On uniqueness and stability for a thermoelastic theory

Quintanilla, R.

doi:10.1177/1081286516634154

Cited by 15 publications

(10 citation statements)

References 23 publications

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“…A type of spatial behavior of the solution for this model has been investigated in detail by Leseduarte and Quintanilla [24]. Some interesting results based on this new heat conduction theory have been reported by Kumari and Mukhopadhyay [25], Quintanilla [26], Kumar and Mukhopadhyay [27], and Kant and Mukhopadhyay [28].…”

Section: Introductionmentioning

confidence: 85%

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

Shivay

Mukhopadhyay

2018

Mathematics and Mechanics of Solids

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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.

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Section: Introductionmentioning

confidence: 85%

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

Shivay

Mukhopadhyay

2018

Mathematics and Mechanics of Solids

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“…In this section, we present a brief description of the model and we obtain its mechanical and variational formulations (details can be found in [22]). We also recall an existence and uniqueness result.…”

Section: The Mechanical and Variational Problems: Existence And Uniquenessmentioning

confidence: 99%

“…This heat conduction equation has been complemented with other equations to obtain a thermoelastic problem and several contributions has been dedicated to study it. We recall some of them [13][14][15][16][17][18]22].…”

Section: Introductionmentioning

confidence: 99%

Numerical Resolution of an Exact Heat Conduction Model With a Delay Term

Campo¹,

Fernández²,

Quintanilla³

2019

jaac

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In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.

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“…In this regard, Quintanilla [21, 22] conducted a very interesting analysis of the well-posed problems of dual- and three-phase-lag models of heat conduction equation. We have to note that some approximations of Tzou’s theory have been studied recently by Quintanilla [23] and Amendola et al [24]. As Tzou [4] observed, the T representation (equation (5)) is not convenient to use for problems involving a flux-specified boundary condition.…”

Section: Introductionmentioning

confidence: 99%

On high-order approximations for describing the lagging behavior of heat conduction

Chiriţă

2018

Mathematics and Mechanics of Solids

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This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) [Formula: see text]; (ii) [Formula: see text]; and (iii) [Formula: see text]. This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for [Formula: see text], while for [Formula: see text] some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

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On uniqueness and stability for a thermoelastic theory

Cited by 15 publications

References 23 publications

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

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

Numerical Resolution of an Exact Heat Conduction Model With a Delay Term

On high-order approximations for describing the lagging behavior of heat conduction

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