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Journal of Elasticity

Erschienen in:

08.08.2016

Classical Thermodynamics of Elastic Solids as Open Systems

verfasst von: Chi-Sing Man

Erschienen in: Journal of Elasticity | Ausgabe 2/2017

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Abstract

Herein we study the classical thermodynamics of multicomponent elastic solids as open systems. By writing down explicit expressions for the heat form and the work form, a simple derivation of the fundamental thermodynamic equations for elastic solids as open systems is presented. In particular a formula is obtained for the heats of mass transfer of the chemical components. This paper ends with a rebuttal to the contention that heat can not be unambiguously defined for open, classical reversible systems.

Vorheriger Artikel Justification of the Asymptotic Expansion Method for Homogeneous Isotropic Beams by Comparison with De Saint-Venant’s Solutions

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This “neoclassical” designation is mine. Serrin never used the term “neoclassical thermodynamics”.

The term \(\boldsymbol{\sigma }\cdot d(V_{R} \boldsymbol{E})\) for mechanical work can appear in many equivalent versions where \(\boldsymbol{\sigma}\) and \(\boldsymbol{E}\) are replaced by other conjugate pairs of stress and strain. See, e.g., McLellan [8] for examples.

This assumption is indirectly justified when we show in Sect. 5 that the Second Law implies \(TdS = q\). Contrary to the position taken here, some hold the opposite view. For instance, Münster [7, p. 46] asserted that “[i]t is …not generally possible to define clearly the ‘volume work’ on an open phase. This removes, at the same time, the basis for the definition …of the heat absorbed.” The concept of work for open systems warrants a direct and more detailed general discussion, which goes beyond the scope of the present note.

Here, for easy comparison with formulas in the physics and chemistry literature, we use the same symbol \(\mu_{i}\) to denote \((\partial G/\partial N_{i})_{T, \boldsymbol{\sigma}, \hat{N}_{i}}\). Since \(M_{i} = m_{i} N_{i}\), where \(m_{i}\) is the molecular weight of the \(i\)-th chemical component, the chemical potential as defined in (35) is equal to \(m_{i}\) times the \(\mu _{i}\) that appears in (32).

Note that the \(\varGamma\) in (37) and (39) is equal to its namesake in (5) times the molecular weight of the single-component fluid. See Footnote 4 for remarks on \(\mu\) in (38).

Serrin, J.: An outline of thermodynamical structure. In: Serrin, J. (ed.) New Perspectives in Thermodynamics. Springer, New York (1986) CrossRef

Serrin, J.: The equations of continuum mechanics and the laws of thermodynamics. Meccanica 31, 547–563 (1996) CrossRefMATHMathSciNet

Man, C.-S., Massoudi, M.: On the thermodynamics of some generalized second-grade fluids. Contin. Mech. Thermodyn. 22, 27–46 (2010) ADSCrossRefMATHMathSciNet

Denbigh, K.: The Principles of Chemical Equilibrium, 3rd edn. Cambridge University Press, Cambridge (1971)

Truesdell, C.: A First Course in Rational Continuum Mechanics: Volume 1, General Concepts, 2nd edn. Academic Press, Boston (1991) MATH

Callen, H.B.: Thermodynamics. Wiley, New York (1960) MATH

Münster, A.: Classical Thermodynamics. Wiley-Interscience, London (1970)

McLellan, A.G.: The Classical Thermodynamics of Deformable Materials. Cambridge University Press, Cambridge (1980)

Truesdell, C., Bharatha, S.: The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines. Springer, New York (1977) CrossRefMATH

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Truesdell, C.: The Tragicomical History of Thermodynamics 1822–1854. Springer, New York (1980) CrossRefMATH

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Denbigh, K.G.: The Thermodynamics of the Steady State. Methuen, London (1951)

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Fitts, D.D.: Nonequilibrium Thermodynamics: A Phenomenological Theory of Irreversible Processes in Fluid Systems. McGraw-Hill, New York (1962)

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Hill, T.L.: An Introduction to Statistical Thermodynamics. Addison-Wesley, Reading (1960)

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Planck, M.: Treatise on Thermodynamics. Dover, New York (1945)

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Serrin, J.: Conceptual analysis of classical second laws of thermodynamics. Arch. Ration. Mech. Anal. 70, 355–371 (1979) CrossRefMathSciNet

Titel: Classical Thermodynamics of Elastic Solids as Open Systems
verfasst von: Chi-Sing Man
Publikationsdatum: 08.08.2016
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 2/2017
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-016-9591-4

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