Inhomogeneous waves at the boundary of a generalized thermoelastic anisotropic medium

The Christoffel equation is derived for the propagation of plane harmonic waves in a generalized thermoelastic anisotropic (GTA) medium. Solving this equation for velocities implies the propagation of four attenuating waves in the medium. The same Christoffel equation is solved into a polynomial equation of degree eight. The roots of this equation define the vertical slownesses of the eight attenuating waves existing at a boundary of the medium. Incidence of inhomogeneous waves is considered at the boundary of the medium. A finite non-dimensional parameter defines the inhomogeneity of incident wave and is used to calculate its (complex) slowness vector. The reflected attenuating waves are identified with the values of vertical slowness. Procedure is explained to calculate the slowness vectors of the waves reflected from the boundary of the medium. The slowness vectors are used, further, to calculate the phase velocities, phase directions, directions and amounts of attenuations of the reflected waves. Numerical examples are considered to analyze the variations of these propagation characteristics with the inhomogeneity and propagation direction of incident wave. Incidence of each of the four types of waves is considered. Numerical example is also considered to study the propagation and attenuation of inhomogeneous waves in the unbounded medium.     

Reflection and transmission in anisotropic dissipative multilayers

A planarly-stratified multilayer is considered with material properties depending on the Cartesian coordinate normal to the layers. Upon the assumption that the time dependence be harmonic, the equations of motion and the constitutive equations (of a viscoelastic solid) are given the form of a first-order system of ordinary differential equations. The propagator of the whole multilayer is determined and hence the reflection and transmission matrices are obtained for different boundary conditions. Next a new algorithm, which avoids some drawbacks of other procedures, is outlined. The reflection and transmission matrices of the multilayer are determined by recursive relations via the matrices, of reflection and transmission, associated with the single layers.     

Inhomogeneous plane waves in viscous fluids

The propagation of elliptically polarised inhomogeneous plane waves in a linearly viscous fluid is considered. The angular frequency and the slowness vector are both assumed to be complex. Use is made throughout of Gibbs bivectors (complex vectors). It is seen that there are two types of solutions—the zero pressure solution, for which the increment in pressure due to the propagation of the wave is zero, and a universal solution which is independent of the viscosity.Since the waves are attenuated in time, the usual mean energy flux vector is not a suitable way of measuring energy flux. A new energy flux vector, appropriate to these waves is defined, and results relating it with energy dissipation and energy density are obtained. These results are related to a result derived directly from the balance of energy equation.     

Inhomogeneous electromagnetic plane waves in crystals

Summary The propagation of inhomogeneous, time harmonic, elliptically polarised, electromagnetic plane waves in non-absorbing, magnetically isotropic, but electrically anisotropic, crystals is considered. The electric displacement and the magnetic induction are assumed to have the forms D exp l(S · x–t) and B exp l(S · x–t), respectively, at the place x and time t, where D, S, B are Gibbs bivectors (complex vectors) and is real. The implications of Maxwell's equations for the various field quantities are interpreted simply and directly through the use of bivectors and their associated ellipses.The propagation of circularly polarised waves is considered in detail. For such waves the electric displacement bivector is isotropic: D · D = 0. In order that such waves may propagate it is found that either (i) D is parallel to the slowness bivector S, so that both D and S are isotropic and coplanar, or (ii) D is parallel to the magnetic induction bivector B, so that both D and B are isotropic and coplanar. It is shown that for type (ii) the secular equation must have a double root for the slowness and conversely if the secular equation has a double root then there exists an isotropic electric displacement right eigenbivector of the optical tensor.Both types of waves are possible in a biaxial crystal. They complement each other in the following way. For type (i) all but two great circles on the unit sphere are possible circles of polarisation for circularly polarised waves with D and S parallel. Each of the exceptional great circles is such that an optic axis is normal to the plane of the circle. These two exceptional circles are the only possible circles of polarisation for circularly polarised waves of type (ii) when D and B are parallel.The situation for uniaxial crystals is similar—the only essential difference being that for uniaxial crystals there is only one exceptional circle since there is only one optic axis.For isotropic crystals the situation is quite different. Circularly polarised waves of type (i) are not possible. All great circles on the unit sphere are possible circles of polarisation for circularly polarised waves of type (ii) with D and B parallel.     

