The results obtained in the previous investigations on the stiffness properties of elastic fibrous composites within the framework of the model of regular structure correspond well with the data available in the literature [1][2][3][4]. In the present study, the methods of [5] are generalized for composites with piezoelectric components.Let us consider an infinite piezoelectric medium reinforced by a biperiodic system of identical fibers with respect to the Cartedsn coordinates Oxlx2x 3. We assume that the fiber croas-section is a simply connected region D with sufficiently smooth closed contour L, while the properties of the reinforced medium (composite) are determined by the structure of the unit cell rl o (Fig. 1) constructed in the periods co I and co 2 (Im o~ I = 0, Im co2/c01 > 0).Let <(Y13 >, <0"23> be the average shear stresses and , the average vector components of the electric induction in the material. Then we obtain a piezoceramic medium regularly reinforced under the conditions of the antiplane strain.For a uniform piezoelectric medium, we have the corresponding material equations [6,7] 0"i3 = c4~aiu3 -elsEi, Dm= elsamU3 + ellEm, a~ = a/ax~ (i, m = 1, 2),the equilibrium equationand the electrostatic equations aid 1+a2D z=0, E,=~ (re=l, 2). (3)Here u3(x 1, x2), ai3(x 1, x2), Em(X 1, x2), and Dm(X 1, x2) are the elastic displacements along the x 3 axis, and the components of the mechanical stress tensor and the vectors of the electric field and induction, respectively, E and Ell s C44 = C44 , els, = ell are the shear modulus at the zero electric field, the piezomodulus, and the dielectric constant at zero strains, while ~o(x I, x2) is the electric potential.

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We consider a symmetric problem of the theory of elasticity for the stressed state of a thick-walled shell whose end faces are covered with a diaphragm. The boundary-value problem is reduced to an infinite system of singular integral equations of the second kind. The expressions for stresses characterizing the stressed state of this shell are deduced. Based on the developed analytic algorithm, we performed numerical experiments whose results are represented in the graphical form and contain new quantitative and qualitative data about the stressed state of a thick-walled shell depending on its geometric parameters and Poisson's ratios of its materials.The problem of stress concentration is quite urgent for the contemporary machine building because it determines the reliability and durability of designed structures or separate structural elements. Stress concentrators either appear in structures due to the imperfect quality of materials used for their manufacturing (cavities, cracks, foreign inclusions) or may be caused by the technological or design needs (holes, notches, etc.). In both cases, it is important to study the influence either of a single stress concentrator on the stressed state of designed products or of several stress concentrators with regard for their mutual influence. The necessity of getting more accurate descriptions of the stressed state of machine components near stress concentrators forces the researchers to try to describe the stressed state of structural elements in the three-dimensional statement [4,[8][9][10][11][12].Among the efficient methods aimed at the solution of three-dimensional problems for a layer (cylinder), we should especially mention the Lur'e method of homogeneous solutions [9]. The problems of stressed state for a layer weakened by various stress concentrators were investigated with the help of this method in [2,8]. Another efficient method for the analysis of three-dimensional problems posed for layers is the method of vector eigenfunctions. In [4], this method was used to study the Kirsch problem. For a cylinder with loaded lateral surface, the problem was solved by the method of eigenfunctions in [11]. An approach based on equations of three-dimensional theory of elasticity was proposed in [3] for the solution of the problem of stressed states of thick-walled orthotropic cylinders. The boundary-and finite-element methods are alternative methods used for the solution of three-dimensional problems of the theory of elasticity. In [19], the procedure of finiteelement discretization was applied to study stress concentration in a finite plate with circular or elliptic holes. Papers [13,17] are devoted to the improvement of the boundary-element method in its application to threedimensional problems of the theory of elasticity.By using the Lur'e method of homogeneous solutions, it is possible to construct a set of partial solutions for a layer (cylinder) under any conditions imposed on its base but, in the case of mixed-type boundary conditions (sliding fastening of the end fa...