磁电弹性圆锥顶端作用集中荷载的解析解

当磁电弹性材料特征根互异时，用5个势函数表示的通解出发，对圆锥顶端作用集中扭矩Mx的扭转、集中力Px和点电荷Q的压缩、集中力Px和集中力矩My的弯曲变形问题，用一些调和函数的线性组合分别构造了势函数，并根据边界条件求出了势函数中的待定系数从而确定势函数，再将势函数代入通解得到磁电弹性圆锥顶端作用集中载荷时的位移、电势、磁势、应力、电位移和磁感应强度的三维解析解。此解形式简单便于应用。当圆锥角2α=π时，可退化得到半空间问题的解。     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere

The dynamic solution of a multilayered spherically isotropic piezoelectric hollow sphere subjected to radial dynamic loads is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state-space method, and the dynamic part is obtained by the method of separation of variables coupled with the initial parameter method as well as the orthogonal expansion technique. By using the quasi-static and dynamic parts, the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered spherically isotropic piezoelectric hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric dynamic loads. Finally, numerical results are presented and discussed.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

A Spectral Theory Formulation for Elastostatics by Means of Tensor Spherical Harmonics

Consider a set of (N+1)-phase concentric spherical ensemble consisting of a core region encased by a sequence of nested spherical layers. Each phase is spherically isotropic and is functionally graded (FG) in the radial direction. Determination of the elastic fields when the outermost spherical surface is subjected to a nonuniform loading and the constituent phases are subjected to some prescribed nonuniform body force and eigenstrain fields is of interest. When the outermost layer is an unbounded medium with zero eigenstrain and body force fields, then an N-phase multi-inhomogeneous inclusion problem is realized. Based on higher-order spherical harmonics, presenting a three-dimensional strain formulation with a robust form of compatibility equations, a spectral theory of elasticity in the spherical coordinate system is developed. Application of the established spectral theory leads to the exact closed-form solution when the elastic moduli of each phase vary as power-law functions of radius.     

A mode III crack crossing the magnetoelectroelastic bimaterial interface under concentrated magnetoelectromechanical loads

A mode III crack cutting perpendicularly across the interface between two dissimilar semi-infinite magnetoelectroelastic solid is studied under the combined loads of a line force, a line electric charge and a line magnetic charge at an arbitrary location. The impermeable conditions are implied on the crack faces. The technique developed in literature for the elastic bimaterial with a crack cutting interface is exploited to treat the magnetoelectroelastic bimaterial. The Riemann-Hilbert problem can be formulated and solved based on complex variable method. Analytical solutions can be obtained for the entire plane. The intensity factors around crack tips can be defined for the elastic, electric and magnetic fields. It shows that, no matter where the load position is, the electric displacement intensity factors (EDIFs), as well as the magnetic induction intensity factors (MIIFs), are identical in magnitude but opposite in sign for both crack tips, on condition that a line force is solely applied. Alternatively, if only a line electric charge is considered, then the stress intensity factors (SIFs) and the MIIFs exhibit the behavior. Likewise, if only a line magnetic charge is applied, it turns to the SIFs and the EDIFs instead. In addition, the dependence of the intensity factors is graphically shown with respect to the location of a line force. It is found that the SIF for a crack tip tends to be infinite if the applied force is approaching the tip itself, but the EDIF, with the complete opposite trend, tends to be vanishing. Finally, focusing on the more practical case of piezoelectric/piezomagnetic bimaterial, variation of the SIF along with the moduli as well as the piezo constitutive coefficients is explored. These analyses may provide some guidance for material selection by minimizing the SIF. It is also believed that the results obtained in this paper can serve as the Green’s function for the dissimilar magnetoelectroelastic semi-infinite bimaterial with a crack cutting the interface under general magnetoelectromechanical loads.     

General solution for the coupled equations of transversely isotropic magnetoelectroelastic solids

IntroductionMechanicsandphysicsofmediapossessingsimultaneouslypiezoelectric ,piezomagneticandmagnetoelectriceffects ,namely ,magnetoelectroelasticsolids,haveattractedmoreandmoreattentionduetotheirgreatpotentialapplicationsinthetechnologiesofsmartandadaptivematerialsystem[1] .Sometheoreticalinvestigationsappearedintheliteratureinclude :1)Theexistenceproblemofsurfacewavesinsemi_infiniteanisotropicmagnetoelectroelasticmediawithvariousboundaryconditions[2 ,3 ] ;2 )Green’sfunctions[4～ 7] ;3)Inho…     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

球面各向同性球体内的动态热应力集中

利用有限克尔变换得到了求面各向同笥球体的热冲击作用下的动态热应力解析表达式。从表达式中,可以看出球中向同性球体的动态热应力集中现象明显不同于各向同性球体。另外,所描述的动态热应力集中现象与文献「１,３」也有一定的区别。     

The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies

First, the general solution for transversely isotropic magnetoelectroelastic media is given concisely in form of five harmonic functions. Second, the extended Boussinesq and Cerruti solutions for the magnetoelectroelastic half-space are obtained in terms of elementary functions by utilizing this general solution. Third, the coupled fields for elliptical Hertzian contact of magnetoelectroelastic bodies are solved in smooth and frictional cases. At last, the graphic results are presented.     

