Delayed behavior of double shear perturbations of isotropic incompressible elastic-viscoplastic materials

A study of the instantaneous and delayed behavior of a double shear perturbation superimposed on an equilibrium state of an isotropic incompressible medium with internal variables is presented. Elastic media with general internal variables and true viscoplastic media where an intermediate configuration and particular internal parameters are chosen, are successively considered. In both cases, conditions on evolution laws and free energy are proposed, and proved to be sufficient to obtain a stable system of differential equations for the perturbations. As a consequence, the two delayed wave speeds are then real and less than or equal to the instantaneous elastic wave speeds. When the equilibrium state lies on a viscoplastic yield surface, delayed wave speeds and loading conditions may be identified with those we obtain in a plastically deforming (rate-independent) medium, with the same surface as a plastic yield surface. The signification of the relaxation time introduced is also discussed.     

Acceleration waves in elastic dielectrics with polarization gradient effects

In this paper the propagation of acceleration waves arbitrary form propagating into a deformed eleastic dielectric with polarization effect is investigated. An acceleration wave is defined as a second order propagating surface of discontinuity on which the position vector, the polarization vector and Maxwell potential, and their first order derivatives with respect to time and space coordinates are continuous while the second order derivatives of these quantities may suffer jumps but are continuous everywhere else. By computing the jumps of the balance equations on the singular surface, implicit equations for wave speeds corresponding to non-zero amplitudes of the acceleration wave are obtained. It is noteworthy that the second order derivatives of Maxwell potential are also continuous across the acceleration wave.The same equation for wave speeds are also derived for isotropic elastic dielectrics. The wave speeds for longitudinal and transverse waves are obtained in explicit forms and the conditions of existence for real wave speeds are investigated.     

Existence of longitudinal waves in anisotropic poroelastic solids

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

Propagation of acceleration waves in micropolar elastic solids

The propagation and growth of acceleration waves of arbitrary form propagating into a deformed micropolar elastic solid are investigated. The speeds of propagation are obtained for isotropic micropolar elastic materials and the growth equation is established. The growth equation of acceleration wave is integrated in the case of principle waves for isotropic materials and the decay conditions are examined for both macro- and micro-amplitudes.     

Subsonic interfacial (Stoneley) waves in anisotropic multiferroic bimaterials with a viscous interface

The problem of subsonic interfacial (Stoneley) wave propagation in anisotropic multiferroic bimaterials with a viscous interface is treated. A concise analytical method is constructed for deduction of possible subsonic interfacial wave with varying viscosity of the interface. A numerical scheme and several calculations are given based on the method, which demonstrate interesting results. For an interface constructed by a piezoelectric half-space and a piezomagnetic half-space, when assumed to be non-viscous, calculation shows that it does not permit any subsonic interfacial wave. Yet when the same interface is assumed to be viscous, at least one possible subsonic interfacial wave speed appears which varies with the viscosity of the interface. By introducing the relation between viscosity of certain adhesives and temperature, the possibility of control of interfacial wave speeds through accommodating the working temperature is put forward.     

Propagation of inhomogeneous plane waves in anisotropic viscoelastic media

A new technique is explained to study the propagation of inhomogeneous waves in a general anisotropic medium. The harmonic plane waves are considered in a viscoelastic anisotropic medium. The complex slowness vector is decomposed into propagation vector and attenuation vector for the given directions of propagation and attenuation of waves in an unbounded medium. The attenuation is further separated into the contributions from homogeneous and inhomogeneous waves. A non-dimensional inhomogeneity parameter is defined to represent the deviation of an inhomogeneous wave from its homogeneous version. Such a partition of slowness vector of a plane wave is obtained with the help of an algebraic method for solving a cubic equation and a numerical method for solving a real transcendental equation. Derived specifications enable to study the 3D propagation of inhomogeneous plane waves in a viscoelastic medium of arbitrary anisotropy. The whole procedure is wave-specific and obtains the propagation characteristics for each of the three inhomogeneous waves in the anisotropic medium. Numerical examples analyze the variations in propagation characteristics of each of the three waves with propagation direction and inhomogeneity strength.     

Dispersive surface waves in anisotropic linear electromagnetoelasticity

The occurrence of surface waves on a dispersive piezoelectric half-space is considered on the basis of a fully electromagnetic approach. A linear constitutive model is adopted which accounts for both dielectric and magnetic time-dispersion and a Coulomb-type gauge condition is introduced to derive a Stroh eigenvalue problem in the frequency domain. Compatibility conditions are obtained for surface modes in the ranges of quasi-acoustic and quasi-electromagnetic speeds. The propagation properties are derived having recourse to an asymptotic analysis for the transformed constitutive tensors and it is found that at most three surface waves can propagate in each range of speed. Dispersive solutions are also discussed in the quasi-static limit.     

