УДК 539.3

   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 G_j(j = 1,2,ј,n₁), strengthened by rigid curvilinear stringer L_m(m = 1,2,ј,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 X_3n = 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 G_j and L_m 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 = cE44¶mu3 - e15Em,   Dm = e15¶mu3 + ne11 Em    (m = 1,2)

(2.2)

equations of electrostatics         div ® D   = 0,    ® E   = - grad f.

(2.3)

   In (2.1) - (2.3) s_m3 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; c^E₄₄, e₁₅ and n^e₁₁ 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:

cE44С2u3+e15С2f = r ¶2u3 ¶t2 ,   e15С2u3 - ne11С2f = 0.

(2.4)

From (2.4) we have the following relations

С2u3 - c-2 ¶2u3 ¶t2 = 0,    С2F = 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 й к л cE44 ж и 1 + k215 ц ш u3 + e15F щ ъ ы , D1 - iD2 = -2ne11 ¶F ¶z ,    E1 - iE2 = -2 ¶ ¶z ж з и F + e15 ne11 ц ч ш ,    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).

С2U3 + g2U3 = 0,    С2F* = 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 = ИL_m as follows

U3± = 0,

(2.8)

Es+ = Es-,    Dn+ = Dn-.

(2.9)

   Here E_s and D_n 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 L_m the moment from its beginning a_m to end b_m (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

ж з и ¶U3± ¶s ц ч ш = 0.

(2.10)

   The mathematical record of the boundary conditions on contour G = ИG_j 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 ж з и F* + e15 ne11 U3 ц ч ш = 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]:

cE44С2U3 + e15С2f* + pw2U3 = -P0d(x1 - x10, x2 - x20), e15С2U3 - ne11Сf* = 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) = ж з и k215Q0 4ie15(1 + k215) - P0 4icE44(1 + k215) ц ч ш H0(1)(gr), f*(x1,x2) = - Q0 2pne11 ln r + i 4(1 + k215)ne11 й к л e15P0 cE44 - k215Q0 щ ъ ы 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) м н о у х L  q(z)H0(1)(gr)ds + у х L  p(z*)H0(1)(gr1)ds ь э ю +        + k152 4ie15(1 + k215) у х G  f(z*)H0(1)(gr1)ds,

(3.3)

F*(x1,x2) = - 1 2pne11 у х G  f(z*)ln r1)ds,   r = |z - z|,    r1 = |z* - z|,    z О L,    z* О G.

Here H_n⁽¹⁾(x) is the Hankel’s 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 О L and z ® z^* О G in boundary conditions (2.10), (2.11), and also (2.13) or (2.14). The given system will be represented in the form: