## Basic Equations

We place a thin-walled electroelastic body in the three-orthogonal coordinate system. We then choose curvilinear coordinates a ¡ and aj so that they coincide with the curvatures of the middle surface of the shell and a linear coordinate 7 along the normal to the middle surface.

FIGURE 1. Strained piezoelement in eleclric field

A complete system of equations in electroelasticity theory consists of the equilibrium equations, the geometrical relationships between the components of the deformation tensor and those of the displacement vector, the constitutive relations, and the electrostatic equations. The equilibrium equations and geometrical formulas are taken from linear elasticity theory [ 105],

Let an electroelastic body be deformed by some forces. As a result its point M with coordinates a i, aj, 7 is displaced. The displacement vector components V|, v2, and V3 are projected on the coordinate lines ai,a2, and 7, i.e.,

Every equation with subscripts / and j contains two equations: one for i - 1 and j = 2, and the other for i = 2 and 7=1. We will use Greek letters for subscripts that assume the values 1, 2, or 3 and Latin letters for subscripts that assume the values 1 or 2.

We say that displacements r,- and V3 are positive if their projections on the coordinate axes are positive. In the linear theory, a deformed state is described by the deformation tensor epi,; the deformations are related to the displacements

1 OHi

THREE-DIMENSIONAL EQUATIONS OE ELECTROELAST1C1T)' e3' Hi da, \HJ +Hid1\H,

The components e\ \, e22, and of the deformation tensor are relative elongations along the three coordinate axes at the point with coordinates a-), aj, 7. The components e¡j and e^ are relative shifts in the planes a,a7 and a,7. The Hj in formulas 2.2 are Lame coefficients. In our three-orthogonal coordinate system, the Lame coefficients are related to the coefficients of the first quadratic form Ai and the principal radius of curvatures of the middle surface i.e.,

From the theory of surfaces we know that the coefficients of the first quadratic form are related to the curvature radius the coordinate lines via Codazzi-Gauss formulas dctj \Ri

Rj daj

Using formulas 2.3 and 2.4 for the derivatives of //, with respect to a-,- and 7, we get dAi

OHj d7

di Ri

We substitute the coefficients 2.3 into the right-hand sides of equations 2.2 and use 2.5 to obtain geometrical formulas that are more suitable for a theory of shells, i.e., en = — a.

To be more concise, we use dVi e> = ~r — + k,Vj + Aj a a -, R i

1 dv

The stresses state that the point M with coordinates «1,02,7 ¡s fully described by the stress tensor opll{p,(i = 1,2,3). Figure 2 shows stresses with positive direction. The components of the stresses should meet the differential equilibrium equations, which assume the form d H d ' o jj

d d d dll — (HiHjan) + —(H3H2au) + — (//j//,^) - H2-^au dH dH dH

07 oa 1 aa2

in our three-orthogonal coordinate system. Here q\,q2, and q3 are the components of the vector of the bulk forces. In order to obtain the differential equations of motion, we introduce the inertial terms d\ J 92V3

PW and into the right-hand sides, where p is the density of the material and t is the time.

The relation between the elastic field tensors (strain epil and stress <rp/1) and the electric field vectors (induction V with components T>i,T>2, and V3 and strength £ with components £\,£2, and £3) can be represented by four equally important systems of piezoeffect equations:

\ O; = elUeu + efa eu = sfjki°ki + duA O, = dik,au + e]k£k e0 = s?jk,ou + gk,jVt

Ei - -hiklekl + PfjVk where sfjkl and s?kl are elastic compliances at constant electric field and induction, respectively; cfjki and c®u are elasticity coefficients under the same conditions; and \$j,Pfj are dielectric permittivities and "nonpermittivi-

ties" for constant strain and elastic stress; dkij, ekjj, gkij, and hkij are piezoelectric constants that are also called a piezoelectric modulus.

