Heat Conduction of Micropolar Solids Sensitive to Mirror Reflections of Three-Dimensional Space
https://doi.org/10.26907/2541-7746.2023.4.389-403
Abstract
This article considers a variant of the heat conduction theory of thermal conductivity, in which the heat flux pseudovector has a weight of 1. The pseudoinvariants associated to the heat flux pseudovector are sensitive to mirror reflections and inversions of threedimensional space. The primary purpose of the study was to find a heat flux vector that is algebraically equivalent to the microrotation pseudovector and to measure elementary volumes and areas using pseudoinvariants that are sensitive to mirror reflections. To represent spinor displacements, a contravariant microrotation pseudovector with a weight of +1 was selected. Thus, the heat flux and mass density were expressed as odd-weight pseudotensors. The Helmholtz free energy per unit doublet pseudoinvariant volume was employed as the thermodynamic state potential of the following functional arguments: absolute temperature, symmetric parts, and accompanying vectors for the linear asymmetric strain tensor and the wryness pseudotensor. The results obtained show that the thermal conductivity coefficient and heat capacity of elastic micropolar solids are pseudoscalars of odd weight, indicating their sensitivity to mirror reflections.
Keywords
heatconduction,
micropolarity,
volumetensorelement,
heatfluxpseudotensor,
mirror reflection,
semi-isotropic solid
Supporting agencies: We thank Professor S.A. Lurie for insightful discussions on a wide range of topics related to the mechanics of deformable solids. This study was supported by the Russian Science Foundation (project no. 23-21-00262)
About the Authors
E. V. Murashkin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
Moscow, 119526
Y. N. Radayev
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
Moscow, 119526
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.G., Lakhtakia A. Negatively refracting chiral metamaterials: A review. SPIE Rev., 2010, vol. 1, no. 1, art. 018003. https://doi.org/10.1117/6.0000003.
3. Tomar S.K., Khurana A. Wave propagation in thermo-chiral elastic medium. Appl. Math. Modell., 2013, vol. 37, no. 22, pp. 9409–9418. https://doi.org/10.1016/j.apm.2013.04.029.
4. Besdo D. Ein beitrag zur nichtlinearen Theorie des Cosserat–Kontinuums. Acta Mech., 1974, vol. 20, no. 1, pp. 105–131. https://doi.org/10.1007/BF01374965. (In German)
5. Nowacki W. Theory of Micropolar Elasticity. Berlin, Springer, 1972. 285 р.
6. Nowacki W. Theory of Asymmetric Elasticity. Oxford, Pergamon Press, 1986. 383 p.
7. Dyszlewicz J. Micropolar Theory of Elasticity. Berlin, Heidelberg, Springer, 2004. 345 p. https://doi.org/10.1007/978-3-540-45286-7.
8. Truesdell C., Toupin R. The classical field theories. In: Flu¨gge S. (Ed.) Principles of Classical Mechanics and Field Theory. Book Ser.: Encyclopedia of Physics. Vol. III/1. Berlin, Go¨ttingen, Heidelberg, Springer, 1960, pp. 226–902. https://doi.org/10.1007/978-3-642-45943-6_2.
9. Schouten J.A. Tenzornyi analiz dlya fizikov [Tensor Analysis for Physicists]. Moscow, Nauka, 1965. 456 p. (In Russian)
10. Sokolnikoff I.S. Tenzornyi analiz. Teoriya i primeneniya v geometrii i v mekhanike sploshnykh sred [Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua]. Moscow, Nauka, 1971. 376 p. (In Russian)
11. Synge J.L., Schild A. Tensor Calculus. New York, Dover Publ., 1978. 334 p.
12. Das A.J. Tensors. The Mathematics of Relativity Theory and Continuum Mechanics. Berlin, Springer Sci. & Bus. Media, 2007. 290 p. https://doi.org/10.1007/978-0-387-69469-6.
13. Gurevich G.B. Osnovy teorii algebraicheskikh invariantov [Foundations of the Theory of Algebraic Invariants]. Moscow, Leningrad, OGIZ, GITTL, 1948. 408 p. (In Russian)
14. Veblen O., Thomas T.Y. Extensions of relative tensors. Trans. Am. Math. Soc., 1924, vol. 26, no. 3, pp. 373–377. https://doi.org/10.2307/1989146.
15. Veblen O. Invarianty differentsial’nykh kvadratichnykh form [Invariants of Quadratic Differential Forms]. Moscow, Izd. Inostr. Lit., 1948. 139 p. (In Russian)
16. Radayev Y.N., Murashkin E.V. Pseudotensor formulation of the mechanics of hemitropic micropolar media. Probl. Prochn. Plast., 2020, vol. 80, no. 4, pp. 399–412. https://doi.org/10.32326/1814-9146-2020-82-4-399-412. (In Russian)
17. Murashkin E.V., Radayev Y.N. On a micropolar theory of growing solids. Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki, 2020, vol. 24, no. 3, pp. 424–444. https://doi.org/10.14498/vsgtu1792.
