Skew-symmetric Wave Modes Research Articles

Analysis of wave motion and deformation in elastic plate based on two temperature theory of thermoelasticity

ABSTRACT This article explores the effects of two temperatures on the combined problem of wave propagation and thermomechanical loading in a homogeneous isotropic plate. The governing equations are converted to non-dimensional form, and the potential functions are introduced to simplify the problem. The plate's surfaces are assumed to be traction-free, rigidly fixed, thermally insulated and isothermal to study the wave propagation phenomenon. Secular equations in closed form and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation are derived. The phase velocity and attenuation coefficient are examined for the first problem. The results for uncoupled, coupled, Lord-Shulman (LS) and Green-Lindsay (GL) theories of thermoelasticity have been attained as special cases and agree with that in literature for one temperature theory. In the second part, thermo-mechanical loading has been considered to obtain the expressions of stress and displacement components, conductive and thermodynamic temperatures. The obtained analytical results are computed numerically as well as displayed graphically for copper material to demonstrate the impacts of two temperature theory and compared theoretically as well as numerically with one temperature Lord and Shulman theory.

Study of Axi-Symmetric Vibrations in a Micropolar Transversely Isotropic Layer

The present investigation deals with the propagation of circular crested Lamb waves in a homogeneous micropolar transversely isotropic medium. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and microrotation are computed numerically for magnesium as a material and the dispersion curves, amplitudes of displacements and microrotation for symmetric and skew-symmetric wave modes are presented graphically to evince the effect of anisotropy. Some special cases of interest are also deduced.

Propagation of waves in the layer of a thermo-viscoelastic transversely isotropic medium

The article is presented to enhance our knowledge about the propagation of Lamb waves in the layer of a viscoelastic transversely isotropic medium in the context of thermoelasticity with GN theory of type-II and III. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and temperature distribution were also obtained. Finally, the numerical solution was carried out for cobalt and the dispersion curves, amplitudes of displacements and temperature distribution for symmetric and skew-symmetric wave modes are presented to evince the effect of anisotropy. Some particular cases are also deduced.

Analysis of wave motion in an anisotropic initially stressed fiber-reinforced thermoelastic medium

The present investigation deals with the analysis of wave motion in the layer of an anisotropic, initially stressed, fiber reinforced thermoelastic medium. Secular equations for symmetric and skewsymmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and temperature distribution were also obtained. Finally, the numerical solution was carried out for Cobalt and the dispersion curves, amplitudes of displacements and temperature distribution for symmetric and skew-symmetric wave modes are presented to evince the effect of anisotropy. Some particular cases are also deduced.

Wave propagation in micropolar thermoelastic layer with two temperatures

The propagation of waves in micropolar thermoelastic homogeneous plate with two temperatures subjected to stress free thermally insulated and isothermal conditions is investigated. The secular equations for both symmetric and skew-symmetric wave mode of propagation have been derived. Helmholtz decomposition technique has been used to simplify mathematical model and frequency equations for different mechanical conditions. The regions of secular equations are obtained and some special cases such as thin plate and short wavelength waves of the secular equations are also discussed. The phase velocity and attenuation coefficient are computed numerically and depicted graphically. The amplitudes of stress components and thermodynamic temperature distribution for the symmetric and skew-symmetric wave modes are computed analytically and presented graphically. Results of some earlier workers have been deduced as particular cases.

Vibration analysis of wave motion in micropolar thermoviscoelastic plate

The aim of the present article is to study the micropolar thermoelastic interactions in an infinite Kelvin-Voigt type viscoelastic thermally conducting plate. The coupled dynamic thermoelasticity and generalized theories of thermoelasticity, namely, Lord and Shulman\'s and Green and Lindsay\'s are employed by assuming the mechanical behaviour as dynamic to study the problem. The model has been simplified by using Helmholtz decomposition technique and the resulting equations have been solved by using variable separable method to obtain the secular equations in isolated mathematical conditions for homogeneous isotropic micropolar thermo-viscoelastic plate for symmetric and skew-symmetric wave modes. The dispersion curves, attenuation coefficients, amplitudes of stresses and temperature distribution for symmetric and skew-symmetric modes are computed numerically and presented graphically for a magnesium crystal.

Propagation of waves in the transversely isotropic thermoelastic Green-Naghdi type II and type III medium

The article is presented to enhance our knowledge about the propagation of the Rayleigh-Lamb waves in the layer of a transversely isotropic medium in the context of thermoelastic model of GN type II and type III. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. Amplitudes of displacements and temperature distributions are also obtained. Finally, a numerical solution is found for cobalt as a medium material, and dispersion curves, amplitudes of displacements, and temperature distributions for symmetric and skew-symmetric wave modes are presented. Some particular cases are also deduced.

