Explicit Secular Equations Research Articles

Rayleigh waves in anisotropic porous media and the polarization vector method

In this paper, the propagation of Rayleigh waves in orthotropic non-viscous fluid-saturated porous half-spaces with sealed surface-pores and with impervious surface is investigated. The main aim of the investigation is to derive explicit secular equations and based on them to examine the effect of the material parameters and the boundary conditions on the propagation of Rayleigh waves. By employing the method of polarization vector the explicit secular equations have been derived. These equations recover the ones corresponding to Rayleigh waves propagating in purely elastic half-spaces. It is shown from numerical examples that the Rayleigh wave velocity depends strongly on the porosity, the elastic constants, the anisotropy, the boundary conditions and it differs considerably from the one corresponding to purely elastic half-spaces. Remarkably, in the fluid saturated porous half-spaces, Rayleigh waves may travel with a larger velocity than that of the shear wave, a fact that is impossible for the purely elastic half-spaces.

A note on paper “Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media”

The paper is devoted to correcting structure of the representation for Rayleigh waves at non-semisimple degeneracy of the fundamental matrix. The incorrect representation was given in a paper “Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media” by Cerv and Plesek (2013) .

Rayleigh waves in a layered orthotropic elastic half-space with sliding contact

The presented paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary thickness. The layer and the half-space may be compressible or incompressible and they are in sliding contact with each other. The main aim of the paper is to derive explicit exact secular equations of the wave for four possible combinations: both the layer and the half-space are compressible or incompressible, one is compressible and the other is incompressible. When the layer and the half-space are both compressible, the explicit secular equation is derived by using the effective boundary condition method. For the three remaining cases, the explicit secular equations are deduced directly from this secular equation by using the incompressible limit technique. Based on the obtained secular equations, the effect of incompressibility and the sliding contact on the Raleigh wave propagation is considered through some numerical examples. It is shown that the incompressibility (of half-spaces and coating layers) and the sliding contact strongly affects the Raleigh wave velocity.

On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

The secular equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this secular equation in both papers. This secular equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explicit secular equation of the wave. The technique makes clear the way from the traction-free and continuity conditions to the secular equation and enables us to recognize the misprints appearing in the reported secular equation. The technique can be employed to obtain explicit secular equations of Rayleigh waves for many other cases. Moreover, the paper introduces a transfer matrix in explicit form for an orthotropic layer that is much simpler in form than the one obtained previously.

Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer

• The propagation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer is investigated. • The half-space and the layer may be compressible or incompressible and they are in welded contact. • For the compressible/compressible case, the exact secular equation is derived by using the effective boundary condition method. • For three remaining cases, the exact secular equations are obtained by the incompressible limit technique. In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer and the half-space may be compressible or incompressible and they are in welded contact with each other. The main aim of the paper is to derive explicit exact secular equations of the wave for four possible combinations of a (compressible/incompressible) half-space coated by a (compressible/incompressible) layer. For the compressible/compressible case, the explicit secular equation is derived by using the effective boundary condition method. For three remaining cases, the explicit secular equations are derived from the secular equation for the compressible/compressible case by using the incompressible limit technique along with the expressions of the reduced elastic compliances in terms of the elastic stiffnesses. Based on the obtained secular equations, the effect of incompressibility on the Raleigh wave propagation is considered numerically. It is shown that the incompressibility strongly affects the Raleigh wave velocity and it makes the Raleigh wave velocity increasing.

Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces

In this paper, the propagation of Rayleigh waves in an incompressible elastic half-space with impedance boundary conditions is investigated. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3=0. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. For the monoclinic case, the method of polarization vector is used for deriving the secular equation. This is an algebraic equation of eighth-order. When the impedance parameters vanish, the equations obtained coincide with the corresponding secular equations of Rayleigh waves with traction-free boundary conditions.

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3=0. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.

Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media

This paper is concerned with the derivation of implicit and explicit secular equations for Rayleigh waves polarized in a plane of symmetry of an anisotropic linear elastic medium. It has been confirmed, in accord with Ting’s paper [2], that the Rayleigh waves propagate with no geometric dispersion. Numerical evaluations of both the implicit and explicit equations give the same values of Rayleigh wave velocities. In the case of orthotropic material (thin composites) it has been found that Rayleigh wave velocity depends significantly (as with bulk waves) on the directions of principal material axes. For the same material model the analytical solutions, based on implicit and explicit secular equations, were compared against the finite element and experimental data that had been published by Cerv et al. [4] in 2010. It emerged that the theory was in accordance with the experiment.

