From the Cosserats mechanics backgrounds to modern field theory

• Autorzy: Badur Janusz , Dudda Waldemar

Identyfikatory

DOI

10.31648/ts.10315

• Języki publikacji: EN

Abstrakty

EN

In the paper, yet weekly known, Cosserats’ original four concepts as follow: the four-time unification of rigid body dynamics, statics of flexible rods, statics of elastic surfaces and 3D deformable body dynamics; the intrinsic formulation based on the local, von Helmholtz symmetry group of monodromy; the invariance under the Euclidean group. The concept of a set of low-dimensional branes immersed into Euclidean space are revalorized and explained in terms of the modern gauge field theory and the extended strings theory. Additionally, some useful mathematical tools that connect the continuum mechanics and the classical field theory (for instance, the convective coordinates, von Mises’ “Motorrechnung”, the Grassmann extensions, Euclidean invariance, etc.) are involved in the historical explanation that how the ideas were developing themself.

Słowa kluczowe

EN

finite; Cosserats continuum; Darboux curvature vector; moving frame; Frenet trihedron; intrinsic coordinates; four-time operators; gauge symmetry flux conservation; gauge potentials; Mauer-Cartan structure equations; von Helmholtz symmetry group; Euclidean group of transformations; weak principle of momentum and angular momentum conservation; Euler laws of dynamics; Cauchy first laws; Cauchy second laws

• Wydawca: Wydawnictwo Uniwersytetu Warmińsko-Mazurskiego w Olsztynie
• Czasopismo: Technical Sciences / University of Warmia and Mazury in Olsztyn
• Rocznik: 2024
• Tom: Vol. 27
• Strony: 211--264
• Opis fizyczny: Bibliogr. 223 poz., rys.

Twórcy

autor

Badur Janusz

• Institute of Fluid Flow Machinery, Polish Academy of Sciences in Gdańsk

• https://orcid.org/0000-0001-7725-6926

autor

Dudda Waldemar

• Department of Mechanics and Basics of Machine Design Faculty of Technical Sciences University of Warmia and Mazury in Olsztyn

