MECHANICS

539.3

D. I. Bardzokas, M. L. Filshtinsky

Oscillations of a Piezoceramic Space with Tunnel Opennings and Rigid
Stringer (Antiplane Deformation)

(Submitted by academician L.A. Agalovian 12/X 2004)

   1. Introdaction. Many actual scientific and technological problems of modern engineering are connected with the investigatins of processes of propagation of waves in piezoelectrics and with the definition of dynamic strength in the vicinity of heterogeneities of various types. Solution of appearing in this case of complicated problems requires the usage of modern mathematical means and, in particular, methods and approaches of the dynamic theory of elasticity. Development of these methods is reflected in monographs [1-5] which appeared during the last decades. The procedure of application of the method of boundary integral equations to investigations of diffraction problems of electroelastic waves is developed in [6].
   In the given article there is constructed an analitical algorithm for investigation of coupled fields in a piezoceramic medium, weakened by heterogenities of tunnel types along the axis of the material symmetry of the opening and rigid linear inclusions (stringer). The excitation of oscillations in the medium takes place due to the harmonically changing with time shear stresses acting on the surfaces of the cavities.
   2. Statement of the problem. Consider the referring to the Cartesian system of coordinates x1x2x3 piezoceramic space, containing tunnel in the direction of axis x3 opening Gj(j = 1,2,,n1), strengthened by rigid curvilinear stringer Lm(m = 1,2,,n2). Excitation of an electroelastic field in the medium takes place under the influence of the prescribed on the surface of the openings harmonically changing with time, not depending on coordinate x3 shear forces X3n = Re(X3e-iwt) (t is the time, is the circular frequency). Assuming that the vector of preliminarily polarization of piezoceramics is directed along axis x3, considering two variants of the electric boundary condition: the surfaces of the openings are electrodized and grounded (variant A); the surfece of the openings are bounded with vacuum (variant B). We will also assume that functions X3 and the curves of contours Gj and Lm satisfy the Holder condition [7].
   Under the given conditions in a piecewise-homogeneous space we have an electroelastic field corresponding to the state of antiplane deformation. The full system of the equations in a quasistatic approximation includes the following relations [5]:
equation of motion        1s13 + 2s23 = r 2u3
t2
,   i =
xi
                
(2.1)
   material equations of a medium
sm3 = cE44mu3 - e15Em,   Dm = e15mu3 + ne11 Em    (m = 1,2)
(2.2)
equations of electrostatics         div

D
 
= 0,   
E
 
= - grad
f.             
(2.3)

   In (2.1) - (2.3) sm3 are the components of the stress tensor, u3 is the component of the elastic displacement vector in the direction of axis x3; and are the vectors of strenght and induction of an electric field; f is the electric potential; cE44, e15 and ne11 are the shear modulus, measured at constant value of an electric field, the piezoelectric constant and dielectric permeability, measured at fixed deformation, respectively; r is the mass density of the material.
   The system of equations (2.1) - (2.3) will be brought to differential equations with respect to displacement u3 and electric potential f:
cE442u3+e152f = r 2u3
t2
,   e152u3 - ne112f = 0.
(2.4)
From (2.4) we have the following relations
2u3 - c-2 2u3
t2
= 0,    2F = 0,
f =

 

(2.5)

where c is the velocity of a shear wave in the piezoceramic medium, k15 is the factor of a mechanical coupling [5].
   Mechanical and electric quantities taking into account (2.2), (2.3) and (2.5) may be expressed as functions u3 and F over formula
s13 - is23 = 2
z


cE44
1 + k215
u3 + e15F

,
D1 - iD2 = -2ne11 F
z
,    E1 - iE2 = -2
z


F + e15
ne11


,    z = x1 + ix2.
(2.6)

   Assuming u3 = Re(U3e-iwt), f = Re(f*e-wt) and F = Re(F*e-iwt) we will write down equations (2.5) with respect to amplitude quantities (where g is the wave number).
2U3 + g2U3 = 0,    2F* = 0,   f* = e15
ne11
U3 + F*,    g = w
c
,
(2.7)

   Assuming that the insert is fixed, let us represent the mechanical and electric boundary conditions on contour L = Lm as follows
U3 = 0,
(2.8)
Es+ = Es-,    Dn+ = Dn-.
(2.9)

   Here Es and Dn are tangential component of an electric strength vector and the normal component of an electric induction vector, respectively sign "plus" and "minus" refer to the left and right edges of inclusion Lm the moment from its beginning am to end bm (Fig.1).
   To obtain an efficient, in the sense of numerical realization of the system of integral equetions, boundary condition (2.8) it is recommended to differentiate over arc coordinates


U3
s


= 0.
(2.10)

