Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Antiplane Strain

1978 ◽  
Vol 45 (2) ◽  
pp. 343-349 ◽  
Author(s):  
T. J. Delph ◽  
G. Herrmann ◽  
R. K. Kaul
Keyword(s):  
Wave Propagation ◽  
Elastic Medium ◽  
Elastic Body ◽  
Dispersion Equation ◽  
Harmonic Wave ◽  
Wave Number ◽  
Harmonic Waves ◽  
Antiplane Strain ◽  
Frequency Wave ◽  
Asymptotic Values

The propagation of horizontally polarized shear waves through a periodically layered elastic medium is analyzed. The dispersion equation is obtained by using Floquet’s theory and is shown to define a surface in frequency-wave number space. The important features of the surface are the passing and stopping bands, where harmonic waves are propagated or attenuated, respectively. Other features of the spectrum, such as uncoupling at the ends of the Brillouin zones, conical points, and asymptotic values at short wavelengths, are also examined.

Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Plane Strain, Analytical Results

1979 ◽  
Vol 46 (1) ◽  
pp. 113-119 ◽  
Author(s):  
T. J. Delph ◽  
G. Herrmann ◽  
R. K. Kaul
Keyword(s):  
Wave Propagation ◽  
Asymptotic Behavior ◽  
Plane Strain ◽  
Elastic Body ◽  
Harmonic Wave ◽  
Wave Number ◽  
Periodic Medium ◽  
Frequency Wave ◽  
Wave Numbers ◽  
Band Characteristic

The problem of harmonic wave propagation in an unbounded, periodically layered elastic body in a state of plane strain is examined. The dispersion spectrum is shown to be governed by the roots of an 8 × 8 determinant, and represents a surface in frequency-wave number space. The spectrum exhibits the typical stopping band characteristic of wave propagation in a periodic medium. The dispersion equation is shown to uncouple along the ends of the Brillouin zones, and also in the case of wave propagation normal to the layering. The significance of this uncoupling is examined. Also, the asymptotic behavior of the spectrum for large values of the wave numbers is investigated.

A Spectral Finite Element Model for Wave Propagation Analysis in Laminated Composite Plate

2006 ◽  
Vol 128 (4) ◽  
pp. 477-488 ◽  
Author(s):  
A. Chakraborty ◽  
S. Gopalakrishnan
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Shear Deformation ◽  
Mechanical Response ◽  
Laminated Composite ◽  
Wave Number ◽  
Single Element ◽  
Element Formulation ◽  
Frequency Wave ◽  
Spectral Finite Element

A new spectral plate element (SPE) is developed to analyze wave propagation in anisotropic laminated composite media. The element is based on the first-order laminated plate theory, which takes shear deformation into consideration. The element is formulated using the recently developed methodology of spectral finite element formulation based on the solution of a polynomial eigenvalue problem. By virtue of its frequency-wave number domain formulation, single element is sufficient to model large structures, where conventional finite element method will incur heavy cost of computation. The variation of the wave numbers with frequency is shown, which illustrates the inhomogeneous nature of the wave. The element is used to demonstrate the nature of the wave propagating in laminated composite due to mechanical impact and the effect of shear deformation on the mechanical response is demonstrated. The element is also upgraded to an active spectral plate clement for modeling open and closed loop vibration control of plate structures. Further, delamination is introduced in the SPE and scattered wave is captured for both broadband and modulated pulse loading.

Wave Propagation and Parametric Instability in Materials Reinforced by Fibers With Periodically Varying Directions

1975 ◽  
Vol 42 (4) ◽  
pp. 825-831 ◽  
Author(s):  
M. Schoenberg ◽  
Y. Weitsman
Keyword(s):  
Composite Material ◽  
Wave Propagation ◽  
Elastic Medium ◽  
Parametric Instability ◽  
Transversely Isotropic ◽  
Harmonic Waves ◽  
Fiber Reinforced ◽  
Frequency Bands ◽  
Structural Periodicity ◽  
Dominant Direction

This paper concerns the propagation of plane harmonic waves in an infinite fiber-reinforced elastic medium. The composite material is represented by an equivalent homogeneous transversely isotropic matter whose preferred directions coincide with the orientations of the fibers. The fibers are assumed to wobble periodically about a dominant direction, all fibers being parallel to each other. This wobbliness endows the material with a structural periodicity which generates dispersion at all frequencies and instability for various frequency bands. The zones of instability are analyzed in some detail.