• https://orcid.org/0000-0003-4083-9479

Bibliografia

• ALBLAS J.B. 1969. Continuum mechanics of media with internal structure. In: Teoria dei continui polari. Istituto Nazionale di Alta Matematica (INDAM), Rome, Italy, April 2-5, 1968. Symposia Mathematica, 1: 229-251. Academic Press Inc., London.
• ANDRADE J. 1898. Leçons de mécanique physiquek. Société d’éditions scientifiques, Paris.
• ARIANO R. 1924. Deformacioni finite di sistemi continui. Annali di Matématica Pura ed Applicata (ser 4o), 2: 216-261.
• ARMERO F., ROMERO I. 2003. Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics of nonlinear Cosserat rods. Computational Mechanics, 31: 3-26.
• ARON H. 1874. Das Geleichgewicht und die Bewegung einer unendlich dünnen beliebig gekrümmten elestischen Schale. Journal fur die Reine und Angewandte Mathematik, 78: 136-174.
• ATLURI S.N., CAZZANI A. 1995. Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2: 49-138. https://doi.org/10.1007/BF02736189
• BADUR J. 1989. A Yang-Mills type of equation for the compatibility conditions. International Journal of Engineering Science, 27: 1439-1442.
• BADUR J. 1990. Quasi-Abelian gauge theory of axisymmetric deformation of shells of revolution. International Journal of Engineering Science, 28: 563-572.
• BADUR J. 1991. Extension of many-time Hamiltonian formalism to the theory of deformable Cosserat bodies. International Journal of Engineering Science, 29: 69-77.
• BADUR J. 1993. Pure gauge theory of the Cosserat surface. International Journal of Engineering, Science, 31: 41-59.
• BADUR J. 1993. Space-time compatibility conditions for strains and velocities. Rendiconti di Matematica, 13: 1-29.
• BADUR J. 2009. Principles of Cosserat p-brane extended mechanics. In: COSSERAT+100, International Conference on legacy of Théorie des corps déformables by E.F. Cosserat. Ed. C. Capriz. M. Brocato. Paris.
• BADUR J. 2021. Eternal symmetries of Noether. IMP Press, Gdańsk.
• BADUR J. 2022. Eternal relativity of Whitehead. IMP Press, Gdańsk.
• BADUR J., CHRÓŚCIELEWSKI J. 1983. Powłokowy element skończony oparty o kinematykę Cosseratów. Konferencja „Metody numeryczne w mechanice”. Białystok.
• BADUR J., CHRÓŚCIELEWSKI J. 2015. On a four-time unification of the Cosserats continua by the intrinsic approach. 3rd Polish Congress of Mechanics, Gdańsk.
• BADUR J., OCHRYMIUK T., KOWALCZYK T., DUDDA W., ZIÓŁKOWSKI P. 2022. From fluid mechanics backgrounds to modern field theory. Acta Mechanica, 223: 3453-3465.
• BADUR J., PIETRASZKIEWICZ W. 1986. On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained microrotations. In: Finite Rotations in Structural Mechanics. Ed. W. Pietraszkiewicz. Springer-Verlag, Wien, p. 19-32.
• BADUR J., POVSTIENKO Y. 1998. Cosserat boundle versus the motor calculus. Archives of Mechanics, 50: 367-376.
• BADUR J., STUMPF H. 1989. On the influence of E. and F. Cosserat on modern continuum mechanics and the field theory. Mitteilungen aus dem Institut für Mechanik, no 72, Ruhr-Universität, Bochum.
• BADUR J., ZIÓŁKOWSKI P., ZIÓŁKOWSKI P.J. 2015. On angular velocity slip in nonoflows. Microfluidics and Nanofluidics, 19: 191-199.
• BASAR Y. 1987. A consistent theory of geometrically non-linear shells with an independent rotation vector. International Journal of Solids and Structures, 23(10): 1401-1415.
• BASAR Y., WEICHERT D. 2000. Nonlinear continuum mechanics of solids. Springer Verlag, Berlin.
• BASSET A.B. 1894. On the deformation of thin elastic plates and shellsk. American Journal of Mathematics, 16: 255-290.
• BASSET A.B. 1895. On the deformation of thin elastic wiresk. American Journal of Mathematics, 17: 281-317.
• BELTRAMI E. 1871. Sur principi fondamentali della idrodinamica. Memorie Reale Accademia Scienze Istituto Bologna, series 3, t. 1, p. 431-476.
• BELTRAMI E. 1872. Sur principi fondamentali della idrodinamica. Memorie Reale Accademia Scienze Istituto Bologna, series 3, t. 2, p. 381-437.
• BELTRAMI E. 1873. Sur principi fondamentali della idrodinamica. Memorie Reale Accademia Scienze Istituto Bologna, series 3, t. 3, p. 349-407.
• BELTRAMI E. 1874. Sur principi fondamentali della idrodinamica. Memorie Reale Accademia Scienze Istituto Bologna, series 3, t. 5, p. 443-484.
• BELTRAMI E. 1911. Sulle equazioni generali dell’elasticità. Opere Matematiche, III: 383.
• BESDO D. 1974. Ein Beitrag zur nichtlinearen theorie des Cosserat-Kontinuums. Acta Mechanica, 20: 105-131.
• BESSAN E. 1963. Sui sistemi continui nel case asimetrico. Annali di Matematica Pura ed Applicata, 62: 169-222.