   The mathematical record of the boundary conditions on contour G = Gj for the considered variants of boundary conditions has the form

n
{ cE44(1 + k215)U3 + e15F*} = X3,
(2.11)
f* = 0,
(2.12)
Dn = -ne11 F*
n
= 0.
(2.13)
   Boundary equalities (2.12) and (2.13) satisfy variants A and B, respectively. Below instead of (2.12) we will use condition

s


F* + e15
ne11
U3

= 0.
(2.14)

   Thus, the problem consist of the determining of functions U3 and F* from differential equations (2.7) and boundary conditions (2.9) - (2.11), and also (2.13) or (2.14).
   3. Solvable system of singular integral equatins of boundary problems of electroelastisity. Constructing the integral representations of functions U3 and F* we will use the fundamental solution of the system of equations (2.4) in case, when the dependence on time has harmonical character. In this case we proceed from the system of equatons [6]:
cE442U3 + e152f* + pw2U3 = -P0d(x1 - x10, x2 - x20),
e152U3 - ne11f* = Q0d(x1 - x10, x2 - x20).
(3.1)

   Here P0 and Q0 are linear densities of concentrated shear conditions and charges, acting at point z0 = x10 + ix20 of the medium; d(x,y) = d(x)d(y) is the Dirac d-function. The solution of equations (3.1) is found simply and is determined by formulas
U3(x1,x2) =

k215Q0
4ie15(1 + k215)
- P0
4icE44(1 + k215)


H0(1)(gr),
f*(x1,x2) = - Q0
2pne11
ln r + i
4(1 + k215)ne11


e15P0
cE44
- k215Q0

H0(1)(gr),
r = |z - z0|,       z = x1 + ix2
(3.2)
   According to (3.2) we will write the representations of the solution in the form
U3(x1,x2) = i
4cE44(1 + k215)





L 
q(z)H0(1)(gr)ds +


L 
p(z*)H0(1)(gr1)ds

+
       + k152
4ie15(1 + k215)



G 
f(z*)H0(1)(gr1)ds,
(3.3)
F*(x1,x2) = - 1
2pne11



G 
f(z*)ln r1)ds,   r = |z - z|,    r1 = |z* - z|,    z L,    z* G.

Here Hn(1)(x) is the Hankels function of the first kind of order n, ds is the element of arc length of the contour, over which the integration is carried out. It is easy to become convinced that the determined in (3.3) functions U3 and F* automatically satisfy electric conditions (2.9) on L and radiation conditions at infinity, and also provide the carrying out of equality [U3] = U3+ - U3- = 0 in (2.8). Unknown "densities" q(z), p(z*) and f(z*) are determined from the complex system of three integral equations, which are obtained as a result of substitution of limiting values corresponding to derivative functions (3.3) at z z L and z z* G in boundary conditions (2.10), (2.11), and also (2.13) or (2.14). The given system will be represented in the form:




L 
q(z)G1(z, z0)ds +


G 
p(z*)G2(z*, z0)ds +


G 
f(z*)G3(z*, z0)ds = 0,
(3.4)
- 1
2
p(z0*) +


L 
q(z)G4(z, z0)ds +


G 
p(z*)G5(z*, z0*)ds +


G 
f(z*)G6(z*, z0*)ds = X3(z0*),
lf(z0*) +


L 
q(z)G7(z, z0*)ds +


G 
p(z*)G8(z*, z0*)ds +


G 
f(z*)G9(z*, z0*)ds = 0,
G1(z, z0) = 1
cE44


gH1(gr0)sin(y0 - a0) - 2i
p
Im eiy0
z - z0


,
G2(z*, z0) = g
cE44
H1(1)(gr0)sin(y0 - a10), G3(z*, z0) = - k215g
e15
H1(1)(gr0)sin(y0 - a10),
G4(z, z0*) = ig
4
H1(1)(gr20)cos(y10 - a20),
G5(z*, z0*) = 1
2p
Re eiy10
z* - z0*
+ ig
4
H1(1)(gr30)cos(y10 - a30),
G6(z*, z0*) = cE44k215
4ie15
gH1(gr30)cos(y10 - a30),    H1(x) = 2i
px
+ H1(1)(x),
r0 = |z - z0|, r10 = |z* - z0|, r20 = |z - z0*|, r30 = |z* - z0*|, k = k215
1 + k215
,
a0 = arg(z - z0), a10 = arg(z* - z0), a20 = arg(z - z0*), a30 = arg(z* - z0*),
y0 = y(z0), y10 = y1(x)0*,    z*, z0* G.
In case when f = 0 on contour G (variant A), we have
l = 0, G7(z, z0*) = kg
4ie15
H1(1)(gr20)sin(y10 - a20),
G8(z*, z0*) = k
4ie15


gH1(gr30)sin(y10 - a30) - 2i
p
Im eiy10
z* - z0*


,
G9(z*, z0) = - 1
2pne11(1 + k215)
Im eiy10
z* - z0*
- kg
4ine11
H1(gr30)sin(y10 - a30).
(3.5)
   Satisfying boundary conditions Dn = 0 on contour (variant B) in (3.4) it is necessary to put
l = - 1
2
,    G7(z, z0*) = G8(z*, z0*) = 0,   G9(z*, z0*) = 1
2p
Re eiy10
z* - z0*
.
(3.6)

