Far Eastern Mathematical Journal

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Beltrami-Mitchell equations in a non-Euclidean continuum model

Guzev M. A., Lyubimova O. N., Pestov K. N.

2024, issue 2, P. 178-186 DOI: https://doi.org/10.47910/FEMJ202416

Abstract

This work contains a generalization in covariant form of the classical Beltrami-Mitchell equations for the case of incompatible deformations. It is shown that in the corresponding relations an additional force appears, characterizing the internal non-Euclidean geometry of the material, the description of which is given in terms of the Ricci tensor.

Keywords: Saint-Venant consistency conditions, Beltrami-Mitchell equations, non-Euclidean continuum model, Ricci tensor.

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References

[1] Godunov S. K., ?lementy mekhaniki sploshno? sredy, Nauka, M., 1978, 304 s. [2] Novikov S. P., Ta?manov I. A., Sovremennye geometricheskie struktury i polia, MT?SNMO, M., 2005, 584 s. [3] CHernyshev G. N., Popov A. P., Kozint?sev V. M., Ponomarev I. I., Ostatochnye napriazheniia v deformiruemykh tverdykh telakh, Nauka, M., 1996, 240 s. [4] Kondo K., “On the geometrical and physical foundations of the theory of yielding”, Proc. 2nd Japan Nat. Congr. Appl. Mech. Tokyo, 231, (1953), 41–47. [5] Bilby B. A., Bullough R., Smith E., “Continuos distributions of dislocations: a new application of the methods of non - Reimannian geometry”, Proc. Roy. Soc. A., 231, (1955), 263–273. [6] Sedov L. I., “Matematicheskie metody postroeniia novykh modele? sploshnykh sred”, UMN, 20:5(125), (1965), 121–180. [7] Kadich A., ?delen D., Kalibrovochnaia teoriia dislokat?si? i disklinat?si?, Mir, M., 1987, 168 s. [8] Panin V. E., Griniaev IU. V., Danilov V. I., Strukturnye urovni plastichesko? deformat?sii i razrusheniia, Nauka, Novosibirsk, 1990, 225 s. [9] Grachev A. V., Nesterov A. I., Ovchinikov S. G., “The Gauge Theory of Point Defects”, Phys.Stat. Sol.(b), 156, (1989), 403–410. [10] Miasnikov V. P., Guzev M. A., “Affinno-metricheskaia struktura uprugoplastichesko? modeli sploshno? sredy”, Sovremennye metody mekhaniki sploshnykh sred, Sbornik state?. K 90-letiiu so dnia rozhdeniia akademika Leonida Ivanovicha Sedova, Trudy MIAN, 223, Nauka, MAIK «Nauka/Interperiodika», M., 1998, 30–37. [11] Guzev M. A., Miasnikov V. P., “Neevklidova struktura polia vnutrennikh napriazheni? sploshno? sredy”, Dal?nevost. matem. zhurn., 2:2, (2001), 29–44. [12] Guzev M. A., Qi Ch., “Vyvod uravneni? gradientno? teorii v krivoline?nykh koordinatakh”, Dal?nevost. matem. zhurn., 13:1, (2013), 35–42. [13] Liakhovski? V. D., Bolokhov A. A., Gruppy simmetrii i ?lementarnye chastit?sy, Izd-vo Leningr. un-ta., L., 1983, 336 s. [14] Burenin A. A., Bykovt?sev G. I., Kovtaniuk L. V., “Ob odno? prosto? modeli dlia uprugoplastichesko? sredy pri konechnykh deformat?siiakh” , DAN, 347, (1996), 199–201. [15] Norden A. P., Prostranstva affinno? sviaznosti, Nauka, M., 1976, 584 s.

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