Existence of Global Weak Solutions for Coupled Thermoelasticity with Barber's Heat Exchange Condition

Bień, Marian

doi:10.1515/jaa.2003.163

Cited by 5 publications

(2 citation statements)

References 3 publications

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“…The thermal interaction between the rod's end and the obstacle is described by Barber's heat exchange condition that takes into account the thermal resistance of the air in the gap when there is no contact and the sharp decrease in the resistance with increase in the contact stress. The condition was introduced in [2] and its various aspects were investigated in [3,5,26,30]. In the quasistatic setting in 1D the condition is very elegant (see [2]), and in the dynamic case here it is formulated as a set-inclusion.…”

Section: Introductionmentioning

confidence: 99%

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

Paoli

Shillor²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This work, motivated by the rapid developments in Micro-Electro-Mechanical Systems (MEMS) structures, especially actuators and grippers, analyses the dynamics of a thermo-mechanical system consisting of a horizontal beam joined at one end to a vertical rod. As a result of thermal expansion or vibration of the rod, the other end may come into contact with another part of the MEMS device and that closes an electrical circuit, which is the actuating or switching function of such a beam–rod system. The interaction between the rod's contacting end and the supporting device is described by a normal compliance contact law for the displacements and by an inclusion-type Barber's heat exchange condition for the temperature. The heat-exchange coefficient is a multi-function taking into account the air resistance in the gap when there is no contact and the contact pressure when contact occurs. The model consists of a nonlinear variational inclusion for the temperature coupled with a nonlinear variational equation for the displacements. The existence of a weak solution to the problem is proved by using the Galerkin method, a regularization of Barber's condition with the Yosida approximation of a maximal monotone operator, and a priori estimates.

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Section: Introductionmentioning

confidence: 99%

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

Paoli

Shillor²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…An existence result for the Signorini contact problem, with either a Dirichlet or a heat exchange condition for the temperature, was proved by Shi and Shillor [17] using truncation and compactness arguments, while uniqueness was established by Ames and Payne [1]. Quasi-static and dynamic problems with Barber's heat exchange condition were studied by Xu [19] and Bien [3], respectively. The papers [2], [5], and [6] provide error analysis for some contact problems in elasticity.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of a Unilateral Problem in Planar Thermoelasticity

Copetti¹

2007

SIAM J. Numer. Anal.

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We consider in this paper the numerical approximation of a quasi-static contact problem in linear thermoelasticity that models the evolution of the temperature and displacement of an elastic, homogeneous, and isotropic body that may come in contact with an elastic obstacle. We propose a finite element method to numerically approximate the continuous solution. Convergence without any regularity assumptions is proved and error estimates are obtained if the continuous solution is sufficiently regular.

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A contact problem of a thermoelastic diffusion rod

Aouadi

2010

Z Angew Math Mech

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We consider a one‐dimensional contact problem in linear thermoelastic diffusion theory. The coupled system of equations consists of a hyperbolic equation and two parabolic equations. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. The existence of a weak solution is proved by the use of Sobolev's basic space energy arguments. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity.

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Existence of Global Weak Solutions for Coupled Thermoelasticity with Barber's Heat Exchange Condition

Cited by 5 publications

References 3 publications

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

Numerical Analysis of a Unilateral Problem in Planar Thermoelasticity

A contact problem of a thermoelastic diffusion rod

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