scholarly journals The influence of heterogeneity and initial stress on the propagation of Love-type wave in a transversely isotropic layer subjected to rotation

2021 ◽  
Vol 104 (3) ◽  
pp. 003685042110414
Author(s):  
Fatimah Salem Bayones ◽  
Nahed Sayed Hussein ◽  
Abdelmooty Mohamed Abd-Alla ◽  
Amnah Mohamed Alharbi
Keyword(s):  
Wave Propagation ◽  
Dispersion Relation ◽  
Dispersion Equation ◽  
Initial Stress ◽  
Transversely Isotropic ◽  
Wave Number ◽  
Isotropic Layer ◽  
Rigid Foundation ◽  
Type Wave ◽  
Transversely Isotropic Layer

Introduction: In this paper, a mathematical model of Love-type wave propagation in a heterogeneous transversely isotropic elastic layer subjected to initial stress and rotation of the resting on a rigid foundation. Frequency equation of Love-type wave is obtained in closed form. The material constants and initial stress have been taken as space dependent and arbitrary functions of depth in the respective media. Objectives: The dispersion equation is determined to study the effect of different types of parameters such as inhomogeneity, initial stress, rotation, wave number, the phase velocity on the Love-type wave propagation. Methods: The analytical solution has been obtained, we have used the separation of variables, method and the numerical solution using the bisection method implemented in MATLAB. Results: We present a general dispersion relation to describe the impacts as the propagation of Love-type waves in the structures. Numerical results analyzing the dispersion equation are discussed and presented graphically. Moreover, the obtained dispersion relation is found in well agreement with the classical case in isotropic and transversely isotropic layer resting on a rigid foundation. Finally, some graphical presentations have been made to assess the effects of various parameters in the plane wave propagation in elastic media of different nature.

Collisionless damping of plasma waves as a real physical phenomenon does not exist

2014 ◽  
Vol 6 (3) ◽  
pp. 1291-1296
Author(s):  
V. N. Soshnikov
Keyword(s):  
Energy Conservation ◽  
Dispersion Equation ◽  
Physical Phenomenon ◽  
Harmonic Wave ◽  
Collisionless Plasma ◽  
Plasma Waves ◽  
Wave Number ◽  
Erroneous Statement ◽  
Complex Dispersion

Trivial logic of collisionless plasma waves is reduced to using complex exponentially damping/growing wave functions to obtain a complex dispersion equation for their wave number 1 k and the decrement/increment 2 k (for a given real frequency  and complex wave number k  k1  ik2 ), whose solutions are ghosts 1 2 k , k which do not have anything to do at 2 k  0 with the real solution of the dispersion equation for the initial exponentially damping/growing real plasma waves with the physically observable quantities 1 2 k , k , for which finding should be added, in this case, the second equation of the energy conservation law. Using a complex dispersion equation for the simultaneous determination of 1 k and 2 k violates the law of energy conservation, leads to a number of contradictions, is logical error, and finally also the mathematical error leading to both erroneous statement on the possible existence of exponentially damping/growing harmonic wave solutions and to erroneous values 1 k and 2 k . Mathematically correct conclusion about the damping/growing of virtual complex waves of collisionless plasma is wrongly attributed to the actual real plasma waves.

PROPAGATION OF LOVE WAVE IN FIBER-REINFORCED MEDIUM OVER A NONHOMOGENEOUS HALF-SPACE

2014 ◽  
Vol 06 (05) ◽  
pp. 1450050 ◽  
Author(s):  
SANTIMOY KUNDU ◽  
SHISHIR GUPTA ◽  
SANTANU MANNA ◽  
PRALAY DOLAI
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Dispersion Equation ◽  
Half Space ◽  
Love Wave ◽  
Graphical Representation ◽  
Separation Of Variables ◽  
Classical Equation ◽  
Wave Number ◽  
Fiber Reinforced

The present paper is devoted to study the Love wave propagation in a fiber-reinforced medium laying over a nonhomogeneous half-space. The upper layer is assumed as reinforced medium and we have taken exponential variation in both rigidity and density of lower half-space. As Mathematical tools the techniques of separation of variables and Whittaker function are applied to obtain the dispersion equation of Love wave in the assumed media. The dispersion equation has been investigated for three different cases. In a special case when both the media are homogeneous our computed equation coincides with the classical equation of Love wave. For graphical representation, we used MATLAB software to study the effects of reinforced parameters and inhomogeneity parameters. It has been observed that the phase velocity increases with the decreases of nondimensional wave number. We have also seen that the phase velocity decreases with the increase of reinforced parameters and inhomogeneity parameters. The results may be useful to understand the nature of seismic wave propagation in fiber reinforced medium.

