Email this article On covariant non-constancy of distortion and inversed distortion tensors
- Authors: Radayev Y.N.1, Murashkin E.V.1, Nesterov T.K.1
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Affiliations:
- Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
- Issue: Vol 26, No 1 (2022)
- Pages: 37-47
- Section: Mechanics of Solids
- URL: https://bakhtiniada.ru/1991-8615/article/view/101395
- DOI: https://doi.org/10.14498/vsgtu1891
- ID: 101395
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Abstract
The paper deals with covariant constancy problem for tensors and pseudotensors of an arbitrary rank and weight in an Euclidean space. Requisite preliminaries from pseudotensor algebra and analysis are given. The covariant constancy of pseudotensors are separately considered. Important for multidimensional geometry examples of covariant constant tensors and pseudotensors are demonstrated. In particular, integer powers of the fundamental orienting pseudoscalar satisfied the condition of covariant constancy are introduced and discussed. The paper shows that the distortion and inversed distortion tensors are not actually covariant constant, contrary to the statements of those covariant constancy which can be found in literature on continuum mechanics.
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##article.viewOnOriginalSite##About the authors
Yuri N. Radayev
Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
Email: y.radayev@gmail.com
ORCID iD: 0000-0002-0866-2151
SPIN-code: 5886-9203
Scopus Author ID: 6602740688
ResearcherId: J-8505-2019
http://www.mathnet.ru/person39479
D.Sc. (Phys. & Math. Sci.), Ph.D., M.Sc., Professor; Leading Researcher; Lab. of Modeling in Solid Mechanics
Russian Federation, 101–1, pr. Vernadskogo, Moscow, 119526Evgenii V. Murashkin
Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
Email: evmurashkin@gmail.com
ORCID iD: 0000-0002-3267-4742
SPIN-code: 4022-4305
Scopus Author ID: 12760003400
ResearcherId: F-4192-2014
http://www.mathnet.ru/person53045
Cand. Phys. & Math. Sci., PhD, MD; Senior Researcher; Lab. of Modeling in Solid Mechanics
Russian Federation, 101–1, pr. Vernadskogo, Moscow, 119526Timofey K. Nesterov
Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences
Author for correspondence.
Email: nesterovtim4@gmail.com
ORCID iD: 0000-0003-0844-0484
http://www.mathnet.ru/person180992
M.Sc. (Applied Mathematics); Postgraduate Student; Lab. of Modeling in Solid Mechanics
Russian Federation, 101–1, pr. Vernadskogo, Moscow, 119526References
- Rozenfel’d B. A. Mnogomernye prostranstva [ Multidimensional Spaces]. Moscow, Nauka, 1966, 648 pp. (In Rissian)
- Gurevich G. B. Foundations of the theory of algebraic invariants. Groningen, P. Noordhoff, 1964, viii+429 pp.
- Synge J. L., Schild A. Tensor Calculus, Dover Books on Advanced Mathematics, vol. 5. New York, Courier Corporation, 1978, ix+324 pp.
- Schouten J. A. Tensor Analysis for Physicist. Oxford, Clarendon Press, 1951, xii+277 pp.
- McConnell A. J. Application of Tensor Analysis. New York, Dover Publ., 1957, xii+318 pp.
- Sokolnikoff I. S. Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, Applied Mathematics Series. New York, John Wiley & Sons, 1964, xii+361 pp.
- 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. https://doi.org/10.14498/vsgtu1635.
- Radayev Yu. N., Murashkin E. V. Pseudotensor formulation of the mechanics of hemitropic micropolar media, Problems of Strength and Plasticity, 2020, vol. 82, no. 4, pp. 399–412 (In Russian). https://doi.org/10.32326/1814-9146-2020-82-4-399-412.
- Murashkin E. V., Radayev Yu. N. On a micropolar theory of growing solids, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 3, pp. 424–444. https://doi.org/10.14498/vsgtu1792.
- Kovalev V. A., Murashkin E. V., Radayev Yu. N. On the Neuber theory of micropolar elasticity. A pseudotensor formulation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 4, pp. 752–761. https://doi.org/10.14498/vsgtu1799.
- Berdichevsky V. L. Variational Principles of Continuum Mechanics. Moscow, Nauka, 1983, 448 pp. (In Russian)
- Truesdell C., Toupin R. The Classical Field Theories, In: Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, III/1; eds. S. Flügge. Berlin, Heidelberg, Springer, 1960, pp. 226–858. https://doi.org/10.1007/978-3-642-45943-6_2.
- Truesdell C., Noll W. The Nonlinear Field Theories of Mechanics. Berlin, Springer, 2004, xxix+602 pp.
- Maugin G. A. Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computation, vol. 3. London, Chapman & Hall, 1993, xii+276 pp.
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