Coupled elastic waves in uniform isotropic media

This article deals with a certain type of wave in an infinite elastic medium. In contrast to ordinary longitudinal and transverse waves, the amplitude of the type of wave in question depends sinusoidally on the coordinates of a plane which is transverse to the direction of propagation of the wave, i.e., the wave is actually a packet of travelling and stationary waves. Longitudinal waves of this type are always coupled with transverse waves, while transverse waves of the given type may be coupled with longitudinal waves or another transverse wave or may exist as a single wave in the form of a packet containing a travelling wave and a stationary wave. The coupled waves have two phase velocities, which depend on the mechanical properties of the medium, the frequency of vibration, and the wave numbers of the stationary waves. Coupled surface waves in an elastic medium are more general in character than Rayleigh waves; they exhibit dispersion, and they can be used to explain certain seismological observations made during earthquakes—the complete absence of vertical displacements in some cases and the frequent occurrence of horizontal displacements parallel to the wave front. Allowing for the coupling of elastic waves in a layer leads to a more general characteristic equation than the equation obtained in the Rayleigh-Lamb problem. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 19–28, September, 1999.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Diffraction of discontinuous waves by ellipsoidal interfaces of transversely isotropic elastic media

Discontinuous waves penetrating the interfaces of transversely isotropic elastic media are studied using the ray-path method. The cases where such waves interact with curvilinear reflectors and with biconvex and biconcave lenses are considered. It is shown that discontinuous waves can focus or scatter depending on the physical properties and sequence of the media under consideration.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 98–106, October 2004.     

Evolution of the fronts of quasi-compressional and quasi-shear discontinuous waves in inhomogeneous transversely isotropic elastic media

The propagation and evolution of the fronts of discontinuous waves in inhomogeneous transversely isotropic elastic media are studied. A method to draw evolving rays and fronts is proposed. Geometrical singularities on the fronts are studied for different parameters of anisotropy and inhomogeneity     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

Surface and interfacial waves in ionic crystals

The continuum linear theory of ionic crystals is applied to develop a two-dimensional eigenvalue problem in the Stroh formalism. An integral approach is exploited to study the occurrence of surface waves along a free boundary of the crystal. Dispersion relations are obtained by separating real and imaginary parts of the governing system and various boundary conditions are examined. The problem of interfacial waves along the separation boundary between two different crystals is also outlined. Numerical computations are performed for a centrosymmetric crystal (KCl) in order to evaluate bulk wave speeds, limiting speed of surface waves and solutions to the dispersion equations for different boundary conditions.     

Calculation of nonstationary elastic waves in an isotropic layer

A plane problem of nonstationary waves in an infinite isotropic layer is considered. A normal force begins to act on the boundary of the layer at the instant t=0. The opposite side of the layer is free from stresses. Using integral transformations, the solution of the problem is obtained in terms of transforms. Expanding the transform solution in a series of exponential powers and inverting each term of the resulting series, the exact solution of the problem is analytically determined. The fields of stresses and velocities in the layer are calculated. The use of analytical relationships for the calculation, in contrast to the calculation with finite-difference methods, allows us to fairly accurately determine the wave pattern and to eliminate the specific effects inherent in the difference equations. The calculation algorithm used in this work allows us to calculate the solutions of the problem at any point of the layer. The results presented give an idea about the distribution of stresses and velocities of particles across the thickness and in the longitudinal direction. The calculation of nonstationary problems by summing over waves, as is done in the present work, side by side with the methods presented in [1, 2], allows transient wave processes in the layer to be represented in a more complete manner.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnlcheskoi Fizikl, No. 4, pp. 148–155, July–August, 1973.     

Epstein transition zones in nonhomogeneous isotropic elastic media

A simple transformation in the independent variable makes it possible to obtain power series solutions of the stress equations of motion of elasticity for inhomogeneous elastic media whose refractive indices are represented by the Epstein profiles. Graphs of reflection and transmission coefficients versus angles of incidence are presented for different frequencies in the case of incident P as well as incident SV waves for some profiles.     

Rayleigh waves of arbitrary profile in anisotropic media

The paper deals with surface wave propagation in an orthorhombic elastic half-plane. The general profile of the wave is considered, incorporating the anisotropy effects within the known representation in terms of a single plane harmonic function.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Analogy between antiplane shear waves in anisotropic and isotropic media

A linear transformation is presented that transforms the fundamental equations of antiplane shear waves in an anisotropic medium into those of an isotropic medium. By this transformation, the solution of the former problem may be easily obtained if the corresponding latter problem has been solved.     

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     

Analogy between elastic and thermoelastic problems for symmetrically loaded isotropic and transverally isotropic media

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 12, pp. 51–60, December, 1995.