Helix-coil transition in terms of Potts-like spins

In the spin model of a helix-coil transition in polypeptides a preferred value of spin has to be assigned to the helical conformation, in order to account for different symmetries of the helical vs. the coil states, leading thus to the Generalized Model of Polypeptide Chain (GMPC) Hamiltonian as opposed to the Potts model Hamiltonian, both with many-body interactions. Comparison of explicit transfer matrix secular equations of the Potts model and the GMPC model reveals that the largest eigenvalue of the Potts model with Δ many-body interactions coincides with the largest eigenvalue of the GMPC model with Δ - 1 many-body interactions, indicating the identity of both free energies. In distinction, the second largest eigenvalues in both models do not coincide, indicating a different behavior for the spatial correlation length that in its turn defines the width of the helix-coil transition interval. We explore in detail the thermodynamic consequences, resulting from spin models with and without the built-in spin anisotropy, that should indicate which model to favour as a more appropriate description of the equilibrium physical properties pertaining to the helix-coil transition.

An explicit secular equation of Rayleigh waves propagating along an obliquely cut surface in a directional fiber-reinforced composite

The propagation of Rayleigh waves along an obliquely cut surface in a directional fiber-reinforced composite was studied recently by Ohyoshi [Ohyoshi T. Compos Sci Technol 2000; 60; 2191-6]. The author derived the secular equation of the wave, however, it is still implicit. In this paper, a fully explicit secular equation of the wave is obtained by using the method of first integrals. From it we immediately arrive at the secular equation of the wave for the case when the cut surface is parallel to the fiber direction. This secular equation is much more simple than the ones obtained recently by Cerv [Cerv J. Int Rev Mech Eng (IREME) 2008; 2; 762-72] and Cerv et al. [Cerv J et al. Composite Structures 2010; 92; 568-77]. Based on the obtained secular equations some approximate formulas for the velocity of Rayleigh waves are established and it is shown that they are good approximations. The explicit secular equations and the approximate formulas for the velocity derived in this paper are useful for analyzing the effect of the material properties and the orientation of the fiber direction on the Rayleigh wave velocity, especially they are powerful tools for solving the inverse problem: determining the material parameters from the measured values of the velocity.

Secular Equations for Rayleigh and Stoneley Waves in Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity

The Stroh formalism is employed to study Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. The 6×6 fundamental matrix N is no longer real. Nevertheless the coefficients of the sextic equation for the Stroh eigenvalue p are real. The orthogonality and closure relations are derived. Also derived are three Barnett-Lothe tensors. They are not necessarily real. Secular equations for Rayleigh and Stoneley wave speeds are presented. Explicit secular equations are obtained when the materials are orthotropic. In the literature, the secular equations for Stoneley waves in orthotropic materials are obtained without using the Stroh formalism. As a result, it requires computation of a 4×4 determinant. The secular equation presented here requires computation of a 2×2 determinant, and hence is fully explicit. A Rayleigh or Stoneley wave exists in the exponentially graded material under the influence of gravity if the wave can propagate in the homogeneous material without the influence of gravity. As the wave number k→∞, the Rayleigh or Stoneley wave speed approaches the speed for the homogeneous material.

Method of first integrals and interface, surface waves

The method of first integrals (MFI) based on the equation of motion for the displacement vector, or based on the one for the traction vector was introduced recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves, the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii) Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.

Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

The problem of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity was studied recently by Abd-Alla and Ahmed [A.M. Abd-Alla, S.M. Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity, Appl. Math. Comput. 135 (2003) 187–200], who derived the secular equation of the wave in the implicit form. In this paper, by using an appropriate representation of the solution, we obtain the secular equation of the wave in the explicit form. Moreover, considering its special cases, we derive explicit secular equations for a number of investigations of Stoneley waves under the influence of gravity, for which only the implicit dispersion equations were previously obtained.

Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999) 61–69], and the secular equation of the wave in the implicit form was derived. However, due to the uncorrect representation of the solution, the secular equation is not right. The main aim of the present paper is to reconsider this problem. We find the secular equation of the wave in explicit form. By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived.

Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity is investigated. Using an appropriate representation of the solution we derive the secular equation of the wave motion in the explicit form. Moreover, following the same approach, we obtain the explicit secular equations for a number of previously investigated Rayleigh wave problems whose dispersion equations were obtained only in the implicit form.

