A Green's function approach to the deflection of a FGM plate under transient thermal loading

Summary  Green's function approach is adopted for analyzing the deflection and the transient temperature distribution of a plate made of functionally graded materials (FGMs). The governing equations for the deflection and the transient temperature are formulated into eigenvalue problems by using the eigenfunction expansion theory. Green's functions for solving the deflection and the transient temperature are obtained by using the Galerkin method and the laminate theory, respectively. The eigenfunctions of Green's function for the deflection are approximated in terms of a series of admissible functions that satisfy the homogeneous boundary conditions of the plate. The eigenfunctions of Green's function for the temperature are determined from the continuity conditions of the temperature and the heat flux at interfaces. Received 9 October 2000; accepted for publication 3 April 2001     

Dyadic Green's functions for cylindrical waveguides with moving media

The dyadic Green's function for cylindrical waveguides of circular or rectangular cross section with a moving, isotropic, homogeneous medium is developed using the method of eigenfunction expansion. The orthogonality properties of the vector mode functions are discussed. In contrast to waveguides with a stationary medium, it is seen that the normalization factor in the case of the E mode introduces a pole in the integral representation for the Green's function which must be excluded from the integration contour.     

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loéve(K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re,= 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90% of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, posses viscous boundary layers at the walls and are much richer in harmonics than the original basis functions. Chaotic temporal behaviour is observed in all modes and increases for higher-order eigenfunctions. The structure and dynamical behaviour of the eigenmodes are discussed as well as their use in the representation of the turbulent flow.     

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.     

Analysis of scattering waves in an elastic layered medium by means of the complete eigenfunction expansion form of the Green’s function

A scattering problem due to an object and a plane incident wave in an elastic layered half space is presented in this paper. The complete eigenfunction expansion form of the Green’s function developed by the author and the boundary integral equation method are introduced into the analysis. First, the complete eigenfunction expansion form of the Green’s function is investigated for its application to the scattering problem. A comprehensive explanation is also given for the fact that the complex Rayleigh wave modes exhibit standing waves. Next, a method for the analysis of scattering waves by means of the Green’s function is presented. The advantage of the present method is that the formulation itself is independent of the number of layers and the scattering waves can be decomposed into the modes for the spectra defined for the layered medium. Several numerical calculations are performed to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the complete eigenfunction expansion form of the Green’s function provides accurate values for application to a boundary element analysis. The spectral structure and radiation patterns of the scattering wave are presented and investigated. The differences in directionality can be found from the radiation patterns of the scattering waves decomposed into the modes for the spectra.     

SPLITTING OF THE SPECTRAL DOMAIN ELECTRICAL DYADIC GREEN’S FUNCTION IN CHIRAL MEDIA

IntroductionAchiralmediumisanewtypeofspecialmediummaterial.Broadapplicationprospectsforchiralmediainmicrowaves,millimeterwaves,electronicdevices,integratedoptics,andsoonfieldhaveattractedconsiderableattention .Theelectromagneticproblemwithchiralmediahasbeenahotresearchtopicoftheelectromagnetictheory .TheeigenfunctionexpansionproblemofthedyadicGreen’sfunctionfortheelectromagneticwavefieldinchiralmediahasbeendeeplyinvestigatedinRefs [1～ 6] .FromHelmholtztheorem ,anarbitraryvectorfieldfcouldb…     

Crack kinking in piezoelectric materials

A solution is presented for a class of two-dimensional electroelastic branched crack problems. Explicit Green's function for an interface crack subject to an edge dislocation is developed using the extended Stroh formulation allowing the branched crack problem to be expressed in terms of coupled singular integral equations. The integral equations are obtained by the method that models a kink as a continuous distribution of edge dislocations, and the dislocation density function is defined on the line of the branch crack only. Competition between crack extension along the interface and kinking into the substrate is investigated using the integral equations and the maximum energy release rate criterion. Numerical results are presented to show the effect of electric field on the path of crack extension. The work was supported by the Australian Research Council through a Queen Elizabeth II fellowship and by the Australian Academy of Science through the J.G. Russell Award.     

A Partially Built-in Plate under Uniform Load

A thin plate has the form of the infinite strip ?∞<x<∞, 0≤y≤aand has the edge y=abuilt-in. The edge y=0 has its right half 0<x<∞ built-in while the left half ?∞<x<0 is free. The whole plate is now subjected to a uniform load p ₀applied to its upper surface. What is the resulting deflection of the plate and what are the induced moment and shear resultants? We present a solution to this classical problem based on eigenfunction expansions. In the right and left halves of the strip, the deflection can be expanded as separate eigenfunction expansion series, but these are difficult to match across the line x=0 because of the singularity at (0,0) induced by the boundary conditions. We adopt the novel technique of expanding the field near the centre of the strip in its correct form as a series of Williams polar eigenfunctions, and then linking this expansion to the right and left eigenfunction expansions by using a special form of elastic reciprocity. These right and left reciprocity conditions give two infinite systems of linear equations satisfied by the polar expansion coefficients, and we prove that these equations are sufficient to determine these coefficients. Further applications of reciprocity give closed form expressions for the right and left eigenfunction expansion coefficients so that the whole solution is then determined. The method yields accurate results using small systems of linear equations. We present numerical results for the deflection of the plate and the induced moment and shear resultants.     