Longitudinal and transverse waves in anisotropic elastic materials

Summary It is shown that a necessary and sufficient condition for a longitudinal wave to propagate in the direction n in an anisotropic elastic material is that the elastic stiffness C ₁₁ (n) is a stationary value (maximum, minimum or saddle point) at n. Explicit expressions of all n and the corresponding elastic stiffness C ₁₁ (n) for which a longitudinal wave can propagate are presented for orthotropic, tetragonal, trigonal, hexagonal and cubic materials. As to longitudinal waves in triclinic and monoclinic materials, only few explicit expressions are possible. We also present necessary and sufficient conditions for a transverse wave to propagate in the direction n. As an illustration, explicit expressions of all n, the polarization vector a and the wave speed c for which a transverse wave can propagate in cubic and hexagonal materials are given. The search for n in hexagonal materials confirms the known fact that a transverse wave can propagate in any direction. A longitudinal wave is necessarily accompanied by two transverse waves. However, a transverse wave can propagate without being accompanied by a longitudinal wave.     

Transient waves in inhomogeneous anisotropic elastic shells

Summary This paper considers the problem of transient wave propagation in linearly elastic Cosserat shells of constant thickness that may be anisotropic and inhomogeneous. The methods of rays and of singular wave curves are combined to find and integrate the transport equations governing growth-decay behaviour of the six possible wave modes. Conditions on material parameters and wave geometry are obtained for various different uncouplings of the wave modes. Some special cases of propagation conditions and of decay equations are worked out in detail.     

A method for multidimensional wave propagation analysis via component‐wise partition of longitudinal and shear waves

An explicit integration algorithm for computations of discontinuous wave propagation in two‐dimensional and three‐dimensional solids is presented, which is designed to trace extensional and shear waves in accordance with their respective propagation speeds. This has been possible by an orthogonal decomposition of the total displacement into extensional and shear components, leading to two decoupled equations: one for the extensional waves and the other for shear waves. The two decoupled wave equations are integrated with their CFL time step sizes and then reconciled to a common step size by employing a previously developed front‐shock oscillation algorithm that is proven to be effective in mitigating spurious oscillations. Numerical experiments have demonstrated that the proposed algorithm for two‐dimensional and three‐dimensional wave propagation problems traces the stress wave fronts with high‐fidelity compared with existing conventional algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

Propagation of inhomogeneous waves in anisotropic piezo-thermoelastic media

A mathematical model for the propagation of harmonic plane waves in an anisotropic piezo-thermoelastic medium is explained through three relations. Two of them relate the stress-induced harmonic variations in temperature and electric potential to mechanical displacement of material particles. The third is a system that defines modified Christoffel equations for wave propagation in the medium. The solution of this system is ensured by a quartic equation whose complex roots explain the existence and propagation of four attenuating waves in the medium. The effects of piezoelectricity and thermoelasticity on the wave propagation are analyzed in the discussion of special cases. An angle between propagation direction and direction of maximum attenuation defines the attenuated wave as inhomogeneous wave. The complex slowness vector for each of the four attenuated waves in the medium is resolved to calculate the phase velocity and the attenuation factor for its propagation as an inhomogeneous wave along a general direction in three-dimensional space. The variations in phase velocities and attenuation factors with propagation direction are computed, for a realistic numerical model.     

Quasi-steady flames on an evolving atmosphere

Summary By setting up equations for the differences between local values of temperature and concentration, and values appropriate to a spatially uniform, but temporally evolving, reactive atmosphere, the propagation of a plane quasi-steady combustion wave (flame) can be studied in an unbounded atmosphere. Confining attention (arbitrarily) at this stage to waves of low speed, comparable with generally accepted thermal flame-speed values, it is found that a continuous spectrum of quasi-steady wave speeds is possible. The structures of these waves are dependent on their speed, and are similar to the ones recently found for half-space burner flames. The actual speed of wave propagation in any particular case must depend upon details of the larger disturbance field into which the quasi-steady flame must fit, with obvious implications for flame acceleration and related matters.     

Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres

An exact analytical solution is obtained for the scattering of electromagnetic waves from a plane wave with arbitrary directions of propagation and polarization by an aggregate of interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes. The expansion coefficients of a plane wave with arbitrary directions of propagation and polarization, for both TM and TE modes, are derived in terms of spherical vector wave functions. The effects of the incident angle α and the polarization angle β on the radar cross sections (RCSs) of several types of collective uniaxial anisotropic spheres are numerically analyzed in detail. The characteristics of the forward and backward RCSs in relation to the incident wavelength are also numerically studied. Selected results on the forward and backward RCSs of several types of square arrays of SiO? spheres illuminated by a plane wave with different incident angles are described. The accuracy of the expansion coefficients of the incident fields is verified by comparing them with the results obtained from references when the plane wave is degenerated to a z-propagating and x- or y-polarized plane wave. The validity of the theory is also confirmed by comparing the numerical results with those provided by a CST simulation.     

