## Ca i V

The number is found by equation 45.11 from the solution of the antiplane problem with conditions For the rigidly fixed edge cv2 a2o without electrodes For defining ftio and pw, we should integrate the homogeneous conditions of the plane boundary layer with the following conditions at the edge 2 0 VII. vb2, 0, VIIL v .+ 0. IX. v t 0, In problems VI and VII, we should find the resultants of the horizontal forces r6 and r7. In problems VIII and IX, we should find (he resultants of the vertical...

## A crC f fc

After some transformations, the conditions in equation 50.14 will be written fr, w(0, C) (p , 0 + t13.(0, < )*(P > 0 + 2aVt,(0, Qs( > , 0 Jo +2(7lAu(0,CW > ,0 < < - COsPn I + ft(p,.,C)V .(0,CWC 0. (50.17) i(P .) 5 Ccosp C + sin > ( (sin2 ,, + g(P ,0 PuCsinpnC - cosp C (cos2 > +

## Jl

The second term in the first condition is 0(r)2s) for k2 aio) 0 and 0(r 2 3'') fork2(aio) 40. The numbers p2,pi, and m are found by formulas (45.7) from the solutions to problems VIII XI. The number Ji is calculated using equation 45.11, where we put i where Sl2 has been found from the homogeneous equations of the antiplane problem with the following conditions at the edge 0 (n, r21 rl3 Dt 0, a 0, I - .y, 6 - I + 2q) THE INTERNA L ELECTRO ELA STIC STA TE A ND THE IiO UN DA RY LAYER 217 Here, J2...

## Aei

Allowing for 22.1 and replacing the sought-for quantities by their asymptotic representations, we rewrite the boundary conditions in the form r 5 vvl0 + r v ie) 0, i ' + V 2+fl w 0 2* + ilU el 0, V 2sl Sr + , 2+7,u) 0 0 ) + 1 2+ ) _ This problem can be solved by taking a s. We preserve only the principal terms and find the boundary conditions for the membrane theory < > 0, 4 0, V(m) V (22.5) If the edges of the shell have no electrodes, conditions for the...

## Afv V l

On the electrode-covered edges a, a,i and a, a,-2 of the plate. For simplicity, we will assume that the mechanical load is absent. Using the respective shell theory formulas, we get the following electro-elasticity relations and strain-displacement formulas As compared to equations 13.29, in equations 13.35 we see additional terms containing electrical quantities. For defining i > (0 we have the differential equation The electroelastic state is calculated in the following order. First, we...

## 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...

## Bimorphic Shells And Plates

Consider a shell that consists of two rigidly connected piezoceramic layers of the same thickness, h. There are electrode coverings on the middle surface of the two-layer shell and on its faces. Both piezoceramic layers are thickness-polarized in one direction. Electric power is supplied to the electrodes (Figure 4). Should the layers be disconnected, one of the layers would suffer elongation, and the other would suffer compression under the action of the electrical load. But since the layers...

## Boundary Conditions In Electroelasticity Theory

The equilibrium equations 2.11, geometrical relations 2.6 or 2.7, piezoeffect equations 2.14 or 2.16, and the equations of electrostatics 2.18 or 2.19 constitute a complete system, where the number of equations is equal to the number of unknowns. To solve problems, we use this system of differential equations supplemented by additional boundary conditions for the sought-for electrical and mechanical quantities. In electroelasticity theory, mechanical boundary conditions are formulated just like...

## C ac x

9V> * A dV> * pa Til, + V-7TT - l*L FIGURE 31. Stresses and electrical potential of the boundary layer at the shell edge as functions of the thickness coordinate FIGURE 31. Stresses and electrical potential of the boundary layer at the shell edge as functions of the thickness coordinate FIGURE 32. The stress T12, on a shell face as a function of the coord nate FIGURE 33. The displacement on a shell face as a function of the coordinate FIGURE 33. The displacement on a shell face as a...

## Cr J i

On the electrodes covering the bar ends fi, the boundary conditions for the electrical quantities have the form for DYNAMIC PROBLEMS IN THEORYOEPIEZOCERAMIC PIATF.S & SHELLS 131 Disconnected electrodes The potential difference on the electrodes is given as In equations 31.1 to 31.4, we use dimensionless quantities, dimensionless variable and dimensionless parameter A 0-, sfjCT, w, Et - djsE D, -j-D, Let us find the solution to the system of equations 31.1 to 31.4 for the different boundary...

## Tf

The quantities A ), A2,1 A i, 1 A 2, 1 jR,, 1 R2, k , and k2 that characterize the geometry of the shell's middle surface are expanded in power series in the variable a i in the neighborhood of a aio- For example, for A i, we can take into account formulas 38.1 and write a series Al0 + Rri(Al),l Aw + T M < i-i Here, A io and (Ai) are coefficients of the Taylor series. For quantities with superscript 1, formulas 38.12 and 38.13 are written up to 0(t) ). For defining an approximate solution of...

## D D r

We have only written the formulas required to derive the boundary conditions. 43 BOUNDARY CONDITIONS IN THE THEORY OF SHELLS WITH THICKNESS POLARIZATION (ELECTRODE-COVERED FACES) We will construct the boundary conditions with the accuracy accepted for the electroelasticity relations in Chapter 6, i.e., An exclusion is made for some boundary conditions whose error is not specially specified. As before, we will consider only two variants of the electrical boundary conditions on the faces (1) when...

## Dij dn

TT-, 2n* -nv - < *' ,, w, + 7H, . (2S.6) The numbers a, A, ci2, t i, and i 2 are taken from Section 27. The dimensionless quantities with asterisks are introduced by formulas 27.5, to which we should add The initial equations are membrane. This means that we do not consider the vibrations described using the moment equations of the theory of piezoceramic shells. We can do this because, as it was shown in Scction 27, the frequency parameters A, for which...

## Dj kPu n I i tis

2 - i 0 + kdi5n. (41.7) In the new notation, the conditions on the faces without electrodes ( 1, ( -1 look like We integrate equations 41.7. Their solutions are < i> ci cosp( + c2 s'wp + nt(p, Qcosp( + bi(p, Qs'mp( U ci cos kp + C4 sin kp - d ci + cos pC, + ( C2 + b (j),(,) ) sin pQ > +a2(p,( )coskp(, + b2 pX) nkp l> 2(P,Q - 2 COSkp(d(. (41.9)

## Fp OK cos K

H(p,Q -z -*(p,Ocosp( + < T< t> (j),OsmpOdC (50.11) 2per Jo We find fl)(p) and a2(p) from the boundary conditions 50.9 i(P,0 ( cos2p - y - pb2(p, 1) p + j Vu(0,I) - -V3.(0,1) -2 a p 2 p We take an inverse Laplace transform and compute the integral with the help of the theorem on residues. The equation > (p) 0 has a three-tuple root if> 0 and infinitely many four-tuple rootsp ,p , - , . Since the problem implies a decreasing solution, the residues with respect to the roots p and p with...

## F[f r

These transformations allow us to include all the boundary layer quantities in the boundary conditions 44.4 and write them in the form 1 2s+c 2 2 3i+rn 1 1 ar2M 2T3,o + V-2j T3,,2 + r M,, + - 0 We pass to the forces and moments in formulas 44.10 as in formulas 36.12. After the transformation, we gel the following boundary conditions at the free electrode-covered edge rvj o-iq The rest of the boundary conditions can be found similarly. We give them without derivation. Rigidly fixed...

## General Theorems On Electroelasticity

Consider a piezoceramic body of volume V bounded by the surface ft in the coordinates Adhere, as before, the Greek indices assume the values 1, 2, or 3 and the Latin indices assume the values 1 or 2). For simplicity, we will only deal with the static case 82 . B I qpvpdv + I (crp ipv - ipnpVp)dQ Here qp are the components of the bulk forces vp are the components of the displacement vector ap,> is the stress tensor ep 1 is the strain tensor Vp are the contravariant components of the electric...

