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Continuum Mechanics Through the Twentieth Century pp 61–77Cite as

Axiomatization and Thermo-Mechanics

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Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 196))

Abstract

In contradistinction with Chapter 4, the present chapter deals with a more voluntary tendency at axiomatization and abstraction, probably inherited from the early writings of Hamel in Germany and Duhem in France at the dawn of the Twentieth century. Such a program was essentially expanded under the magisterial leadership of Clifford Truesdell in the USA, after his careful historical perusal of mechanics from the origin to the 1940s. The pursued aim was a rational reconstruction of the whole of continuum mechanics in a somewhat Bourbakian style. Impressive encyclopedic contributions by Truesdell, Toupin and Noll were the lighthouses that “illuminated” the world community of mechanics. Simultaneously, a scientific journal (the A.R.M.A.) set forth standards and a definite style. A rather strict thermodynamic frame work was proposed by B.D. Coleman and W. Noll. The notions of fading memory and the required satisfaction of the Clausius-Duhem (thermodynamic) inequality are fundamental ingredients in this presentation. However, attractive as it was, some parts of this true credo imposed too much constraint on the thermomechanical modelling so that some freedom had to be granted and some generalization were necessary in a too much corseting frame work. As dutifully exposed in this chapter, this led to the conception of a rational extended thermodynamics (in particular by I. Müller) as also a less revolutionary but very efficient thermo-mechanics with well-chosen internal state variables.

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References

  • Bataille J, Kestin J (1979) Irreversible processes and physical interpretation of rational thermodynamics. J Non-Equilibr Thermod. 4:229–258

    Google Scholar 

  • Beris AN, Edwards BJ (1994) The thermodynamics of flowing systems. Oxford University Press, New York

    Google Scholar 

  • Bridgman PW (1945) The nature of thermodynamics. Harvard University Pres, Cambridge

    Google Scholar 

  • Brillouin M (1900) Théorie moléculaire des gaz. Diffusion du mouvement et de l’énergie. Ann Chimie (7), 20, 440–485

    Google Scholar 

  • Coleman BD (1964) On thermodynamics of materials with memory. Arch Rat Mech Anal 17:1–46

    Google Scholar 

  • Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys. 47:597–613

    Article  Google Scholar 

  • Coleman BD, Noll W (1961) Foundations of linear viscoelasticity. Rev Mod Phys 33:239–249

    Article  MathSciNet  MATH  Google Scholar 

  • Coleman BD, Owen DR (1974) A mathematical foundation for thermodynamics. Arch Rat Mech Anal 54:1–104

    Article  MathSciNet  MATH  Google Scholar 

  • Day WA (1972) The thermodynamics of simple materials with fading memory. Springer, Berlin

    Book  MATH  Google Scholar 

  • Day WA (1976) Entropy and hidden variables in continuum thermodynamics. Arch Rat Mech Anal 62:367–369

    Article  MATH  Google Scholar 

  • de Groot SR, Mazur P (1962) Non-equilibrium thermodynamics. North-Holland, Amsterdam (Dover reprint, New York, 1986)

    Google Scholar 

  • Dill EH (1975) Simple materials with memory. In: Eringen AC (ed) Continuum physics, vol II. Academic Press, New York, pp 284–403

    Google Scholar 

  • Domingos JJD, Ninas MRN, Whitelaw JH (ed) (1973) Foundations of continuum thermodynamics. In: Proceeding of Bussaco Meeting. Wiley, New York

    Google Scholar 

  • Duhem P (1911) Traité d’énergétique ou de thermodynamique générale. Two volumes. Gauthier-Villars, Paris

    Google Scholar 

  • Eckart C (1940) The thermodynamics of irreversible processes I. The simple fluid, II. Fluid mixtures. Phys. Rev 58, 267–275

    Google Scholar 

  • Eckart C (1948) The thermodynamics of irreversible processes III. The theory of elasticity and anelasticity. Phys Rev 73:373–382