Inhomogeneous plane waves in compressible viscous fluids

The propagation of inhomogeneous plane waves in a compressible viscous fluid is considered. The frequency and the slowness vector are both allowed to be complex. There are seen to be two types of solutions: (a) two transverse waves, which involve no density or pressure fluctuations, (b) a longitudinal wave, which involves no fluctuations in vorticity. For each type, a propagation condition is obtained giving the (complex) squared length of the slowness vector as a function of frequency. Each depends also on the viscosities. It is seen how to recover the incompressible case as the limit in which the inviscid acoustic wave speed tends to infinity. Each wave is shown to be linearly stable for real frequencies. These waves are attenuated in space and time but nevertheless it is possible to define constant weighted mean values (over a cycle of the propagating part of the wave) of the energy density, energy flux and dissipation. The energy-dissipation equation and the propagation conditions are used to derive relationships between these constant weighted means, some of which are generalizations to compressible fluids of previously known results for incompressible fluids. Explicit expressions in terms of frequency are given for the weighted means.     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

Non-dispersive waves in nonlinear anisotropic dielectrics

The propagation and interaction of transverse electro-magnetic disturbances in a half space of anisotropic non-dispersive dielectric and the associated reflection problem are discussed. For certain types of material symmetry, solutions are obtained in the nearly linear approximation. It is shown that an incident pulse produces a reflected disturbance during a time interval larger than the duration of the pulse. In the case of a sinusoidal incident signal the spectrum of the reflected wave contains non-harmonic frequencies.     

Guided electromagnetic waves in anisotropic media

Summary The propagation of guided waves in anisotropic media has recently become of interest in two fields, viz. in the interpretation of ferromagnetic resonance experiments and in the construction of microwave fourpoles which violate the reciprocity relation. In both cases we are faced with the solution of Maxwell's equations in a volume which is enclosed by perfectly conducting walls and which is completely or partially filled with a medium whose magnetic permeability is described by a second order tensor. An account is given here of some work, both theoretical and experimental, on this subject. Chapter I is an introduction, containing a short survey of the theory of guided waves in isotropic media and of the problems arising in anisotropic media, together with a historical synopsis. Chapter II gives a general formulation of the theory of guided waves in anisotropic media, comprising the existing theories, and also deals with some new applications. In Chapter III a cavity technique for measuring Faraday rotations is described which has several advantages over older techniques. In Chapter IV experimental results obtained for the series of Ferroxcubes IVA, B, C, D, E are collected. Chapter V finally deals with the physical interpretation of these results. In particular the experimental data are compared with Rado's theory of the permeability tensor in non-saturated ferromagnetics.     

On the dissipative zone in anisotropic damage models for concrete

This paper presents a three-dimensional model to simulate the behavior of plain concrete structures that are predominantly tensile loaded. This model, based on continuum damage mechanics, uses a symmetric second-order tensor as the damage variable, which permits the simulation of orthotropic degradation. The validity of the first and the second law of thermodynamics, as well as the validity of the principle of maximum dissipation rate, are required. That is attained by defining the loading functions in quantities that are thermodynamically conjugated to the damage variables. Furthermore, the evolution rule is derived by maximizing the energy dissipation rate. This formulation is regularized by means of the fracture energy approach by introducing a characteristic length. The basic and new idea in this paper is that the characteristic length should always coincide with the width of the dissipative zone appearing in the simulation. The integration points with increasing damage in one loading increment are the dissipative zone in this loading increment. The main objective of this paper is the convenient formulation of approaches for the characteristic length in order to attain the coincidence of the characteristic length with the width of the dissipative zone appearing in the simulation. It is shown that simulations are objective and yield good results if the requirement is fulfilled that the characteristic length in the constitutive law coincides with the width of the dissipative zone in the simulation.     

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.     

Signal velocity for transient waves in linear dissipative media

The aim of this paper is to discuss the concept of signal velocity for transient sinusoidal waves in linear dissipative media. The method of the steepest descent path, first introduced by Brillouin in this respect, is used to show whether the signal velocity may equal the group velocity or the phase velocity.     

Theory of elasticity of anisotropic bodies

International Applied Mechanics -