## Hi if

U sn 6< Vi 12 < , 2 C2f 31ll2 ( 22 33) The condition in 25.4 should hold at the electrode-covered edge. Those in 25.5 and 25.6 should hold at the edges without electrodes, q, m and 2 2o, respectively. Consider the vibrations of a piezoceramic shell with electrode-covered faces to which we apply a potential difference 2Ve ,u harmonically varying with time 1. The electroelasticity relations for the shell are given in Section 7 and have the form 7.23 and 7.24. The normal component 3 of the...

## Ao I dHu

We transform the formulas to a more convenient form 1 d v 4 THE METHOD OF PARTITIONING A STATIC ELECTRO ELASTIC STATE 69 _ 2A3 ,22 I d V 21 1 d , d3w If the faces are electrode-covered, formulas 21.7 and 21.9 remain valid. For the edge a2 a2o formulas 21.8 are replaced by n22e2+n2 e 0, T 2 ( i i + 21 2) Calculations are made by 21.11 supplemented by the formulas 22 BOUNDARY CONDITIONS FOR THE PRINCIPAL ELECTROELASTIC STATE AND SIMPLE EDGE Let us consider an example of dividing the boundary...

## D r

We use equations 34.19 and 34.18, for shells without electrodes on the faces, to obtain the following differential relation, which will be needed for further investigation da A dcn da2A2da2) 3 + 2 7e x ( do, + da2 x 2 + TT Al ( - ) V' > 0. (34.2 ) The terms in the braces are small. They are on the order of 0(V) compared to the principal terms. Just as in nonelectrical shells 22 , the accuracy of equation 34.8 is maximal for hypotheses of the Kirchhoff-Love type. For more accurate...

## If [U n

In our case, the problem is divided into two simpler ones mechanical and electrical. The mechanical problem coincides with that of the theory of nonelectrical shells within the coefficients. It is described by a system of differential equations that contains the equilibrium equations 5.3, strain-displacement formulas 7.19, and electroelasticity equations (equations 7.23,7.24 and 7.27). The electrical quantity in the electroelasticity relations is expressed through the electric potential found...

## In V

Respectively, and take their sum to obtain the solution to the system of equations 45.4 under conditions I III. The solution of problem III is multiplied by 771 because the terms which correspond to the conditions 45.5 in the right-hand sides of equations 45.4 have this order. The stresses at the infinitely distant edge of the half-band are equal to zero. For the band to be at equilibrium, we should equate the resultant of the horizontal (directed along the lines) stresses at the edge to zero,...

## Info

72 + (.f,G,+5f3C2) 0 ' 2 + ( 2 12 + sf3 22) 2 - (rf31 - 4c, - sf3c2) < > (0) 0 7 '-5 (if212 + *B 22) 2 ( 31 - if2ci - ifjcz) V(0) - f ' (46.1 1 ) For defining and 14, we should solve three auxiliary plane problems In problems X, XI, and XII, we calculate (he horizontal resultants Tio, T 1, and Ti2 of the forces at the edge 0 and the resultants D)0, D 1, and D 2 of the electric induction vector component normal to the edge surface A, A'13 and A'u are the vertical resultants at the edge,...

## L

V TT ( T - ) . (23.20) The auxiliary problem equations coincide with those for simple edge effect 20.8 to 20.13. The partition of the boundary conditions at the free edge leads to two boundary conditions for the principal problem, which are a combination of the edge forces and moments, and to nonhomogeneous nontangential conditions for the auxiliary problem as in the theory of nonelectric shells 15 . The principal problem asymptotics is described by 23.9 for < 7 0, and r b 2 4i. Its error is...

## Pjc

The constants ci, ,C4 are found in conditions 41.8. We assume that the functions 41.2 specified at the edge are even functions of We set Co -1 and Ci 0 to get d bi(p. )cosp + (I Ci sinp + kb2Qh I)cost . (41.10) We substitute 41.10 into 41.9 and find the residues of the functions U and Then, by using the inversion theorem, we get the final formulas for v2 and i > . For these formulas to have no exponentially increasing terms, we equate the residues to zero at the points p - 0 and get the...

