Higher order approximate dynamic models for layered composites

Based on a higher order dynamic approximate theory developed in the present study for anisotropic elastic plates, two dynamic models, discrete and continuum models (DM and CM), are proposed for layered composites. Of the two models, CM is more important, which is established in the study of periodic layered composites using smoothing operations. CM has the properties: it contains inherently the interface and Floquet conditions and facilitates the analysis of the composite, in particular, when the number of laminae in the composite is large; it contains all kinds of deformation modes of the layered composite; its validity range for frequencies and wave numbers may be enlarged by increasing, respectively, the orders of the theory and interface conditions. CM is assessed by comparing its prediction with the exact for the spectra of harmonic waves propagating in various directions of a two-phase periodic layered composite, as well as, for transient dynamic response of a composite slab induced by waves propagating perpendicular and parallel to layering. A good comparison is observed in the results and it is found that the model predicts very well the periodic structure of spectra with passing and stopping bands for harmonic waves propagating perpendicular to layering. In view of the results, the physical significance of Floquet wave number is also discussed in the study.     

Dispersion relations and mode shapes for waves in laminated viscoelastic composites by variational methods

The propagation of oscillatory waves through periodic elastic composites has been analysed on the basis of the Floquet theory. This leads to self-adjoint differential equation systems which it was proved convenient to solve by variational methods. Many composites, such as the light-weight high-strength boron-epoxy material, consist of strong reinforcing components in a plastic matrix. The latter can exhibit viscoelastic properties which can have a significant influence on wave propagation characteristics. Replacement of the elastic constant by the viscoelastic complex modulus changes the mathematical structure so that the differential equation system is no longer self-adjoint. However, a modification of the variational principles is suggested which retains formal self-adjointness, and yields variational principles which contain additional boundary terms. These are applied to the determination of wave speeds and mode shapes for a laminated composite made of homogeneous elastic reinforcing plates in a homogeneous viscoelastic matrix for plane waves propagating normally to the reinforcing plates. These results agree well with the exact solution which can be evaluated in this simple case. The variational principles permit solutions for periodic, but otherwise arbitrary variation of material properties.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

Waves, normal modes and frequencies in periodic and near-periodic rods. Part II

A systematic approach to the study of normal modes and frequencies of disordered periodic rods is presented within a new transfer matrix framework proposed earlier by the authors. The normal frequency structure and mode localization of multiply-disorder periodic rods are investigated. The Monte Carlo and the perturbation method are applied to study the effects of material parameter uncertainties on normal modes and frequencies of randomly-disordered periodic rods. Some intricate aspects are investigated statistically, and it is shown that for this strongly-coupled elastic system, multiple and/or random disorders lead to more localized modes in or near stop-bands in a more complex way. In addition, high frequency wave localization is a typical feature of such a strongly-coupled but randomly-disordered periodic rod system.     

Dynamics of spiral waves driven by a dichotomous periodic signal

We study the effects of a dichotomous periodic force on meandering and rigidly rotating spiral waves. For a meandering state, the periodic forcing induces more modulating frequencies according to the rules of frequency-locked relations and linear combinations. It can also generate some unique closed tip orbits. On the modulating period T-axis, there exist all kinds of resonant entrainment bands. Arnold tongues exist in the period-amplitude space. The width of entrainment bands is affected by the symmetry of positive and negative parts in each signal unit. In addition, appropriate choices of T-value can be used to eliminate spiral waves. For a rigidly rotating state, the periodic forcing can induce a transition toward meandering spiral waves via generating a transitive bidirectional spiral wave. It is very interesting that, after the transition, the meandering spiral wave has the same primary rotating period as the free meandering states.     

Propagation of shear elastic waves in composites with a random set of spherical inclusions (effective field approach)

The work is dedicated to the problem of plane monochromatic shear wave propagation through elastic matrix composite materials with a homogeneous random set of spherical inclusions. The effective field method (EFM) and quasi-crystalline approximation are used for the calculation of phase velocity and attenuation factor of the mean wave field propagating through the composite. The version of the method developed in the work allows us to obtain the dispersion equation for the wave vector of the mean wave field that serves for all frequencies of the incident field, properties and volume concentrations of the inclusions. The long- and short-wave asymptotic solutions of the dispersion equation are found in closed analytical forms. Numerical solutions of this equation are constructed in a wide region of frequencies that covers the long-, middle- and short-wave regions of the propagating waves. The phase velocities and attenuation factors of the mean wave field in the composites are analyzed for various elastic properties, density and volume concentrations of the inclusions. Comparisons of the predictions of the method with some numerical computation of the effective parameters of matrix composites are presented; possible errors in predictions of the velocities and attenuation factors of the mean wave field in the composites are indicated and discussed.     

