T_{ij}^{(0)} = 2hc_{ijkl}^{**}S_{kl}^{(0)} - e_{kij}^{**}E_k^{(0)}, \\T_{ab}^{(1)} = \frac{2}{3}{h^3}{\gamma _{abcd}}S_{cd}^{(1)} - {\psi _{iab}}E_i^{(1)}, \\\end{array}[/itex]$$

((14.3))

$$\begin{gathered}D_i^{(0)} = e_{iq}^{**}S_q^{(0)} + (1/2h)\varepsilon _{ij}^*E_j^{(0)} - (15/8{h^3}){\varepsilon _{i2}}E_2^{(2)}, \hfill \\D_i^{(1)} = {\psi _{iab}}S_{aa}^{(1)} + (3/2{h^3}){\zeta _{ij}}E_j^{(1)}, \hfill \\D_2^{(2)} = (15/8{h^5})(3{\varepsilon _{22}}E_2^{(2)} - {h^2}{\varepsilon _{2k}}E_k^{(0)}), \hfill \\\end{gathered} $$

((14.4))

and all the quantities have been defined previously in Chapter 13.