   In (3.4) - (3.6) by quantities y = y(z) and y1 = y1(z*) are designated the angles between axis x1 and normals to contour L and G, respectively.
   Having determined functions q(z), p(z*) and f(z*) over formulas (2.6) taking into account integral representations (3.3) we may calculate all the components of the electroelastic field in the field. At e15=0 system (3.4) will correspond to a piezopassive (isotropic) space.
   4. Determination of the concentration of stresses in a piecewise-homogeneous space. Calculate shear stress ss = s23cosy1 - s13siny1 on the surface of an opening. Taking into account (2.2) we find
ss = Re(Tse-iwt),    Ts(z*) = c44(1 + k215) U3
s
+ e15 F*
s
,    z* G.
(4.1)

   Substituting into (4.1) the limiting values of derivatives U3/s, F*/s at z z0* G, calculated with the help of representations (3.3) we will obtain the expression for amplitude of shear stress Ts
Ts(z0*)
= e15
k





G 
p(z*)G8(z*, z0*)ds +


G 
q(z)G7(z, z0*)ds

-
- e15g
4ine11



G 
f(z*)sin(y10 - a30)H1(gr30)ds
(4.2)

Appearing in (4.2) function G7(z, z0*), G8(z*, z0*) are determined in (3.5).
   Formula (4.2) permits to investigate the concentration of stresses in the space according to the frequency of excitation, position and configuration of heterogenities. Here we also should mention the circumstance concerning the behaviour of electroelastic quantities in the vicinity of the inclusion. From the integral representations of the displacement amplitude in (3.3) we obtain equality
q(z) = cE44(1 + k215)

U3
n


,
(4.3)

where the square brackets designate the jump of the corresponding quantity on L. From relations (2.2), (2.7) and (2.9) it follows that

sn = Re(Tne-iwt),  [Tn] = cE44

U3
n


+ e15

F
n


,  

F
n


= e15
ne11


U3
n


.
(4.4)
   From expressions (4.3), (4.4) we obtain equality
q(z) = [Tn].
(4.5)

Thuse, on the basis of (4.5) function q(z) may be interpreted as intensity of contact forces of interchange of the rigid inclusion and medium. From here it follows that for equilibrium of the inclusion there should be performed equality




L 
q(z)ds = 0.
(4.6)

Condition (4.6) should be considered as an additional one when solving the system of singular integral equations (3.4) in the class of functions, not restrained on tips L [5]. Due to (2.2), (2.9) and (2.10) we have

[Ds*] = ne11[Es*] = 0,  [En*] = - e15
ne11


U3
n


= - k
e15
q(z).
(4.7)

   On the basis (4.7) we may conclude, that electric induction vector is continuous in the area of a cylinder, and electric stress vector undergoes on the inclusion. If we consider a crack contour (mathematical cut) as L, in case when the prescribed on its edges stresses are self-balancing, vector undergoes a jump on L, and is continuous [6].
   5. Results of calculations. As an example consider a space with circular opening and linear inclusions, orientated under angle J to axis Ox1 (material is ceramics PZT-4 [8]). Parametric equations of contour L has the form
Rez = gd cosJ,  Imz = gd sinJ + h    (-1 d 1)
(5.1)

   Solution of system (3.4) together with additional condition (4.6) taking into account (5.1) was carried out numerically by the method of quadratures [9, 10].
   In Fig.2 there is shown the change of quantity m = |Ts/Z| at point of the contour of opening b = p in the function of normalized wave number g*R = gRat J = 0, h/R = 3, g/R = 1.5 (b is the polar angle, R is the radius of the opening). The curve with number m is given for loading X3 = Zsin(mb) (m = 1,2,3). The full lines conform to variant A, the dashed ones to variant B. It is seen that by increasing parameters in peak values g*R dispeace to the right.
   Concluding remarks. The represented approach to the solution of the stationary dynamic problem of electroelasticity permits to investigate the influence of the inertial effect on the behaviour of the components of the electric field in a piezoceramic space with tunnel heterogenities of a rather arbitrary configuration. As it follow from Fig. 2 under dynamic loading quantity m may exceed its static analogue almost by 2.5 times (curve 3).

Fig. 1                                               Fig. 2

   From the represented result of the calculations it follows that the behaviour of the electric and mechanical quantities considerably depend on the frequency of the harmonic loading, mutual position and configuration of heterogenities.
   The work was carried out in the framework of an agreement on scientific cooperation between the National Technical University of Athens and the Institute of Mechanics of the National Academy of Sciences of Armenia.

   National Technical University of Athens, Greece
   Sumy State University, Ukraine


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