Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

Three-dimensional exact solutions for free vibrations of simply supported magneto-electro-elastic cylindrical panels

This work investigates the free vibrations of magneto-electro-elastic cylindrical panels based on three-dimensional theory. Firstly, the general solutions for transversely isotropic magneto-electro-elastic materials are introduced and the displacement functions in the general solutions are expanded in trigonometric functions along the circumferential and axial directions. Then an ordinary differential equation of the displacement functions in radial direction is derived and solved. As a result, the frequency equations are obtained through the traction-free conditions on the cylindrical surfaces of the panel as well as the electric and magnetic conditions. For the torsion and thickness-shear modes, the frequency equations in simpler forms are presented. It is found that the magneto-electro-elastic coupling effects disappeared in torsion vibration. Meanwhile, the frequencies of pure elastic materials and magneto-electro-elastic materials have an explicit relation for the thickness-shear modes. The aforementioned solutions satisfy all the governing equations and boundary conditions point by point and they are three-dimensionally exact. Finally the numerical example demonstrates the present method and is compared with those from finite element method. Parametric investigation is also conducted to show the behavior of free vibrations of cylindrical panels.     

Normal point force and point electric charge in a piezoelectric transversely isotropic functionally graded half-space

Normal point force and point electric charge acting on surface of a transversely isotropic piezoelectric half-space with a functionally graded transversely isotropic piezoelectric coating are considered. Elastic moduli, piezoelectric constants and dielectric permeabilities of the coating vary with depth according to arbitrary functions. Analytical expressions for the elastic displacements and potential of the electrostatic field are derived for a fixed value of the depth coordinate. Asymptotical analysis of these formulas is derived for small and big values of a radial coordinate. An equilibrium of the half-space under the action of axisymmetric mechanical (normal and tangential) and electric loading is studied and a scheme of reducing the solutions of mixed boundary value problems to integral equations is obtained. As an illustration of the obtained solution, the PZT-5H piezoceramics with typical examples of functionally graded and homogeneous coatings are considered. The results include computations of the profiles of displacements and electric potential for different types of variation of electro-elastic properties in the coating.     

任意荷载作用下层状横观各向同性弹性地基的直角坐标解

首次建立了在直角坐标系下层状地基力学问题的通用解法,改变了过去仅能在柱坐标系下进行求解此类的状况。首先将坐标系的原点选在荷载影响范围以外足够远处,从直角坐标系下的横观各向同性弹性问题的基本方程出发,利用Laplace变换及其微分性质,建立了单层横观各向同性弹性地基的状态控制方程,并利用状态空间理论给出了单层地基的解答。然后再利用传递矩阵技术,给出了任意荷载作用下的层状横观各向同性弹性地基的解析解。用提供的方法求解层状横观各向同性地基的非轴对称问题比在极坐标下求解简单、快捷。     

A refined theory of transversely isotropic piezoelectric plates

Summary. A refined theory for transversely isotropic piezoelectric plates is derived from the general solution of three-dimensional transversely isotropic piezoelasticity by means of Lure operator method. As a special case, the governing differential equations for transversely isotropic elastic plates are obtained directly.     

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

The general solution of three-dimensional problems in magnetoelectroelastic media

The general solution of three-dimensional problems in transversely isotropic magnetoelectroelastic media is obtained through five newly introduced potential functions. The displacements, electric potential, magnetic potential, stresses, electric displacements and magnetic inductions can all be expressed concisely in terms of the five potential functions, all of which are harmonic. The derived general solution is then applied to find the fundamental solution for a generalized dislocation and also to derive Green's functions for a half-space magnetoelectroelastic solid.     

Three-dimensional fracture analysis in transversely isotropic solids

In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.     

Bending of piezoelectric plates with a circular hole

Departing from the refined theory of transversely isotropic piezoelectric plates, an analytical solution for the bending of an infinite piezoelectric plate with a circular hole is obtained. Expressions of moment, stress and electric displacement concentration factors are presented in closed form. When the piezoelectric coupling is absent, the results reduce to the corresponding solutions for the transversely isotropic elastic plates. Some numerical results for PZT-6B piezoelectric ceramics are obtained and illustrated by figures. These results show that the effect of piezoelectric coupling on the concentration factors is not negligible.     

An exact Peano Series solution for bending analysis of imperfect layered functionally graded neutral magneto-electro-elastic plates resting on elastic foundations

In this article, a three-dimensional solution is presented for the bending analysis of functionally graded and layered neutral magneto-electro-elastic plates resting on two-parameter elastic foundations, considering imperfect interfacial bonding. The equations of motion, Gauss's equations for electrostatics and magnetostatics, and boundary and interface conditions are satisfied exactly regardless of the number of layers. No assumptions on deformations, stresses, and magnetic and electric fields along the thickness direction are introduced. The interfacial imperfection is modeled using a generalized spring layer. The state-space method is employed for solving the governing partial differential equations. Effects of a two-parameter elastic foundation, gradient index, bonding imperfection, and applied mechanical and electrical loads on the response of the functionally graded magneto-electro-elastic plate are discussed. The obtained exact solution can serve as a benchmark for assessing the accuracy of layered functionally graded magneto-electro-elastic plate theories.     

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

Dynamic potentials and Green’s functions of a quasi-plane magneto-electro-elastic medium with inclusion

The dynamic potentials of a quasi-plane magneto-electro-elastic medium of transversely isotropic symmetry with an inclusion of arbitrary shape are derived, and the dynamic potentials are finally governed by six scalar equations which can be regarded as inhomogeneous wave equations, Laplace and Helmholtz equations. The explicit expressions of the dynamic Green’s functions of this medium are also obtained both in the space–time domain and in the space–frequency domain. Closed-form expressions for the space–frequency representation of the dynamic potentials are given for the case when the inclusion is circular. The results are employed to obtain the generalized displacement fields of a circular inclusion undergoing uniform eigenstrain, eigenelectric field and eigenmagnetic field. In contrast to the corresponding static Eshelby inclusion problem the magneto-electro-elastic fields (i.e. strain, electric and magnetic field) inside the inclusion are non-uniform in the space–frequency domain.     

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).