?/'D3, = rfEU + T]l~2st2(T ,,„<?, + T22,(l2) + v2_2S'3T-33,

We add equations 34.5 and mechanical conditions 34.6 to equations 35.2 to 35.4.

Consider equations 35.3 and 35.4 as equations with independent variable We integrate them with respect to £ up to t/2_2s and get v3* = 1'3 o + ?? -7— I ¿31 Jo

We continue the integration with respect to £ of the electroelasticity equations. By neglecting the small terms (within our approximation) in the obtained relations, we get the formulas for the needed quantities:

»'3* = »'3,0 + VC^.I Vi* - v„o + //1_2t<v;,i gi* = gift,

Tii* = 7/1_2lr,vio + <t,;,| + t/'c2r,7,2, (ty>) r,3» = t-,3,0 + //i_2iCt-,3,i + C2t"<3,2 + i/C3r<3,3, (r33, , £,„ , £>,„) £3, = ^£3.0 + ly'-^Cfs,! + K27?2-2i-"£3,2 />3. = ^3,0 + /•r?2_2S""+"(Cf3,. + T,'-2iC203,2 + C303,3 + i/V £>3,4) V». = Vx~2s-bipft + T/XVM + V~2*~'C2V>.2 + n/2-2'-" cV.3- (35.6)

Here, the number /' is chosen to be zero if the shell faces are electrode-covered and to be one if there are no electrodes on the faces. In the parentheses on the right, we give the quantities whose expansions in ( have similar forms. We substitute the expansions 35.6 into equations 35.2 to 35.4, equate the coefficients at the same powers of and use the notation accepted in the theory of shells.

We start with the formulas for shells with faces completely covered with electrodes. Instead of the electroelasticity relations for forces, just as in the theory for nonelectrical shells with purely moment stressed state, we get from the first two formulas 35.2 the approximate equations

The electroelasticity relations for moments are written as




11 33

The term in the square brackets is of the order rfs compared to the principal terms. For small values of the variability index s, it may introduce an essential contribution; but as it grows, the contribution becomes much less. For the electrical quantities, we will get

3dls 4/i3


We do not write the formulas for strains and displacements here because they are the same as those in equations 7.19.

The forces 7} and S,y are found from the equilibrium equations. With the forces known, we can find the sums of stresses (equations 35.6) using the formulas


Ri du

¿31 di A

^33 ;





The quantities and r® are found from equations 35.11 after T",-,

have been determined. As in the theory of nonelectrical shells with pure moment stressed-strained state, the transition formulas from forces to stresses differ from the Kirchhoff-Love formulas

For electrode-covered faces with given potential, we can divide the problem into mechanical and electrical. The complete system for the mechanical problem comprises equations 35.7 and 35.8, the equilibrium equations, and strain-displacement formulas. Having solved the mechanical problem, we use equations 35.9 and 35.10 to find the electrical quantities.

For a shell without electrodes on the faces, we should take E

0 in equations 35.7, 35.8, and 35.12. Equations 35.10 do not change. We add the following differential equation for defining i]){0):


daj A 2 da2

We will show in the following section that the decision whether the problem can be divided into the mechanical and electrical parts is based on the type of electrical conditions on the shell's edge. This question will be discussed when we obtain the boundary conditions.


For definiteness, we will assume, as before, that the shell is polarized along the a2-lines. The constitutive relations 2.16 and 2.19 for this type of shell are given in Section 2.

We will investigate shells with and without electrodes on the faces separately. We start with shells with electrode-covered faces for which conditions 34.6 are specified. For simplicity, we assume that there is no mechanical surface load:

We represent the needed quantities of the three-dimensional elasticity theory in the form

By formulas 34.1, we extend the scale in the electroelasticity relations 2.16 and 2. J 9 and .substitute the sought-for quantities by the quantities with asterisks in equation 36.2 to get a1 l-2i+r l-c r33* 2-2s r-

