Three-dimensional elastic displacements induced by a dislocation of polygonal shape in anisotropic elastic crystals

Dislocations and the elastic fields they induce in anisotropic elastic crystals are basic for understanding and modeling the mechanical properties of crystalline solids. Unlike previous solutions that provide the strain and/or stress fields induced by dislocation loops, in this paper, we develop, for the first time, an approach to solve the more fundamental problem—the anisotropic elastic dislocation displacement field. By applying the point-force Green’s function for a three-dimensional anisotropic elastic material, the elastic displacement induced by a dislocation of polygonal shape is derived in terms of a simple line integral. It is shown that the singularities in the integrand of this integral are all removable. The proposed expression is applied to calculate the elastic displacements of dislocations of two different fundamental shapes, i.e. triangular and hexagonal. The results show that the displacement jump across the dislocation loop surface exactly equals the assigned Burgers vector, demonstrating that the proposed approach is accurate. The dislocation-induced displacement contours are also presented, which could be used as benchmarks for future numerical studies.     

On the theory of plane inhomogeneous waves in anisotropic elastic media

In the theory of plane inhomogeneous elastic waves, the complex wave vector constituted by two real vectors in a given plane may be described with the aid of two complex scalar parameters. Either of those parameters may be taken as a free one in the characteristic condition assigned to the wave equation. This alternative underlies the two fundamental approaches in the theory, namely, one associated with the Stroh eigenvalue problem and the other with the generalized Christoffel eigenvalue problem. The two approaches are identical insofar as a partial nondegenerate wave solution (partial mode) is concerned, but they differ in the fundamental solution (wave packet) assembling, and their dissimilarity is also revealed in the presence of degeneracies, which may involve either of the two governing parameters or both of them. Therefore, use of both approaches is essential for studying the degeneracy phenomenon in the theory of inhomogeneous waves. The criteria for different types of degeneracy, related to a double eigenvalue of the Stroh matrix or the Christoffel matrix and at the same time to a repeated root of the characteristic condition, are formulated by appeal to the matrix algebra and to the theory of polynomial equations. On this basis, dimensions of the manifolds, associated with degeneracy of different types in the space of variables, are established for elastic media of unrestricted anisotropy. The relation to the boundary-value problems is discussed.     

Saint-Venant's problem for inhomogeneous and anisotropic elastic bodies

A method of solving Saint-Venant's problem for inhomogeneous and anisotropic elastic bodies is presented.     

The displacements solution for steady-state longitudinal vibrations of segment-wise inhomogeneous elastic rods

Summary The Laplace Integral Transform is used to construct the displacements, solution for steady-state vibrations of an inhomogeneous elastic rod made up of n, (n2) homogeneous elastic segments of the same cross section. The method consists in determining the displacements function of the inhomogeneous rod for zero initial conditions and for displacements type superimposed boundary conditions, with the residues calculated for the poles of the Laplace images of the ends boundary conditions. The correctitude of the constructed solution is then proved and several examples are given to illustrate the method (harmonic end excitations are included).

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Sommario Si usa la trasformata integrale di Laplace per costruire gli spostamenti soluzione del problema delle vibrazioni stazionarie di una barra elastica non omogenea fatta di n, (n2) segmenti elastici omogenei di uguale sezione. Il metodo consiste nel determinare la funzione per gli spostamenti della barra non omogenea in corrispondenza di condizioni iniziali nulle e per sovrapposte condizioni al contorno del tipo spostamento, coi residui calcolati per i poli delle immagini di Laplace delle condizioni al contorno agli estremi. Si dimostra poi la correttezza della soluzione costruita e si danno diversi esempi per illustrare il metodo (comprese le armoniche e le eccitazioni).

     

Further criteria for elastic waves in anisotropic media

Two additional criteria for the existence of cusp points on elastic wave surfaces are developed.A previously published method [1] is extended to give a simple necessary and sufficient condition for cusps about (1, 1, 0) axes in cubic and tetragonal media. This criterion is plausibly adapted to provide a simple inequality applicable to any section of slowness surface represented by separable quadratic and quartic equations.Two tables of numerical examples are presented.     

On wave propagation in inhomogeneous elastic media

Wave propagation in an inhomogeneous elastic rod or slab is considered. The governing equations are written in a matrix form and transformations are sought which reduce the system to a form associated with the wave equation. Integration of the system is then immediate. It is shown that such reduction may be achieved subject to a function involving the density and elastic parameters of the material adopting certain multi-parameter forms. These parameters are available for fitting to the behaviour of a variety of inhomogeneous elastic materials. A specific initial boundary value problem is solved by utilising the present method.     

Plane SH-waves in harmonically inhomogeneous elastic media

This paper considers the oblique propagation of a plane SH-wave in an inhomogeneous elastic medium whose material properties vary harmonically with a space variable. Assuming a small deviation in the harmonic variation, the method of multiple scales is applied and approximate solutions are obtained for three types of the medium.Effects of oblique propagation of SH-waves on a resonant frequency and on an unstable region of wave propagation are discussed. Effects on the reflection coefficient at a stress-free boundary when the wave propagates in a semi-infinite space are also discussed.     

Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

Three-dimensional Green’s functions for infinite anisotropic piezoelectric media

A direct, effective and concise method is adopted in this paper to find out the Green’s functions for infinite anisotropic piezoelectric media. The partial differential equations satisfied by the Green’s functions turn into a set of inhomogeneous algebraic equations after by using Fourier transform. Then inverse transform the solutions of the algebraic equations, the Green’s functions can be expressed by contour integral. Finally, the explicit expression can be obtained for the Green’s functions by using residual theory. The method demonstrated in this paper is easier to follow by people without knowledge of Radon transform, which has been used to obtain the Green’s functions by others.     

Criteria for elastic waves in anisotropic media — a consolidation

The geometry relating to the tangent plane at a stationary point on a surface has been used to re-examine various criteria for the existence of parabolic points (inflexions in 2D-sections) on the outermost sheet of the slowness surface for elastic waves in anisotropic media.Previous results obtained by the authors, exact [4,5], and approximate [6], are related in detail. Further approximations, based on geometrical properties, are derived; one of these proves equivalent to a sufficient condition first applied by McCurdy [8].Numerical investigation shows that, over a wide range of anisotropy, the simply applied approximate criterion of [6] is sufficient and within the accuracy of observation of the elastic stiffnesses.     

The Kirchhoff–Helmholtz integral for anisotropic elastic media

The Kirchhoff–Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth interface in acoustic or isotropic elastic media due to a given incident wavefield and observation points sufficiently far away from the interface. This integral makes use of the Kirchhoff approximation of the unknown scattered wavefield and its normal derivative at the interface in terms of the corresponding quantities of the known incident field. An attractive property of the Kirchhoff–Helmholtz integral is that its asymptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. Here, we extend the Kirchhoff–Helmholtz modeling integral to general anisotropic elastic media. It uses the natural extension of the Kirchhoff approximation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff–Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff–Helmholtz integral, we also derive an extension to anisotropic media of a useful decomposition formula of the geometrical spreading of a primary reflection ray.