Evaluation of Green's function derivatives for exponentially graded elasticity

Effective formulas for computing Green's function of an exponentially graded three‐dimensional material have been derived in previous work. The expansion approach for evaluating Green's function has been extended to develop corresponding algorithms for its first‐ and second‐order derivatives. The resulting formulas are again obtained as a relatively simple analytic term plus a single double integral, the integrand involving only elementary functions. A primary benefit of the expansion procedure is the ability to compute the second‐order derivatives needed for fracture analysis. Moreover, as all singular terms in this hypersingular kernel are contained in the analytic expression, these expressions are readily implemented in a boundary integral equation calculation. The computational formulas for the first derivative are tested by comparing with results of finite difference approximations involving Green's function. In turn, the second derivatives are then validated by comparing with finite difference quotients using the first derivatives. Published in 2010 by John Wiley & Sons, Ltd.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407–26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.     

Green's function for anisotropic dielectric halfspace

A spatial representation of the Green's function for anisotropic halfspace is considered. This representation can be used for analyzing an electric field connected with planar metal structures applied in surface-acoustic-wave devices     

Green's function expansion for exponentially graded elasticity

New computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical Green's function for multiple cracks in anisotropic bodies

The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary-integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.     

Green's function for two-and-a-half dimensional elastodynamic problems in a half-space

 This paper presents an analytical solution, together with explicit expressions, for the steady state response of a homogeneous three-dimensional half-space subjected to a spatially sinusoidal, harmonic line load. These equations are of great importance in the formulation of three-dimensional elastodynamic problems in a half-space by means of integral transform methods and/or boundary elements. The final expressions are validated here by comparing the results with those obtained with the boundary element method (BEM) solution, for which the free surface of the ground is discretized with boundary elements.     

Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates

In this paper, a three dimensional vibration analysis of nano-plates is studied by decoupling the field equations of Eringen theory. Considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. In order to find the solution for a nano-plate based on the presented formulation, one of the three equations in terms of displacement components and corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. A Navier-type method for finding the exact three dimensional solution of a nano-plate is presented using the Fourier series technique. Exact natural frequencies of nano-plates are presented and compared with the results of nonlocal first order and third order shear deformation theories.     

Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis

The boundary-integral equation medthod is particularly well suited for solution of stress concentration and elastic fracture mechanics problems. The method was not previously applicable to anisotropic three dimensional problems because no efficient technique existed for calculation of the required point load solution for an infinite body. A technique has been developed to evaluate numerically the anisotropic point load soultions, and used to generate data bases for various materials. An intrpolation technique is used to evaluate the point load solutions efficiently within a higher order boundary-intgral equation code. The effectiveness of the technique is verified by solution of problems involving both uniaxial stress states and stress concentrations.     

Real-space Green's function of an isolated point charge in an unbounded anisotropic dielectric

In modern electronic devices the finite size of the substrates requires three-dimensional analysis. To this end, the boundary element method (BEM) may be utilized. The BEM involves problem-specific Green's functions (GFs), which are to be constructed first. This work has derived a GF which is the electric potential response to an isolated point charge in an unbounded anisotropic dielectric.     

Green's function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface

The most recent theoretical studies have shown that three-dimensional (3-D) radiation effects play an important role in the optical remote sensing of atmospheric aerosol and land surface reflectance. These effects may contribute notably to the error budget of retrievals in a broad range of sensor resolutions, introducing systematic biases in the land surface albedo data sets that emerge from the existing global observation systems. At the same time, 3-D effects are either inadequately addressed or completely ignored in data processing algorithms. Thus there is a need for further development of the radiative transfer theory that can rigorously treat both 3-D and surface anisotropy effects and yet be flexible enough to permit the development of fast forward and inversion algorithms. We describe a new theoretical solution to the 3-D radiative transfer problem with an arbitrary nonhomogeneous non-Lambertian surface. This solution is based on an exact semianalytical solution derived in operator form by the Green's function method. The numerical implementation is based on several parameterizations that accelerate the solution dramatically while keeping its accuracy within several percent under most general conditions.     

