Elastostatic Green's function for advanced materials subject to surface loading

This paper gives a brief review of the Green's function method for solution of the general three-dimensional Boussinesq problem for advanced materials that are highly anisotropic. The Boussinesq problem refers to calculation of stress and/or strain fields in semi-infinite solids, subject to surface loading by solving the equations of elastostatic equilibrium. Analytical and semi-analytical expressions are derived for the elastostatic Green's functions based upon the delta-function representation developed earlier. The Green's function provides a computationally efficient method for solving the anisotropic Boussinesq problem. The Green's function should be useful for modeling physical systems of topical interest such as nanostructures in semiconductors, interpretation of nanoindentation measurements, and application to the boundary-element method of stress analysis of advanced materials. Numerical results for displacement and stress fields are presented for carbon-fiber composites having general orthotropic, tetrahedral, and hexagonal symmetries, and single-crystal silicon having cubic symmetry.     

Green's function for steady-state heat conduction in a bimaterial composite solid

The derivation of a Green's function for steady-state heat conduction in anisotropic bimaterials is presented. The Green's function is obtained through a Fourier representation to obtain both free-space, singular parts and region-dependent, regular parts. To obtain the region-dependent parts of the Green's function, the homogeneous solution is written using the virtual force method. Full details of the necessary inversion integrals are provided. The Green's function is shown to degenerate to the usual logarithmic potential for steady-state heat conduction in isotropic solids. The normal derivatives necessary for implementation of the Green's function in boundary integral equations are provided, and an example calculation of the Green's function in a quartz-copper material system is presented.     

An integral-equation solution for cracked half-planes bonded together and containing debondings along their interface

The general solution of an arbitrary system of microdefects (i.e. cracks and/or holes) in an isotropic elastic half-plane bonded partially, along an infinite number of straight line segments to another half-plane consisting of a different isotropic elastic material, is formulated in this paper using the complex variable technique. The solution in terms of complex potentials is given by integrals over the cracks and/or holes with integrands expressed in terms of Green's functions and an unknown complex density function. Finally, the problem is reduced to the solution of a singular integral equation for the complex density function only along the microdefects. The appropriate Green's functions are derived from the solution of the problem of a concentrated force or a dislocation existing in either of the two half planes. Numerical results are presented for the stress intensity factors in three different cases.     

An integral representation of the stress intensity factors for three-dimensional static problems

For finding suitable expressions for the stress intensity factors (SIFs) under a general three-dimensional condition, the first stress invariant and the displacement tangent to a crack edge are analyzed. By using Green's theorem, the SIFs are expressed by integrals for the most general situations. K _I and K _II are expressed by integrals of the first stress invariant and its partial derivative. K _III is expressed by an integral of the displacement tangent to the crack edge and its partial derivative. The integrals include a surface integral on a smooth surface of arbitrary shape, and a line integral along part of the surface's boundaries. The expressions are valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. The expressions provide a new approach for the determination of the SIFs.     

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.     

Boundary element formulation for 2D solids with stiff and soft thin inclusions

In this work, a linear boundary element formulation is presented for analysing solids containing stiff and soft thin inclusions. A particular sub-region technique, in which the equilibrium is preserved along interfaces without traction approximation, is adapted to model thin inclusions as fibres and beams immersed in a solid. An alternative formulation, in which tractions along interfaces are preserved as unknowns, eliminating therefore displacements, is also derived and applied to the analysis of inclusions in 2D solids in general. For the case of thin inclusions, fibres and beams, the displacement field is properly approximate over the cross-sections. The quasi-singular integrals appearing in all presented formulations are computed by using closed expressions or employing a numerical scheme with sub-elements.     

