For citation:
Radayev Y. N. Representation of Waves of Displacements and Micro-rotations by Systems of the Screw Vector Fields. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 4, pp. 468-477. DOI: 10.18500/1816-9791-2020-20-4-468-477, EDN: LJYZXJ
Representation of Waves of Displacements and Micro-rotations by Systems of the Screw Vector Fields
The present study concerns the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and micro-rotations in the case of a harmonic dependence of the physical fields on time. The system is known from many previous discussions on the micropolar elasticity. A new analysis aimed at uncoupling the coupled vector differential equation of the linear theory of micropolar elasticity is carried out. A notion of proportionality of the vortex parts of the displacements and microrotations to a single vector, which satisfies the screw equation, is employed. Finally the problem of finding the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. Corresponding representation formulae are given. Obtained results can be applied to problems of the linear micropolar elasticity concerning harmonic waves propagation along cylindrical waveguides.
- Cosserat E., Cosserat F. Theorie des corps deformables. Paris, Herman et Fils, 1909. 226 p.
- Southwell R. V. An Introduction to the Theory of Elasticity for Engineers and Physicists. Oxford Engineering Science Series. Oxford, Clarendon Press; London, Oxford University Press, 1936. 510 p.
- Lurie A. I. Theory of Elasticity. Berlin, Heidelberg, Springer, 2005. 1050 p.
- Nowacki W. Theory of Asymmetric Elasticity. Oxford, New York, Toronto, Sydney, Paris, Frankfurt, Pergamon Press, 1986. 383 p.
- Nowacki W. Theory of Elasticity. Moscow, Mir, 1975. 872 p. (in Russian).
- Radayev Yu. N. The Lagrange multipliers method in covariant formulations of micropolar continuum mechanics theories. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, vol. 22, no. 3, pp. 504–517 (in Russian). DOI: https://doi.org/10.14498/vsgtu1635
- Achenbach J. D. Wave Propagation in Elastic Solids. North-Holland Series in Applied Mathematics and Mechanics. Vol. 16. Amsterdam, London, North-Holland; New York, American Elsevier, 1973. 425 p.
- Kovalev V. A., Radayev Yu. N. Volnovye zadachi teorii polia [Wave Problems of the Field Theory]. Saratov, Izd-vo Saratovskogo universiteta, 2010. 328 p. (in Russian).
- Truesdell C., Toupin R. The Classical Field Theories. In: S. Flugge (ed.). Encyclopedia of Physics. Vol. III/1. Principles of Classical Mechanics and Field Theory. Berlin, Gottingen, Heidelberg, Springer, 1960. P. 226–902.
- Gunter W. Zur Statik und Kinematik des Cosseratschen Kontinuums. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 1958, Bd. 10, s. 195–213 (in Germany).
- Kessel S. Lineare Elastizitatstheorie des anisotropen Cosserat-Kontinuums. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 1964, Bd. 16, s. 1–22 (in Germany).
- Pal’mov V. A. Fundamental equations of the theory of asymmetric elasticity. J. Appl. Math. Mech., 1964, vol. 28, iss. 3, pp. 496–505. DOI: https://doi.org/10.1016/0021-8928(64)90092-9
- Neuber H. U¨ ber Probleme der Spannungskonzentration im Cosserat-Ko¨rper. Acta Mechanica, 1966, vol. 2, iss. 1, pp. 48–69 (in Germany). DOI: https://doi.org/10.1007/BF01176729
- Dyszlewicz J. Micropolar Theory of Elasticity. Lecture Notes in Applied and Computational Mechanics. Berlin, Heidelberg, Springer, 2004. 345 p.
- 878 reads