## Plates With Tangential Polarization

As in the theory of shells with tangential polarization, we will assume, for definiteness, that the plate has been polarized along the a2-lines. The equilibrium equations and strain-displacement formulas for the plane and bending problems are given in the previous section. Here we will only write the electroelasticity and electrostatics equations obtained as a special case of the shell theories given in Sections 9 and 10. PLATES WITH FACES NOT COVERED BY ELECTRODES The Plane Problem The...

## QcnV aa V

Is given on the electrode-covered faces of the edges. For simplicity, we will assume that the mechanical load is absent. In order to construct the shell theory consisting of the previous four hypotheses, we should reconsider the second Kirchhojf's hypothesis and replace equation 7.13 with an approximate formula We add the fifth hypothesis in that a,3 is neglected in equation 7.5 The sixth hypothesis will discard D for n 1 in equation 8.7 Using these hypotheses after transformations, we get the...

## S rU s

In the formulas in 20.8 to 20.11, we passed from the coordinates to the coordinates a, using 20.2 and omitted the subscript 0 that denotes the number of the approximation. We stress that in the zeroth approximation theory for simple edge effect, the coefficients characterizing the metric and curvature of the middle surface are treated as constants with respect to near the edge I Q 0- After some transformations, we get the equation for the simple edge effect with respect to the displacement w...

## 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 is given on the electrodes. Under the effect of the electric field, the domains oriented along the...

## Shells With Thickness Polarization Electrodecovered Faces

D3 e 3 3+< 3 (crn +cr22) + 33 33 (7.6) We assume that the electrical potential is defined on the electrode-covered surfaces. The theory of shells with thickness polarization can be based on hypotheses similar to those proposed by Kirchhoff, which are generalized to the electroe-lastic case as follows. In the electroelasticity relations 7.2 and 7.6, the stresses < 733 can be neglected as compared to the principal stresses o ,o22. Therefore, we can replace 7.2 and 7.6 by the approximate...

## Si

The equations of the shell have the form dS FIGURE 28. Piezoceramic cylindrical shell with electrode-covered edges FIGURE 28. Piezoceramic cylindrical shell with electrode-covered edges The first resonance frequency as a function of cylindrical shell sizes in vacuum and in glycerine The first resonance frequency as a function of cylindrical shell sizes in vacuum and in glycerine Here, the v-line coincides with the generatrix of the cylindrical surface and v is the displacement in the circular...

## T gj sj

These vibrations are characterized by the relations The principal problem asymplotics for quasitangential vibrations can be obtained by putting a 2s, b - 0, and r s in formulas 23.9. The principal error is a quantity of order e, where The equations of quasitangential vibrations coincide with those of the plane problem and have the form The principal problem system 23.19 is integrated, taking into account the tangential conditions. The auxiliary problem equation consists of integrating 20.8 to...

## Tfu C o Dl

44 BOUNDARY CONDITIONS FOR A SHELL WITH THICKNESS POLARIZATION (FACES WITHOUT ELECTRODES) Consider a free electrode-covered edge qj qio. We will assume that the shell has other edges which are well fixed so that the shell is in a membrane electroelastic state. The three-dimensional boundary conditions at the edge are r, 0, r21 0, t3, 0, ip V. (44.1) As before, we represent the total electroelastic state as a sum r + V-2i+fC r + rf -* (r,-1 + '') + Crft r2 .< 0*W T2I,I+7 y 21,0 ' 21,1J

## The Theory Of Piezoceramic Plates With Thickness Polarization

The theory of piezoceramic plates can be constructed from the three-dimensional electroelasticity relations and the hypotheses used for the derivation of the equations of shell theory. We will not repeat our argument and will obtain the equations for the plates as a special case of the respective equations of the shell theory introduced in Sections 7 and 8. For this purpose, we use the plate geometry and choose the zero curvatures in the shell theory equations, i.e., and in all formulas of the...