• BORST R. DE. 1991. Simulation of strain localization: a reappraisal of the Cosserat continuum. Engineering with Computers, 8: 317-332.
• BROCATO M., CAPRIZ G. 2001. Gyrocontinua. International Journal of Solids and Structures, 38: 1089-1103.
• CAPRIZ G. 2008. On ephemeral continua. Physical Mesomechanics, 11: 285-298.
• CAPRIZ G. 2010. Hypocontinua. In: Continuous Media with Microstructure. Ed. B. Albers. Springer, Berlin.
• CAPRIZ G., PODIO-GUIDUGLI P. 1977. Formal structure and classification of theories of oriented media. Annali di Matematica Pura ed Applicata, Ser. IV, 115: 17-39.
• CAPRIZ G., VIRGA E. 1994. On singular surfaces in the dynamics of continua with microstructure. Quarterly of Applied Mathematics, 52: 509-517.
• CARNOT L. 1793. Les Principes fondamentaux de l’équilibre et du movement. De l’imprimerie de Crapelet chez Deterville, Paris.
• CARTAN E. 1923. Sur les variétés à connexion affine et la theorie de la relativité généralisée (premièrepartie). Annales Scientifiques de l’École Normale Supérieure, Serie 3, 40: 325-412.
• CARTAN E. 1924. Sur les variétés à connexion affine et la theorie de la relativité généralisée (première partie) (Suite). Annales Scientifiques de l’École Normale Supérieure, Serie 3, 41: 1-25.
• CARTAN E. 1925. Sur les variétés à connexion affine et la theorie de la relativité généralisée (deuxième partie). Annales Scientifiques de l’École Normale Supérieure, 42: 17-88.
• CARTAN E. 1935. La méthóde du repére mobile, la théorie des groupes continus et les espaces generalisés. Hermann, Paris.
• CAUCHY A.-L. 1823. Recherchessur l’équilibre et le mouvement des corpes solides ou fluides, élastiques ou non élastiques. Bulletin de la Societe Philomatique de Paris, p. 9-13.
• CESARO E. 1926. Vorlesugen über Natüraliche Geometre. 2nd ed. Trans. G. Kowalewski. Verlag und Druck von B.G. Teubner, Leipzig.
• CHAICHIAN M., NELIPA N.F. 1984. Introduction to the Gauge Field Theories. Springer, Berlin.
• CHEN W.Z. 1944. The intrinsic theory of thin shells and plates. I. General theory. Quarterly of Applied Mathematics, 1: 297-327.
• CHRÓŚCIELEWSKI J., MAKOWSKI J., PIETRASZKIEWICZ W. 2004. Statyka i dynamika powłok wielopłatowych. Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk, Warszawa.
• CHRÓŚCIELEWSKI J., MAKOWSKI J., STUMPF H. 1992. Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. International Journal for Numerical Methods in Engineering, 35: 63-94.
• CHRÓŚCIELEWSKI J., PIETRASZKIEWICZ W., WITKOWSKI W. 2010. On shear correction factors in the nonlinear theory of elastic shells. International Journal of Solids and Structures, 47: 3537-3445.
• CLAYTON J.D. 2022. Finsler differential geometry in continuum mechanics: Fundamental concepts, history, and renewed application to ferromagnetic solids. Mathematics and Mechanics of Solids, 27(5). https://doi.org/10.1177/10812865211049468
• COSSERAT E., COSSERAT F. 1909. Note sur la théorie de l’action euclidienne. In: Traité de mécanique rationelle. Ed. P. Appell. T. III, p. 557-629. Gauthier-Villars, Paris.
• COSSERAT E., COSSERAT F. 1909. Théorie des corps déformables. Hermann, Paris.
• COSSERAT E., COSSERAT F. 1896. Sur la theorie de l’elasticite. Premier mémoire. Annales de la Faculté des sciences de Toulouse, Mathématiques, 10(3-4): 1-116.
• COSSERAT E., COSSERAT F. 1907. Sur la mécanique générale. Comptes Rendus, 145: 1139-1142.
• CRAIG T. 1898. Displacement depending on one, two and three parameters in a space of four dimensions. American Journal of Mathematics, 20: 135-156.
• CRISFIELD M.A., JELENIĆ G. 1998, Objectivity of strain measures in geometrically exact 3D beam theory and its finite element implementation. Proceedings of the Royal Society of London, 455: 1125-1147.
• DANIELSON D.A., HODGES D.H. 1984. Nonlinear beam kinematics by decomposition of the rotation tensor. ASME Journal of Applied Mechanics, 54: 258-262.
• DARBOUX G. 1890. Leçons sur la théorie générale des surfaces. Gauthier Villars Et Fils, Paris.
• DARBOUX G. 1900. Sur les déformations finites et sur les systèmes triples de surfaces orthogonales. Proceedings of the London Mathematical Society, 32: 377-383.
• DE LEÓN M., EPSTEIN V., JIMÉNEZ V. 2021. Material geometry: groupoids in continuum mechanics. Pergamon, New York.
• DELENS P.-C. 1927. Méthods et problèmes des géométries différentielles, Euclidienne et conforme. Gauther-Villars, Paris.
• DELL’ISOLA F., DELLA CROTE A., GIROGIO I. 2015. Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Tupin and some future research perspectives. Mathematics and Mechanics of Solids, 20(8): 887-928.
