Displacement potentials for functionally graded piezoelectric solids

Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.     

Stress and free vibration analysis of functionally graded beams using static Green's functions

In this paper static Green's functions for functionally graded Euler-Bernoulli and Timoshenko beams are presented. All material properties are arbitrary functions along the beam thickness direction. The closed-form solutions of static Green's functions are derived from a fourth-order partial differential equation presented in [2]. In combination with Betti's reciprocal theorem the Green's functions are applied to calculate internal forces and stress analysis of functionally graded beams (FGBs) under static loadings. For symmetrical material properties along the beam thickness direction and symmetric cross-sections, the resulting stress distributions are also symmetric. For unsymmetrical material properties the neutral axis and the center of gravity axis are located at different positions. Free vibrations of functionally graded Timoshenko beams are also analyzed [3]. Analytical solutions of eigenfunctions and eigenfrequencies in closed-forms are obtained based on reference [2]. Alternatively it is also possible to use static Green's functions and Fredholm's integral equations to obtain approximate eigenfunctions and eigenfrequencies by an iterative procedure as shown in [1]. Applying the Sensitivity Analysis with Green's Functions (SAGF) [1] to derive closed-form analytical solutions of functionally graded beams, it is possible to modify the derived static Green's functions and include terms taking cracks into account, which are modeled by translational or rotational springs. Furthermore the SAGF approach in combination with the superposition principle can be used to take stiffness jumps into account and to extend static Green's functions of simple beams to that of discontinuous beams by adding new supports. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

The minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implications for whether low‐rank structures exist in the linear system after numerical discretization. The Green's function for a coercive elliptic differential operator in divergence form was shown to be highly separable [2], and efficient numerical algorithms exploiting low‐rank structures of the discretized systems were developed. In this work, a new approach to study the approximate separability of the Green's function of the Helmholtz equation in the high‐frequency limit is developed. We show (1) lower bounds based on an explicit characterization of the correlation between two Green's functions and a tight dimension estimate for the best linear subspace to approximate a set of decorrelated Green's functions, (2) upper bounds based on constructing specific separable approximations, and (3) sharpness of these bounds for a few case studies of practical interest. © 2018 Wiley Periodicals, Inc.     

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.     

Three-dimensional dynamic Green’s functions in transversely isotropic tri-materials

An analytical derivation of the elastodynamic fundamental solutions for a transversely isotropic tri-material full-space is presented by means of a complete representation using two displacement potentials. The complete set of three-dimensional point-load, patch-load, and ring-load Green’s functions for stresses and displacements are given, for the first time, in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Selected numerical results for point-load and patch-load Green’s functions are presented to portray the dependence of the response on layering, the frequency of excitation, and type of loading.     

Aufspaltungen linearer gewöhnlicher Differentialoperatoren und der zugehörigen Greenschen Funktionen

Let M be a regular linear ordinary differential operator of the n-th order, associated with certain homogeneous boundary conditions. Suppose that M is invertable. We provide sufficient conditions to split M into a product of lower order operators M_k, which may be singular at the endpoints of the given interval. To these splittings-which depend on the given boundary conditions-there corresponds a splitting of the associated Green's function. The results are applied in the theory of inverse-positive operators and the theory of totally positive Green's functions. These applications, in general, require the operators M_k to be singular. Moreover, for special classes of operators the splittings can effectively be used for solving boundary value problems numerically.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.     

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Two Green's function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Green's second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Green's function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.     

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

Transient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary-domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace-transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dispersion relations of elastic waves in two-dimensional tessellated piezoelectric phononic crystals

A type of two-dimensional tessellated piezoelectric phononic crystal is theoretically studied in this paper, formed by homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Firstly, the propagation properties of in-plane and anti-plane Bloch waves in each piezoelectric rectangular column are systematically investigated. Furthermore, the total transfer matrices of Bloch waves are obtained based on transfer matrices of homogeneous piezoelectric and inhomogeneous functionally graded rectangular columns. Finally, the Bloch theorem is used to obtain the dispersion relations of in-plane and anti-plane Bloch waves. The influences of non-dimensional geometrical parameters and gradient profile functions on the dispersion relations are discussed based on the graphically numerical results. Under a special value of modal parameter, the dispersion surfaces and curves of in-plane Bloch waves approximatively have plane and axis symmetries respectively, and an approximate total band-gap of anti-plane Bloch waves arise. The theoretical models and numerical discussions will provide a direct guidance of multi-material 3D printing for inhomogeneous periodic structures with dispersion and band-gaps properties.     

Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

Green's function of two-dimensional time-fractional diffusion equation using addition formula of Wright function

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

A modified Green's function for the internal gravitational wave equation in a layer of a stratified medium with a constant shear flow

The construction of a modified Green's function for the internal gravitational wave (IGW) equation in a layer of a stratified medium when there are constant mean shear flows is considered and the basic properties of the corresponding eigenvalue problems and the modified eigenfunctions and eigenvalues are investigated. It is shown that each mode of the modified Green's function consists of a sum of three terms describing (1) the IGWs that propagate from the source, (2) the effects of a time varying source, localized in a certain neighbourhood of it, and (3) the effects of the displacement of the fluid (an internal discontinuity) caused by the source. The resulting expressions are analysed out for a constant and oscillating source of the generation of IGWs in which each of the terms of Green's function is represented in the form of simple quadratures.     

The diffraction in a class of unbounded domains connected through a hole

In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi‐infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov–Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic modeling for rotating composite Timoshenko beam and analysis on its bending-torsion coupled vibration

A formulation is presented for steady-state dynamic responses of rotating bending-torsion coupled composite Timoshenko beams (CTBs) subjected to distributed and/or concentrated harmonic loadings. The separation of cross section's mass center from its shear center and the introduced coupled rigidity of composite material lead to the bending-torsion coupled vibration of the beams. Considering those two coupling factors and based on Hamilton's principle, three partial differential non-homogeneous governing equations of vibration with arbitrary boundary conditions are formulated in terms of the flexural translation, torsional rotation and angle rotation of cross section of the beams. The parameters for the damping, axial load, shear deformation, rotation speed, hub radius and so forth are incorporated into those equations of motion. Subsequently, the Green's function element method (GFEM) is developed to solve these equations in matrix form, and the analytical Green's functions of the beams are given in terms of piecewise functions. Using the superposition principle, the explicit expressions of dynamic responses of the beams under various harmonic loadings are obtained. The present solving procedure for Timoshenko beams can be degenerated to deal with for Rayleigh and Euler beams by specifying the values of shear rigidity and rotational inertia. Cantilevers with bending-torsion coupled vibration are given as examples to verify the present theory and to illustrate the use of the present formulation. The influences of rotation speed, bending-torsion couplings and damping on the natural frequencies and/or shape functions of the beams are performed. The steady-state responses of the beam subjected to external harmonic excitation are given through numerical simulations. Remarkably, the symmetric property of the Green's functions is maintained for rotating bending-torsion coupled CTBs, but there will be a slight deviation in the numerical calculations.