    Article  MathSciNet  MATH  Google Scholar 

  • Epstein M (2012) The elements of continuum biomechanics. Wiley, Chichester

    Book  Google Scholar 

  • Ericksen JL (1991) Introduction to the thermodynamics of solids. Chapman & Hall, London

    MATH  Google Scholar 

  • Eringen AC (1960) Irreversible thermodynamics and continuum mechanics. Phys Rev 117:1174–1183

    Article  MATH  Google Scholar 

  • Eringen AC (1962) Nonlinear theory of continuous media. McGraw Hill, New York

    Google Scholar 

  • Eringen AC (1967) Mechanics of continua. Wiley, New York

    MATH  Google Scholar 

  • Eringen AC, Maugin GA (1990) Electrodynamics of continua, vol II. Springer, New York

    Book  Google Scholar 

  • Fabrizio M, Morro A (2003) Electromagnetism of continuous media. Oxford University Press, UK

    Book  MATH  Google Scholar 

  • Grmela M (1984) Bracket formulation of dissipative fluid mechanics equations. Phys Lett. A102:355–358

    MathSciNet  Google Scholar 

  • Grmela M, Öttinger HC (1997) Dynamics and thermodynamics of complex fluids.I. Development of a general formalism. Phys Rev E56:6620–6632

    Google Scholar 

  • Gurtin ME, Williams WO (1971) An axiomatic foundation for continuum thermodynamics. Arch Rat Mech Anal. 26:83–117

    Article  MathSciNet  Google Scholar 

  • Halphen H, Nguyen Quoc Son (1975) Sur les matériaux standards généralisés. J Mécanique (Paris) 14:39–63

    MATH  Google Scholar 

  • Hutter K (1977) The foundations of thermodynamics. Its basic postulates and implications: a review of modern thermodynamics. Acta Mechanica 27:1–54

    Article  MathSciNet  Google Scholar 

  • Ignatieff YA (1996) The mathematical world of Walter Noll. Springer, Berlin

    Book  MATH  Google Scholar 

  • Jaumann G (1911) Geschlossenes System physikalischer und chemischer Differentialgesetze. Akad. Wiss. Wien Sitzber (IIa) 120:385–530

    MATH  Google Scholar 

  • Jou D, Casas-Vasquez J, Lebon G (1993) Extended irreversible thermodynamics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Leigh DC (1968) Nonlinear continuum mechanics. McGraw Hill, New York

    Google Scholar 

  • Liu I-S (1972) Method of Lagrange multipliers for exploitation of the entropy principle. Arch Rat Mech Anal 46:131–146

    MATH  Google Scholar 

  • Machlup S, Onsager L (1953) Fluctuations and irreversible processes II: systems with kinetic energy. Phys Rev 91:1512–1515

    Article  MathSciNet  MATH  Google Scholar 

  • Maugin GA (1980) The method of virtual power in continuum mechanics. Application to coupled fields. Acta Mechanica 35:1–70

    Article  MathSciNet  MATH  Google Scholar 

  • Maugin GA (1990) Internal variables and dissipative structures. J Non-Equilibr Thermod. 15:173–192

    Google Scholar 

  • Maugin GA (1992) Thermomechanics of plasticity and fracture. Cambridge University Press, UK

    Book  MATH  Google Scholar 

  • Maugin GA (1999) The thermomechanics of nonlinear irreversible behaviours. World Scientific, Singapore

    Google Scholar 

  • Maugin GA, Drouot R (1983) Internal variables and the thermodynamics of macromolecules in solution. Int J Engng Sci. 21:705–724

    Article  MathSciNet  MATH  Google Scholar 

  • Maugin GA, Muschik W (1994) Thermodynamics with internal variables, I. General concepts. J Non-equilibr Thermod. 19:217–249

    MATH  Google Scholar 

  • Meixner J (1972) The fundamental inequality in thermodynamics. Physica 59:305–313