Direct use of discrete complex image method for evaluating electric field expressions in a lossy half space

A modelling technique is proposed for direct use of the discrete complex image method (DCIM) to derive closed-form expressions for electric field components encountered in the electric field integral equation (EFIE) representing a lossy half space problem. The technique circumvents time consuming numerical computation of Sommerfeld integrals by approximating the kernel of the integrals with appropriate mathematical functions. This is done by appropriate use of either the least-square Prony (LS-Prony) method or the matrix pencil method (MPM) to represent electric field expressions in terms of spherical waves and their derivatives. A comparison is made between the two methods based on the computation time and accuracy and it is shown that the LS-Prony method performs two?three times faster than the MPM in approximating the integral kernels depending on the platform. The main feature of the proposed technique is its ability for direct inclusion in the kernel of computational tools based on the method of moments solution of the EFIE. This can be viewed as an advantage over the conventional DCIM approximation of spatial Green's functions for mixed potential integral equation for cases where the problem in hand can be more efficiently represented by the EFIE (e.g. the thin-wire EFIE). The accuracy of the proposed technique is validated against numerical integration of Sommerfeld integrals for an arbitrary electric dipole inside a lossy half space.     

Green's functions for transversely isotropic piezoelectric multilayered half-spaces

This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.     

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. K _I and K _II are represented by path integrals of the first stress invariant and its partial derivative. K _III is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for K _I. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Diffraction of water waves by slender barriers

Linear water-wave theory is used to tackle the problem of diffraction of surface waves by a fixed slender barrier in deep water for two basic situations: (i) when the barrier is partially immersed, and (ii) when the barrier is completely submerged. Analytical expressions for the first-order corrections to the reflection and transmission coefficients are derived in terms of integrals involving the shape functions describing the two sides of the slender barrier. A relatively straightforward perturbation technique is used along with the application of Green's theorem in the fluid region. Corresponding analytical expressions representing the reflection and transmission coefficients are also deduced, (i) for a nearly vertical barrier and (ii) for a vertically symmetric slender barrier, as special cases for both the problems. For a nearly vertical barrier it is observed, analytically, that there is no first-order correction to the transmitted wave at any frequency. Computations for the reflection and transmission coefficients up to O(), where ; is a small nondimensional number, are also performed and presented here.     

Transient waves in two semi infinite viscoelastic media separated by a plane interface

We consider the reflection and transmission of a transient, cylindrical, longitudinal wave by a plane interface between two semi infinite linear viscoelastic media.The four relaxation functions are assumed to be synchronous, but no further restrictions are placed on the viscoelastic behaviour of the two materials.By means of Laplace and Fourier transformation together with Cagniard-de Hoop's way of integrating we obtain expressions for the displacement components in terms of one- or three-dimensional integrals. The integrands contain a function which in general has to be found numerically from its Laplace transform.Computations are carried out for elastic materials and for two different standard linear solids. The displacement of the incident wave is found for different radii, and at two observation points the displacement components of the reflected longitudinal wave with its faster head wave and of the reflected transverse wave are computed.     

Formulation of the plastic source method for plane inelastic problems Part 1: Green's functions for plane inelastic deformation

Summary Green's function representation of the residual stress caused by any plane inelastic strain is given for the infinite region with a simple defect such as a crack or an elliptic hole. The plane inelastic strain developed within an infinitesimal region is represented by a double couple whose Green's functions (or complex potentials) are derived using the analytic function theory. The inelastic strain over a finite region is, then, represented by a continuous distribution of such double couples and its complex potentials by area integrals. Closed form expression for the stress field arising from the uniform inelastic strain distribution is given for the general polygonal region withn sides.With 5 FiguresThis paper is dedicated to the memory of Aris Phillips, founding Co-Editor of Acta Mechanica, and was presented at the Aris Phillips Memorial-Symposium, Gainesville, Fla., 1987.     

