Abstract
By equating to zero all components of the Kröner incompatibility tensor of rank 2n − 4 or of the Riemann tensor dual to the Kröner tensor, n 2(n 2 − 1)/12 independent consistency equations for the stresses in an n-dimensional isotropic elastic medium are derived. The problem concerning the equivalence of the system of these equations to systems following from equating to zero either all n(n + 1)/2 components of the Ricci tensor or only one curvature invariant is investigated. It is shown that the answer to this question depends on the dimension of the space. Three cases are singled out: n = 2 (plane problem of elasticity theory), n = 3 (spatial problem of elasticity theory), and n ⩾ 4.
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References
B. E. Pobedrya, Lectures on Tensor Analysis (Moscow: Izd-vo NGU, 1986) [in Russian].
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, “General Solutions of Systems in the Theory of Elasticity in Terms of Stresses That Are Not Equivalent to the Classical System,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), 26–32 (2012) [in Russian].
P. K. Rashevskii, Riemannian Geometry and Tensor Analysis (Moscow: Izd-vo URSS, 2003) [in Russian].
D. V. Georgievskii and B. E. Pobedrya, “The Number of Independent Compatibility Equations in the Mechanics of Deformable Solids,” Prikl. Mat. Mekh. 68(6), 1043–1048 (2004) [in Russian].
B. E. Pobedrya, “Static Problem in Stresses,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (3), 61–67 (2003) [Mosc. Univ. Mech. Bull. 58 (3), 6–12 (2003)].
N. M. Borodachev, “Solutions of Spatial Problem of Elasticity Theory in Stresses,” Prikl. Mekh. 42(8), 3–35 (2006) [Internat. Appl. Mech. 42 (8), 849–878 (2006)].
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Georgievskii, D.V., Pobedrya, B.E. On the compatibility equations in terms of stresses in many-dimensional elastic medium. Russ. J. Math. Phys. 22, 6–8 (2015). https://doi.org/10.1134/S1061920815010021
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DOI: https://doi.org/10.1134/S1061920815010021