Study on wave localization in disordered periodic layered piezoelectric composite structures

The two-dimensional wave propagation and localization in disordered periodic layered 2-2 piezoelectric composite structures are studied by considering the mechanic-electric coupling. The transfer matrix between two consecutive sub-layers is obtained based on the continuity conditions. Regarding the variables of mechanical and electrical fields as the elements of the state vector, the expression of the localization factors in disordered periodic layered piezoelectric composite structures is derived. Numerical results are presented for two cases—disorder of the thickness of the polymers and disorder of the piezoelectric and elastic constants of the piezoelectric ceramics. The results show that due to the piezoelectric effects, the characteristics of the wave localization in disordered periodic layered piezoelectric composite structures are different from those in disordered periodic layered purely elastic ones. The wave localization is strengthened due to the piezoelectricity. And the larger the piezoelectric constant is, the larger the wave localization factors are. It is found that slight disorder in the piezoelectric or elastic constants of the piezoelectric ceramics can lead to more prominent localization phenomenon.     

Two dimensional wave problems in rotating elastic media

The rotation of an elastic medium makes it act anisotropically and dispersively. The eigenvectors for plane wave propagation are in general complex and thus the waves are elliptically polarized. In general the waves are neither pure shear nor pure compressional waves, and their speeds depend on the ratio of rotational frequency of the medium and the angular frequency of the wave.The class of problems discussed here involves waves propagating perpendicularly to the axis of rotation and in particular we discuss plane strain modes. The reflection and refraction of plane waves is considered.The plane waves are used to construct a general solution in cylindrical coordinates. The solution is given in terms of Bessel functions. The cylindrical solution is applied to scattering by circular cylinders. The problem of free oscillations is mentioned briefly.     

Features of plane wave propagation along the layers of a prestrained nanocomposite

Features of the propagation of longitudinal and transverse plane waves along the layers of nanocomposites with process-induced initial stresses are studied. The composite has a periodic structure: it is made by repeating two highly dissimilar layers. The layers exhibit nonlinear elastic behavior in the range of loads under consideration. A Murnaghan-type elastic potential dependent on the three invariants of the strain tensor is used to describe the mechanical behavior of the composite constituents. To simulate the propagation of waves, finite-strain theory is used for developing a problem statement within the framework of the three-dimensional linearized theory of elasticity assuming finite initial strains. The dependence of the relative velocities of longitudinal and transverse waves on two components of small initial stresses in each layer and on the volume fraction of the constituents is studied. It is established that there are thickness ratios of layers in some nanocomposites such that the wave velocities are independent of the initial stresses and equal to the respective wave velocities in composites without initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 3–26, April 2007.     

The coupling between ocean waves and rectangular ice sheets

This paper describes a semi-analytic approach to problems involving rectangular elastic plates of shallow draft floating on water. Specifically, two problems are considered: the scattering of plane monochromatic incident waves by a single elastic plate and the propagation/attenuation of waves through a periodic rectangular arrangement of plates. The approach combines Fourier methods with Rayleigh–Ritz methods for free modes of rectangular plates which reduces each problem to an algebraic system of equations which are numerically accurate and efficient to compute. A selection of results are given to illustrate the work. The approach can be applied to many problems in hydroelasticity including the seakeeping of large flat-bottomed marine vessels, deflections in very large floating structures such as offshore airports and wave propagation through areas of broken sea ice.     

