## If [U n

In our case, the problem is divided into two simpler ones: mechanical and electrical. The mechanical problem coincides with that of the theory of nonelectrical shells within the coefficients. It is described by a system of differential equations that contains the equilibrium equations 5.3, strain-displacement formulas 7.19, and electroelasticity equations (equations 7.23,7.24 and 7.27). The electrical quantity in the electroelasticity relations is expressed through the electric potential found from the conditions 7.10. The other electrical quantities can be found after solving the mechanical problem; we find £3" and D(30> from equations 7.26 and 7.25. We use equations 7.15, 7.8, 7.10 and 7.16 to find

For shells with thickness polarization and electrode-covered faces, the boundary conditions are purely mechanical and have a typical form. We write the boundary conditions for the edge a, = a,o restricting ourselves to

Aj oaj

We note that the second hypothesis does not imply that the elongation of the normal element to the middle surface is small. The relative elongation of the normal element is comparable to that along the coordinate lines ei and ej of the middle surface. When the mechanical problem is solved, the relative elongation can be found by the formula a 7 2/¡ h

The stresses <7,-3 and 0-33 can be neglected in a similar way only in the elec-troelasticity relations. We can see from the three-dimensional equilibrium equations that if a„ and <jy vary with thickness by a linear law, then o^ and <733 vary with respect to 7 by square and cubic laws, respectively. Just as in the theory of nonelectrical shells, <7,3 can be found after the two-dimensional problem has been solved from the formulas

where the right-hand sides contain only known quantities, i.e., the components of the surface load (equations 3.1) and the transverse forces Nj. The stresses <733 are found from the three-dimensional equilibrium equations and conditions 3.1 on the faces:

(2)= If 1 flAzgff) 1 djAio^) | a',',' | o™ °33 2V/M2 da\ A,A2 da2 + tf, R2

Consider different electrical conditions on the faces of an electrode-covered shell. Since the electrodes are equipotential surfaces, the potential difference is a constant quantity independent of the coordinates. In some cases, the constant V is known. For example, V is known in equations 7.10, 7.16, and 7.23 when an electric power is supplied to the electrodes. If the electrodes are short-circuited, the potential difference at the electrodes is zero and we can use V = 0 in equations 7.16 and 7.23.

If the shell's faces are completely covered by the electrodes and shorted by a contour with known complex conductivity (equation 3.5), we should add an equation integral to the system of differential equations of the shell theory. Recalling our hypotheses, we can write the later in two ways, i.e., for forces or strains:

2hY 2kl

Here / is an imaginary unit, Q is the square of one of the electrodes, w is the angular frequency of vibrations, operating by the law e~'u'. If the electrodes are disconnected, we should put Y = 0 in equation 7.35.

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