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

Trigonometric Integrals for the magnetic field of the coil of rectangular cross section

Formulas are derived giving the vector potential and the magnetic field components of a general coil of rectangular cross section and constant winding density. The solution is given in a cylindrical coordinate system in terms of trigonometric integrals. The formulas presented have been cross-checked and validated against alternative expressions giving the various field components as integrals of expressions containing Bessel and Struve functions. The trigonometric integrals for the fields can be evaluated easily to several hundred significant figures using mathematical packages such as Maple or Mathematica. Alternatively, they can be evaluated with a small FORTRAN program. Sample results and field line plots obtained with the method are given, and the field of a coil of rectangular cross section is examined in some detail. A comparison with the results of a finite-element method is also given.     

Element differential method and its application in thermal‐mechanical problems

In this paper, a new numerical method, element differential method (EDM), is proposed for solving general thermal‐mechanical problems. The key point of the method is the direct differentiation of the shape functions of Lagrange isoparametric elements used to characterize the geometry and physical variables. A set of analytical expressions for computing the first‐ and second‐order partial derivatives of the shape functions with respect to global coordinates are derived. Based on these expressions, a new collocation method is proposed for establishing the system of equations, in which the equilibrium equations are collocated at nodes inside elements, and the traction equilibrium equations are collocated at interface nodes between elements and outer surface nodes of the problem. Attributed to the use of the Lagrange elements that can guarantee the variation of physical variables consistent through all elemental nodes, EDM has higher stability than the traditional collocation method. The other main features of EDM are that no mathematical or mechanical principles are required to set up the system of equations and no integrals are involved to form the coefficients of the system. A number of numerical examples of 2‐ and 3‐dimensional problems are given to demonstrate the correctness and efficiency of the proposed method.     

Application of the Green and the Rayleigh-Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity

Summary An elementary but quite general method for the construction of path-independent integrals in plane and three-dimensional elasticity is suggested. This approach consists simply in using the classical Green formula in its reciprocal form for harmonic functions and, further, the more general Rayleigh-Green formula also in its reciprocal form, but for biharmonic functions. A large number of harmonic and biharmonic functions appears in a natural way in the theory of elasticity. Therefore, the construction of path-independent integrals (or, probably better, surface-independent integrals in the three-dimensional case) becomes really a trivial task. An application to the determination of stress intensity factors at crack tips is considered in detail and only the sum of the principal stress components is used in the path-independent integral. Further applications of the method are easily possible.     

Differentiation of finite element approximations to harmonic functions (EM field computation)

Derivatives of finite-element solutions are essential for most postprocessing operations, but numerical differentiation is an error-prone process. High-order derivatives of harmonic functions can be computed accurately by a technique based on Green's second identity, even where the finite element solution itself has insufficient continuity to possess the desired derivatives. Data are presented on the sensitivity of this method to solution error as well as to the numerical quadratures used. The procedure is illustrated by application to finding second and third derivatives of a first-order finite-element solution.<>     

Flat crack under shear loading

Summary A solution is called complete when the explicit expressions are derived for the field of displacements as well as stresses in an elastic body. A new method is proposed here which allows us to obtain exact and complete solutions to various crack problems in elementary functions; no integral transforms or special function expansions are involved. The method is based on the new results in potential theory obtained earlier by the author. The method is applied to the case of a concentrated tangential loading of a penny-shaped crack. The main potential function and the relevant Green's functions are derived. An approximate analytical solution is obtained for a flat crack of general shape. A new set of asymptotic expressions is presented for the field of stresses and displacements near the crack tip in a transversely isotropic space. The use of the method is illustrated by examples.     

Transmission of a Gaussian beam into a biaxial crystal.

We derive integral representations that are suitable for studying the transmission of an electromagnetic Gaussian beam through a plane interface that lies between an isotropic medium and a biaxially anisotropic crystal for the case in which the interface normal is along one of the principal axes of the crystal. To that end, we use recently developed exact solutions for the transmitted fields to derive explicit expressions for the corresponding dyadic Green's functions as well as integral representations that are suitable for asymptotic analysis and efficient numerical evaluation.