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Lyapunov Functions for Thermomechanics with Spatially Varying Boundary Temperatures

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Analysis and Continuum Mechanics

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Abstract

Consider a continuous body subjected to conservative body and surface forces, with a part ∂Ω 2 of the boundary maintained at a temperature θ = θ 0(X) and with the remainder of the boundary thermally insulated. A calculation of Duhem [1911] shows that if θ 0 is constant then the equations of motion possess a Lyapunov function, the equilibrium free energy, given in a standard notation (see Section 2) by

$$E = athop {nt ll }imits_mega Reft( {rac{1}{2}{{eft| psilon ight|}^2} + U + si - {heta _0}ta } ight)dX - athop {nt {{t_R} dot xdA.} }imits_{artial mega ackslash artial {mega _1}} $$
(1.1)

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland

    J. M. Ball & G. Knowles

  2. Department of Electrical Engineering, Imperial College of Sciene and Technology, London, UK

    J. M. Ball & G. Knowles

Authors
  1. J. M. Ball
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  2. G. Knowles
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Dedicated to James Serrin on the occasion of his 60th birthday

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© 1989 Springer-Verlag Berlin Heidelberg

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Ball, J.M., Knowles, G. (1989). Lyapunov Functions for Thermomechanics with Spatially Varying Boundary Temperatures. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_4

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  • DOI: https://doi.org/10.1007/978-3-642-83743-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50917-2

  • Online ISBN: 978-3-642-83743-2

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