Geodesic
On rationally complete algebraic systems of finite strain tensors of complex continua
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 1, pp. 71-84 
V. A. Kovalev; Yu. N. Radayev. On rationally complete algebraic systems of finite strain tensors of complex continua. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 1, pp. 71-84. http://geodesic.mathdoc.fr/item/ISU_2017_17_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the mathematical description of complex continua and the systematic derivation of strain tensors by the notion of isometric immersion of complex continuum in a plane space of higher dimension. Problem of establishing of complete systems of irreducible objective strain and extra-strain tensors for complex continuum immersed in an external plane space is considered. The solution to the problem is given by methods of the field theory and the theory of algebraic invariants. Strain tensors are obtained as irreducible algebraic invariants of contravariant vectors of the external space emerging in the complex continuum action density. Considerations are restricted to rational algebraic invariants. Completeness criteria for systems of rational algebraic invariants and rational syzygies are discussed and applied to strain tensors of micropolar elastic continua. Objective strain tensors of micropolar continuum are alternatively obtained by combining multipliers of polar decompositions of strain and extra-strain gradients.

[1] Sedov L. I., An Introduction to Continuum Mechanics, Fizmatgiz, M., 1962, 284 pp. (in Russian) | MR

[2] Illyushin A. A., Continuum Mechanics, Moscow University Press, M., 1978, 287 pp. (in Russian)

[3] Cosserat E. et F., Théorie des corps déformables, Librairie Scientifique A. Hermann et Fils, P., 1909, 226 pp.

[4] Rashevskii P. K., Riemannian Geometry and Tensor Calculus, Nauka, M., 1967, 664 pp. (in Russian) | MR

[5] Eisenhart L. P., Riemannian Geometry, Izd. Inostr. Lit., M., 1948, 316 pp. (in Russian)

[6] Kovalev V. A., Radayev Yu. N., Elements of the Field Theory: Variational Symmetries and Geometric Invariants, Fizmatlit, M., 2009, 156 pp. (in Russian)

[7] Kovalev V. A., Radayev Yu. N., Wave Problems of Field Theory and Thermomechanics, Saratov Univ. Press, Saratov, 2010, 328 pp. (in Russian)

[8] Gurevich G. B., Elements of Theory of Algebraic Invariants, Gostechteretizdat, M.–L., 1948, 408 pp. (in Russian) | MR

[9] Berdichevskii V. L., Variational Principles of Continuum Mechanics, Nauka, M., 1983, 448 pp. (in Russian) | MR

[10] Green A. E., Adkins J. E., Large Elastic Deformations and Non-linear Continuum Mechanics, Oxford Univ. Press, London, 1960, 348 pp. | MR | Zbl

[11] Horn R., Johnson H., Matrix Analysis, Cambridge University Press, Cambridge, 1990, 561 pp. | MR | MR | Zbl