Generalized complex potentials in a plane problem of thermoelectroelasticity are introduced. Expressions for the basic characteristics of the thermoelectroelastic state are derived. Boundary conditions for determining the complex potentials and the general form of these functions for a multiply connected plate are obtained. The potentials are used to solve a specific problem Keywords: thermoelectroelasticity, plane problem, generalized complex potentials, multiply connected plate Extensive studies have been carried out to determine the thermoelastic [2] and electroelastic [12] states of multiply connected anisotropic plates. However, thermoelectroelastic problems have not yet been analyzed. Note that the theory of electroelasticity [1] was used in [13,14] to study the stress-strain state and longitudinal vibrations of rectangular piezoceramic plates with sections polarized in two directions. In the present paper, to solve such problems in a new formulation, we introduce complex potentials and derive the governing equations of thermoelectroelasticity for multiply connected anisotropic plates. We will also solve a specific problem.

Generalized complex potentials, their expressions for a domain with elliptic holes, and the discrete least-squares method are used to analyze the generalized plane electroelastic state of a piezoelectric plate having holes and cracks with electric potentials applied to their boundaries. There are no mechanical loads. A numerical analysis is conducted. The effect of the applied voltage on the electroelastic state of the plate is examined Introduction. The principles of the theory of electroelasticity for piezoelectric bodies [5,7,8] and various methods for the analysis of their electroelastic state are outlined in [2,9,10,13,18,22,23]. Of special interest is the electroelastic state of piezoelectric bodies with holes and cracks. There are numerous approaches to problems for simply connected domains with a hole [3,6,16,20] or a crack [3,10,16,19]. Methods of solving two-dimensional problems for multiply connected domains based on complex variable theory [1,3,11] are best developed. In [3], generalized complex potentials for two-dimensional static problems of electroelasticity were studied, their expressions for bodies with holes and cracks were derived, and a wide class of problems of practical importance for plates with various geometric characteristics and unelectroded boundaries was solved using the least-squares method to satisfy the boundary conditions. Green's function and boundary integral equations were used in [11] to solve two-dimensional static and dynamic problems of electroelasticity for multiply connected piezoelectric bimorphs, infinite and semi-infinite bodies with unelectroded surfaces and some dynamic problems for bodies with partially electroded surfaces. The methods proposed in [3] were used in [1] to study the generalized plane electroelastic state of a finite piezoelectric plate having electroded holes and cracks with electric potentials applied to their boundaries.In the present paper, we extend this approach to an infinite piezoelectric plate with holes and cracks. We will present numerical results and analyze the solutions of problems for plates with two identical circular holes or cracks.

a b s t r a c tSolutions of the two-dimensional electroelasticity problems for infinitely long piezoelectric cylinders with cavities and cracks under electrical potential difference action are obtained using the Lekhnitskii's generalized complex potentials. Herewith, piezoelectric bodies under consideration are free from mechanical loads, some (or all) openings and maybe the exterior boundary of the cylinders are fully covered by thin electrodes. Voltage differences are supplied on the electroded surfaces, the rest boundary surfaces are charge-free. In this case, the complex potentials are investigated. Boundary conditions are satisfied by the least-squares method. There are carried out numerical investigations of behaviour of some basic electroelastic characteristics in a circular hollow cylinder and in an infinite body with two longitudinal cavities and a charge-free plane crack. New electromechanical regularities of influence of material properties, geometrical characteristics of considered regions on values of electroelastic state characteristics and the stress, electric displacements and tensions intensity factor k 1 are determined.

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