Substituting 13.1 and 13.2 into the equations of the theory of piezoceramic shells, we get the equations of the theory of electroelastic plates that, just as in the case of the theory of elastic plates, can be divided into simpler subsystems: one describes the plane stressed-strained state and the other describes the bending stressed-strained state. We write the equations of the plate theory for different electrical conditions on the faces separately.

Suppose that a mechanical load an\-i=±h = ±q?, <73j|7=±/i = ±qf (13.3)

is given on the plate's faces.

For plates with electrode-covered faces, the system of the plane problem includes:

Equilibrium Equations

Y + T o— + ¥Ti - TJ) + 2k-S + 2/ipw2«, + X,- = 0. (13.4) A,- Oil; A; OCij

Here, and below, A; are coefficients of the first quadratic form on the plane, and X, and Z are components of the surface load related to qf and qf through the formulas

Electroelaslicity Relations

Strain-Displacement Formulas

The system of equations 13.4, 13.6, and 13.7 differs from the respective equations of the theory of nonelectrical plates by the presence of the component £3 of the electric field strength in the electroelasticity relations for the forces 7,. The value of E|0) depends on the electrical conditions 011 the faces of the plate.

If we are given the electrical potential

on the electrodes, then in the electroelasticity relations of the plane problem (equations 13.6) is a known quantity defined by

If the electrodes are connected by an external contour with complex conductivity Y and give off electrical energy, the resultant strain causes a potential difference 2V. This is an unknown constant that can be found by integrating D(30) over the surface Q of one of the electrodes (the point denotes the time derivative):

This formula can be transformed into

If the electrodes are disconnected, we should set the conductivity Y equal to zero in 13.10 and 13.11. Then

Jn Jn d 31

If the electrodes are short-circuited, we should set the electrical potential V equal to zero in formulas 13.8 and 13.9:

Just as in the theory of nonelectrical plates, two mechanical boundary conditions should be met at every edge of the plate. For different fixation types, the conditions are as follows for: A free edge a,■ = a,o

A hinge-supported edge a, = a,o

Having solved the plane problem, we can find the electrical quantities

where the electrical quantities vary with thickness of the plate 7 as r(0 -3

Equations for Bending Plates

Equilibrium equations

Elasticity relations

2h B Ah

where B and a are found from 7.27. Strain-displacement formulas

Aj oaAi oaAi oa-,

In our case, the equations for bending plates completely coincide with those for bending nonelectrical plates, because the bending equations 13.19 to 13.21 do not contain electrical quantities.

When the bending problem is solved, we can find the electrical quantities:

V; = V;(0) + 7 2V>(2\ El = 74", Ei = £f > + 7 2£,(2)



- enCj

- c-T pW _1_ il i +

= -/>2i//2\


aa\ oa2

The mechanical conditions at the edge of a plate are the same as in the theory of nonelectric plate for


A hinge-supported edge a, = a,q w = 0, G, = 0 (13.25)

A rigidly fixed edge a -, = a,q lv = 0, 7, = 0 (13.26)

Since the equilibrium equations, strain-displacement formulas 13.4, 13.7, 13.19, and 13.21, mechanical boundary conditions 13.14 to 13.16, 13.24 to 13.26, and formulas for computing the three-dimensional stresses from forces and moments 7.22, 7.33, and 7.34 remain valid for all the electrical conditions on the plate faces considered below, we will only write the electroelasticity relations and formulas for computing the electrical quantities.

Plates with Faces Having No Electrodes

The edges of the plate may or may not have electrodes.

The elasticity relations for the plane problem have the form r, = 2hB(£j + a £j), c 2h

Polynomials 13.18 in the thickness coordinate 7 remain valid for the electrical quantities of the plane problem. The coefficients at different powers of 7 are found from

The problem for a bending plate can be formulated differently depending on the electrical conditions at the edges. As in the theory of shells, we will deal with two cases.


We assume that there is no electrical load at the electrodes. The elasticity relations will have the form

Note that in this case the bending equations, just as in the plane problem, do not contain unknown electrical quantities. Therefore, the bending problem is a mechanical one.

If necessary, the electrical quantities can be found by equations 13.22 and the formulas

For ?/)(0> we get a second-order differential equation 13.31. In order to find the arbitrary integration functions for this equation, we set one condition for every edge.

At the edge a, = a,0 which has no electrodes and is in contact with air or vacuum, the component of the electric induction vector normal to the edge should be equal to zero:


On two electrode-covered edges q, = a,, and a, = a a connected by an external contour with complex conductivity Y, the condition

should hold, where /, is the line of the electrode-covered edge a, = con the middle surface.

For disconnected electrodes, the condition 13.33 should be satisfied where Y = 0, while for short-circuited electrodes, the condition 13.13 should be met.


We are given a potential difference

0 0

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