Abstract
Relativity theory as understood by Einstein is a true Twentieth century development. After the introduction of the four-dimensional version of special relativity by Minkowski and that of energy-moment tensor, to which must be added the fact that general relativity is per se a continuum theory, there was need for a true relativistic theory of the continuum. The present chapter reports in a critical manner the progress made in this theory in two distinct periods, one extending before World War II, and the second in the rough time interval 1950–1980, when solutions were finally proposed in an inclusive way. The first period dealt with attempts at discussing the ad hoc introduction of classical concepts in this new landscape. This included the notion of perfect fluids and a debated discussion of the possible generalization of the notion of rigid-body motion—without which the notion of elasticity could not be introduced. A breakthrough is represented by Eckart’s introduction of a systematic covariant space-and-time resolution of four-dimensional objects and of early elements of continuum thermodynamics. This, combined with the natural influence of the then new trends in classical continuum mechanics (rationalization à la Truesdell), then led to a modern, more axiomatic, formulation that allowed a rational construct of relativistic elasticity, and its generalization to more complex thermomechanical schemes (including generalized continua) and electromagnetic deformable bodies, a development in which the author has been more than a passive witness.
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References
Abraham M (1909) Zur elektrodynamik bewegter Körper. Rend Circ Mat Palermo 28:1–28
Abraham M (1910) Sull’eletrodinamica di Minkowski. Rend Circ Mat Palermo 30:33–46
Bennoun JF (1964) Sur les représentations hydrodynamique et thermodynamique des milieux élastiques en relativité générale. C R Acad Sci Paris 259A:3705–3708
Born M (1911) Elastizitätstheorie und relativitätstheorie. Phys Zeot 12:569–575
Bressan A (1963) Onde ordinarie di discontinuità nei mezzi elastici con deformazioni finite in relatività generale. Riv Mat Univ Parma(2), 4:23–40
Bressan A (1978) Relativistic theories of materials, of springer tracts in natural philosophy, vol 29. Springer, New York-Berlin
Carter B (1973) Speed of sound in a high-pressure general relativistic solid. Phys Rev D7:1590–1593
Carter B, Quintana H (1972) Foundations of general relativistic high- pressure elasticity theory. Proc Roy Soc Lond A331:57–83
Cattaneo C (1962) Formulation relative des lois physiques en relativité générale, Multigraphed Notes of Lectures at Collège de France, Paris, Year 1961–1962
Cattaneo C (1980) Teori macroscopia dei continui relativistici. Pittagora Editrice, Rome
Cattaneo C, Gerardi A (1975) Un problema di equilibrio elastico in relatività generale. Rendiconti Mat (6), 8:187–200
De Donder T Dupont Y (1932–1933), Théorie relativiste de l’élasticité et de l’électromagnétostriction. Bull Sci, Acad Belg (5), 18, 680–691, 762–790, 899–010; 19, 370–378
De Groot SR, Support LG (1972) Foundations of electrodynamics. North- Holland, Amsterdam
Eckart CH (1940) The thermodynamics of irreversible processes III: Relativistic theory of the simple fluid. Phys Rev 58:919–924
Ehlers J (1961) Contributions to the relativistic mechanics of continuous media. Abh Math Akad W Mainz 11:793–837
Einstein A (1905) Zür elektrodynamik bewegter Körper. Ann der Phys. 17:891–921 [English translation in: Lorentz H.A., Einstein A., Minkowski H. and Weyl H. (1923). The principle of relativity: A collection of original memoirs, Methuen, London; reprinted by Dover, New York, 1952]
