Abstract
The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.
Similar content being viewed by others
References
D. V. Georgievskii and M. V. Shamolin, “Levi-Civita Symbols, Generalized Vector Products, and New Integrable Cases in Mechanics of Multidimensional Bodies,” J. Math. Sci. 187 (3), 280–299 (2012).
D. V. Georgievskii and B. E. Pobedrya, “On the Compatibility Equations in Terms of Stresses in Many-Dimensional Elastic Medium,” Russ. J. Math. Phys. 22 (1), 6–8 (2015).
D. V. Georgievskii, “Generalized Compatibility Equations for Tensors of High Ranks in Multidimensional Continuum Mechanics,” Russ. J. Math. Phys. 23 (4), 475–483 (2016).
W. Nowacki, Teoria Spreżystości (Warszawa: PWN, 1970).
H.G. Hahn, Elastizitätstheorie (Stuttgart: Teubner, 1985).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Georgievskii, D. Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions. Russ. J. Math. Phys. 24, 322–325 (2017). https://doi.org/10.1134/S1061920817030050
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920817030050