T\u = —q\eu +r)' qie2* - V Pi--V nE2, a i a, l-2t+ral 1 —c n T33* 2-2i r-

166 THEORY OF PIEZOELECTRIC SHELLS AND PLATES r2i. = qs 111u + Tj —m2t - rf V3—Eu

C»C \<?4 «2 <75 «1 / 96«1«2 «15 "2

All the quantities in equations 36.3 and 36.4 are on the same order. When deriving equations 36.3, we used the notation

<71 = —- <72 = —, <73 = -. <74 = -p-

4"3"22 ^lV'22 46"22

When constructing the equations and boundary conditions of the shell theory in question, we will neglect small quantities of order e where e = 0(771 + ?72~2s). (36.6)

In order to approximately reduce the three-dimensional equations to two-dimensional ones, we integrate equations 36.3 to 36.5 with respect to C- After integration all the needed quantities will be represented as polynomials in Q. For example, we write the asymptotic expansion for the stress r(( t (which holds for tijt)

Til, = 771~2i+fr,v,0 + Cr/7,1 + Vl-C<?Tii,2 + 7?2"2SC3T,,3 + • • •

Within the quantities 0(e) defined by formula 36.6, we should have neglected all the terms starting with r„ 2 in the formula above. But for small variability indexes s, the quantities r,,^ and r,/i2 are comparable and give contributions of the same order into the force T¡. Therefore, we will keep three terms in the expansions of the principal stresses r,-,, and r,y» and the quantities e¡* and m¡,. Taking this remark into account, we obtain the following formulas:

= V^2s*Ce\fi + <>1,1 +'?'C2í'l,2 (niu,T¡j,) e2* = <?2,o + <V~2i+re2,! + V2s~'X2C2,2 (mi*) T¡¡* = 771~2s+'r,vo + (Tu,i + Vl~'X2Vu Trw = T¡jfi + C2r/3i2

T33. = T33.0 + ÍJ'-2,+CC^3,I + C2n.3,2 + »/'"^CVsj.S

168 THEOR Y OF PIEZOELECTRIC SHELLS A ND PIA TES We represent i'2,i as a sum of two quantities:

"2,1 - "2,1 +J/2_3l+Ml- "2,1 = *rfis£3.0, (36.8)

where v'2 | does not depend on the variables Q. Comparing formulas 36.7 to similar expressions in the theory of nonelectrical shells [15, 21], we will see that our expressions for the stresses t,¡» and ry» coincide with the respective formulas for the pure moment stressed state. The formulas for the displacements v2* and v3t have no analogs in the theory of nonelectrical shells. The greatest term in the displacement formulas is the rotation angle v2,i; the largest principal stresses are t¡¡j and r^i that define the moments. The displacement v3, varies by a square law; and for the small variability v3,2, it is comparable to v3,o. This means that in our case using the Kirchhoff hypothesis, the normal has constant length and cannot be applied even in the roughest approximation. The formulas for e,,o,..., «12,1 are as follows

1 dvifi rf e¡,a = — -TTT- + rfRkiVjfi + — i'3,0 A¡ d£i Ri

1 dvifi 2s n 1

1 9"2I é?2,i = — -FTT- +rfRk2\'u A 2 <9£2

We substitute the expansions 36.7 into equations 36.3 and 36.4 and equate the coefficients at the same powers of ( and get

T| 1,0 = 9le1.0 + <72^2,0 ~ V2S~2CPIT33.0 7*22,0 = ^2,0 + 92^1,0 - V S~ CP2Tnfi

7|I,1 = 9i«u + v iCq2e2A 722,1 = rj2-4"2^,! + <72*1,1

7y,0 = 93('"i,0 + "V.o)' 77/,1 = <?3("'1,1 + i?2"4l+2f«l2,l)

Equations 36.8 to 36.10, together with the equations of motion written in the same form, constitute a complete system where the number of unknowns is equal to the number of equations. This verifies the validity of the asymptotic representation 36.2.

