Issue 52

O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04 34 Therefore, an important scientific and technical task for of predicting the possible destruction of structures during their operation and design is a study of the stress-strain state of bodies in the presence of interphase heat-emitting or heat- insulated inclusions in anisotropic piecewise homogeneous media. In particular, the study of the effects on deformation and the distribution of stresses around the specified inclusions of thermoelastic parameters of anisotropic media and the type of interaction contact of inclusions with the medium. In the general this kind of finite bodies problems in the presence of many inclusions, can have only a numerical solution, . However, the presence of exact solutions for single inclusions allows not only to study the stress concentration around the defect, but also improve the efficiency of numerical methods. Concentration of tensions near defects such as cracks or inclusions in the composite media have been investigated by many authors. In particular, in [5] non-axisymmetric problems on various interfacial defects in a composite isotropic space are reduced to systems of two-dimensional singular integral equations (SIE), and for circular defects a method for their exact solutions is proposed. In [6], the axisymmetric problem of the theory of elasticity for a rigid circular inclusion located at the interface between two perfectly connected dissimilar isotropic elastic half-spaces is considered, the solution for which was obtained using the Hankel transform. In papers [7–9], using the method of potential and the Fourier transform, the problems of stationary thermoelasticity for absolutely rigid inclusion in unbounded isotropic space, and in [6] in transversely isotropic space under the action of a heat flux, are reduced to a system of related two-dimensional singular integral equations for jumps of stress. In [1, 2], consider stationary thermal elasticity problems for the body with a thermally permeable disk inclusion, between the surfaces where there is an imperfect thermal contact, as well as with a thin heat-active disk inclusion. The posed problems are reduced to hypersingular integral equations of the first and second kind, for which an exact solution has been obtained. In works [11-14], problems on of nonaxisymmetric interphase defects such as cracks or absolutely rigid inclusions, for different types of contact interaction (full coupling, smooth contact, mixed conditions) with various transversely isotropic half-spaces using the method of singular integral relations (SIR) [15] are being reduced to two-dimensional SIE systems. A method of constructing exact solutions of these SIA systems for circular defects has been proposed, which has allowed to determine the singularity of the stress and displacement fields around the cracks and inclusions under arbitrary load. An analysis of well-known publications shows that nonaxisymmetric problems of thermoelasticity for piecewise homogeneous transversely isotropic media in the presence of interfacial defects are not well-studied.have not been extencively studied. Especially it concerns a smooth (sliding) contact of the inclusion with the medium, despite the importance of such a solution [16]. In this paper, for the first time, we consider the non-axisymmetric stationary thermoelasticity problem for a composite transversely isotropic space containing a heat radiating circular inclusion having a smooth contact with the medium. The problem, using the SIR method, is reduced to the SIE system. An exact solution to the indicated SIE system was obtained, which made it possible to study the fields of stresses, strains, and temperatures around the inclusion. F ORMULATION OF THE PROBLEM et in the plane everywhere except the circular area have the perfect contact of two different transversely isotropic half-spaces. For stress tensor components: vector of movements temperature and heat flux , we introduce the notation:   8 8 1,8 0 { } { } , { , , } { , , , , , , , } k k k z yz xz z k z x y z u v w T q             v (1) In view of the notation (1), the conditions for perfect contact of various transversely isotropic half-spaces outside the area can be written as ( , ) 0, ( 1, 7), k x y k     3 2 7 3 2 7 ( , , 0) ( , , 0), x y x y            ( , ) , x y  (2) where       , , , , 1,8 k k k k x y x y x y k             (3)  0 z    2 2 :{ }, x y a    , , , z yz xz , , , u v w T z q  L