Abstract
Using the support operator technique for two-dimensional problems of the elasticity theory we constructed integrally consistent approximations of the components of the strain tensor and the elastic energy of the medium for the equations of the elasticity theory in terms of displacements. Approximations are constructed for the case of irregular difference grids in the R–Z plane of a cylindrical coordinate system. We use the limiting process assuming that the azimuthal angle tends to zero for passing from the full three-dimensional approximations to the two-dimensional approximations in the R–Z plane. The used technique preserves the divergent form, self-adjointness, and sign-definiteness of the two-dimensional approximations. These properties are inherent in their 3D predecessors corresponding to the operators in the governing differential equations.
Similar content being viewed by others
REFERENCES
O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, Vol. 2: Solid and Structural Mechanics (Butterworth Heinemann, London, 2000).
O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals (Elsevier, Amsterdam, 2005).
V. D. Ivanov, R. A. Pashutin, I. B. Petrov, A. G. Tormasov, and A. S. Kholodov, “Grid characteristic method of dynamic deformation calculation on irregular grids,” Mat. Model. 11 (7), 118–127 (1999).
I. E. Kvasov, I. B. Petrov, and F. B. Chelnokov, “Computation of wave processes in nonuniform 3D constructions,” Mat. Model. 21 (5), 3–9 (2009).
A. A. Samarskii, A. V. Koldoba, Yu. A. Poveshchenko, V. F. Tishkin, and A. P. Favorskii, Difference Schemes on Irregular Grids (Kriterii, Minsk, 1996) [in Russian].
A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and A. Yu. Shashkov, “Using the support operator method for constructing difference analogues of tensor analysis operations,” Differ. Uravn. 18, 1251–1256 (1982).
A. V. Koldoba, Yu. A. Poveshchenko, I. V. Gasilova, and E. Yu. Dorofeeva, “Numerical schemes of the support operators method for elasticity theory equations,” Mat. Model. 24 (12), 86–96 (2012).
V. A. Gasilov, A. Yu. Krukovskii, Yu. A. Poveshchenko, and I. P. Tsygvintsev, “Homogeneous difference schemes for solving conjugate problems of hydrodynamics and elasticity,” KIAM Preprint No. 13 (Keldysh Inst. Appl. Math., Moscow, 2018).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity (Pergamon, Oxford, 1970).
A. J. McConnell, Applications of Tensor Analysis (Dover, New York, 1957).
G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).
S. S. Sokolov et al., “TIM-2D method for calculating the problems of continuum mechanics on irregular polygonal grids with an arbitrary number of bonds in nodes,” Vopr. At. Nauki Tekh., Ser.: Mat. Model. Fiz. Protses., No. 4, 29–44 (2006).
A. B. Novikov and S. A. Glushak, “Resistance of plastic deformation metals under high speed compression,” Khim. Fiz. 19, 65–69 (2000).
G. E. Mase, Theory and Problems of Continuum Mechanics (McGraw-Hill, New York, 1970).
Funding
This work was supported by the Russian Science Foundation (project no. 16-11-00100p).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Poveshchenko, Y.A., Gasilov, V.A., Podryga, V.O. et al. Difference Schemes of Consistent Approximation of the Stress-Strain State and Energy Balance of a Medium. Math Models Comput Simul 12, 99–109 (2020). https://doi.org/10.1134/S2070048220020131
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070048220020131