Green's function boundary conditions in two-dimensional and three-dimensional atomistic simulations of dislocations

1998 ◽  
Vol 77 (1) ◽  
pp. 231-256 ◽  
Author(s):  
S. Rao ◽  
C. Hernandez ◽  
J. P. Simmons ◽  
T. A. Parthasarathy ◽  
C. Woodward
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Atomistic Simulations ◽  
Two Dimensional

Three-Dimensional Green’s Functions in an Anisotropic Half-Space With General Boundary Conditions

2003 ◽  
Vol 70 (1) ◽  
pp. 101-110 ◽  
Author(s):  
E. Pan
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Different Boundary Conditions

This paper derives, for the first time, the complete set of three-dimensional Green’s functions (displacements, stresses, and derivatives of displacements and stresses with respect to the source point), or the generalized Mindlin solutions, in an anisotropic half-space z>0 with general boundary conditions on the flat surface z=0. Applying the Mindlin’s superposition method, the half-space Green’s function is obtained as a sum of the generalized Kelvin solution (Green’s function in an anisotropic infinite space) and a Mindlin’s complementary solution. While the generalized Kelvin solution is in an explicit form, the Mindlin’s complementary part is expressed in terms of a simple line-integral over [0,π]. By introducing a new matrix K, which is a suitable combination of the eigenmatrices A and B, Green’s functions corresponding to different boundary conditions are concisely expressed in a unified form, including the existing traction-free and rigid boundaries as special cases. The corresponding generalized Boussinesq solutions are investigated in details. In particular, it is proved that under the general boundary conditions studied in this paper, the generalized Boussinesq solution is still well-defined. A physical explanation for this solution is also offered in terms of the equivalent concept of the Green’s functions due to a point force and an infinitesimal dislocation loop. Finally, a new numerical example for the Green’s functions in an orthotropic half-space with different boundary conditions is presented to illustrate the effect of different boundary conditions, as well as material anisotropy, on the half-space Green’s functions.

Wave arrival singularities at cuspidal points in the acoustic wave surfaces of anisotropic solids and their unfolding under weak spatial dispersion

2007 ◽  
Vol 463 (2087) ◽  
pp. 2983-3000
Author(s):  
A.G Every ◽  
J.D Kaplunov ◽  
A.V Pichugin ◽  
G.A Rogerson
Keyword(s):  
Acoustic Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Spatial Dispersion ◽  
Three Dimensional ◽  
Step Function ◽  
Two Dimensional ◽  
Displacement Response ◽  
Wave Surfaces ◽  
Wave Arrival

This paper is concerned with wave arrival singularities in the elastodynamic Green's functions of infinite anisotropic elastic solids, and their unfolding into smooth wave trains, known as quasi-arrivals, through spatial dispersion. The wave arrivals treated here are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function, , and (ii) the displacement response to an impulsive line force or two-dimensional Green's function, . These arrivals take on various analytical forms, including step function and logarithmic and power-law divergences. They travel outwards from the source at the group velocities in each direction, and their locus defines the three- and two-dimensional acoustic wave surfaces, respectively. The main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces, and how these arrivals unfold into quasi-arrivals under the first onset of spatial dispersion. This regime of weak spatial dispersion, where the acoustic wavelength, λ , begins to approach the natural length scale, l , of the medium, is characterized by a correction to the phase velocity, which is quadratic in the wavevector, k , and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. Integral expressions are established for the quasi-arrivals near to cuspidal points, involving the Airy function in the case of and the Scorer function in the case of . Numerical results are presented, illustrating the oscillatory nature of the quasi-arrivals and the interference effects that occur near to cuspidal points in the wave surface.

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

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