The static equilibrium of an electroelastic transversely isotropic space with a plane crack under antisymmetric mechanical loads is studied. The crack is located in the plane of isotropy. Relationships are established between the stress intensity factors (SIFs) for an infinite piezoceramic body and the SIFs for a purely elastic body with a crack of the same form under the same loads. This makes it possible to find the SIFs for an electroelastic body without the need to solve specific electroelasitc problems. As an example, the SIFs are determined for a piezoelastic body with penny-shaped and elliptic cracks under shear Keywords: piezoelectricity, plane crack, antisymmetric load, elliptic crack, stress intensity factor Introduction. Various transducers and sensors are often made of piezoelectric ceramic materials (where the mechanical and electric fields are coupled) that are highly brittle. This necessitates a detailed study of the concentration of mechanical and electric fields in piezoceramic bodies with imperfections such as cavities, inclusions, and cracks. However, solving three-dimensional problems of electroelasticity involves significant mathematical difficulties, because the original equations of electrostressed state constitute a complicated system of partial differential equations [1,4]. Plane problems of electroelasticity and magnetoelasticity were studied more fully in [2, 13, 15, 20, 21, etc.]. These studies address both the two-dimensional electroelastic state near single cavities, inclusions, and cracks and the interaction of concentrators of electric and mechanical fields. Three-dimensional problems of electroelasticity for an infinite medium with cavities, inclusions, and cracks were solved in [5-7, 10, 16, 19]. The papers [5,11,19] propose approaches to find general solutions to coupled equations of electroelasticity for a transversely isotropic body. Exact solutions to problems of electroelasticity for spheroidal cavities and inclusions were found in [6,16]. The electrostressed state and stress intensity factors (SIFs) of an infinite medium with penny-shaped and elliptic cracks were analyzed in [1, 10, 18] and [7], respectively.
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.
The paper establishes a relationship between the solutions for cracks located in the isotropy plane of a transversely isotropic piezoceramic medium and opened (without friction) by rigid inclusions and the solutions for cracks in a purely elastic medium. This makes it possible to calculate the stress intensity factor (SIF) for cracks in an electroelastic medium from the SIF for an elastic isotropic material, without the need to solve the electroelastic problem. The use of the approach is exemplified by a penny-shaped crack opened by either a disk-shaped rigid inclusion of constant thickness or a rigid oblate spheroidal inclusion in an electroelastic medium Keywords: transversely isotropic piezoceramic medium, plane crack, rigid inclusion, isotropy plane, penny-shaped crack, disk-shaped rigid inclusion of constant thickness, oblate spheroidal inclusion, stress intensity factorIntroduction. Methods to solve three-dimensional problems of elasticity for bodies with cracks are well developed. Results on stress intensity factors (SIFs) for cracks in an elastic medium are presented in numerous publications [3, 7, 8, 15, 19, 21, etc.]. The brittle fracture of prestressed bodies is studied in [2]. The extensive use of piezoceramic materials, which are distinguished by considerable brittleness, necessitates studying the mechanical and electric fields around stress concentrators such as cavities, inclusions, and cracks in electroelastic bodies [1, 4-6, 9-14, 6-18, 20, 22-24]. However, solving electroelastic problems involves severe mathematical difficulties compared with elastic problems because the original equations for electric and strain states constitute a more complicated system of differential equations [1,4]. The homogeneous equations of electroelasticity for piezoceramic bodies are addressed in [5,22].The present paper establishes a relationship between the solutions of the three-dimensional problem for plane cracks opened by rigid inclusions in an elastic isotropic space and in a transversely isotropic electroelastic medium. It is assumed that the crack is in the isotropy plane of the transversely isotropic electroelastic material and there is no friction between the crack and the inclusion. Thus, we can calculate the stress intensity factor (SIF) K I in a piezoceramic space from the expressions for the SIF in an elastic isotropic medium with a plane crack and a rigid inclusion of the same shape, without the need to solve the electroelastic problem. As examples of using such a relationship, we will consider a penny-shaped crack opened by either a rigid disk-shaped inclusion of constant thickness or a rigid inclusion in the form of an oblate spheroid in an electroelastic space. Problem Formulation and Basic Equations.Consider a three-dimensional transversely isotropic electroelastic space with a system of plane cracks (occupying a domain S ) located in a plane perpendicular to the polarization axis and opened by rigid inclusions (occupying a domain S 1 , i.e., S S 1 Ì ). The free portions of the cracks are not sub...
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