    Article  Google Scholar 

  • Müller I (1973) Thermodynamik, die Grundlage der Materialtheorie. Bertelsmann Universitätsverlag

    Google Scholar 

  • Müller I (1985) Thermodynamics. Pittman, London

    MATH  Google Scholar 

  • Müller I, Ruggeri T (1993) Extended thermodynamics. Springer, New York

    Book  MATH  Google Scholar 

  • Muschik W (1986) Thermodynamical theories: survey and comparison. J Appl Sci. 4:189–200

    Google Scholar 

  • Noll W (1955) On the continuity of the solid and fluid states. J Rat Mech Anal 4(1):3

    MathSciNet  MATH  Google Scholar 

  • Noll W (1958/1959) A mathematical theory of the mechanical behavior of continuous media. Arch Rat Mech Anal 2, 197–226

    Google Scholar 

  • Noll W (1967) Uniform simple bodies with inhomogeneities. Arch Rat Mech Anal 27:1–32

    Article  MathSciNet  Google Scholar 

  • Noll W (1972) A new mathematical theory of simple materials. Arch Rat Mech Anal 48:1–50

    Article  MathSciNet  MATH  Google Scholar 

  • Noll W (2002) The genesis of the “Non-linear Field Theories of Mechanics” (text of a lecture given at the Meeting of the Society for Natural Philosophy in the memory of Clifford Truesdell, Pisa, Italy, Nov. 2000)

    Google Scholar 

  • Öttinger HC (2005) Beyond equilibrium thermodynamics. Wiley-Interscience, Hoboken

    Book  Google Scholar 

  • Rivlin RS (1984) Forty years of non-linear continuum mechanics, Conference at the Ninth International Congress of Rheology. In the Proceedings: pp 2783–2811

    Google Scholar 

  • Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Smith DR (1993) An introduction to continuum mechanics—after Truesdell and Noll. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Truesdell CA (1952) The mechanical foundations of elasticity and fluid dynamics. J Rat Mech Anal 1(1):125–300

    MathSciNet  MATH  Google Scholar 

  • Truesdell CA (1955) Hypo-elasticity. J Rat Mech Anal 4:83–133

    MathSciNet  MATH  Google Scholar 

  • Truesdell CA (1957) Sulle basi della termomeccanica. Rend Acad Lincei (8), 22, 33–88, 158–166

    Google Scholar 

  • Truesdell CA (1961) General and exact theory of waves in finite elastic strain. Arch Rat Mech Anal 8:263–296

    Article  MathSciNet  MATH  Google Scholar 

  • Truesdell CA (1969) Rational thermodynamics. McGraw Hill, New York

    Google Scholar 

  • Truesdell CA (1984a) An idiot’s fugitive essays on Science. Springer, New York

    Book  MATH  Google Scholar 

  • Truesdell CA (1984b) Rational thermodynamics, Second enlarged edition. Springer, New York

    Google Scholar 

  • Truesdell CA, Noll W (1965) The non-linear field theories of mechanics, Handbuch der Physik, Bd.III/3. In: Flügge S (ed) Springer, Berlin [Referred to for short as NFTM]

    Google Scholar 

  • Truesdell CA, Toupin RA (1960) The classical field theories, Handbuch der Physik, Bd. III/1. Flügge S (ed), Springer, Berlin [Referred to for short as CFT]

    Google Scholar 

  • Zaremba S (1903) A series of five papers in French. Bull Int Acad Sci Cracovie 85–93, 380–403, 403–423, 594–614, 614–621

    Google Scholar 

Download references

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  1. Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, Place Jussieu 4, Paris Cedex 05, 75252, France

    Gérard A. Maugin

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Maugin, G.A. (2013). Axiomatization and Thermo-Mechanics. In: Continuum Mechanics Through the Twentieth Century. Solid Mechanics and Its Applications, vol 196. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6353-1_5

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  • DOI: https://doi.org/10.1007/978-94-007-6353-1_5

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