On the Guided Waves in Generalized Dynamical Theories of Thermoelasticity
K. L. Verma
Dean Basic & Applied Sciences, Career Point University, Hamirpur (H.P.) 176041 India
*Corresponding Author E-mail :kl.verma@rediffmail.in, klverma@cpuh.edu.in
ABSTRACT:
Based on the generalized dynamical theories of thermo elasticity and potential functions representation dispersion relations are derived for guided waves propagation in heat conducting thermoplastic isotropic plates. The second sound effect is included to eliminate the thermal wave travelling with infinite velocity as predicted by the diffusion heat transfer model.
KEYWORDS: Guided wave, thermo elasticity, dispersion, thermal relaxation time, coupling.
1. INTRODUCTION:
Energy transport and information over long distances are vital characteristics of waves; guided waves are a kind of elastic wave propagation in such natural wave-guides as plates and pipes. Propagation characteristics of guided waves depend on a structural boundary and have an infinite number of modes associated with propagation. They can be used for many NDE applications, one major benefit of guided wave inspection is its swiftness and low cost. Guided wave technology can be used for testing pipes and tubes much faster than conventional methods. Basic information on the subject and fundamental surveys of guided waves and solids can be found in the books Achenbach, (1984), Viktorov (1967) and Miklowitz, (1978). Future advanced aerospace composite materials will be required to respond well to combined thermal and mechanical loading. These materials will be used for the next generation of aerospace structures, such as the proposed high-speed civil transport. The use of elastic waves to measure elastic properties as well as flaws in solid specimens has received much attention, and many important applications have been developed recently. Under the normal use environment for many aerospace structures, thermal degradation as well as damage due to mechanical fatigue may occur. While studies have been conducted using Lamb waves to examine fatigue damage, few studies have been conducted which monitor either thermal degradation or thermal-mechanical aging. In ideal solids, thermal energy is transported by quantized electronic excitations, which are called free electrons, and the quanta of lattice vibrations, which are called phonons. These quanta undergo collisions of a dissipative nature, giving rise to thermal resistance in the medium. A relaxation time t0 is associated with the average communication time between these collisions for the commencement of resistive flow. The main advantages of thermal and laser based NDE include noncontact evaluation, freedom for complex surface geometry, high spatial and temporal resolution, easy access to cavities, and fast scanning. Two major disadvantages are that the method requires a good physical understanding of thermo elastic wave propagation in solids, which is considerably more complicated than elastic wave propagation, and more complicated instrumentation is needed for data collection. The classical linear theory of thermo elasticity is based Fourier heat conduction law, predicts finite propagation speeds for elastic waves but an infinite speed for thermal disturbance, which is physically unrealistic. The generalized thermo elastic theories proposed by Lord and Shulman (1967) (LS) and Green and Lindsay (GL) (1972) based on a new law of heat conduction to replace Fourier’s law have aroused much interest in recent years. The LS theory introduced a single time constant to dictate the relaxation of thermal propagation while in GL theory, thermal, thermal mechanical relaxations are governed by two different time constants, and the temperature rates are considered among the constitutive variables. In these theories heat equations are hyperbolic one that ensures finite speed of propagation for heat and elastic waves. This model admit second sound even without violating the classical Fourier’s law. Verma and Hasebe (1999) discussed the propagation of thermo elastic vibrations in plates in the context of generalized theories of thermo elasticity. Chandrasekharaiah (1986, 1998) reported review of this literature. The thermo elastic theories proposed by Lord and Shulman (1967) and Green and Lindsay (1972) (here in after called LS and GL theories) have aroused much interest in recent years. These theories are generalization of the coupled thermo elasticity theory [4] and are formulated by using a form of the heat conduction equation that includes the time needed for acceleration of the heat flow. These theories eliminate the paradox of infinite velocity of heat propagation. Recently, the theory of thermo elasticity without energy dissipation, which provide sufficient basic modification in the constitutive equation that permit treatment of much wider class of flow problems, is proposed by Green and Naghdi (1993) (here in after called GN theory). The discussion presented in Chandrasekharaiah (1996) includes the derivation of a complete set of governing equations of the linearized version of the theory for homogeneous and isotropic materials in terms of displacement and temperature fields and a proof of the uniqueness of the solution of the corresponding initial mixed boundary value problem. Chandrasekharaiah (1996) investigated the one dimensional wave propagation in the context of GN theory. Verma (2014) studied the phase and group velocity of thermo elastic Rayleigh waves in transversely isotropic materials the field equations of linear theories of thermo elasticity. Rayleigh wave speed is computed the medium and compared the results with already existing in elasticit and it is shown that thermal relaxation time effect plays a significant role thermo elastic speed of Rayleigh waves at the low values of wave number limits. Recently, relevant theoretical developments on the subject of finite velocity of heat propagation are due to Green and Naghdi [6-8], which provide sufficient basic modification in the constitutive equations that permit treatment of a much wider class of heat flow problems. Green and Nagdhi [6-8] have formulated three different models of thermo elasticity in an alternative way. Among these, in one of the models (Green and Nagdhi, [8]) the most significance is that the internal rate of production of entropy is identically zero, i.e., there is no dissipation of thermal energy. This model is known as thermo elasticity without energy dissipation theory (TEWOEDT) or GN theory of thermo elasticity. In the development of this theory the thermal displacement gradient is considered as a constitutive variable, whereas in the conventional development of a thermo elasticity theory, the temperature gradient is taken as a constitutive variable. In this paper propagation of guided waves in heat conducting isotropic plate in the generalized theory of thermo elasticity is studied. Potential functions are used to derive dispersion relations for the propagation of symmetric and ant symmetric mode. Relevant results of previous investigations are deduced as special cases. The effects of the thermo-mechanical coupling, thermal relaxation times of the plate on the dispersion behavior are examined. Finally numerical solution of the frequency equation is carried out to present free wave dispersion curves.
CONCLUSIONS:
Propagation of guided thermo elastic waves in a homogeneous, isotropic, thermally conducting plate was investigated within the framework of the generalized theory of thermo elasticity proposed by Lord and Shulman and Green and Nagdhi. These theories includes a thermal relaxation time in the heat conduction equation in order to model the finite speed of the thermal wave.
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Received on 14.11.2016 Modified on 28.11.2016 Accepted on 08.12.2016 ©A&V Publications All right reserved DOI: 10.5958/2349-2988.2017.00032.8 Research J. Science and Tech. 2017; 9(1):189-194.
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