Comments on real-space Green's function of an isolated point-chargein an unbounded anisotropic medium

The electric potential of an isolated point-charge in an unbounded anisotropic dielectric can be derived by an approach which merely involves a coordinate transformation and the knowledge of the solution to the corresponding isotropic problem     

A numerical integration Green's function model for ultrasonic field profiles in mildly anisotropic media

A numerical integration technique utilizing a point source Green's function is introduced to analyze the wave behavior in transversely isotropic-type anisotropic media allowing us to make fast and accurate computations of the acoustic field. The centrifugally cast stainless steel (CCSS) used in nuclear power plants is chosen as a sample medium because of its columnar grain character leading to material anisotropy. A representative number of field profiles are computed and plotted to illustrate the quasi-longitudinal, quasi-transverse, and horizontally-polarized shear wave propagation in a transversely-isotropic medium. Phenomena such as beam skewing, beam splitting, beam focusing, unsymmetrical beams, and other anisotropic effects, some of which are already known from earlier experimental observations, emerge as a computational result of the introduced technique.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

Spectral decomposition of anisotropic elasticity

Summary A newly developed approach, based on the spectral decomposition principle, which is especially useful in crystallography, is applied in this paper. The compliance fourth-rank tensor of crystalline media belonging to the monoclinic system is spectrally decomposed, its eigenvalues are evaluated, together which its elementary idempotent tensors, which expand uniquely the fourth-rank tensor space into orthogonal subspaces. Next, the compliance tensor is spectrally analysed for anisotropic media of the orthorhombic, tetragonal, hexagonal and cubic crystal systems, by regarding these decompositions as particular cases of the spectral decomposition for monolinic media. Consequently, the characteristic values and the idempotent fourth-rank tensors are derived, as well as the stress and strain second-rank eigentensors for all the above mentioned symmetries.     

Screw dislocation in nonlocal anisotropic elasticity

Based on Eringen’s model of nonlocal anisotropic elasticity, new solutions for the stress fields of screw dislocations in anisotropic materials are derived. In the theory of nonlocal anisotropic elasticity the anisotropy is twofold. The anisotropic material behavior is not only included in the anisotropy of the elastic stiffness properties, but also in the anisotropy of the nonlocality which is expressed by the anisotropy of the length scale parameters, which is incorporated in the anisotropy of the nonlocal kernel function. Particularly, a new two-dimensional anisotropic kernel which is the Green function of a linear differential operator with three length scale parameters is derived analytically. New solutions for the stresses of straight screw dislocations in anisotropic (monoclinic and hexagonal) materials are found. The stresses do not have singularities and possess interesting features of anisotropy, which are presented and discussed.     

Efficient evaluation of three-dimensional Green's functions in anisotropic elastostatic multilayered composites

The three-dimensional Green's functions in anisotropic elastostatic multilayered composites (MLCs) obtained within the framework of generalized Stroh formalism are expressed as two-dimensional integrals of Fourier inverse transform over an infinite plane. Their numerical evaluations involve tremendous computational efforts in particular in the presence of various singularities and near-singularities due to the presence of material mismatches across interfaces. The present paper derives the complete set of the Green's functions including displacement, stress and their derivatives with respect to source coordinates using a novel and computationally efficient approach. It is proposed for the first time that the Green's functions in the MLCs are expressed as a sum of a special solution and a general-part solution, with the former consisting of the first few terms of the trimaterial expansion solution around a source load. Since the zero-order term contains the singularity corresponding to the homogeneous full-space solution and can be evaluated analytically, and the other higher-order terms contain most of the near-singular behaviors and can be reduced to a line integral over a finite interval, the general-part solution becomes regular and the Green's functions overall can be evaluated efficiently. As an example, the Green's functions in a five-layered orthortropic plate are evaluated to demonstrate the efficiency of the proposed approach. Also, the detailed characteristics of these Green's functions are examined in both the transform- and physical-domains. These Green's functions are essential in developing the boundary-integral-equation formulation and numerical boundary element method for composite laminate problems involving regular and cracked geometries.     

Quasiclassical Green's function in the BCS pairing theory

We study a generalization of the quasiclassical Green's function which allows us to include the transfer of momentum to the particles. In this approach, we may handle Galilei transformations, rotations, and gauge transformations in a systematic way. As an example, we calculate the quasiparticle flow pattern which arises during the motion of the orbital vector in the ABM phase, and discuss the meaning of the intrinsic angular momentum of the Cooper pairs. Finally, we consider charged particles in a magnetic field, and derive a Boltzmann equation for a superconductor which applies to the Hall effect in the case of moving vortices.On leave from Institut für Théorie der Kondensierten Materie, Universität Karlsruhe.