Aspects of energy propagation in highly anisotropic elastic solids

Summary Anomalies in the theory of wave propagation in constrained materials may be reconciled with the standard theory of wave propagation in unconstrained materials by relaxing the constraint slightly and then taking the limit as the constraint is obeyed exactly. In this paper the same method is employed in an attempt to reconcile anomalies in the propagation of energy in a constrained material with the known propagation propertics for unconstrained materials. On relaxing the constraint in a singly constrained material, it is found that the energetics associated with two of the three propagating waves tend to the appropriate known forms for the corresponding constrained material in the limit where the constraint holds exactly. The third wave has no counterpart in the constrained theory and it is conjectured that both the total energy density and the energy flux vector tend to zero as the constrained limit is approached. This conjecture is shown to be true for two simple boundary value problems involving incompressible, and inextensible, elastic half-spaces.     

Energy flux characteristics of inhomogeneous waves in anisotropic thermoviscoelastic media

This study aims to calculate the wave-field characteristics of four attenuating waves in anisotropic thermoviscoelastic medium. An energy balance equation relates the complex-valued energy flux vector to the time-averaged densities of kinetic energy, strain energy and dissipated energy of plane harmonic waves in the medium. A complex slowness vector defines the inhomogeneous propagation of an attenuating wave in the medium. This slowness vector is specified with the phase velocity and the two non-dimensional attenuation parameters of the wave. One of the attenuation parameter defines the inhomogeneity strength of the wave as a measure of its deviation from homogeneous propagation. The phase velocity, attenuation parameters, polarizations of particles, propagation direction are combined to define the group velocity, ray direction and quality factor of attenuation of an inhomogeneous wave in the medium. Numerical examples are considered to study the variations of these characteristics of energy flux with propagation direction and inhomogeneity strength for each of the four attenuating waves in the medium. The effects of anisotropic symmetries are analyzed on the velocities of waves. The decay-rate of energy densities is exhibited with offset in the propagation-attenuation plane.     

PC software for SAW propagation in anisotropic multilayers

A software package that provides an interactive and graphical environment for surface acoustic wave (SAW) and plate-mode propagation studies in arbitrarily oriented anisotropic and piezoelectric multilayers is described. The software, which runs on an IBM PC with math coprocessor, is based on a transfer-matrix formulation for calculating the characteristics of SAW propagation in multilayers that was originally written for a mainframe computer. The menu-driven software will calculate wave velocities and field variable variations with depth for any desired propagation direction: the graphics capability provides a simultaneous display of slowness or velocity and of SAW Deltav/v coupling constant curves, and their corresponding field profiles in either polar or Cartesian coordinates, for propagation in a selected plane or as a function of one of the Euler angles. The program generates a numerical data file containing the calculated velocities and field profile data. Examples illustrating the usefulness of the software in the study of various SAW and plate structures are presented.     

Transient waves in inhomogeneous anisotropic elastic plates

Summary This paper considers the problem of transient wave propagation in elastic Cosserat plates that may be anisotropic, inhomogeneous, or of variable thickness. The methods of rays and of singular wave curves are combined to find and integrate the transport equations governing growth-decay behaviour of the extensional and bending wave modes to derive a common general formula involving the material parameters and wave geometry. For inhomogeneous isotropic plates of variable thickness, conditions for the uncoupling of the wave modes are obtained and some special cases are worked out in detail.     

Lamb waves in a thin isotropic layer between two anisotropic layers

Attenuative Lamb wave propagation in adhesively bonded anisotropic composite plates is introduced. The isotropic adhesive exhibits viscous behavior to stimulate the poor curing of the middle layer. Viscosity is assumed to vary linearly with frequency, implying that attenuation per wavelength is constant. Attenuation can be implemented in the analysis through modification of elastic properties of isotropic adhesive. The new properties become complex, but cause no further complications in the analysis. The characteristic equation is the same as that used for the elastic plate case, except that both real and imaginary parts of the wave number (i.e., the attenuation) must be computed. Based on the Lowe's solution in finding the complex roots of characteristic equation, the effect of longitudinal and shear attenuation coefficients of the middle adhesive layer on phase velocity dispersion curves and attenuation dispersion curves of Lamb waves propagating in bonded anisotropic composites is visualized numerically.     

A continuum approach to magnon-phonon couplings—II: Wave propagation for hexagonal symmetry

The wave propagation problem is envisaged in an elastic ferromagnet with viscous and spinrelaxation processes. The propagation takes place in the hexagonal symmetry induced by an initial configuration which corresponds to a one-ferromagnetic domain state obtained by appropriate magnetic and mechanical loadings. Two cases are considered depending on whether the propagation takes place parallelly or orthogonally to the preferred axis of symmetry. A particular attention is given to the qualitative discussion of the dispersion relation in the cross-over region where magnetoacoustic resonance occurs and for small wave numbers. To that purpose, analytical asymptotic expressions are obtained in terms of three small parameters, the first two measuring the alteration in the usual elastic speeds and the third one being related to the piezomagnetic effect induced by the initial state. A similar asymptotic expression is obtained for the magnetoacoustic Faraday effect, and the possibility of a magnetoacoustic “dichroism” is pointed out.