Elastodynamic Behavior of Axisymmetric Vibrations in Micropolar Thermoelastic Cubic Crystal Plate

The present paper is aimed at studying elastodynamic behavior of axisymmetric vibrations in a homogenous isotropic micropolar thermoelastic cubic crystal plate in context of Lord and Shulman (L-S) and Green and Lindsay (G-L) theories of thermoelasticity. The upper and lower surfaces of the plate are subjected to stress free thermally insulated conditions. The secular equations for symmetric and skew-symmetric wave modes are derived. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew-symmetric motion of the plate. Finally, in order to illustrate and verify the analytical developments, numerical solution of secular equation corresponding to stress free thermally insulated micropolar thermoelastic cubic crystal plate is carried out for magnesium crystal material.

Elastodynamic Behavior of Waves in Thermo-Microstretch Elastic Plate Bordered with Layers of Inviscid Liquid

The elastodynamic behavior of waves in a thermo-microstretch elastic homogeneous isotropic plate bordered with layers of inviscid liquid on both sides subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman and Green and Lindsay theories of thermoelasticity. The mathematical model has been simplified by using the Helmholtz decomposition technique, and the frequency equations for different mechanical situations are obtained and discussed. The special cases such as short wavelength waves and regions of the secular equations are also discussed. Finally, the numerical solution is carried out for a magnesium crystal composite material plate bordered with water. The dispersion curves, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch, and temperature distribution for the symmetric and skew-symmetric wave modes are presented graphically.

Wave Propagation in Microstretch Thermoelastic Plate Bordered with Layers of Inviscid Liquid

The propagation of free vibrations in microstretch thermoelastic homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid on both sides subjected to stress free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman (L‐S) and Green and Lindsay (G‐L) theories of thermoelasticity. The secular equations for symmetric and skewsymmetric wave mode propagation are derived. The regions of secular equations are obtained and short wavelength waves of the secular equations are also discussed. At short wavelength limits, the secular equations reduce to Rayleigh surface wave frequency equations. Finally, the numerical solution is carried out for magnesium crystal composite material plate bordered with water. The dispersion curves for symmetric and skew‐symmetric wave modes are computed numerically and presented graphically.

Analysis of free vibrations for Rayleigh — Lamb waves in a microstretch thermoelastic plate with two relaxation times

The propagation of free vibrations in a microstretch thermoelastic homogeneous isotropic plate subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Green and Lindsay (G—L) theories of thermoelasticity. The secular equations for the microstretch thermoelastic plate in closed form for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. At short wavelength limits, the secular equations for both modes in a stress-free thermally insulated and isothermal homogeneous isotropic microstretch thermoelastic plate reduce to the Rayleigh surface wave frequency equation. The results for symmetric and skew-symmetric wave modes are computed numerically and presented graphically. The theory and numerical computations are found to be in close agreement.

On the propagation of Lamb waves in viscothermoelastic plates under fluid loadings

In this paper the propagation of plane and circular crested viscothermoelastic waves in a homogeneous isotropic, Kelvin–Voigt type viscoelastic thermally conducting, plate sandwiched between inviscid liquid layers is investigated in the context of classical and non-classical theories of thermoelasticity. The secular equations for the symmetric and skew-symmetric modes of plane and circular crested waves are derived in closed form and isolated mathematical conditions. It is noticed that the motion for both the plane and cylindrical waves in plates is governed by Rayleigh–Lamb-type secular equations. The secular equations for thin plate and short wave length waves are also obtained and discussed. The results in the absence of fluid loading, coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. The dispersion curves, attenuation profiles and specific loss in case of symmetric and skew-symmetric wave modes are also presented graphically for a polymethyl methacrylate material plate under fluid loadings. The effect of dissipation due to viscosity is noticed to be quite significant and clearly visible from various curves in the graphs.

Free vibration of microstretch thermoelastic plate with one relaxation time

The propagation of waves in microstretch thermoelastic homogeneous isotropic plate subjected to stress free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skew-symmetric wave mode propagation have been obtained. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical computations are found to be in close agreement.

Effect of rotation on generalized thermo-viscoelastic Rayleigh–Lamb waves

The present paper is aimed at studying the effects of rotation on the thermoelastic interaction in an infinite Kelvin–Voigt-type viscoelastic, thermally conducting plate rotating about the normal to its faces with uniform angular velocity. This facilitates the decoupling of anti-plane/in-plane motion which is not possible, in general. The upper and lower surfaces of the plate are subjected to stress-free, thermally insulated or isothermal conditions. The formulation is applied according to three theories of the generalized thermoelasticity: Lord-Shulman with one relaxation time, Green–Lindsay with two relaxation times, as well as the coupled theory. Secular equations are derived for the plate in closed form isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms. In the absence of mechanical relaxations (Rotation and viscous effects), the results for generalized and couple theories of thermoelasticity were obtained as particular cases from the derived secular equations. In the absence of thermomechanical coupling, the analysis for a viscoelastic plate can be deduced from the present one. Finally the numerical solution is carried out for copper material. The function iteration numerical scheme is used to solve the complex secular equations, in order to obtain phase velocity and attenuation coefficients of propagating wave mode. The dispersion curves and attenuation coefficients profiles so obtained for symmetric and skew-symmetric wave modes are presented graphically to illustrate and compare the theoretical results in the presence and absence of rotation. The study may be useful in the construction and design of gyroscopes and rotation sensors as well as in the application in diverse fields.