## Ti S

The number J is found from the solution to the antiplane problem 38.15 with the homogeneous conditions 38.17 on the faces and the edge condition Hinge-supported edge fV aio without electrodes u2 0, u> 0, Gv 0 WdsU 72- 1 0, tVl- (43.11) We find the number nj by the formula from the solution of the plane problem 38.16 with the homogeneous conditions 38.18 and 38.19 on the faces and the edge conditions

## Tiiad Tiaioi TnaV hJhJh

These conditions imply that the edge mechanical load satisfying conditions 41.4 gives a solution that damps down in the narrow edge zone and cannot be found from the equations of shell theory. Let us find the damping conditions for the antiptane layer. The equations of the antiplane problem in dimensionless coordinates will have the form System 41.5 can be reduced to two equations with respect to v2 and ijr. When solving the system of equations 41.6, the first two conditions in 41.3 should be...

## Tk J

In a similar way, we calculate the resultants of the moments acting on the element's side and get the formulas for the bending moment G and the twisting FIGURE 3. Forces and moments acting on an element of shell surface moment H2 The positive directions of forces and moments are shown in Figure 3. In order to get the equilibrium equations in the terms of forces and moments, we equate to zero the principal vector and the principal moment of all the forces acting on an element of the shell's...

## Twodimensional Equilibrium Equations

We use the surfaces a const, and a2 const, a + dat const, a2 + da2 const to single out an infinitesimally small element of the shell and choose positive stresses as shown in Figure 2. Consider the element's side lying in the plane a2 + da2 const (Figure 2). The length of the arc ab is and the area of the hatched element bounded by the curves 7 const and 7 + 7 const is equal to Here H and H2 are Lame coefficients. The resultants of the stresses acting on the element's side will be o2 H2da2d*y -...

## U pujslb

U C5 cos A 4iai + C(, sin AjAiai 2 C7CosA2-4iai + eg sin A2A1Q1. T 2 h 1 d,U T2 vTx 1 if,(l - u2)A da ' '' y -s-66 We find a particular integral. The asymptotics of the particular solution is (s 6 0) The greatest quantities in equations 26.8 are the normal forces and the deflection. In the roughest approximation, they are defined as We break the boundary conditions into those for the quasitransverse vibrations and those for the quasitangential vibrations. As in the theory of nonelectrical...

## Uf f euT eydSe

We find Condition 27.15 holds on the electrodes. Allowing for the r(l,,> - c- t M) p(si') _ * 2* -eu+f2 2. , u - - (2 .2U) We solve this equation meeting the conditions 27.15 and find lieeting the cc and Then we calculate U sh) as < > J (euIT + + E D dS. The computation was performed for a shell made of PZT-4 of length 21 2R,h r 0.025. Figure 5 gives the amplitude values of forces and displacements near (a) the first resonance for a shell and (b) the second resonance as functions of the...

## Uw J eegTdn

Here, U(d) is the internal energy of the plate with disconnected electrodes, and Uw is the internal energy of the plate with short-circuited electrodes. In both cases of computing the internal energy in equation 30.15, we use the strains e, and eg that were found from the solution of the initial problem. Let us find the internal energy of the plate with disconnected electrodes. The potential difference V on the disconnected electrodes is found from 124 THEORY OF PIEZOELECTRIC SHELLS AND PLATES...

## Rto Ct CMt

+ ri2-s*aW +riu0ipli t V,. (44.2) The problem is physically and mathematically noncontradictory if a 1 - 2x + c, ,6 0, 2s. (44.3) We allow for 44.3 and introduce the notation Then the conditions 44.2 can be written in the form r io + j 2s+C -,. , .,1 2s+c rt . . t s b _ r 731,0 + Cr3t,l +V 731,2+7 7JU + J T3U 0 tf.o + Ctf.i + u'- C2 + + V,. (44.4) THE INTERNAL ELECTRO ELASTIC STATE AND THE BOUNDARY IAYER 205 When deriving the boundary conditions of the shell theory, we use the generalized Saint...