• DILL E.H. 1992. Kirchhoff’s theory of rods. Archive for History of Exact Sciences, 44(1): 1-23.
• DUHEM P. 1893. Le potentel thermodynamique et la pression hydrostatique. Annales Scientifiques de l’École Normale Supérieure, Ser. 3, 10: 187-230.
• DUHEM P. 1901a. Reserches sur l’hydrodynamique. Annales de la Faculté des sciences de Toulouse. Mathématiques, 3(3): 315-377.
• DUHEM P. 1901b. Reserches sur l’hydrodynamique. Annales de la Faculté des sciences de Toulouse, 2e série, 3(4): 379-431.
• DUHEM P. 1902. Reserches sur l’hydrodynamique. Annales de la Faculté des sciences de Toulouse, 2e série, 4: 101-169.
• DUHEM P. 1903. Reserches sur l’hydrodynamique. Annales de la Faculté des sciences de Toulouse, 2e série, 5(2): 5-61, 197-255, 353-404.
• DUHEM P. 1904. Recherches sur l’elasticite. Annales scientifiques de l’École Normale Supérieure, (3)2: 99-139, 375-414.
• DUHEM P. 1905. Recherches sur l’elasticite. Annales scientifiques de l’École Normale Supérieure, 2: 143-217.
• DUHEM P. 1906. Recherches sur l’elasticite. Annales scientifiques de l’École Normale Supérieure, 23: 169-223.
• EDELEN D.G.E., LAGOUDAS D.C. 1988. Gauge Theory of Defects in Solids. North-Holland, Amsterdam.
• EHLERS W., RAMM E., DIEBELS S., D’ADDETTA G.A. 2003. From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. International Journal of Solids and Structures, 40: 6681-6702.
• EL NASCHIE M.S. 2016. Cosserat-Cartan and de Sitter-Witten spacetime setting for dark energy. Quantum Matter, 5: 1-4. https://doi.org/10.1166/qm.2016.1247
• EPSTEIN M., DE LEON M. 1998. Geometrical theory of uniform Cosserat media. Journal of Geometry and Physics, 26(1-2): 127-170.
• ERICKSEN J.L., TRUESDELL C. 1958. Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics and Analysis, 1: 295-323.
• ERINGEN A.C., SUHUBI E.S. 1964. Nonlinear theory of simple microelastic solids. Part I. International Journal of Engineering Science, 2: 189-203.
• ERINGEN A.C., SUHUBI E.S. 1964. Nonlinear theory of simple microelastic solids. Part II. International Journal of Engineering Science, 2: 389-404.
• EULER L. 1752. Découverte d’un nouveau principle de mécanique. Mémoires de l’académie des sciences de Berlin, 6: 185-217.
• FERRARESE G. 1959. Sulla velocita angolare nei moti rigidi e la rotazione locale nelle deformazioni finite. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Serie 8, 36(5): 629-638,
• FERRARESE G. 1971. Sulla compatibilita dei continui alla Cosserat. Annali di Matematica Pura ed Applicata, 108: 109-124.
• FERRARESE G. 1972. Intrinsic formulation of Cosserat continua dynamics. In: Trends in Applications of Pure Mathematics to Mechanics, t. II. Ed. H. Zorski. Pitman, London.
• FERRARESE G. 1976. Sulla formulazione intrinseca della dinamica dei continui alla Cosserat. Annali di Matematica Pura ed Applicata, 108: 109-124.
• FINZI B. 1932. Equazioni intrinseche della meccanica dei sistemi continui perfettamente od imperfettamente flessibili. Annali di Matematica Pura ed Applicata, 11: 215-245.
• FOREST S., CAILLETAUD G., SIEVERT R. 1997. A Cosserat theory for elastoviscoplastic single crystals at finite deformation. Archives of Mechanics, 49(4): 705-736.
• FOREST S., SIEVERT R. 2003. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 160: 71-111.
• FOREST S., SIEVERT R. 2006. Nonlinear microstrain theories. International Journal of Solids and Structures, 43: 7224-7245.
• FORTUNE D., VALLEE C. 2001. Bianchi identities in case of large deformations. International Journal of Engineering Science, 39: 113-123.
• FRANKE J.N. 1889. Mechanika teoretyczna. T. X. Biblioteka Matematyczno-Fizyczna, Kasa J. Mianowskiego, Warszawa.
• FRENET F. 1847. Sur les courbes a double courbure. Thése, Toulouse.
• GOSIEWSKI W. 1877. O zasadach teorii bezwzględnej zjawisk materialnych. Pamiętnik Towarzystwa Nauk Ścisłych w Paryżu, 10: 1-6.
• GREEN A.E., LAWS N. 1966. A general theory of rods. Proceedings of the Royal Society of London, 293: 145-155.
• GREEN A.E., NAGHDI P.M., WAINWRIGHT W.L. 1965. A general theory of Cosserat surfaces. Archive for Rational Mechanics and Analysis, 20: 287-308.
• GRIOLI G. 1960. Elasticità asimmetrica. Annali di Matematica Pura ed Applicata, 50: 389-417.
• GRIOLI G. 1968. Questioni di compatibilità per continui di Cossarat. Sumposia Mathemetica, I: 271-287.
• GRUTTMAN F., SAUER R., WAGNER W. 1998. A geometrically nonlinear eccentric 3D-beam element with arbitrary cross sections. Computer Methods in Applied Mechanics and Engineering, 160: 383-400.
