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

Part of: Special kinds of problems Coupling of solid mechanics with other effects Equations and systems of special type Thin bodies, structures Existence theories

Published online by Cambridge University Press:  27 May 2020

L. Paoli  and
M. Shillor
L. Paoli
Affiliation:
Univ Lyon, Université Jean Monnet Saint-Etienne, CNRS UMR 5208, Institut Camille Jordan, F-42023Saint-Etienne, France
M. Shillor
Affiliation:
Department of Mathematics and Statistics, Oakland University, 146 Library Drive, Rochester, MI48309, USA ([email protected])
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Abstract

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.

Keywords

Dynamic contactNormal complianceBarber's heat exchangeVariational inclusionMaximal monotone operator

MSC classification

Primary: 35M87: Systems of variational inequalities of mixed type
Secondary: 49J40: Variational methods including variational inequalities 74F05: Thermal effects 74K10: Rods (beams, columns, shafts, arches, rings, etc.) 74M15: Contact
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 734 - 760
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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