## Vfi v R

As we pass from stresses to moments these relations assume the form Ei d3lEf UJ 0 G, FIGURE 4. Section of a bimorphic shell element by he ya, - plane FIGURE 4. Section of a bimorphic shell element by he ya, - plane The numbers B and a are found from 7.27. We note that, just like in the theory of nonelectrical shells, there are no elasticity relations for the forces 7, and Sy, and they can only be found from the equilibrium equations. The formulas for other quantities coincide with equations...

## Vo VIniVi o I rr

Where the numbers IY Tg are found by solving problems VII-Vlll. Thus, after eliminating the quantities of the boundary layer in the boundary conditions, we get We find the constants nn,pi, and p2 that enter the terms allowing for the boundary layer corrections by solving the five auxiliary problems I III, VII, and VIII. In the resultant boundary conditions, we turn to the notation typical for shell theory by th& foFmuIasof Section 7. At the rigidly fixed edge ai al0 of a shell with...

## F V V Vj J

V3.2 D,,o D,1' > +r 2- Dl 'o We can obtain the system of equations for defining the quantities i'i*, V3*. by summing up the equations of the principal subsystem for the plane boundary layer and the auxiliary subsystem for the antiplane boundary layer up to the quantities 0(tj1) + i T U +i r22. - T33, -F3 1 ch'3* i(Ti 1* + T33 ) + CT3T22 P - (45.4) The quantities X P ' are found from the formulas 38.12. We will treat the first and third conditions 45.3 as the boundary conditions for the plane...

## TnAhkT [Vl

We should solve auxiliary problem VI with the following conditions at the edge 6 0 Free edge ai io without electrodes A2da2 3 2 ( -if eijii.o J U J' Rigidly fixed edge Qi Qio without electrodes h Qu Mi+m22fi 0 i _I , M2 0 A oal n- + p6 'n i 0 771-f (44.16) where ri3 r7 rg has been determined from the solution of plane problems VII and VIII with the following conditions at the edge 0 VIII. vl 0, + < 0, Dl 0. The numbers p and are found by solving auxiliary problems IX XI XI. v t 0, v . + C2 o,...

## RJ Rj

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. an -i h q , < 73j 7 i qf (13.3) For plates...

## Hypotheses Of The Theory Of Nonelectric Shells And The Saint Venant Principle

In the previous section, we wrote the equilibrium equations of the theory of shells. The equations directly follow from statics laws and no additional assumptions are required. Three-dimensional constitutive relations (piezoeffect equations), strain-displacement formulas, and electrostatic equations should also be transformed so that the sought-for quantities i.e., the vector components of the induction, electric field strength, stresses, strains, and displacements- were referred to the middle...

## F

'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 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* -...

## Shells With Thickness Polarization Faces Without Electrodes

Hold on the faces that are not covered with electrodes and are in contact with vacuum or air. Consider two cases depending on the electrical conditions on the edges. The shell's edges may have electrodes, but no electric power is supplied to them. Here the Kirchhoff-Love hypotheses for mechanical quantities and the third hypothesis for electrical quantities given in Section 7 remain true. The fourth hypothesis implies that we can neglect D3 in equation 7.6 as compared to 33 13 to get 3 - 3i(<...

## Free Shells With Thickness Polarization

We know that a stressed-strained state appears in nonelectrical shells with all edges unfixed (free shells) under the effect of an external load. In this case, the extending strains and shears are small compared to the fiexural strains the largest stresses are defined by the moments. A similar stressed state is typical for free piezoceramic shells. Nevertheless, although the elasticity relations for free nonelectrical shells have a form similar to that for shells with fixed edges, we observe...

## Nellya N Rogacheva

The theory of piezoelectric shells and plates Nellyn N. Rogacheva. p. cm. Includes bibliographical references and index. 1. Piezoelectric devices Mathematical models. 2. Piezoelectric ceramics. 3. Shells Engineering Mathematical models. 4. Plates Engineering Mathematical models. I. Title. TK7872.P54R64 1993 621.381'4 dc20 93-11621 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide...