The stress-concentration problem for an elastic transversely isotropic medium containing an arbitrarily oriented spheroidal inclusion (inhomogeneity) is solved. The stress state in the elastic space is represented as the superposition of the principal state and the perturbed state due to the inhomogeneity. The problem is solved using the equivalent-inclusion method, the triple Fourier transform in space variables, and the Fourier-transformed Green function for an infinite anisotropic medium. Double integrals over a finite domain are evaluated using the Gaussian quadrature formulas. In special cases, the results are compared with those obtained by other authors. The influence of the geometry and orientation of the inclusion and the elastic properties of the medium and inclusion on the stress concentration is studiedThe stress state of an elastic isotropic medium with spherical, spheroidal (ellipsoids of revolution), and ellipsoidal cavities and inclusions and systems of such cavities was analyzed in [2-4, 7-9, 11, 15-18, etc.]. The stress concentration at thin inclusions was studied in [10]. The stress distribution in a transversely isotropic material containing spherical or spheroidal cavities or inclusions was studied in [2, 3, 8, 9, etc.]. In these studies, the cavity or inclusion in question was assumed to be oriented so that its axis of revolution coincides with the axis of transtropy, and the other possible orientations of the stress concentrator were not considered. The exact solutions found are based on the general solutions of three-dimensional static problems for isotropic and transversely isotropic media. If the spheroidal inclusion is oriented differently, then it is necessary to introduce a new (local) inclusion-fixed coordinate system in which the surface of the inclusion is described quite simply and the transversely isotropic material behaves as anisotropic. In this case, the available general solutions for a transversely isotropic material [14] do not help to solve specific stress-concentration problems; therefore, solutions for more complex kind of anisotropy have to be found. A comprehensive review and an analysis of the solutions to static elastic problems of anisotropic bodies are given in [6]. However, up to now, there have been no analytical solutions found to describe the stress distribution around spherical, spheroidal, and ellipsoidal cavities and inclusions in orthotropic materials (without special restrictions on the nine independent elastic constants) even for elementary loads and the stress distribution around spheroidal and ellipsoidal inclusions in transversely isotropic materials with the axes of transtropy different from the axes of revolution of the inclusions.The present paper deals with the stress-concentration problem for a transversely isotropic medium containing a spheroidal inclusion under tension. The axis of revolution of the inclusion forms an angle α with the axis of transtropy of the material. We will use the triple Fourier transform in combination with Eshelby's equiv...
The static equilibrium of an electroelastic transversely isotropic space with a paraboloidal cavity under axisymmetric mechanical and electric loads is analyzed. Paraboloidal coordinates and special harmonic functions are used to obtain an exact solution. The distribution of stresses and electric-flux density over the surface of the cavity subject to internal pressure is analyzed as an example Keywords: piezoelectricity, three-dimensional problem, paraboloidal cavity, force and electric fields Introduction. Various transducers and sensors are often made of piezoceramic 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 electroelastic bodies with defects 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 [2,5]. This is why plane problems of electroelasticity were studied more adequately (see, e.g., [3,12,18]). 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 [1,2,7,[9][10][11][12][13][15][16][17][18][19][20][21]. The papers [7,17,20] propose approaches to find general solutions to coupled equations of electroelasticity for a transversely isotropic body. The electrostressed state and intensity factors for stress and electric-flux density in an infinite medium with penny-shaped and elliptic cracks were analyzed in [9, 10] and [14,15,17], respectively. Purely elastic problems for isotropic and transversely isotropic media with a paraboloidal inclusion were solved in [6,8].
The static equilibrium of an elastic orthotropic medium with an elliptic crack subject, on its surface, to linearly varying pressure is studied. The stress state of the elastic medium is represented as a superposition of the principal and perturbed states. Use is made of Willis' approach based on the triple Fourier transform in spatial variables, the Fourier-transformed Green's function for an anisotropic material, and Cauchy's residue theorem. The contour integrals are evaluated using Gaussian quadratures. The results for particular cases are compared with those obtained by other authors. The influence of orthotropy on the stress intensity factors is studied
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