SomeConsiderations on the Rayleigh–Lamb Wave Propagation in Visco-thermoelastic Plates

The aim of present paper is to study the thermoelastic interaction in an infinite Kelvin–Voigt-type viscoelastic, thermally conducting plate. The upper and lower surfaces of the plate are subjected to stress-free/rigidly fixed, thermally insulated or isothermal boundary conditions. Coupled dynamical thermoelasticity is employed to study the problem. We derive complex secular equations for the plate in closed from and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. In general, the obtained secular equations are complex each of these on separating real and imaginary parts leads to two real frequency equations. These equations contain complete information regarding wavenumber, phase velocity, group velocity and attenuation coefficients of different propagating modes. The regions and the corresponding forms of Rayleigh–Lamb-type secular equations have been obtained and discussed in addition to Lame modes, decoupled shear horizontal modes and thin plate results. At short wavelength limits, the secular equations for symmetric and skew-symmetric waves in stress-free insulated and stress-free isothermal plate reduce to Rayleigh surface wave frequency equations. The amplitudes of temperature and displacement components during symmetric and skew-symmetric motion of the plate have been computed and discussed. In order to illustrate and compare the theoretical results, the numerical solution is carried out for copper material by using the functional iteration method. The dispersion curves, attenuation coefficients, amplitudes of temperature change, and displacements in case of symmetric and skew-symmetric wave modes are presented graphically.

PROPAGATION OF GENERALIZED VISCOTHERMOELASTIC RAYLEIGH–LAMB WAVES IN HOMOGENEOUS ISOTROPIC PLATES

The present paper is aimed at studying the thermoelastic interaction in an infinite Kelvin–Voigt-type viscoelastic, thermally conducting plate. The upper and lower surfaces of the plate are subjected to stress-free, thermally insulated or isothermal conditions. The coupled dynamic thermoelasticity and generalized theories of thermoelasticity, namely, Lord and Shulman's, Green and Lindsay's, and Green and Nagdhi's are employed to understand the thermomechanical coupling and thermal and mechanical relaxation effects. Secular equations for the plate in closed from and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. In the absence of mechanical relaxations (viscous effect), the results for generalized and coupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. In the absence of thermomechanical coupling, the analysis for a viscoelastic plate can be deduced from the present one. The various forms and regions of Rayleigh–Lamb-type secular equation have been obtained and discussed in addition to Lame modes, decoupled shear horizontal (SH) modes, and thin-plate results. At short-wavelength limits, the secular equations for symmetric and skew-symmetric waves in a stress-free insulated and stress-free isothermal plate reduce to the Rayleigh surface wave frequency equation. The amplitudes of temperature and displacement components during symmetric and skew-symmetric motion of the plate have been computed and discussed. Finally, the numerical solution is carried out for copper material. The dispersion curves, and amplitudes of temperature change and displacements for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results.

CIRCULAR CRESTED THERMOELASTIC WAVES IN HOMOGENEOUS ISOTROPIC PLATES

The propagation of circularly crested thermoelastic waves in a homogeneous isotropic cylindrical plate subjected to stressfree and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT), Lord-Shulman (LS), Green-Lindsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity. The secular equation for the circular plate in closed form and isolated mathematical conditions for symmetric and skew symmetric wave mode propagation in completely separate terms are derived. It is shown that the motion for SH modes gets decoupled from the rest of the motion and remains unaffected due to thermomechanical coupling and thermal relaxation effects. The phase velocities for SH modes have also been obtained. It is noticed that the rest of the motion for circular crested waves is again governed by Rayleigh-Lamb-type secular equations. The secular equations for these plate and Lamé modes are also obtained and discussed for different regions. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. At short wavelength limits, the secular equations for symmetric and skew symmetric waves in stressfree insulated and isothermal circular plate reduces to Rayleigh surface wave frequency equations. Finally, the numerical solution is carried out for aluminium-epoxy composite material, and dispersion curves for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.

Generalized thermoelastic waves in homogeneous isotropic plates

The propagation of thermoelastic waves in homogeneous isotropic plate subjected to stress-free and rigid insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT), Lord-Shulman (LS), Green-Lindsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. It is shown that the motion for SH modes gets decoupled from the rest of the motion and remains unaffected due to thermo-mechanical coupling and thermal relaxation effects. The phase velocities for SH modes have also been obtained. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. At short wavelength limits the secular equations for symmetric and skew-symmetric waves in a stress-free insulated and isothermal plate reduce to Rayleigh surface waves frequency equations. Finally, the numerical solution is carried out for aluminum-epoxy composite material and the dispersion curves for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results.