Three-dimensional Green's Functions for Two-phase Transversely Isotropic Piezothermoelastic Media

As in the case of time-harmonic waves (see Chapter 1), we begin with the simplest model, replacing a ship by a point source in the uniform forward motion in calm water. The corresponding velocity potential is sometimes referred to as the Kelvin source, but to keep the terminology unified we call it the Green's function in what follows. Similar to the theory of time-harmonic waves developed in Part 1, the theory of ship waves presented here relies essentially on Green's functions. They are of importance not only for proving solvability theorems (see Chapters 7 and 8) but also for constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems) in Section 8.4. The three-dimensional Green's function of a point source in deep water is considered in detail in Sections 6.1 and 6.2. General facts about the three-dimensional Green's function are considered in Section 6.1 and the far-field expansions for Green's function and the corresponding elevation of the free surface are obtained in Section 6.2. Two-dimensional Green's functions are treated in Section 6.3, which we begin with the simpler case of deep water (Subsection 6.3.1). For water of finite depth, which will be referred to as shallow water, we consider Green's function in Subsection 6.3.2.

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested “slicing” procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

Green's functions play an important role in electroelastic analyses of pyroelecric media. However, most works available on this topic are on case of identical temperature. Based on the 2D and 3D general solution of pyroelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the 2D and 3D Green's function for a steady line heat source and a steady point heat source in semi-infinite, infinite and two-phase pyroelectric plane and media is presented by newly induced harmonic functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the 2D and 3D general solution, and the undetermined constants can be obtained by boundary conditions, continuous conditions and equilibrium conditions.

Knowing Green's function and the charge density found on different conductors, the diverse capacities can eventually be calculated by solving an integral equation. This has been dealt with only for simple dielectric-conductor configurations. In Coen's article, the integral representation of log (Z) is employed in calculating Green's function for microstrips (with or without an upper ground plane). Electrostatically speaking, the boundary conditions along conductors or dielectric interfaces are represented by means of infinite charge series.

The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues pα (α=1,2,3) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

It is well known that the conventional diagonalization used in the multilevel fast multipole algorithm (MLFMA) suffers from low-frequency breakdowns, which limit the application of this powerful method to multiscale problems involving large objects with small (but important) details in terms of wavelength. Using multipoles explicitly or resorting to alternative expansions incorporating evanescent waves for subwavelength interactions are common techniques for broadband implementations of MLFMA, while all these techniques need extensive efforts and re-programming. Recently, we presented an approximate diagonalization of the three-dimensional Green's function using scaled Hankel functions and plane waves (O. Ergul and B. Karaosmanoglu, IEEE Antennas Wireless Propag. Lett., 13, 2014, pp. 1054–1056). While its accuracy is limited (e.g., 3–4% in the worst case for the challenging one-box-buffer scheme), the approximate diagonalization provides stable computations of interactions at arbitrarily short distances. As a major advantage, the approximate diagonalization needs only minor modifications in the existing codes, including the parallel programs, to stabilize the conventional implementations for low frequencies.

The commenters state that the one- and two-dimensional frequency domain Green's functions for isotropic chiral media derived by Engheta and Bassiri (see ibid., vol.37, p.512-5, Apr. 1989) were reported first by the commenters. The authors, in a paper submitted in December 1986 which was published in February 1988, point out that the one- and two-dimensional dyadic Green's functions for isotropic chiral materials, derived by them, extend and continue their original work on three-dimensional chiral dyadic Green's function, which first appeared in March 1986, and was used and acknowledged by the commenters in some of their work.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fundamental solutions play an important role in the analyses of coupled fields in electro-magneto-thermo-elastic material. However, most works available on this topic address the case of uniform temperature. Based on the compact general solution of transversely isotropic electro-magneto-thermo-elastic material, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional fundamental solution for a steady point heat source in an infinite transversely isotropic electro-magneto-thermo-elastic material is presented by five newly induced harmonic functions. Numerical results are given graphically by contours.

Three-dimensional static Green's function for a half space over mixed PEC and dielectric wedges

The three-dimensional electric Green function for uniaxial dielectric media has been given by Chen (1973, 1983) Here we give the magnetic Green's function, whose form suggests that magnetic field formulations may be more tractable than electric field formulations. Using the electric and the magnetic Green's functions, canonical sources for the independent generation of the ordinary and the extraordinary fields are also obtaned.

The elastic response of a vacancy in a semi-infinite fcc copper lattice containing a free surface is calculated by using a new multiscale Green's function method. The method treats the lattice distortion near the vacancy at the atomistic level and the free surface at the macroscopic continuum level in the same formalism. The lattice is modeled using the lattice statics Green's function that fully accounts for the discrete atomistic structure of the lattice and can model a large crystallite containing a million atoms without excessive CPU effort. The method is especially useful for modeling the elastic response of nanocrystals containing point defects in which surfaces and interfaces play important roles. The method bridges the length scales seamlessly by relating the microscopic lattice distortion near a point defect to measurable macroscopic parameters of the solid such as the strain and the displacement field at a free surface. Using the interatomic potential derived by Cleri and Rosato, the lattice distortion, relaxation energy, and relaxation volume due to a vacancy are calculated in an otherwise perfect copper lattice for a million-atom model containing a free (100) surface. The calculated value of the relaxation volume is in excellent agreement with the observed value. Numerical results are also presented for the strain and the displacement fields at the free surface due to a vacancy and the interaction energy between a vacancy and the free surface in anisotropic semi-infinite copper.

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.

Three-dimensional Green's functions of thermoporoelastic axisymmetric cones

We use the 2D general solutions of orthotropic electro-magneto-thermo-elastic material to construct the 2D Green's function for a steady line heat source in the interior of two-phase orthotropic electro-magneto-thermo-elastic plane by ten newly introduced harmonic functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions and equilibrium conditions. All components of coupled field are expressed in terms of elementary functions and are convenient to use.