On interface waves in misoriented pre-stressed incompressible elastic solids

Some relationships, fundamental to the resolution of interface wave problems, are presented. These equations allow for the derivation of explicit secular equations for problems involving waves localized near the plane boundary of anisotropic elastic half-spaces, such as Rayleigh, Sholte, or Stoneley waves. They are obtained rapidly, without recourse to the Stroh formalism. As an application, the problems of Stoneley wave propagation and of interface stability for misaligned predeformed incompressible half-spaces are treated. The upper and lower half-spaces are made of the same material, subject to the same prestress, and are rigidly bonded along a common principal plane. The principal axes in this plane do not however coincide, and the wave propagation is studied in the direction of the bisectrix of the angle between a principal axis of the upper half-space and a principal axis of the lower half-space.

An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain

Surface and interfacial impedance matrices play an important role in the construction of Green's functions, the analysis of surface and interfacial waves and the stability assessment of pre-stressed half-spaces or joined half-spaces. This paper studies these matrices for generally anisotropic pre-stressed incompressible elastic materials. It is shown that the surface-impedance matrix satisfies a simple matrix equation which, for plane-strain deformations, can be solved exactly. As a result, explicit secular equations for surface and interfacial wave speeds and explicit wrinkling/buckling conditions for pre-stressed half-spaces and joined half-spaces are obtained. It is also shown that the plane-strain surface-wave problem is mathematically identical to the edge-wave problem for thin elastic plates. Thus, the uniqueness of surface-wave speed is settled by drawing upon a recent proof of the uniqueness of edge-wave speed. Examples are used to show that it is straightforward to solve the secular equations based on the given formulae either exactly (where possible) or numerically.

Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds

Rayleigh waves are considered for crystals possessing at least one plane of symmetry. The secular equation is established explicitly for surface waves propagating in any direction of the plane of symmetry, using two different methods. This equation is a quartic for the squared wave speed in general, and a biquadratic for certain directions in certain crystals, where it may itself be solved explicitly. Examples of such materials and directions are found in the case of monoclinic crystals with the plane of symmetry at x 3=0. The cases of orthorhombic materials and of incompressible materials are also treated.

An Explicit Secular Equation for Surface Waves in an Elastic Material of General Anisotropy

An explicit secular equation for surface waves in an elastic half-space was obtained by Rayleigh in 1885 (Proc. London Math. Soc. 17) for isotropic materials and by Stoneley in 1963 (Geophys. J. Astron. Soc. 8) for orthotropic materials. Since then no explicit secular equation was found for a material more general than orthotropic materials. Recently, Destrade (J. Acoust. Soc. Amer. 109, 2001) employed the first integrals (originally proposed by Mozhaev) to obtain an explicit secular equation for monoclinic materials with the symmetry plane at x 3 = 0, which is the plane of motion. The first integral approach does not work for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. Ting (Proc. R. Soc. 32, 2002) proposed a new approach that not only recovers Destrade's secular equation easily, but also provides an explicit expression of the Stroh eigenvalues p and the eigenvectors a, b without solving the quartic equation for p. These eigenvalues and eigenvectors are needed in the surface wave solution. Employing this new approach, Ting also obtained explicit secular equations for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. This new approach is extended here to derive an explicit secular equation for surface waves in a general anisotropic elastic material. The secular equation for a general anisotropic material is then specialized to monoclinic materials with the symmetry plane at x 1 = 0, x 2 = 0 or x 3 = 0. It does recover the secular equations of Destrade, Stoneley and Rayleigh.

Self-consistent electromagnetic modes at a metal slab

Maxwell's equations are solved in the presence of a metal slab. The response of the electrons in the slab to an electromagnetic wave is a current charge density which appears to be a non-local pseudopotential in the Helmholtz equation. This pseudopotential comprises Lindhard's longitudinal and transverse dielectric functions internally, the scattering at the surfaces, and the vacuum dielectric function externally. External magnetic modes couple to internal magnetic modes; external electric modes couple to internal electric and longitudinal modes; bound longitudinal modes and bound electric and magnetic modes are shown to arise as well. Explicit secular equations for the frequency of longitudinal and transverse bulk modes are derived which include the scattering at the surface. The phase shift of the free transverse modes at the slab is calculated and related to the reflectivity and the surface energy of the slab. The phase shift of the free electric modes is governed by the internal longitudinal dielectric function and that of the free magnetic modes by the internal transverse dielectric function.