Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

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This paper investigates a thermoelasticity theory based on the recent heat conduction model proposed by Quintanilla ( Mech Res Commun 2011; 38: 355–360). Taylor’s expansion of this model leads to an interesting problem of heat conduction. Serious attention has been paid by researchers in the last few years to investigating various heat conduction models. We have considered this newly proposed model of heat conduction given by Quintanilla and employed for coupled thermoelastic problems. We derive the basic governing equations for a homogeneous and isotropic medium and aim to derive some important theorems. Firstly, the uniqueness theorem of a mixed initial and boundary value problem of linear thermoelasticity in the present context is proved. A variational principle is derived for the basic governing equations of motion on the basis of a functional in the context of the present problem. A reciprocity theorem is established by using Laplace transformation. Furthermore, generalization of Somigliano and Green’s theorem for this model is proved on the basis of our reciprocity relation.

Section: Introductionmentioning

confidence: 85%

Some basic theorems on a recent model of linear thermoelasticity for a homogeneous and isotropic medium

Shivay

Mukhopadhyay

2018

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“…It is worth noting that our uniqueness theorem applies to a large class of problems to the one proposed at [10]. In fact the new aspect is that we here do not assume that the elasticity tensor is positive definite.…”

Section: Uniqueness and Instabilitymentioning

confidence: 99%

“…The aim of this contribution is the study of this new thermoelastic problem. It is worth noting that in a recent paper Kumari and Mukhopadhayay [10] considered this coupled theory and proved some basic theorems. One of the results were the uniqueness of solutions using the standard energy arguments.…”

Section: Introductionmentioning

confidence: 99%

“…To be precise, we first use the logarithmic convexity argument to obtain the uniqueness and the instability of solutions. As we do not assume positiveness of the elasticity tensor our uniqueness result is new and it applies to a larger class of problems than the ones analysed in [18]. The logarithmic convexity argument is a powerful tool for studying the qualitative properties of the solutions of several thermoelastic problems [19, 20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On uniqueness and stability for a thermoelastic theory

Quintanilla¹

2016

In this paper we investigate a thermoelastic theory obtained from the Taylor approximation for the heat flux vector proposed by Choudhuri. This new thermoelastic theory gives rise to interesting mathematical questions. We here prove a uniqueness theorem and instability of solutions under the relaxed assumption that the elasticity tensor can be negative. Later we consider the one-dimensional and homogeneous case and we prove the existence of solutions. We finish the paper by proving the slow decay of the solutions. That means that the solutions do not decay in a uniform exponential way. This last result is relevant if it is compared with other thermoelastic theories where the decay of solutions for the one-dimensional case is of exponential way.Peer ReviewedPostprint (author's final draft

“…The obtained results are extended to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction to the delay term and a Phragmén-Lindelöf type alternative is obtained for equations acting both forward and backward in time. Recently, Kumari and Mukhopadhyay [24] established the uniqueness of the solution of this theory [22,23]. They also discussed the variational principle and reciprocity theorem of this model.…”

Section: Introductionmentioning

confidence: 99%

Investigation of harmonic plane waves: detailed analysis of recent thermoplastic model with single delay term

Kumari

Kumar

Mukhopadhyay

2018

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This work is concerned with a recent thermoelastic model. We investigate the propagation of plane harmonic waves in the context of this very recently proposed heat conduction model, an exact heat conduction model with a single delay term, established by Quintanilla. This model attempted to reformulate the heat conduction model that takes into account microstructural effects in heat transport phenomena and provided an alternative heat conduction theory with a single delay term. We aim to study the harmonic plane waves propagating in a thermoelastic medium by employing this new model and derive the exact dispersion relation solution. We mainly focus on a longitudinal wave coupled to a thermal field and find two different modes of this wave. We derive asymptotic expressions for several important characterizations of the wave fields: phase velocity, specific loss, penetration depth, and amplitude ratio. Analytical expressions for these wave characteristics are obtained for different cases of very-high- and low-frequency regions for elastic- and thermal-mode longitudinal waves. To verify the analytical results, we also carry out computational work to obtain numerical results of the wave characterizations for intermediate values of frequency and illustrate the results graphically. We show that our analytical and numerical results are in perfect match. On the basis of the analytical and numerical results, a thorough analysis of the effects of the single delay parameter on various wave characteristics is presented. We highlight several characteristic features of the new thermoelastic model, as compared with other models.