Shells With Tangential Polarization Electrodecovered Faces

As before we will assume that the shell is polarized along the a2-lines.

The electroelastic state considered in this section does not resemble the stressed and strained state of nonelectrical shells [80]. We will show that this electroelastic state cannot arise due to mechanical loads. Consider two cases depending on the electrical conditions on the faces.


Suppose that an electrical potential t/>|7=±/,= ±V (10-D

is given on the electrodes. Under the effect of the electric field, the domains oriented along the c*2-lines tend to become normal to the middle plane. Therefore, the largest displacement is vi,". and the largest stresses are (jJ," and cr^", which are used to determine the moments. We do not need to solve the problem to obtain the formulas t<°) -

To be more exact in defining the needed quantities, we construct a theory of shells based on the following hypotheses.

First Hypothesis

We can neglect the quantities 033, E1, and E2 as compared to the principle quantities cr,-, and a,j in the electroelasticity relations 9.2 and write the latter in a simplified form

Second Hypothesis

The electrical potential varies with the thickness coordinate as We find from 10.1 that

It is clear from these formulas that £3 is a constant quantity, i.e., dtp V

Third Hypothesis

We replace the relations 9.3 by the approximate formulas

from which we get the three-dimensional strains e,, mf and the stresses <7,,, <r, e[=£i+l(Kl+kldi5E3), e2 =£2+1*2 m\ - u>2 + 7(r - kid^E-i), m2 = u>i + 7r ou = oT + 1o»\ ai} = of+1af. (10.5)

We use the formulas 5.1 to pass from stresses to forces and moments and

34 THEORY OF PIEZOELECTRIC SHELLS AND PLATES get from 10.3 to 10.5 the electroelasticity relations Tj = 2/i(/i„e, + n.jEj), < c 2h


To define the electrical quantities, we need one more hypothesis. Fourth Hypothesis

We will neglect the stresses <723 and 033 and the components e[|£i and £^£2 of the electric field intensity in 9.4 as compared to the components D3 and D2 of the electric induction. Then

Since the component £3 is constant and <7,y varies with 7 by a linear law, a simple transformation of 9.4 will give n - r>(0) - J V

D\ = D(,0) + 7D1", D2 = Dj01 + 7D(2", (10.7)


Electroelasticity relations 10.6 for forces and moments do not contain unknown electrical quantities; therefore, just like in the case of a shell with thickness polarization and electrode-covered faces, the problem can be divided into mechanical and electrical ones. The first coincides to constant factors with the theory of nonelectrical shells, and the second gives the electrical quantities directly from equations 10.7 and 10.8 after the mechanical problem has been solved.

Let us write the boundary conditions. Since the directions cvi and a2 are not equivalent for the tangential polarization, we will consider the edges ai = qiq and a2 = a2o, separately. At the hinge-supported and free edges cvi — a,o and o?2 = «20. the boundary conditions have the usual form 7.30 and 7.31. At the rigidly fixed edge a2 - «20, the condition for the rotation angle 72 should be changed to take into account the angle that is constant over the entire shell and is due to the electrical load

If the geodesic curvature k\ = (1 /A\A2){dA\/da2) of the c*i-line is equal to zero at the rigidly fixed edge c*i = c*io, the boundary conditions have the usual form 7.32. If k\ J 0, the boundary conditions of the shell theory cannot be found without calculating the three-dimensional electroelastic state localized at the edge (boundary layer). We will discuss this problem later using an asymptotic approach.


Consider an electroelastic state of a shell with electrode-covered faces connected by an external contour with conductivity Y. We supply electric power to the electrodes

Jo where V is an unknown constant.

For this electroelastic state, the first hypothesis of this section and the second Kirchhoff's hypothesis remain valid. By assuming these hypotheses we arrive at the laws for variation with thickness of the displacements and stresses that were used in the Kirchhoff theory.

Third Hypothesis

The electrical potential varies with thickness by a cubic law xl> = V(0) + 7 V->(l) + 12H>(2) + (10.11)

Fourth Hypothesis

Using our hypotheses, we find that E3 and D3 vary with respect to 7 by a square law; D, varies by a linear law:

£, = £f+ 7£<3,) + 72£<32) D3=D<0) + 7O(3,, + 72of D, = D;0) + 7DS1). (10.12)

The system obtained with the help of the hypotheses breaks into two subsystems. One is mechanical and coincides with the Kirchhoff theory, and the other is electrical.

Let us write the elasticity relations

T, - 2/i(/!„£, + n¡j£j), G, - - — (/I,,K., + njjKj)


After the mechanical problem has been solved, we can find the electrical quantities from the equations n(0) _ «15 „

2h3 1

0 0

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