Topological aspects of linear elastisity theory (methodological notes)

doi:10.26565/2222-5617-2018-28-1

V. D. Natsyk V. N. Karazin Kharkiv National University, sq. Svoboda 4, Kharkiv 61077, Ukraine
I. M. Pakhomova V. N. Karazin Kharkiv National University, sq. Svoboda 4, Kharkiv 61077, Ukraine

DOI: https://doi.org/10.26565/2222-5617-2018-28-1

Keywords: topology, Young's moduli, shear, bulk moduli, Poisson's ratio

Abstract

A comparative discussion is given of the deformation properties of three-dimensional (3D) and two-dimensional (2D) solids, which are considered in the approximation of continuum mechanics as elastic continua with three and two spatial dimensions.
Attention is drawn to the effectiveness of attracting concepts and methods of geometry to establish the general patterns of deformation of such systems without taking into account the physicochemical properties of atoms and the interatomic forces. In a geometric description, these continua are elastic spaces with different topological properties, which leads to significant differences in the relationships between the characteristics of their elasticity: Young's moduli, shear, bulk moduli, and Poisson's ratio. Deformation characteristics that can be considered as unique topological invariants of 3D and 2D elastic continua are established

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References

1. L. D. Landau. E.M. Lifshits. Teoriya uprugosti. Nauka. M.(1978). 730 s.
2. D.Ya. Stroyk. Kratkiy ocherk istorii matematiki. Nauka. M.(1978). 335 s.
3. K. F. Kleyn. Elementarnaya matematika s tochki zreniya vysshey. Nauka. M. (1987). 431 s.
4. V.S. Lukianets. Fiziko-matematicheskiye prostranstva i realnost. Naukova dumka. Kiyev (1971). 109 s.
5. I. F. Lyuksutov. A. G. Naumovets. V. L. Pokrovskiy. Dvumernyye kristally. Naukova dumka. Kiyev (1988). 320s.
6. A. M. Kosevich. Teoriya kristallicheskoy reshetki (fizicheskaya mekhanika kristallov). KhGU Vishcha shkola. Kharkov (1988). 327 s.
7. A. M. Kosєvіch. Mekhanіka kristalіchnoї gratki. serіya «Unіversitetskі lektsії z fіziki tverdogo tіla». Nauk. vid. Akta. Kharkіv (2005). 305 s.
8. K. S. Novoselov. UFN 181. 1299 (2011).
9. V.D. Natsik. S.N. Smirnov. FNT. 39. 690 (2013)
10. V. D. Natsik. S. N. Smirnov. V. I. Belan. FNT. 44. 877 (2018).
11. G. Korn. T. Korn. Spravochnik po matematike (dlya nauchnykh rabotnikov i inzhenerov). Nauka. M. 1977. 831s.
12. M. Nakahara. Geometry. Topology and Physics. IOP Publishing. Bristol 1990
13. A.M. Kosevich. FNT 30. 135 (2004).