Two-Dimensional Electroelastic Problem for a Multiply Connected Piezoelectric Body

A general method based on complex variable theory is proposed to determine the magnetic and elastic fields of a piezomagnetic body. This method is used to derive the basic relations for complex potentials in the two-dimensional problem of magnetoelasticity, their general representations for a multiply connected domain, expressions for stresses, displacements, vectors of magnetic field intensity and magnetic flux density, and magnetic field potential. A closed-form solution is obtained for a body with an elliptic (circular) hole or crack subjected at infinity to the action of a constant magnetoelastic field. Numerical results for a piezomagnetic plate with a circular hole are presented Keywords: anisotropic body, complex potentials, crack, hole, inclusion, magnetic field intensity, magnetic flux density, magnetoelasticity, plane problemIntroduction. In recent years, interest in piezomagnetic materials has heightened. Development of analytic and numerical methods for solving specific classes of problems is still one of the urgent tasks in the theory of magnetoelasticity of anisotropic piezomagnetic bodies. The governing equations of magnetoelasticity were derived in [6], internal and external two-dimensional problems of magnetostatics were solved in [1], and the interaction of mechanical strains in solids with an electromagnetic field was investigated in [12]. The great prospects for piezomagnetic materials in modern electronics and engineering generate interest in their effective properties [3-5], interaction of magnetic and mechanical fields [9], and magnetoelastic problems for piezomagnetic plates [10,11], bodies with inclusions [7], holes, and cracks [2].In the present paper, we extend the general approaches to the solution of two-dimensional electroelastic problems for multiply connected bodies [8] to magnetoelastic problems for piezomagnetic bodies with holes and cracks. We will introduce complex potentials for a two-dimensional magnetoelastic problem, derive formulas for the basic magnetoelastic characteristics, formulate boundary conditions for the potentials, obtain their general representations for multiply connected domains, find a magnetoelastic solution for a body with an elliptic (circular) cavity or a crack, and present numerical results.1. Problem Formulation. Let us consider a multiply connected cylindrical anisotropic piezomagnetic body weakened by L longitudinal cavities with generatrices parallel to the cylinder axis. We will use a rectangular coordinate frame Oxyz with the z-axis directed along the cavity generatrices. The cross section of the body by the plane Oxy is a multiply connected domain S bounded by the external boundary L 0 and the outlines L l ( , ) l L = 1 of the holes. As a special case where the outside surface is at

Section: Introductionmentioning

confidence: 99%

“…(1.5) and the second equation in (1.2) with (1.8), we obtain a system of differential equations [8]: …”

Section: Introductionmentioning

confidence: 99%

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Two-Dimensional Magnetoelastic Problem for a Multiply Connected Piezomagnetic Body

Kaloerov

Boronenko

2005

“…This is why plane problems of electroelasticity have recently been studied in more detail. Noteworthy are the papers [2,11,14,17,18] that address the two-dimensional electroelastic state around a single cavity, inclusion, and crack and the interaction of concentrators of electric and mechanical fields. Three-dimensional problems of electroelasticity for an infinite medium with cavities, inclusions, and cracks are solved in [5-7, 9, 10, 13, 15, 16].…”

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confidence: 99%

On the stress state of a piezoceramic body with a flat crack under symmetric loads

Kirilyuk

2005

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface Keywords: piezoelectricity, flat crack, elliptical crack, stress intensity factor, electric-displacement intensity factorIntroduction. The wide use of piezoelectric ceramic materials, which are highly brittle, in various transducers (based on the coupling of mechanical and electric fields) necessitates a careful study into the concentration of mechanical and electric fields in electroelastic bodies with imperfections such as cavities, inclusions, and cracks. However, the solution of three-dimensional problems of electroelasticity involves severe mathematical difficulties since the original system of equations describing the electrostressed state of a body consists of complicated coupled differential equations [1,4]. This is why plane problems of electroelasticity have recently been studied in more detail. Noteworthy are the papers [2,11,14,17,18] that address the two-dimensional electroelastic state around a single cavity, inclusion, and crack and the interaction of concentrators of electric and mechanical fields. Three-dimensional problems of electroelasticity for an infinite medium with cavities, inclusions, and cracks are solved in [5-7, 9, 10, 13, 15, 16]. The papers [5,15,16] propose approaches to finding the general solutions of coupled equations of electroelasticity for a transversely isotropic body. The exact solutions of electroelastic problems for spheroidal and hyperboloidal cavities and inclusions have been found in [6,13]. The electrostressed state and stress intensity factors (SIFs) and electric-displacement intensity factors (EDIFs) for an infinite medium with penny-shaped and elliptic cracks are studied in [1, 9, 10] and [7, 15, 16], respectively.

“…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.…”

mentioning

confidence: 99%

Thermoelectroelastic state of a multiply connected anisotropic plate

Kaloerov

Khoroshev

2005

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.