• GRUTTMANN F., STEIN E., WRIGGERS P. 1989. Theory and numerics of thin elastic shells with finite rotations. Ingenieur-Archiv, 59: 54-67.
• GÜNTHER W. 1958. Zur Statik und Kinematik des Cosseratschen Kontinuum. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 10: 195-213.
• GÜNTHER W. 1961. Analoge Systeme von Schalengeleichungen. Ingenieur-Archiv, 30: 160-186.
• HAY G.E. 1942. The finite displacement of thin rods. Transactions of the American Mathematical Society, 51: 65-102.
• HEHL F., KRÖNER E. 1965. Über den Spin in der allgemeinen Relativitätstheorie Eine notwendige Erweiterung der Einsteinschen Feldgleichungen. Zeitschrift für Physik, 187: 478-489.
• HEHL F., KRÖNER E. 1965. Zum materialgesetz eines elastischen Medius mit Momentenspannungen. Zeitschrift für Naturforschung, 20: 336-350.
• HEHL F.W. 1973. Spin and torsion in general relativity. I. Foundations. General Relativity and Gravitation, 4: 333-349.
• HEHL F.W. 2017. Gauge theory of gravity and spacetime. In: Towards a theory of spacetime theories. Eds. D. Lehmkuhl, G. Schiemann, E. Scholz. Springer, Berlin.
• HEHL F.W., OBUKHOV Y.N. 2007. Élie Cartan’s torsion in geometry and in field theory, an essay. General Relativity and Quantum Cosmology. arXiv preprint arXiv:0711.1535. https://doi.org/10.48550/arXiv.0711.1535
• HELLINGER E. 1914. Die allgemein ansätze der mechanik der kontinua. Enzyklopädie der Mathematischen Wissenschaften, virter Tailband – Mechanik, part 4, Artikel 30, p. 601-694. Eds. F. Klein, C.H. Müller. B.G. Teubner Verlag, Leipzig.
• HELMHOLTZ H. 1868. Über die Tataschen die Geometrie zugrunde liegen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 9: 193-221.
• HENCKY H. 1915. Űber den Spannungszustand kreisrundem platten. Zeitschrift für Angewandte Mathematik und Physik, 63: 311-317.
• HESS W. 1884. Ueber die Biegung und Drillung eines unendlich dünnen elastischen Stabes, dessen eines Ende von einem Kräftepaar angegriffen wied. Mathematische Annalen, 23: 181-212.
• HODGES D.H. 1990. A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. International Journal of Solids and Structures, 26: 1253-1273.
• HODGES D.H., ATILGAN A.R., DANIELSON D.A. 1993. A geometrically nonlinear theory of elastic plates. Journal of Applied Mechanics, 60: 109-116.
• IBRAHIMBEGOVIC A. 1994. Stress resultant geometrically nonlinear shell theory with drilling rotations. Part 1. A consistent formulation. Computer Methods in Applied Mechanics and Engineering, 118: 265-284.
• JAUMANN R.G. 1918. Physik der kontinuierlichen Medien. Denkschriften by Akademie der Wissenschaften in Wien, 95: 461-562.
• KADIĆ A., EDELEN D.G. 1983. A gauge theory of dislocations and disclinations. Lecture Notes in Physics, 174.
• KAFADAR C., ERINGEN A.C. 1971. Micropolar media. Part I. The classical theory. International Journal of Engineering Science, 9: 271-329.
• KESSEL S. 1970. Spannungsfelder einer Schraubenversetzung und einer Stufenversetzung im Cosseratschen Kontinuum. ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 50: 547-553.
• KIRCHHOFF G. 1850. Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik, 40: 51-88.
• KIRCHHOFF G. 1852. Über die Gleichungen des Gleichgewichts eines elastischen Körpers bei nicht unendlich kleinen Verschiebungen seiner Teile. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, 9: 762-773.
• KIRCHHOFF G. 1859. Űber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. Journal für die reine und angewandte Mathematik, 56: 285-313.
• KIRCHHOFF G. 1876. Vorlesungen über mathematische Physik: Mechanik. B.G. Teubner, Leipzig.
• KIRCHHOFF G. 1883. Vorlesungen über mathematische Physik: Mechanik. B.G. Teubner, Leipzig.
• KLINGER F. 1942. Die Statik und Kinematik des räumlich gekrümmten elastischen Stabes. Sitzungsberichte, Akademie der Wissenschaften in Wien, IIa, 151: 13-79.
• KLUGE G. 1969. Zur Dynamik der allgemeinen versetzungstheorie bei berücksichtigung von momentenspannungen. International Journal of Engineering Science, 7: 169-182.
• KOITER W.T. 1964. Couple-stresses in the theory of elasticity. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Amsterdam, 64: 17-44.
• KRAUSS F. 1929. Űber die Grundleichungen der Elastizitätstheorie scheach deformirter Schalen. Mathematische Annalen, 101: 61-92.
• KRÖNER E. 1960. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational Mechanics and Analysis, 4: 273-334.
• LACHNER D., LIPPMANN H., TOTH L.S. 1994. On Cosserat plasticity and plastic spin for isotropic materials. Archives of Mechanics, 46: 531-539.