18. Murashkin E.V., Radayev Y.N. On the theory of linear micropolar hemitropic media. Vestn. ChGPU im. I.Ya. Yakovleva. Ser. Mekh. Predel’nogo Sostoyaniya, 2020, no. 4(46), pp. 16–24. https://doi.org/10.37972/chgpu.2020.89.81.031. (In Russian)
19. Murashkin E.V., Radaev Y.N. Coupled thermoelasticity of hemitropic media. Pseudotensor formulation. Mech. Solids, 2023, vol. 58, no. 3, pp. 802–813. http://dx.doi.org/10.3103/s0025654423700127.
20. Murashkin E.V., Radaev Y.N. A negative weight pseudotensor formulation of coupled hemitropic thermoelasticity. Lobachevskii J. Math., 2023, vol. 44, no. 6, pp. 2440–2449. https://doi.org/10.1134/S1995080223060392.
21. Radaev Y.N. Tensors with constant components in the constitutive equations of hemitropic micropolar solids. Mech. Solids, 2023, vol. 58, no. 5, pp. 1517–1527. https://doi.org/10.3103/S0025654423700206.
22. Murashkin E.V., Radaev Y.N. Schouten’s force stress tensor and affinor densities of positive weight. Probl. Prochn. Plast., 2022, vol. 84, no. 4, pp. 545–558. https://doi.org/10.32326/1814-9146-2022-84-4-545-558. (In Russian)
23. Murashkin E.V., Radayev Y.N. The Schouten force stresses in continuum mechanics formulations. Mech. Solids, 2023, vol. 58, no. 1, pp. 153–160. http://dx.doi.org/10.3103/s0025654422700029.
24. Murashkin E.V., Radayev Y.N. Algebraic algorithm for the systematic reduction of onepoint pseudotensors to absolute tensors. Vestn. ChGPU im. I.Ya. Yakovleva. Ser. Mekh. Predel’nogo Sostoyaniya, 2022, no. 1(51), pp. 17–26. https://doi.org/10.37972/chgpu.2022.51.1.002.
25. Murashkin E.V., Radayev Y.N. Covariantly constant tensors in Euclid spaces. Elements of the theory. Vestn. ChGPU im. I.Ya. Yakovleva. Ser. Mekh. Predel’nogo Sostoyaniya, 2022, no. 2(52), pp. 106–115. https://doi.org/10.37972/chgpu.2022.52.2.012. (In Russian)
26. Murashkin E.V., Radayev Y.N. Covariantly constant tensors in Euclid spaces. Applications to continuum mechanics. Vestn. ChGPU im. I.Ya. Yakovleva. Ser. Mekh. Predel’nogo Sostoyaniya, 2022, no. 2(52), pp. 118–127. https://doi.org/10.37972/chgpu.2022.52.2.013.
27. Rosenfeld B.A. Mnogomernye prostranstva [Multidimensional Spaces]. Moscow, Nauka, 1966. 648 p. (In Russian)
28. Poincar´e H. Sur les r´esidus des int´egrales doubles. Acta Math., 1887, t. 6, pp. 321–380. (In French)
29. Poincar´e H. Analysis situs. J. ´Ecole Polytech., 1895, t. 2, no. 1, pp. 1–123. (In French)
30. Murashkin E.V., Radayev Y.N. On a ordering of area tensor elements orientations in a micropolar continuum immersed in an external plane space. Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki, 2021, vol. 25, no. 4, pp. 776–786. https://doi.org/10.14498/vsgtu1883.
31. Murashkin E.V., Radaev Y.N. On theory of oriented tensor elements of area for a micropolar continuum immersed in an external plane space. Mech. Solids, 2022, vol. 57, no. 2, pp. 205–213. https://doi.org/10.3103/s0025654422020108.
32. Murashkin E.V. On the boundary conditions formulation in the problems of synthesis of woven 3D materials. Vestn. ChGPU im. I.Ya. Yakovleva. Ser. Mekh. Predel’nogo Sostoyaniya, 2021, no. 1(47), pp. 114–121. (In Russian)
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For citations:
Murashkin E.V., Radayev Y.N. Heat Conduction of Micropolar Solids Sensitive to Mirror Reflections of Three-Dimensional Space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2023;165(4):389-403. (In Russ.) https://doi.org/10.26907/2541-7746.2023.4.389-403
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