Elasticity solutions for a piezoelectric cone under concentrated loads at its apex

IntroductionTheproblemofaconesubjectedtoconcentratedloadsatitsapexisaclassicalprobleminthetheoryofelasticity.AnumberofscholarshavestUdiedtheproblem.LovereportedthesolutionstotheproblemofanisotropicconeunderconcentfatCdforcesatitsapex['].Lur'estudiedthisclassofproblemssystematicallybymeansofPapkovich-Neubergeneralsolution[2].LekniskiiandHu,byusingtheirrespectivegeneralsolutions,studiedcompressionandbendingproblemsofatransverselyisotropicconesubjectedtoaxialconcentfatedforcesandtfansverseconc…     

THE REMARKABLE NATURE OF RADIALLY SYMMETRIC DEFORMATION OF ANISOTROPIC PIEZOELECTRIC INCLUSION

The present paper deals with spherically symmetric deformation of an inclusion- matrix problem, which consists of an infinite isotropic matrix and a spherically uniform anisotropic piezoelectric inclusion. The interface between the two phases is supposed to be perfect and the system is subjected to uniform loadings at infinity. Exact solutions are obtained for solid spherical piezoelectric inclusion and isotropic matrix. When the system is subjected to a remote traction, analytical results show that remarkable nature exists in the spherical inclusion. It is demonstrated that an infinite stress appears at the center of the inclusion. Furthermore, a cavitation may occur at the center of the inclusion when the system is subjected to uniform tension, while a black hole may be formed at the center of the inclusion when the applied traction is uniform pressure. The appearance of different remarkable nature depends only on one non-dimensional material parameter and the type of the remote traction, while is independent of the magnitude of the traction.     

The action of a concentrated force and a concentrated electric charge in an infinite transversally isotropic piezoceramic medium

An explicit solution of the problem of electroelasticity is constructed for a transversally isotropic medium under the action of a concentrated force and a concentrated electric charge. The cases where the force acts along the anisotropy axis and along the isotropy axis are considered. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 80–87, April, 2000.     

Spherical ceramic pebbles subjected to multiple non-concentrated surface loads

This paper presents an analytical solution for the stress distributions within spherical ceramic pebbles subjected to multiple surface loads along different directions. The method of solution employs a displacement approach together with the Fourier associated Legendre expansion for piecewise boundary loads. The solution corresponds to spherically isotropic elastic spheres. The classical solution for isotropic spheres subjected diametral point loads is recovered as a special case of our solution. For the isotropic pebbles under consideration, stresses within spheres are numerically evaluated. The results show that the number of loads does have significant influence on the maximum tensile stress inside the sphere. Moreover, the applicability of solutions using the series expansion method for stresses near surface load areas is also examined. The stresses evaluated with large enough number of terms agree quite well with those derived from FEM simulations, except around the edge of circle load area.     

Circular crack in a transversely isotropic piezoelectric space under point forces and point charges

In this paper, two kinds of circular crack including external circular crack and penny-shaped crack in a transversely isotropic piezoelectric space are considered. Firstly, we obtain the solution to the problem of an external circular crack in a transversely isotropic piezoelectric space subjected to antisymmetric normal point forces and point charges. Based on this, the solution of one-sided loading of an external circular crack is constructed. Secondly, the real shape of an external circular crack and the opening displacement of a penny-shaped crack under an arbitrary point force and point charge are further obtained. At last, the results are presented in a graphical form. The project supported by the National Natural Science Foundation of China (19872060 and 69982009) and the Postdoctoral Foundation of China     

层合球面各向同性热释电空心球的瞬态响应

运用叠加原理，将层合球面各向同性热释电空心球的球对称动力学问题的解分成准静态和动 态两部分，准静态部分首先运用状态空间法给出了显式表达式，然后运用分离变量法、初参 数法和特征函数展开技术，给出了动态部分的表示式，再结合内外表面上的电学边界条件和 界面上的电学连续条件，导出一个关于时间函数的第二类Volterra积分方程，运用插值法 可成功地给出此积分方程的高精度数值解，最终可求得原问题的位移、应力、电位移以及电 势的响应. 此方法适用任意层数且各层是任意厚度的层合热释电空心球作用随时间以任意形 式变化的球对称温度场. 文中还给出了数值结果.     

On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B _r of radius r centered at the point where the force is applied is statically equivalent in the limit as r0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid.The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B _r are statically equivalent in the limit as r0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B _r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.     

Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium

The hyper-singular boundary integral equation method of crack analysis in three-dimensional transversely isotropic magnetoelectroelastic media is proposed. Based on the fundamental solutions or Green’s functions of three-dimensional transversely isotropic magnetoelectroelastic media and the corresponding Somigliana identity, the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces. The extended displacement discontinuities include the displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, and correspondingly the extended tractions on crack face represent the conventional tractions, the electric displacement and the magnetic induction boundary values. The near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach. A solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials. The influence of different electric and magnetic boundary conditions, i.e., electrically and magnetically impermeable and permeable conditions, electrically impermeable and magnetically permeable condition, and electrically permeable and magnetically impermeable condition, on the solutions is studied. The crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical-electric-magnetic loadings. An iteration approach is presented for the solution of the non-linear model. The exact solution is obtained for the case of uniformly applied loadings on the crack faces. Numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method.     

An electric point charge moving along the poling direction of a transversely isotropic piezoelectric solid

The problem of an electric point charge moving constantly along the poling direction of a transversely isotropic piezoelectric solid is considered in a moving coordinate system, which moves together with the electric point charge. A general solution in the moving coordinate system is given, and all the field components, such as displacements, electric potential, stresses and electric displacements, can be concisely expressed in terms of four quasi-harmonic functions. We also present two examples to demonstrate the effect of the moving velocity on the values of _i. Once the general solution is given, the axisymmetric problem of a moving electric point charge can be easily solved. The explicit expressions of all the field components caused by the moving electric charge are presented, and the effect of the moving velocity on these field components is numerically investigated.     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.