We pass in the obtained equations to the notation typical for shell theories. The displacements and vv of the middle surface of the shell are related to the three-dimensional displacements v, and i>3 by equations 36.2 and 36.7 as

The forces and moments are expressed through the stresses using equations 5.1,5.2, 36.2, 36.10. We get

r+h lh2

Omitting the cumbersome, but simple, transformations, we write some intermediate results:

0 = V u)j gifi = 77 7/, vu = -r/_1+3i_2c7i \'2i] = i/isf/'ft - i/72,

m,,i = 77'/?^ - Mis£3), m2,i = r?'l+4i"2t'/?r r)'R\ . „ V* . _ 1 d 1

From the third equilibrium equation 2.11 and the second condition in 36.1, we can get

Into formulas 36.12, we substitute 36.10 that was transformed using equations 36.11,36.13, and 36.14; and within the quantities 0(t]x +i]2"~2s), we obtain the following electroelasticity relations:

For electroelastic states with great variability, the terms with coefficients /, and by can be neglected. Relations 36.15 together with the equilibrium equations and strain-displacement formulas, which coincide with the respective formulas from the theory of nonelectrical shells, form a complete system of eight-order differential equations. By formula 34.1, we return to the variable 7 and can write the quantities we are seeking as polynomials in degrees of 7:

Having found the forces, moments, and displacements of the middle surface, we define the expansion coefficients in 36.16 by the formulas where

v<° = rfls£j - 72, v<° = rfls£j - 72, u) 'm"ij r-


We should note that the problem consists of mechanical and electrical parts. After solving the mechanical problem, all the electrical quantities can be found from equations 36.18. We stress that the mechanical problem differs from the theories of Kirchhoff-Love in that the principal stresses and normal displacements vary by square laws; therefore, the transition from forces to stresses is done by more complicated formulas 36.17. Recall that in the Kirchhoff-Love theory, the transition formulas are as simple as


Consider a shell whose faces are not covered with electrodes. Let a mechanical surface load be defined as in 3.1. The electrical conditions on the faces without electrodes are written as those in 3.4.

We choose the following asymptotic representation for the sought-for quan-


Tjj = n22Tij*, r33 = 7?'~"f/222T/3<., 7/3 = V "1/i22r,3,

Quantities s, 77, and c have the same sense, as before.

We integrate the electroelasticity equations with respect to £ within 36.6. Before the integration, we change the sought-for quantities by formulas 37.1 and the independent variables by formulas 34.1 to get the following laws of variation of the sought-for quantities to

Vi* = v,,0 + r?l_2l+rCv/,i t,,»,t,■,,,£>,»)

733, = 733,0 + <t33.i + t?'c2733,2 + v2"^^^

Passing to the notation of the shell theory, we get the equations which will be divided into two groups. The first group will be comprised of

544 2h3

ocvi oa 2

In addition to equations 37.3, we should include the equilibrium equations and strain-displacement formulas in the first group. We then get a closed system of tenth-order differential equations with respect to the unknown mechanical and electrical quantities. All other quantities will be included in the second group:

Having found the solution to the system of the first group, we use formulas 37.4 to find the electrical quantities that did not enter the first group. Note that the complete problem, generally speaking, cannot be divided into the mechanical and electrical problems with the exception of some special cases. Say, for an axially symmetric problem [26], that all the electrical quantities in equations 37.3 can be expressed in terms of the forces. This allows us to divide the problem into the mechanical and electrical parts. Here, the equations of the mechanical problem differ from those of the nonelectrical shell theory by the sense of the coefficients before the strain components in the constitutive relations.

Chapter 7



We will assume that just as for nonelectrical shells [22, 43] the complete elec-troelastic state can be represented as a sum of two electroelastic states: the inner electrostatic state and the boundary layer. The inner electroelastic state varies relatively slowly along the coordinate lines of the middle surface and is described by the equations of piezoceramic shell theory. The boundary layer damps down quickly in the direction perpendicular to the edge and is described by three-dimensional electroelasticity equations.