• LAGRANGE J.L. 1762. Application de la méthode exposée précédente a la solution de différmes problémes de dynamique. Mélanges de Philosophie et de Mathématiques de la Société Royale de Turin, 2: 196-298.
• LAKES R. 1995. Experimental methods for study of Cosserat elastic solids and other generalized continua. In: Continuum models for materials with micro-structure. Ed. H. Mühlhaus. Ch. 1, p. 1-22. J. Wiley, New York..
• LAMÉ G. 1852. Leçons sur la Théorie Mathématique de l’Élasticité. Bachelier, Paris.
• LAME G., CLAPEYRON E. 1833. Mémoire sur l’équilibre intérieur des corps solides homogènes. Mémoires l’Acad. Royale des Sciences de l’Institut de France, 4: 465-562.
• LANGE L. 1885. Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes. Philosophische Studien, 2: 266-297.
• LAZAR M., HEHL F.W. 2010. Cartan’s spiral staircase in physics and, in particular, in the gauge theory of dislocations. Foundations of Physics, 40: 1298-1325.
• LE CORRE Y. 1965. La dissymétrie du tenseur des efforts et ses conséquences. Journal de Physique et Le Radium, 17: 934-939.
• LE K.C., STUMPF H. 1998. Strain measures, integrability condition and frame indifference in the theory of oriented media. International Journal of Solids and Structures, 35(9-10): 783-798.
• LECORNU M.L. 1880. Sur l’équilibre des surfaces flexibiles at inextensibiles. Journal de l’École polytechnique, 29: 1-100.
• LEHMANN TH. 1964. Formäderungen eines klassischen Kontinuum in vierdimensionaler Darstellung. International Congress for Applied Mechanics, Ed. H. Görtler, p. 376-382.
• LIPPMANN H. 1969. Eine Cosserat-Theorie des plastischen Fließens. Acta Mechanica, 8: 255-284.
• LOVE A.E.H. 1888. The small free vibrations and deformations of a thin elastic shell. Philosophical Transactions of the Royal Society, A, 179: 491-546.
• LUO A.C.J. 2010. On a nonlinear theory of thin rods. Communications in Nonlinear Science and Numerical Simulation, 15: 4181-4197.
• MAC CULLAGH J. 1839. An essay towards a dynamical theory of crystalline reflexion and refraction. Transactions of the Royal Irish Academy, 21: 17-50.
• MAKOWSKI J., STUMPF H. 1990. Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation. International Journal of Solids and Structures, 26(3): 353-368.
• MALCOLM D.J., GLOCKNER P.G. 1972. Nonlinear sandwich shell and Cosserat surface theory. Journal of the Engineering Mechanics Division, 98(EM5): 1183-1203.
• MAUGIN G.A. 1998. On the structure of the theory of polar elasticity. Philosophical Transactions of the Royal Society A, 356: 1367-1395.
• MAUGIN G.A. 2014. Continuum Mechanics Through the Eighteenth and Nineteenth Centuries 2014 Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914). Springer, Cham.
• MEISSNER K. 2013. Classical Field Theory. Wydawnictwo Naukowe PWN, Warszawa.
• MINDLIN R.D. 1964. Microstructure in linear elasticity. Archive for Rational Mechanics and Analysis, 16: 51-78.
• MINDLIN R.D., TRIESTEN H. 1962. Effects of complex-stress in linear elasticity. Archive for Rational Mechanics and Analysis, 11: 415-448.
• MISES R. VON. 1924. Motorrechnung, ein neues Hilfsmittel der mechanic. ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 4: 155-181.
• NADLER B., RUBIN M.B. 2003. A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. International Journal of Solids and Structures, 40: 4585-4614.
• NEFF P. 2006. A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations. International Journal of Engineering Science, 44: 574-594.
• NEFF P. 2019. Cosserat Theory by Prof. Dr. Patrizio Neff. Lehrstuhl für Nichtlineare Analysis, Universität Duisburg-Essen. Retrieved from http://www.uni-due.de/mathematik/ag_neff/cosserat
• NOWACKI W. 1966. Couple-stresses in the theory of thermoelasticity. Bulletin de L’Academie Polonaise des Sciences, 14: 97-106, 203-212.
• NOWACKI W. 1986. Theory of Asymmetric Elasticity. Pergamon-Press, Oxford.
• O’REILLY O.M., TURCOTTE J.S. 1997. Elastic rods with moderate rotation. Journal of Elasticity, 48: 193-216.
• OPOKA S., PIETRASZKIEWICZ W. 2004. Intrinsic equation of nonlinear deformation and stability of thin elastic shells. International Journal of Solids and Structures, 41: 3275-3292.
• PAPENFUSS C., FOREST S. 2006. Thermodynamical frameworks for higher grade material theories with internal variables or additional degrees of freedom. Journal of Non-Equilibrium Thermodynamics, 31: 319-353.
• PASTORI M. 1934. Equilibro di lastre a membrane elastiche. Rendiconti del Circolo Matematico di Palermo, 58: 1-48.