Rheological behaviour and applicability of viscoelastic model for wave propagation in non-homogeneous polymer rods

2014 ◽  
Vol 10 (1) ◽  
pp. 36-58
Author(s):  
Rajneesh Kakar ◽  
Shikha Kakar ◽  
Kanwaljeet Kaur
Keyword(s):  
Wave Propagation ◽  
Harmonic Wave ◽  
Viscoelastic Model ◽  
Homogeneous Medium ◽  
Rheological Behaviour ◽  
Harmonic Waves ◽  
Newtonian Viscosity ◽  
Content Type ◽  
Viscosity Coefficients ◽  
Polymer Rods

Purpose – This purpose of this paper is to discuss certain aspects of five-parameter viscoelastic models to study harmonic waves in the non-homogeneous polymer rods of varying density. There are two sections of this paper, in first section, the rheological behaviour of the model is discussed numerically and then it is solved analytically with the help of Friedlander Series using Eikonal equation of optics. In another section, the applicability of the developed model is studied through harmonic wave propagation in polymer non-homogeneous rods. The authors have used linear partial differential equations for finding the dispersion equation of harmonic waves in the polymers. All the cases taken in this study are discussed analytically and numerically with MATLAB. Design/methodology/approach – A five-parameter viscoelastic model constituting of three dash-pots D 2(η 2), D 2′(η 2′), D 3(η 3) and two springs S 1(G 1), S 2(G 2) is considered. Where, G 1, G 2 are the modulli of elasticity and η 2, η 2′, η 3 are the Newtonian viscosity coefficients of the considered model, respectively. The parameters of the model are non-homogeneous in nature, i.e. module of elasticity and viscosity coefficients are space dependent. 1D problem is formed by taking the material in the form of rod of inhomogeneous polymer material by taking one end at x=0. The co-ordinate x is measured positive in the direction of the axis of the filament. Findings – When the density, rigidity and viscosity all are equal for the first material specimen, the sound speed is constant, i.e. non-homogeneous has no effect on speed and phase of the wave is given. So it becomes the case of semi non-homogeneous medium (a medium when characteristics are space dependent while the speed is independent of space variable). The use of five-parameter models is mostly restricted in the field of rock mechanics. Thus, these models can be used in determining the time-dependent behaviour of a polymer medium. Originality/value – The paper is original and it is not published elsewhere.

Wave Propagation Analysis in Anisotropic Plate Using Wavelet Spectral Element Approach

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
Mira Mitra ◽  
S. Gopalakrishnan
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Differential Equations ◽  
Laminated Composite ◽  
Composite Plates ◽  
Spectral Element ◽  
Wave Number ◽  
Frequency Wave ◽  
Propagation Analysis ◽  
Spectral Finite Element

In this paper, a 2D wavelet-based spectral finite element (WSFE) is developed for a anisotropic laminated composite plate to study wave propagation. Spectral element model captures the exact inertial distribution as the governing partial differential equations (PDEs) are solved exactly in the transformed frequency-wave-number domain. Thus, the method results in large computational savings compared to conventional finite element (FE) modeling, particularly for wave propagation analysis. In this approach, first, Daubechies scaling function approximation is used in both time and one spatial dimensions to reduce the coupled PDEs to a set of ordinary differential equations (ODEs). Similar to the conventional fast Fourier transform (FFT) based spectral finite element (FSFE), the frequency-dependent wave characteristics can also be extracted directly from the present formulation. However, most importantly, the use of localized basis functions in the present 2D WSFE method circumvents several limitations of the corresponding 2D FSFE technique. Here, the formulated element is used to study wave propagation in laminated composite plates with different ply orientations, both in time and frequency domains.

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