In the theory of nonelectrical shells, the boundary layer plays a secondary role and is rarely resorted to for a strict calculation of the inner stressed-strained state. The corrections due to allowing for the boundary layer are rather small. We will show that this situation can also be observed in the theory of piezoceramic shells but with some exceptions: the calculation should be started with the boundary layer because it defines the greatest stresses. Also, the corrections due to the boundary layer under the conditions of the shell theory introduce qualitative changes into the description of the internal electroelastic state [81, 82],

For constructing the boundary conditions in the theory of electroelastic shells, we should use the Saint Venant principle generalized to the case of electroelasticity [83]. In what follows the generalized Saint Venant principle is obtained from the solutions of three-dimensional electroelastic problems.


We assume that the investigated edge of a shell coincides with the surface 0| = r»io- As in Part I, we introduce the electroelasticity equations by using equations 2.12 and 2.13, the nonsymmetric tensor rw instead of the symmetric stress tensor opfl, and the vectors D and E instead of the electric induction vector P and electric strength vector £.

We asymptotically extend the scale near the edge a, -o„> = /??/'£,, a2 = Rt?£,2, 7 = (38.1)

This means that the sought-for electroelastic state has the same great variability with respect to the variables ai and 7 and much less variability with respect to the variable tv2, which is equal to the variability of the inner electroelastic state.

The equations for a piezoceramic preliminarily polarized in the thickness direction will have the form

1 1 dr2

' V + *21'1 + fy) + +-snfl2r22 + if3r33] - d}iE3 = 0

V ft-gç — [-Sl3«l t + S^a2T22 + S^Tii] - (liiEi =0

= eTuu2E I +i/|5fliTi3 a2D2 = e7uf\E2 + </i5ti2T23

The equations of the electroelastic boundary layer, like those of the boundary layer of nonelectrical shells, can be divided into the equations for the plane and antiplane boundary layers. In the first approximation, the equations of the plane boundary layer are the equations of a plane electroelastic problem. The equations of the antiplane boundary layer coincide within the coefficients with the equations of the antiplane problem of the theory of elasticity.

We assume that the asymptotics of the plane and antiplane layers leads to a noncontradictory iterative process:

ç£ Tk J? Tk E k E k _2 'Ji! nk \ _ r, k k k k k rf s ■S11T21'S11TI2,S11T23,S11T32, fr < ¿T U2 j ~ V (T2\*,T\2*,T23*,T32*,V2*.U2*)

( E k E k E k E k E k V\ v3 , ^ . r-k . r-k ¿31 nk ¿31 nk\ ^llrM,5llT22,Jllr33,sllTl3,JllT31,p J > ¿31 y , ¿31 E,, ¿31 £3 - ¿rD\<^TD3 )

As usual, the degrees of rj are chosen so that the dimensionless quantities with asterisks are of the same order.

Formulas 38.5 and 38.6 unite two asymptotics for the plane and antiplane boundary layers. The superscript k should be replaced by a if the quantity belongs to the antiplane boundary layer and by b if it belongs to the plane boundary layer. For the antiplane problem, we choose r - 0. (38.7)

We change the variables 38.1 in the three-dimensional electroelasticity equations, taking into account the asymptotics 38.5 to 38.8, and break the resultant system into two subsystems. In the first subsystem, the quantities 38.5 and 38.7 of the antiplane layer are principal. In the second subsystem, we find the quantities 38.6 and 38.8 of the plane layer. We write out the equations of the antiplane and plane boundary layers uniting them for brevity:

A io

^-^2^ + ^3 + 7/^=0 (38.9) 02. -'|T2*3.+ff23 + '7lK23 = 0 (38.10)

^ - (^i. - ^12. - 3J - £$. + IVf + VIVf = 0 ^fw - t22, + - El + + IV*1 = o ^ - (a3T<3, - ,,rfu - „,T<2J - d^El + V < = 0 d^l i , i/,5 , ,

Dl - El - r2(Tf3. + r2*2.) - f3T}3, + 7/K31 = 0 1 <9£>* eL dDl , . ,.

In the formulas above, we used (he notation 34.7. The notations ... ,¿3' follow from the formulas

=-C>2~tÎU-02T"22, x\ = = W\ = Wh2 = Ai6 = 0, tW) _ c-b K21 - -^l*

0 0

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