Conference Proceeding

Mathematics in Space and Applied Sciences (ICMSAS-2023)
ICMSAS-2023

Subject Area: Mathematics
Pages: 331
Published On: 03-Mar-2023
Online Since: 04-Mar-2023

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Author(s): Sanjeev Kumar, Virender Singh

Email(s): sanjeevrananit@gmail.com , rana.vr4@gmail.com

Address: Dr Sanjeev Kumar, Virender Singh Departmentof Mathematics, Govt. College Dharamshala, H.P Department of Mathematics Govt. College Khundian, HP
*Corresponding Author

Published In:   Conference Proceeding, Mathematics in Space and Applied Sciences (ICMSAS-2023)

Year of Publication:  March, 2023

Online since:  March 04, 2023

DOI:




Analysis of Homogenous Isotropic Plate Waves without Energy Dissipation Byasymptotic Method

 

Dr Sanjeev Kumar, Virender Singh

Departmentof Mathematics, Govt. College Dharamshala, H.P

Department of Mathematics Govt. College Khundian, HP

*Corresponding Author E-mail:  sanjeevrananit@gmail.com, rana.vr4@gmail.com

 

ABSTRACT:

The present investigations are concerned wave motion in an infinite homogenous isotropic thermoelastic stress free, thermally insulated plate in the context of Green and Naghdi (GN) theory of thermoelasticity by asymptotic method. The governing equations for the extensional, transversal and flexural motion are derived from the system of three-dimensional dynamical equations of linear theory of thermoelasticity. All coefficients of the differential operator are obtained and found to be explicit functions of the material parameters. The velocity dispersion and frequency equations for the extensional and flexural wave motions are deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation. The asymptotic differential equations and group velocity expression in terms of phase velocity as well as frequency for flexural wave motions are also derived. The secular equations have also been derived for long and short wavelengths. The dispersion curves for phase and group velocity of various flexural wave modes are shown graphically for aluminum-epoxy material thermoelastic plates.

 

KEY WORDS: Flexural, Asymptotic, Phase Velocity, Group Velocity, GN theory.

 

1. INTRODUCTION:

The conventional theory of thermoelasticity based on Fourier’s heat conduction law has parabolic heat conduction equation which predicts an infinite speed for the propagation of heat. The concept of the hyperbolic nature of heat conduction equation involvinga finite speed of thermal disturbances isfirst time reported by Maxwell [1], known as the second sound. Chester [2] provides some justification to the fact that the so-called second sound must exist in solid. A wave-like thermal disturbance is referred to as ‘second sound’ by Chandrasekharaiah[3]. Ackerman and Overtone [4], Ackerman and Bentman [5]experimentally exhibited the actual occurrence of second sound at low temperatures and small intervals of time. Green and Naghdi [6] postulated a new concept in generalized thermoelasticity in which equation don’t involve temperature rate, which motivates the name of GN theory as“thermoelasticity without energy dissipation”. Lord and Shulman [7] formulated a generalized dynamical theory of thermoelasticity by introducing a single relaxation time needed for acceleration of heat flow. Green and Lindsay [8] modified entropy production inequality and formulated another generalized dynamical theory that involves two relaxations times constrained by inequality . Their analysis modified constitutive relations along with governing equations. Strunin [9] considered the entropy production inequality and proved that no such constrained arise on relaxation times. Authors Chandrasekharaiah[10], Chandrasekharaiah and Srinath [11], Othman and Song [12] studied interesting problems in the context of GN theory. Roychoudhuri and Bandyopadhyay[13] investigated the propagation of time-harmonic plane thermoelastic waves of assignedfrequency in an infinite rotating medium using Green-Naghdi model [6] oflinear thermoelasticity and deriveda more general dispersion equationto examine the effect of rotation on the phase velocity of the modified coupledthermal dilatational shear waves. Chandrasekharaiah and Srinath [14] investigated the waves emanating from the boundary of a spherical cavity in a homogeneous andisotropic unbounded thermoelastic body in context of GN theory.

 

Zelentsov[15] proposed an asymptotic method for solving transient elastic problem of thin strips by employing combination of Laplace and Fourier transform and obtained asymptotic solutions for large values of Laplace transform parameter. Kirova et al. [16] have studied the asymptotic behavior for linear and non-linear waves in viscoelastic materials. Ryabenkov and Faizullina [17] proved that asymptotic method is identical with method of hypothesis and successive approximations for slabs and plates.Aghalovyan [18-21] investigated the boundary value problems of vibrations of elastic structures such as plates and shells with the help of asymptotic methods. Aghalovyan and Gevorgyan [22], Aghalovyan and Aghalovyan[23], Aghalovyan and Ghulghazaryan [24] and Agalovyan andGevorgyan [25] solved both classical and non-classical boundary value problems of vibrations in beams, plates and shells by asymptotic methods.  Gales [26] studied asymptotic spatial behavior of solutions in a mixture consisting of two thermoelastic solids. Gevorgyan[27] investigated the thermoelastic wave propagation in a transversely isotropic heat conducting as well as non-heat conducting elastic materials by using asymptotic method. Losin[28,29] studied the asymptotic of flexural and extensional waves in homogeneous isotropic elastic plate by using asymptotic method. Losin[30] established the equivalence of dispersion relations obtained from operator plate model and Rayleigh-Lamb frequency equation. Sharma and Kumar [31] extended the work of Losin[28, 29] for transversely isotropic elastic plates. Sharma et al. [32] investigated the flexural and transversal wave motions in homogeneous isotropic thermoelastic plates by using asymptotic method.

 

Owing to the technological advances in recent years, plate elements are commonly selected as design components in many engineering structures, especially in the aerospace, marine and construction sectors, because of their ability to resist loads. With the evolution of light plate-structures, tremendous research interests in the vibration of the plates are generated.  The negligence of considering vibration as a design factor can lead to excessive deflections and failures. The vibration design aspect is even more important in micro-machines such as electronic packaging, micro- robots, etc. because of their enhanced sensitivities to vibrations.  The dynamical problems of the theory of elasticity become increasingly important due to their application in diverse fields. The high velocity of modern aircrafts gives rise to aerodynamic heating, which produces intense thermal stresses that reduce the strength of the aircraft structure. The present work is an attempt to find a frequency and velocity dispersion relation from three-dimensional analog of the Rayleigh-Lamb frequency equation that would be sufficient for extensional, transversal and flexural wave motion in generalized thermoelastic plates. The analysis is based on the approach and asymptotic method of Prosenko [33] used in Losin [28, 29] and Sharma et al. [34] with modification that the approximate matrix inversion by Neumann’s series has been replaced by actual matrix inversion. This modification is found very effective as it eliminates restrictions due to the convergence interval for the infinite matrix series and permits the model to be applicable for long and short wave as well in any material Losin[28].  

 

2. FORMULATION OF THE PROBLEM:

We consider free wave motion in homogenous isotropic thermoelastic plate of thickness 2h initially at uniform temperature in the undistributed state. The origin of Cartesian co-ordinate system  is taken at any point  in the middle plane of the plate and z-axis is pointed along the thickness of the plate. We assume that the plate is infinite in  and directions which thus occupies the region         



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Author/Editor Information

Dr. Sanjay Kango

Department of Mathematics, Neta Ji Subhash Chander Bose Memorial, Government Post Graduate College, Hamirpur Himachal Pradesh-177 005, INDIA