Water surface wave radiation generated by multiple cylinders oscillating with identical frequency

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Reflection and transmission of obliquely incident Rayleigh surface waves by a viscoelastic interphase

The reflection and transmission of obliquely incident Rayleight surface waves by an interphase between two quarter spaces of identical or different materials, have been investigated. The mechanical behavior of the interphase is represented by a thin viscoelastic layer. By using the full space Green's functions due to a spatially harmonic line load, the mathematical statement of the 3-dimension problem is reduced to a 2-dimension system of singular integral equations. The far-field behavior of the scattered waves leads to the definition of reflection and transmission coefficients,R andT. The system of the singular integral equations are solved forR andT with the boundary element method. The results are presented for selected values of the elastic constants of the joined quarter spaces, the parameters of the interphase and the incident angles of Rayleigh surface waves.     

THE ELASTIC FIELD INDUCED BY A HEMISPHERICAL INCLUSION IN THE HALF—SPACE

The elastic field induced by a hemispherical inclusion with uniform eigeustralns in asemi-infinite elastic medium is solved by using the Green‘s function method and series expansion tech-nique. The exact solutions axe presented for the displacement and stress fields which can be expressedby complete elliptic integrals of the first, second, and third kinds and hypergeometric functions. Thepresent method can be used to determine the corresponding elastic fields when the shape of the inclusionis a spherical crown or a spherical segment. Finally, numerical results axe given for the displacementand stress fields along the axis of symmetry (x3-axis).     

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J-integral) and the T-stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.     

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.     

矩形中厚板Hamilton正则方程的解析解

运用Fourier分析方法,建立了对边简支的矩形中厚板弯曲问题的完备的辛本征展开. 借助于Mathematica软件的帮助,得到了来源于矩形中厚板问题的Hamilton算子的本征函数. 接着证明了本征函数系的完备性,这为使用分离变量法求解相应问题提供了理论保证;进而运用完备性定理,得到了问题的解析解;一个数值算例验证了结果的正确性.     

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In this paper we study the antiplane problem of concentrated point force moving with constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium.The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too.It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers.The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.     

On Low-Dimensional Galerkin Models for Fluid Flow

In this paper some implications of the technique of projecting the Navier–Stokes equations onto low-dimensional bases of eigenfunctions are explored. Such low-dimensional bases are typically obtained by truncating a particularly well-suited complete set of eigenfunctions at very low orders, arguing that a small number of such eigenmodes already captures a large part of the dynamics of the system. In addition, in the treatment of inhomogeneous spatial directions of a flow, eigenfunctions that do not satisfy the boundary conditions are often used, and in the Galerkin projection the corresponding boundary conditions are ignored. We show how the restriction to a low-dimensional basis as well as improper treatment of boundary conditions can affect the range of validity of these models. As particular examples of eigenfunction bases, systems of Karhunen–Loève eigenfunctions are discussed in more detail, although the results presented are valid for any basis. Received 10 September 1999 and accepted 13 December 1999     

HALF-SPACE GREENS'S FUNCTION DUE TO A SPATIALLY HARMONIC LINE LOAD AND SCATTERED FIELDS IN THE FAR FIELD REGION

Half-space Green's function due to a spatially harmonic line load has been expressed asa sum of the full-space Green's functions and a 2-D integral representation of the reflected waves bythe free surface of the half-space.By using the obtained half-space Green's function,an integral rep-resentation of the scattered waves by a cylindrical obstacle is then derived.Finally,by analyzing thefar-zone behavior of the integrands of the integral representation.the far-field pattern of the scatteredwaves in a half-space obtained.     

A generalized force measure of conditions at crack tips

A tentative measure of the forces tending to cause crack growth—the apparent crack extension force—is defined within the framework of continuum mechanics. By an associated fracture criterion initiation of growth may be predicted as well as the direction of preferred growth. The theory is specialized to elastic, viscoelastic and elastic-plastic materials. Under specified conditions the apparent crack extension force may be expressed by surface integrals over the boundary of an arbitrary part of the body for quasi-static deformation and for steady-state propagation of the crack. For plane deformation and for infinitesimal deformation under plane stress conditions these surface integrals reduce to path independent line integrals which include the J integral by Rice[1] and the G integral by Sih[2] as special cases.     

Green's functions,bi-linear forms,and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r,r) for the infinite strip region -1y1, -<x<, with the boundary conditionsG=G/y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on sufficient to ensure that is exponentially small asx. In particular it is proved that this is so, solely under the condition that be bounded asx.A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with free-free boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).The methods used here extend to all other linear homogeneous boundary conditions for these geometries.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.