FIGURE 2. A shell element with applied stresses

We use a brief notation of tensor theory, where the pairs of subscripts that can be interchanged are replaced by the scheme 11 —» 1,22 —► 2,33 —' 3,32 —► 4,31 —»5,21 —> 6. By using this notation, we can write the equations as square matrices. For example, the second system in matrix form looks like

 e\ E J11 E 12 13 0 0 0 1 0 0 ¿13 CTll ei ?£ 12 £ ..£ 13 0 0 0 1 0 0 ¿13 <722 ei vE 13 V£ '13 ^ 44 0 0 0 1 0 0 ¿44 ^33 e23 0 0 0 £ 44 0 0 0 ¿15 0 f 23 - 0 0 0 0 iE ■>44 0 ¿15 0 0 o 13 e\i 0 0 0 0 0 4 1 0 0 0

The relations between the stress tensors, deformation tensors, and electrical quantities for piezoceramics polarized along the 7-lines can be written as piezo-effect equations:

eu = .ff|<7„ + SEn(7jj + if3(T33 + ¿31 £3 £?33 = if3((T| I + CT22) + if3°'33 + i/3.^3

Table 1 gives physical constants for the piezoceramics PZT-4 and PZT-5 that will be needed for our computation [8].

We use formulas 2.3 and 2.5 to transform the equilibrium equations 2.9 and the relations 2.10 to a more convenient form, vis.

{ë + Wj§+ kjiTii - Tjj) + k,(Tii + T,j) + + aia2Ch =

-— - — + —• — + — -— +k2Tl} + kiT2i + — +ala2qi = 0. R1 R2 A\ oa 1 A2 oa2 07

Here the asymmetric stress tensor tpii is related to the symmetric stress tensor CTp/i. i-e.,

T,i - CI ¡a,,, T,J - (IjOjj, T, 3 = T3/ = rtyO-,-3. T33 = «t«2<733- (2.12)

We will need this notation when considering forces and moments in shell theory.

We change the components T>p,£p for the quantities Dp and Ep according to the formulas

£,- = £3 = ci\a2£i, Di = UjVh />) = clidiDi (2.13)

which will simplify some of the equations.

The polarization that makes the domains perpendicular to the middle surface along the 7-lines is usually termed thickness polarization. We solve the first two equations 2.10 for o\\ and a22, introduce the asymmetric stress tensor 2.12

THREE-DIMENSIONA L EQUATIONS OE ELECTR0ELASTIC1TY 9 TABLE 1

Some physical properties of piezoceramics PZT-4, PZT-58

THREE-DIMENSIONA L EQUATIONS OE ELECTR0ELASTIC1TY 9 TABLE 1

Some physical properties of piezoceramics PZT-4, PZT-58

 Quantity PZT-4 PZT-5 if,, 10-12 m2/H 12.3 16.4 sE 12 -4.05 -5.74 sE 13 -5.31 -7.22 33 15.5 18.8 39.0 47.5 SE 66 32.7 44.3

e0 = 8.85 x I0~12 F/m e0 = 8.85 x I0~12 F/m and the electric vector components Ep and Dp 2.13. Formulas 2.10 will then be written as

 Ty 1 — mi \aj + ttlj ) V«2 7~22 «1 / ™ +<'33 37 II Cli

We write the piezoeffect equations for a piezoceramics with preliminary tangential polarization. We imply a polarization for which the domains are arranged along one family of coordinate lines in parallel with the middle surface, i.e., e\\=Su(J\\+sEncr22 + sEnon + dT,\£2

en = if3o-n + S33CT22 + + ¿33^2 en = .v^^ii + s^3a22 + in<T33 + ¿31 £2 e\2 = 1 +^15^1

Here we assume that the polarization orients the domains along the family of Q2-lines.

We solve 2.15 for crn and <722> taking into account equations 2.12 and 2.13, to get the required piezoeffect equations:

T33 aj

 T.v J?"k=? sl4 V«; 3V3 07 - c£ T" «2 3vi ai «1 Di - er ^£ d2 where we introduced and c, by the formulas
0 0