On the constitutive pseudoscalars of hemitropic micropolar media in inverse coordinate frames

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The paper is devoted to the constitutive pseudoscalars associated with the theory of hemitropic micropolar continuum. The basic concepts of pseudotensor algebra are presented. The pseudotensor form of the hemitropic micropolar elastic potential is given, based on 9 constitutive pseudoscalars (3 are pseudoscalars and 6 are absolute scalars). The weights of the constitutive pseudoscalars are calculated. The fundamental orienting pseudoscalar of weight \(+1\) is used to formulate transformation rules for constitutive pseudoscalars. The governing equations of the hemitropic micropolar elastic continuum are derived. The equations of the dynamics of the hemitropic micropolar continuum are discussed in terms of pseudotensors in right- and left-handed Cartesian coordinate systems. The presence of inverse modes along with normal ones is shown for wave propagation across the hemitropic micropolar continuum.

About the authors

Eugenii 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

101–1, pr. Vernadskogo, Moscow, 119526, Russian Federation

Yuri N. Radayev

Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Author for correspondence.
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

101–1, pr. Vernadskogo, Moscow, 119526, Russian Federation

References

  1. Lakes R. Elastic and viscoelastic behavior of chiral materials, Int. J. Mech. Sci., 2001, vol. 43, no. 7, pp. 1579–1589. https://doi.org/10.1016/S0020-7403(00)00100-4
  2. Mackay T., Lakhtakia A. Negatively refracting chiral metamaterials: a review, SPIE Reviews, 2010, vol. 1, no. 1, 018003. https://doi.org/10.1117/6.0000003
  3. Tomar S., Khurana A. Wave propagation in thermo-chiral elastic medium, Appl. Math. Model., 2013, vol. 37, no. 22, pp. 9409–9418. https://doi.org/10.1016/j.apm.2013.04.029
  4. Truesdell C., Toupin R. The Classical Field Theories, In: Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, III/1; ed. S. Flügge. Berlin, Göttingen, Heidelberg, Springer, 1960, pp. 226–902. https://doi.org/10.1007/978-3-642-45943-6_2
  5. Gurevich G. B. Foundations of the Theory of Algebraic Invariants. Groningen, P. Noordhoff, 1964, viii+429 pp.
  6. McConnell A. J. Application of Tensor Analysis. New York, Dover Publ., 1957, xii+318 pp.
  7. Schouten J. A. Tensor Analysis for Physicists. Oxford, Clarendon Press, 1954, xii+277 pp.
  8. Sokolnikoff I. S. Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, Applied Mathematics Series. New York, John Wiley & Sons Inc, 1964, xii+361 pp.
  9. Synge J. L., Schild A. Tensor Calculus, Dover Books on Advanced Mathematics, vol. 5. New York, Courier Corporation, 1978, ix+324 pp.
  10. Vekua I. N. Osnovy tenzornogo analiza i teorii kovariantov [Foundations of Tensor Analysis and the Theory of Covariants]. Moscow, Nauka, 1978, 411 pp. (In Russian)
  11. Kochin N. E. Vektornoe ischislenie i nachala tenzornogo ischisleniya [Foundations of Tensor Analysis and Covariant Theory]. Moscow, Nauka, 1951, 427 pp. (In Russian)
  12. 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
  13. 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
  14. 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
  15. Veblen O., Thomas T. Y. Extensions of relative tensors, Trans. Amer. Math. Soc., 1924, vol. 26, no. 3, pp. 373–377. https://doi.org/10.1090/S0002-9947-1924-1501284-6
  16. Veblen O. Invariants of Quadratic Differential Forms, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 24. Cambridge, Univ. Press, 1927, viii+102 pp.
  17. Rozenfel'd B. A. Mnogomernye prostranstva [Multidimensional Spaces]. Moscow, Nauka, 1966, 648 pp. (In Russian)
  18. Murashkin E. V., Radaev Yu. N. On a generalization of the algebraic Hamilton–Cayley theory, Mech. Solids, 2021, vol. 55, no. 6 (to appear).
  19. Nowacki W. Theory of Micropolar Elasticity. Course held at the Department for Mechanics of Deformable Bodies, July 1970, Udine, International Centre for Mechanical Sciences. Courses and Lectures, vol. 25. Wien, New York, Springer, 1972, 286 pp. https://doi.org/10.1007/978-3-7091-2720-9
  20. Nowacki W. Theory of Asymmetric Elasticity. Oxford, Pergamon Press, 1986, viii+383 pp.
  21. Besdo D. Ein Beitrag zur nichtlinearen Theorie des Cosserat–Kontinuums [A contribution to the nonlinear theory of the Cosserat–continuum], Acta Mechanica, 1974, vol. 20, no. 1, pp. 105–131 (In German). https://doi.org/10.1007/BF01374965
  22. 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). https://doi.org/10.14498/vsgtu1635
  23. Kopff A. The Mathematical Theory of Relativity. London, Methuen, 1923, viii+214 pp.
  24. Radayev Yu. N. Prostranstvennaya zadacha matematicheskoy teorii plastichnosti [The Spatial Problem of the Mathematical Theory of Plasticity]. Samara Univ., Samara, 2006, 340 pp. (In Russian)

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