Einstein A (1916) Die Grundlage der allgemeine Relativitätstheorie. Ann der Phys 49:769–822 [English translation in: Lorentz H.A., Einstein A., Minkowski H. and Weyl H. (1923). The principle of relativity: A collection of original memoirs, Methuen, London; reprinted by Dover, New York, 1952]
Einstein A, Laub J (1908) Ueber die elektromagnetischen Felde auf ruhende Körper ausgeübten ponderomotorischen Kräfte. Ann der Phys 26(541–550):3
Eringen AC, Maugin GA (1990) Electrodynamics of continua, vol 2. Springer, New York
Glass EN, Winicour J (1972) Elastic general relativistic systems. J Math Phys 13:1934–1940
Glass EN, Winicour J (1973) A geometrical generalization of Hooke’s law. J Math Phys 14:1285–1290
Grot RA (1968) Relativistic theory of the propagation of wave fronts in nonlinear elastic materials. Int. J. Engng. Sci 6:295–307
Grot RA, Eringen AC (1966a) Relativistic continuum mechanics -I- mechanics and thermodynamics. Int J Eng Sci 4:611–638
Grot RA, Eringen AC (1966b) Relativistic continuum mechanics -II- Electromagnetic interactions with matter. Int J Eng Sci 4:639–670
Halbwachs F (1960) Théorie relativiste des fluides à spin. Gauthier-Villars, Paris
Hehl FW (1969) Spin und Torsion. Universität Clausthal, Germany, Habilitationschrift
Hehl FW, Von der Heyde P, Kerlick GD, Nester JL (1976) General relativity with spin and torsion: Foundations and prospects. Rev Mod Phys 48(3):393–416
Herglotz G (1911) Ueber die mechanik des deformerbaren Korpers vom Standpunkte der relativitätstheorie. Ann. der Phys 36:493–533
Hernandez WC (1970) Elasticity in general relativity. Phys Rev D1:1013–1018
Herrmann H, Muschik W, Ruckner G, von Borzeszkowski HH (2004) Spin axioms in different geometries of relativistic continuum mechanics. Found Phys 34(6):1005–1021
Ignatowsky WV (1911) Sur Elastizitätstheorie vom Standpunke der relativitätstheorie. Phys Zeit 12:1013–1018
Kafadar CB, Eringen AC (1972) Polar media – The relativistic theory. Int J Eng Sci 27:307–329
Lamla E (1912) Ueber die Hydrodynamik des Relativitätsprinzip. Ann der Phys 37:772–796
Lianis G (1973a) General form of constitutive equations in relativistic physics. Nuovo Cimento 14B(1):57–103
Lianis G (1973b) Formulation and application of relativistic constitutive equations for deformable electromagnetic materials. Nuovo Cimento 16B(1):1–43
Lianis G (2000) Relativistic approach to continuum physics. J Mech Behav Materials 11(1–3):105–119
Lichnerowicz A (1955) Théories relativistes de la gravitation et de l’électromagnétisme. Masson, Paris
Lichnerowicz A (1967) Relativistic hydrodynamics and magneto- hydrodynamics. Benjamin, New York
Lichnerowicz A (1971) Onde de choc, ondes infinitésimales et rayons en hydrodynamique et magnétohydrodynamique, in: Relativistic fluid dynamics, Ed. C. Cattaneo, pp 87–203, Cremonese, Rome
Lichnerowicz A (1976) Shock waves in relativistic magnetohydrodynamics under general assumptions. J Math Phys 17:2135–2142
Maugin GA (1971) Magnetized deformable media in general relativity. Ann Inst Henri Poincaré A15:275–302
Maugin GA (1972a) Relativistic theory of magnetoelastic interactions. J Phys A: Gen Phys A5:786–802
Maugin GA (1972b) An action principle in general relativistic magneto- hydrodynamics. Ann Inst Henri Poincaré A16:133–169
Maugin GA (1973) Harmonic oscillations of elastic continua and detection of gravitational waves. Gen Relativ Gravit Jl 4:241–272
Maugin GA (1974) Sur les fluides relativistes à spin. Ann Inst Henri Poincaré A20:41–68