• PIETRASZKIEWICZ W. 1979. Finite rotation and lagrangean description in the non-linear theory of shells. Państwowe Wydawnictwo Naukowe, Warszawa.
• PIETRASZKIEWICZ W. 1988. Geometrically non-linear theories of thin elastic shells. Mitteilungen aus dem Institut für Mechanik, Ruhr-Universität, Bochum.
• PIETRASZKIEWICZ W., BADUR J. 1983a. Finite rotations in the description of continuum deformation. International Journal of Engineering Science, 21: 1097-1115.
• PIETRASZKIEWICZ W., BADUR J. 1983b. On non-classical forms of compatibility conditions in continuum mechanics. Trends in Applications of Pure Mathematics to Mechanics, IV: 197-227.
• PIOLA G. 1833. La meccanica dei’corpi naturalmente esteci trattata col calcolo delle variazioni. Opuscoli matematici e fisici di diversi autori, 1: 201-236. Giusti, Milano.
• PIOLA G. 1848. Intorno alle equazioni fondamentali del movimento di copri qualsivoglino, considerati second la naturale loro forma e costituzione. Memorie di matematica e fisica della Società italiana delle scienze, 24: 1-186.
• POINCARÉ H. 1892. Leçons sur la théorie de l’Élasticité. Georges Carré, Paris.
• POISSON S.-D. 1831. Mémoire sur la equations generales de la l’équilibre et du mouvement des corps solides élastiques et des fluides. Journal de l’École polytechnique, 13(20): 1-174.
• POISSON S.-D. 1833. Traité de Mécanique. Courcier, Paris.
• POMMARET J.-F. 1997. E. and F. Cosserat et le secret de la théorie mathématique de l’élasticité. Annales des ponts et chaussées, Nouvelle série, 82: 59-66.
• POMMARET J.-F. 2010. Parametrization of Cosserat Equations. Acta Mechanica, 215: 43-55.
• POMMARET J.-F. 2014. The mathematical foundations of gauge theory revisited. Journal of Modern Physics, 5: 157-170.
• POMMARET J.-F. 2016. Deformation Theory of Algebraic and Geometric Structures. LAP-publishing, Saarbrucken.
• POMMARIET J.-F. 1989. Gauge Theory and General Relativity. Reports on Mathematical Physics, 3(27): 313-344.
• RANKINE W.J.M. 1851. Laws of the elasticity of solids bodies. Cambridge and Dublin mathematical Journal, 6: 41-80, 178-181, 185-186.
• REECH F. 1852. Cours de mécanique, d’après la nature généralement flexible et élastique des corps. Carilian-Goeury, Paris.
• REISSNER E. 1950. On axisymmetrical deformation of thin shells of revolution. Proceedings of Symposia in Applied Mathematics, 3: 27-52.
• REISSNER E. 1972. On finite symmetrical strain in thin shells of revolution. Journal of Applied Mechanics, 39: 1137-1138.
• REISSNER E. 1974. Linear and nonlinear theory of shells. In: Thin Shell Structures, p. 29-44. Prentice-Hall, Englewood Cliffs.
• REISSNER E. 1981. On finite deformation of space-curved beams. Journal of Applied Mathematics and Physics, 32: 734-744.
• REISSNER E., WAN F.M. 1968. A note on Günther’s analysis of couple stress. In: Mechanics of Generalized Continua. Ed. E. Kröner. Springer-Verlag, Berlin.
• RUBIN M.B. 2000. Cosserat Theories: Shells, Rods and Points. Kluwer Academic Publishers, Dordrecht.
• SANSOUR C. 1998. A unified concept of elastic-viscoplastic Cosserat and micromorphic continua. Journal de Physique IV Proceedings, 8: 341-348.
• SANSOUR C. 1998. A theory of the elastic-viscoplastic Cosserat continuum. Archives of Mechanics, 50: 577-597.
• SANSOUR C., BUER H. 1992. An exact finite rotation shell theory, its mixed variational formulation and its fnite element implementation. International Journal for Numerical Methods in Engineering, 34: 73-115.
• SANSOUR C., SKATULLA S. 2008. A non-linear Cosserat continuum-based formulation and moving least square approximations in computations of size-scale effects in elasticity. Computational Materials Science, 41: 589-601.
• SAWCZUK A. 1967. On the yielding of Cosserat-Continua. Archiwum Mechaniki Stosowanej, 19: 471-492.
• SCHAEFER H. 1967. Analysis der Motorfelder im Cosserat-Kontinuum. ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 47: 319-328.
• SCHAEFER H. 1967. Das Cosserat-Kontinuum. ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 47: 485-498.
• SCHOUTEN J.A. 1954. Calculus Ricci. 2nd ed. Springer Verlag, Berlin.
• SHIELD R.T. 1973. The rotation associated with large strains. SIAM Journal on Applied Mathematics, 25: 483-491.
• SIGNORINI A. 1943. Transformazioni termoelastiche finite. Annali di Matematica Pura ed Applicata, 22: 33-143.
• SIMMONDS J.G., DANIELSON D.A. 1972. Nonlinear shell theory with finite rotation and stress function vectors. Journal of Applied Mechanics, 39: 1085-1090.
• SIMO J.C. 1992. The (symmetric) hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation. Computer Methods in Applied Mechanics and Engineering, 96: 189-200.
