H. J. Ding

On free vibration of a functionally graded piezoelectric rectangular plate

SummaryOn the basis of three-dimensional theory equations of transversely isotropic piezoelasticity, two independent state equations with variable coefficients are derived. To this end, separation formulae for displacements and shear stresses are employed. A laminated approximation is used to transform the state equations to the ones with constant coefficients in each layer. The free vibration problem of a piezoelectric rectangular plate with a functionally graded property is then investigated. Discussion on the boundary conditions is presented.

Degenerate scales and boundary element analysis of two dimensional potential and elasticity problems

Some widely used conventional boundary integral equations yield non-equivalent solutions compared to their corresponding boundary value problems when the scales in the fundamental solutions reach their degenerate scale values. The effects of such degenerate scales on a two-dimensional boundary element solution of potential and plane elasticity problems are investigated for different geometries. Moreover, a simple convenient method is suggested to evaluate these degenerate scales. On the other hand, using the necessary and sufficient boundary integral formulation can eliminate the non-equivalence of these conventional boundary integral formulations.

Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method

A general non-axisymmetric exact analysis of the statics of a laminated piezoelectric hollow sphere is presented in the paper by using a state space method. To select a proper set of state variables, three displacement functions and two stress functions are introduced. It is found that the basic equations of a spherically isotropic piezoelectric medium are eventually turned to two separated state equations with constant coefficients, the solutions of which are then obtained by virtue of matrix theory. The continuity conditions at each interface are then used to derive two relationships between respective boundary variables at the inner and outer spherical surfaces. No matter how many layers the sphere contains, the orders of the final solving equations remain unaltered.

THE FUNDAMENTAL SOLUTIONS FOR TRANSVERSELY ISOTROPIC PIEZOELECTRICITY AND BOUNDARY ELEMENT METHOD

Abstract In this paper, we first supplement two groups of simplified general solution based on previous work. Those results are in terms of harmonic functions and fit for the cases of multiple eigenvalues. Then by trial and error, we obtain the fundamental solutions for three cases for a piezoelectric infinite media by giving the expressions of harmonic functions. Finally, by use of those solutions, we implement a boundary element method program to perform numerical calculations. The numerical results agree well with the analytical ones.

THREE-DIMENSIONAL FREE VIBRATION ANALYSIS OF A FLUID-FILLED PIEZOCERAMIC HOLLOW SPHERE

Abstract This article presents an exact three-dimensional analysis of the free vibration of a piezoceramic hollow sphere filled with a compressible fluid medium. Three displacement functions are introduced to rewrite three components of the mechanical displacement in spherical coordinates. For the problem of general nonaxisymmetric free vibration, it is shown that the controlling equations are finally simplified to an uncoupled second-order ordinary differential equation and a coupled system of three such equations. Solutions to these differential equations are obtained. The coupled vibration problem of piezoceramic hollow sphere is then considered. It is found that there are two independent classes of vibrations. The first class is independent of the contained fluid and is not affected by the electric field, while the second is related to both the fluid parameter and the electric field. Exact frequency equations are derived and numerical results are finally presented.

Free vibration of transversely isotropic piezoelectric circular cylindrical panels

This paper presents an exact three-dimensional free vibration analysis of a transversely isotropic piezoelectric circular cylindrical panel. The general solution for coupled equations for piezoelectric media that was recently proposed by Ding et al. (Int. J. Solids Struct. 33 (1996) 2283) is employed. By using the variable separation method, three-dimensional exact solutions are obtained under several boundary conditions. Numerical results are finally presented and compared with available data in literature. The results show the non-dimensional frequencies of the piezoelectric panel are bigger than that of the non-piezoelectric one.

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

Abstract Compared with a given boundary value problem of plane elasticity, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions which are dependent upon Poissons ratio and geometry. Such a non-equivalence of solutions of boundary integral equations can be eliminated by using a necessary and sufficient boundary integral formulation proposed by He [Necessary and sufficient BIE-BEM: its theory and practice. Ph.D. Dissertation, Zhejiang University, Hangzhou, China (1993)]. Numerical analysis shows that the conventional boundary integral equation yields incorrect non-equivalent results when the scale in the fundamental solution is near its degenerate scale value. Also, this non-equivalence can be remedied by using the necessary and sufficient boundary integral equation.

A necessary and sufficient boundary integral formulation for plane elasticity problems

With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poissons ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem.

A three-dimensional formula for determining stress intensity factors in finite element analysis of cracked bodies

Abstract A three-dimensional crack opening displacement correlation formula to calculate stress intensity factors is derived for 20-node collapsed isoparametric quarter-point elements. This formula has one order higher accuracy than the one usually used. Numerical results show that the new formula is insensitive to the variation of element size as compared to the old one for the finite element analysis of three-dimensional fracture problems.

Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems

Abstract For a given mixed type potential problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions. Numerical results show that the conventional boundary integral formulation yields incorrect potential and flux results when the distance scale in the fundamental solution approaches its degenerate value. Such a kind of non-equivalence of the conventional boundary integral equation can be eliminated by the use of the necessary and sufficient boundary integral formulation which always ensures the equivalence of solutions.