Maugin GA (1975) Sur la formulation des lois de comportement en mécanique relativiste des milieux continus, Multigraphed main document of Doct. d’Etat ès Sciences Mathématiques, Université de Paris-6, Paris
Maugin GA (1976a) Conditions de compatibilité pour une hypersurface singulière en mécanique relativiste des milieux continus. Ann Int Henri Poincaré A24:213–241
Maugin GA (1976b) Un principe variationnel pour le schéma fluide relativiste à spin, Ann. di Mat. Pura ed Applicata (Italia) (4)110:247–277
Maugin GA (1977) Infinitesimal discontinuities in initially stressed relativistic elastic solids. Commun Math Phys 53:233–256
Maugin GA (1978a) On the Covariant Equations of the Relativistic Electrodynamics of Continua-I- General equations. J Math Phys 19:1198–1205
Maugin GA (1978b) On the covariant equations of the relativistic electrodynamics of continua-III- elastic solids. J Math Phys 19:1212–1219
Maugin GA (1978c) On the covariant equations of the relativistic electrodynamics of continua-IV-media with spin. J Math Phys (USA) 19:1220–1226
Maugin GA (1978d) Exact relativistic theory of wave propagation in prestressed nonlinear elastic solids. Ann. Inst. Henri Poincaré A28:155–185
Maugin GA (1978e) Relation between wave speeds in the crust of dense magnetic stars. Proc. Roy. Soc. Lond. A354:537–552
Maugin GA (1979) Nonlinear waves in relativistic continuum mechanics [on the occasion of Einstein’s Centenary]. Helv Phys Acta 52:149–170
Maugin GA (1981) Ray theory and shock formation in relativistic elastic solids. Phil. Trans. Roy. Soc. Lond 302:189–215
Maugin GA, Eringen AC (1972) Relativistic Continua with Directors. J Math Phys. 13:1788–1798
Maugin GA, Trimarco C (1992) Pseudo-momentum and Material Forces in Nonlinear Elasticity: Variational Formulations and Application to Brittle Fracture. Acta Mech 94:1–28
Minkowski H (1908) Die grundgleichungen für die elektromagnetischen vorgänge in bevegten körpern. Gottinger Nachrichen 53–111 [Also: “Raum und Zeit”. Address delivered at the 80th Assembly of German Natural scientists and Physicians, Köln, September 21, 1908]
Misner CW, Thorne KS, Wheeler JA (1970) Gravitation. Freeman, San Francisco
Murnaghan FD (1937) Finite deformation of an elastic solid. Amer J Math 59:235–260
Noether E (1911) Zur Kinematik des Starren Körpers in der Relativitätstheorie. Ann der Phys. 31:919–944
O’Brien S , Synge J.L. (1953) Jump conditions at discontinuities in general relativity. Commun Dublin Inst Adv Studies, Series A, No.9
Oldroyd JG (1970) Equations of state of continuous matter in general relativity. Proc Roy Soc London A316:1–28
Papapetrou A (1972) Vibrations élastiques excitées par une onde gravitationnelle. Ann H Poincaré A16:63–78
Rayner CB (1963) Elasticity and general relativity. Proc Roy Soc. London A272:44–53
Schöpf HG (1964) Allgemeinrelativistische prinzipien der kontinuumsmechanik. Ann Phys (Leipzig) 12:377–395
Souriau J.-M. (1958), in: Alger Mathématiques, no.2, 1958 [and in: Géométrie et relativité, Hermann, Paris 1964, Section 36]
Synge JL (1959) A theory of elasticity in general relativity. Math Zeit 72:82–87
Taub AH (1948) Relativistic Rankine-Hugoniot equations. Phys Rev 74:328–334
Taub AH (1957) Singular hypersurfaces in general relativity, Illinois J. Math 1:370–388
Weyssenhhoff J, Raabe A (1947) Relativistic theory of spin-fluids and spin-particles. Acta Phys Pol 9:7–53
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Maugin, G.A. (2013). Relativistic Continuum Mechanics: A 20th Century Adventure. 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_15
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