• SIMON E.R., DELL’ISOLA F. 2017. Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua” by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 97(4): 477-506. https://doi.org/10.1002/zamm.201600108
• SIMON E.R., DELL’ISOLA F. 2018a. Exegesis of Sect. II and III.A from “fundamentals of the mechanics of continua by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 98(1): 36-68. https://doi.org/10.1002/zamm.201600293
• SIMON E.R., DELL’ISOLA F. 2018b. Exegesis of Sect. III.B from from “fundamentals of the mechanics of continua by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 98(1): 69-105. https://doi.org/10.1002/zamm.201700112
• SKIBA E. 1874. Przyczynek do teorii strun. Pamiętnik Akademii Umiejętności w Krakowie, 3: 130-154.
• STAZI L. 1976. Sulla mechanica intrinseca dei continui iperelastici. Rendiconti del Circolo Matematico di Palermo.
• STEINMANN P. 1994. A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. International Journal of Solids and Structures, 31(8): 1063-1084.
• STEINMANN P., STEIN E. 1997. A uniform treatment of variational principles for two types of micropolar continua. Acta Mechanica, 121: 215-232.
• STOJANOVIĆ R. 1972. Nonlinear micropolar elasticity. In: Micropolar Elasticity. Eds. W. Nowacki, W. Olszak. International Centre for Mechanical Sciences, Udine.
• STUMPF H., BADUR J. 1990. On the non-Abelian motor calculus. ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 70: 551-555.
• SUDRIA J. 1925. Contribution à la théorie de l’action euclidienne. Annales de la Faculté des Sciences de Toulouse, 3ème série, 17: 63-152.
• SUDRIA J. 1935. L’action éuclidéenne de déformation et de mouvement. Mémoires de mathématique et de physique, 29: 56.
• SYNGE J.L., CHIEN W.Z. 1941. The intrinsic theory of elastic shells and plates. In: Theodore von Kármán, anniversary volume; contributions to applied mechanics and related subjects by the friends of Theodore von Kármán on his sixtieth birthday. California Institute of Technology, Pasadena.
• THOMSON W., TAIT P.G. 1879. Treatise on Natural Philosophy. Vol. I. Cambridge University Press, Cambridge.
• THOMSON W., TAIT P.G. 1883. Treatise on Natural Philosophy. Vol II. Cambridge University Press, Cambridge.
• TONOLO A. 1930. Equaqzioni intrinseche di equilibro dell’elasticità negli spazî a curvatura costante. Rendiconti del Seminario Matematico della Università di Padova, 1: 73-84.
• TONTI E. 1976. On the formal structure of Physical Theories. Dipartimento di Matematica del Politecnico di Milano.
• TOUPIN R. 1962. Elastic materials with couple stresses. Archive for Rational Mechanics and Analysis, 11: 385-413.
• TOUPIN R. 1964. Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis, 17: 85-112.
• TRUESDELL C. 1953. The mechanical foundation of elasticity and fluid dynamics. Journal for Rational Mechanics and Analysis, 1: 125-300, errata 2: 593-616.
• TRUESDELL C. 1960. The Rational Mechanics of Elastic or Flexible Bodies. Leonhardi Euleri Opera Omnia, II: 1-435.
• TRUESDELL C., TOUPIN R.A. 1960. The Classical Field Theories. In: Handbuck der Physik. Vol. III/I. Ed. S. Flugge. Springer-Verlag, Berlin
• VALID R. 1979. An intrinsic formulation for the nonlinear theory of shells and some approximations. Computers and Structures, 10: 183-194.
• VARDOULAKIS I. 2019. Cosserat Continuum Mechanics. With Applications to Granular Media. Lecture Notes in Applied and Computational Mechanics, 87. https://doi.org/10.1007/978-3-319-95156-0
• VOIGT W. 1887. Teoretische Studien über Elasticitätverhältinsse der Krystalle. I. II. Abh K Ges Wissen Göttingen, 34: 3-52, 53-100.
• WEATHEBURN C.E. 1927. On small deformation of surfaces and of thin elastic shells. The Quarterly Journal of Pure and Applied Mathematics, 50: 272-296.
• WILSON E.B. 1913. An advance in theoretical mechanics: Théorie des corps déformables by E. and F. Cosserat. Bulletin of the American Mathematical Society, 19(5): 242-246.
• WIŚNIEWSKI K. 1998. A shell theory with independent rotations for relaxed Biot stress and right stretch strain. Computational Mechanics, 21(2): 101-122.
• ZERNA W. 1950. Beitrag zur allgemeinen Schalenbiegetheore. Ingenieur-Archiv, 17: 147-164.
• ZHONG-HENG G. 1963. Homographic representation of the theory of finite thermoelastic deformations. Archives of Mechanics (Archiwum Mechaniki Stosowanej), 15: 475-505.
• ZHOUNG-HENG G., DUBEY R.N. 1983. “Method of principal axes” in nonlinear continuum mechanics. Advances in Mechanical Engineering, 13: 1-17.