Abstract
Present work aims to develop a geometrization of thermodynamics of continua within the classical approximation where the velocity of the light is considered infinite, but nevertheless in the spirit of relativity. The connection on the manifold represents the gravitation. The temperature has the status of a vector and its gradient, called friction, merges the temperature gradient and strain velocity. We claim that the energy-momentum-mass tensor is covariant divergence free. It is a geometrized version of the first principle. The modeling of the dissipative continua is based on an additive decomposition of the momentum tensor into reversible and irreversible parts. The second principle is based on a tensorial expression of the local production of entropy which provides its Galilean invariance. On this ground, we propose a relativistic version of the second principle compatible with Poincare’s group.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bargmann V (1954) On unitary representation of continuous groups. Ann Math 58:1–46
Cartan É (1923) Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Ann de l’École Normale Supérieure 40:325–412
de Saxcé G, Vallée C (2003) Affine tensors in shell theory. J Theor Appl Mech 41:593–621
de Saxcé G, Vallée C (2010) Construction of a central extension of a Lie group from its class of symplectic cohomology. J Geometry Phys 60:165–174
de Saxcé G, Vallée C (2011) Affine tensors in mechanics of freely falling particles and rigid bodies. Math Mech Solid J 17:413–430
de Saxcé G, Vallée C (2012) Bargmann group, momentum tensor and Galilean invariance of Clausius-Duhem inequality. Int J Eng Sci 50:216–232
de Saxcé G, Vallée C (2016) Galilean mechanics and thermodynamics of continua. Wiley-ISTE
Iglesias P (1981) Essai de thermodynamique rationnelle des milieux continus. Ann de l’Institut Henri Poincaré 34:1–24
Künzle HP (1972) Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Ann de l’Institut Henri Poincaré 17:337–362
Misner C, Thorne K, Wheeler J (1973) Gravitation. Freeman
Noll W (1973) Lectures on the foundations of continuum mechanics and thermodynamics. Arch Ration Mech Anal 52:62–92
Souriau JM (1964) Géométrie et relativité. Collection Enseignement des sciences. Hermann, Paris
Souriau JM (1970) Structure des systèmes dynamique. Dunod, Paris
Souriau JM (1976/1977). Thermodynamique et géométrie. Lecture Notes in Mathematics 676:369–397
Souriau JM (1978) Thermodynamique relativiste des fluides. Rendiconti del Seminario Matematico Università Politecnico di Torino 35:21–34
Souriau JM (1997) Structure of dynamical systems. A symplectic view of physics. Birkhäuser, New York
Souriau J (1997) Milieux continus de dimension 1, 2 ou 3: statique et dynamique. In: Proceeding of the 13\(^{eme}\) Congrès Français de Mécanique, pp 41–53, Poitiers-Futuroscope
Toupin R (1957/1958) World invariant kinematics. Arch Ration Mech Anal 1:181–211
Truesdell C (1952) The mechanical foundation of elasticity and fluid dynamics. J Ration Mech Anal 1:125–171
Truesdell C, Toupin R (1960) The classical field theories. In: Encyclopedia of Physics, S. Flügge, Vol II/1, Principles of classical mechanics and field theory. Springer, Berlin
Vallée C (1978) Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste. Ph.D. thesis, University of Poitiers
Vallée C (1981) Relativistic thermodynamics of continua. Int J Eng Sci 19:589–601
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
de Saxcé, G. (2017). 5-Dimensional Thermodynamics of Dissipative Continua. In: Frémond, M., Maceri, F., Vairo, G. (eds) Models, Simulation, and Experimental Issues in Structural Mechanics. Springer Series in Solid and Structural Mechanics, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-48884-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-48884-4_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-48883-7
Online ISBN: 978-3-319-48884-4
eBook Packages: EngineeringEngineering (R0)