Inhomogeneous wave propagation in partially saturated soils

In the present study, inhomogeneous plane harmonic waves propagation in dissipative partially saturated soils are investigated. The analytical model for the dissipative partially saturated soils is solved in terms of Christoffel equations. These Christoffel equations yields the existence of four wave (three longitudinal and one shear) modes in partially saturated soils. Christoffel equations are further solved to determine the complex velocities and polarizations of four wave modes. Inhomogeneous propagation is considered through a particular specification of complex slowness vector. A finite non-dimensional inhomogeneity parameter is considered to represent the inhomogeneous nature of these four waves. Impact of tortuosity parameter on the movement of pore fluids is considered. Hence, the considered model is capable of describing the wave behavior at high as well as mid and low frequencies. Numerical example is considered to study the effects of inhomogeneity parameter, saturation of water, porosity, permeability, viscosity of fluid phase and wave frequency on the velocity and attenuation of four waves. It is observed that all the waves are dispersive in nature (i.e., frequency dependent).     

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.     

Gap in a continuous spectrum of an elastic waveguide with a partly clamped surface

A periodic elastic waveguide whose continuous spectrum contains a gap (interval that can contain a discrete spectrum only) is constructed. The gap prevents wave propagation in the corresponding range of frequencies, which can be used in designing filters and dampers of elastic waves.     

WAVE LOCALIZATION IN RANDOMLY DISORDERED PERIODIC PIEZOELECTRIC RODS

The wave propagation in periodic and disordered periodic piezoelectric rods is studied in this paper. The transfer matrix between two consecutive unit cells is obtained according to the continuity conditions. The electromechanical coupling of piezoelectric materials is considered. According to the theory of matrix eigenvalues, the frequency bands in periodic structures are studied. Moreover, by introducing disorder in both the dimensionless length and elastic constants of the piezoelectric ceramics, the wave localization in disordered periodic structures is also studied by using the matrix eigenvalue method and Lyapunov exponent method. It is found that tuned periodic structures have the frequency passbands and stopbands and localization phenomenon can occur in mistuned periodic structures. Furthermore, owing to the effect of piezoelectricity, the frequency regions for waves that cannot propagate through the structures are slightly increased with the increase of the piezoelectric constant.     

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.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

Propagation of bending waves in phononic crystal thin plates with a point defect

In this paper, the band structures of bending waves in a phononic crystal thin plate with a point defect are studied using an improved plane wave expansion method combined with the supercell technique. In particular, a phononic crystal thin plate composed of an array of circular crystalline Al₂O₃ cylinders embedded in an epoxy matrix with a square lattice is considered in detail. Full band gaps are shown. When a point defect is introduced, the bending waves are highly localized at or near the defect, resulting in defect modes. The frequency and number of the defect modes are strongly dependent on the filling fraction of the system and the size of the point defect. The defect bands appear from the upper edge of the gap and move to the middle of the gap as the defect size is reduced. For a given defect, the frequencies of the defect bands increase as the filling fraction increases.     

单向纤维增强复合材料中剪切波多重散射问题的复变函数法

基于弹性波的多体散射理论和复变函数方法,利用波函数展开法和保角映射方法,研究了任意形纤维增强复合材料中剪切波的传播。根据相应的边界条件确定弹性波模式系数。给出了复合材料中基体区和纤维核区的波场。作为算例,分析了相同入射频率,不同纤维核的尺寸和形状的情况下,弹性波的传播特性。通过分析发现,低频长波区,弹性波以近似正弦函数的形式传播,纤维核的尺寸和形状对波场影响不明显;随着频率的增加,影响变得明显,大于一定的频率,纤维核的形状可明显地辨别出。最后对结果进行了分析讨论。     

A new approach for developing dynamic theories for structural elements : Part 2: Application to thermoelastic layered composites

In this study a continuum theory is proposed which predicts the dynamic behavior of thermoelastic layered composites consisting of two alternating layers. In constructing the theory, it is noted that the governing equations for a single layer, derived in Part I, hold in each phase of the layered composite. The theory is completed by supplementing these equations with continuity conditions and using a smoothing operation. The derivation of the continuity conditions is based on the assumption that the layers are perfectly bonded at interfaces. To assess the theory, spectra from the exact and the derived theory are compared for waves propagating in various directions of the composite. The match between the two is excellent. For waves propagating normal to layering the theory predicts both the banded and periodic structure of the spectra. The region of validity of the theory on the wave number-frequency plane can be enlargened by increasing the orders of the theory and the continuity conditions.