## Free Shells With Thickness Polarization

We know that a stressed-strained state appears in nonelectrical shells with all edges unfixed (free shells) under the effect of an external load. In this case, the extending strains and shears are small compared to the fiexural strains; the largest stresses are defined by the moments.

A similar stressed state is typical for free piezoceramic shells. Nevertheless, although the elasticity relations for free nonelectrical shells have a form similar to that for shells with fixed edges, we observe quite a different situation in the theory of piezoelectrical shells.

Let us formulate hypotheses for the theory of free electroelastic shells with electrode-covered faces. Note that this theory can be used to describe slowly-varying electroelastic states at a distance from the edges. Besides, in contrast to the previous cases, we cannot assume that a, in 2.8 is equal to unity, even in the roughest approximation, because the principal terms in some equations cancel each other while the less significant terms may become principal.

First Hypothesis

We assume that the principal stresses in the constitutive relations 2.14 vary with respect to 7 as

Second Hypothesis

We can neglect the stresses T33 compared to the principal stresses r„ in the constitutive relations

Third Hypothesis

As a result of the strain, an element perpendicular to the middle surface remains perpendicular, i.e.,

dVl dVl

and its elongation is

Fourth Hypothesis

The electrical potential is a quadratic function of 7

THEORY OF PIEZOELECTRIC SHELLSAND PIATF.S

where the second term is much greater than the other terms. From equation 11.5 we get

Fifth Hypothesis

The quantity Z)3 is independent of 7:

It follows from equations 11.3 and 11.4 that the displacements v, and V} are linear functions of 7:

We represent the sought-for quantities as functions of 7 (equations 11.1, 11.6, and 11.8) and substitute them into the first equation of 11.2 and the second equation of 2.14. Equating the coefficients at the same powers of 7, we get

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