## A

and mechanical conditions 3.1 hold on the faces of the shell. We write three-dimensional constitutive relations on - »¡¡e; + mjej - ~ c,E2, aiS = -jr('», + mj) - ^£f, (9.2)

S44 ^44

dv dv

D3 = eTllEi+dl5a2i, D i = e[,£, +dxsa2u D2 - e\iE2 + d}] (<7|, + a33) + ¿33f722

d d dD3

In order to construct the theory of shells, we use the following simplifying hypotheses [80].

### First Hypothesis

Neglecting the stresses (733 as compared to cr„ in the constitutive relations 9.2 and 9.4, we obtain a,, = n^e, + n^ej - c,£2, D2 - £33£2 + 1 + d^a22. (9.5)

Second Hypothesis

It coincides with the Kirchhoff-Love second hypothesis (Section 6) and leads to the relations 7.14.

Third Hypothesis

We assume that the components £, do not depend on 7; therefore, i]) does not change with thickness, i.e., iP = r\ = = (9.6)

Ai oai

EQUATIONS OF THE THEORY OF PIEZOCERAMIC SHELLS 31 Fourth Hypothesis

We neglect D(3'' in the last formula 9.4 as compared to Df]. Using these hypotheses and the usual scheme, we find the electroelasticity relations

'/; = 2h(n,r£, + mjej) - 2/îc,-40> S = Sy= - di5E\Q))

S44 2/i3

If we add the equilibrium equation 5.3 and strain-displacement relations 7.19 to 9.7 and 9.8, we will get a complete system of tenth-order differential equations for the sought-for quantities.

Having solved the problem, we can, if necessary, define less important electrical and mechanical quantities. The stresses an and a33 are calculated by equations 7.33 and 7.34. Since <t,7 and cr,y vary with respect to 7 by a linear law, 9.4 implies that D, also varies by a linear law. According to equation 7.7, D} changes by a square law. We write the formulas for the coefficients D, and D3 in the polynomial in 7, taking into account the conditions 9.1:

The coefficients £3 and i]> in the polynomial in 7 can be found using the corresponding formulas in equations 9.4:

32 THEORY OF PIEZOELECTRIC SHELLS AND PLATES Fm _ 1 n<2> 3d 15 (N2 \

In our case, the system of equations of the shell theory is of order ten; therefore, four mechanical and one electrical condition should be satisfied at each edge. Using the Saint Venant principle, generalized to the electroelasticity case, we find the boundary conditions

at the edge without electrodes with a,- = a,o, and

at the electrode-covered edges with a, = a-,\ and a, = a(2. The mechanical conditions have the form in equations 7.30 to 7.32.

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