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  4. The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

MMC.
2020;
Volume 7, Number 2
: pp. 259–268
https://doi.org/10.23939/mmc2020.02.259
Received: January 28, 2020
Revised: May 27, 2020
Accepted: May 30, 2020

Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 259–268 (2020)

Authors:
  1. V. V. Pabyrivskyi
  2. N. V. Pabyrivska
  3. P. Ya. Pukach
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University

In this paper, the approaches to the study of mathematical models of the theory of linear elasticity are developed.  The general formulation of the 3-dimensional problem based on the representation of the fundamental solution in the form of V. P. Revenko in terms of spatial harmonic functions is considered.  The formulation in the harmonic potentials of the 3-dimensional problem of elasticity in a cylindrical coordinate system for bodies bounded by the canonical surface is done.  The boundary-value problems of pure rotation and circular symmetry in harmonic potentials are formulated.  The mentioned approaches make it possible to obtain analytical solutions to these problems and are the theoretical basis for calculating the strength parameters of technical systems in a way of analysis of their mathematical models.

elasticity theory
problem of torsion
fundamental solution
harmonic potentials
deformation tensor
stress tensor
  1. Timoshenko S. P., Gudyer D.  Teoriya uprugosti. Moscow, Nauka (1975), (in Russian).
  2. Papkovich P. F.  Teoriya uprugosti. Moscow, Oborongiz (1939), (in Russian).
  3. Neyber G.  Kontsentratsiya napryazheniy. Moscow, Gostekhizdat (1947), (in Russian).
  4. Revenko V. P.  Solving the three-dimensional equations of the linear theory of elasticity.  Int. Appl. Mech. 45 (7), 730-741 (2009).
  5. Revenko V. P.  Construction of the general solution to the Lame three-dimensions equations of elasticity theory.  Matematychnyi visnyk Naukovoho tovarystva im. Shevchenka. 2, 185–198 (2005), (in Ukrainian).
  6. Revenko V. P.  Solution of 3d boundary value problem of elasticity theory for bodie of rotation.  Prykladni problemy mekhaniky i matematyky. 12, 130–136 (2014), (in Ukrainian).
  7. Podilchuk Yu. N.  Prostranstvennyye zadachi teorii uprugosti i plastichnosti (pod obshchey redaktsiyey A. N. Guz').  Krayevyye zadachi statiki uprugogo tela. Kyiv, Naukova Dumka (1984), (in Russian).
  8. Mors P. M., Feshbakh H.  Methods of Theoretical Physics. Part 2. McGraw-Hill Science/Engineering/Math (1953).
  9. Pukach P. Ya., Kuzio I. V., Nytrebych Z. M., Ilkiv V. S.  Analytical methods for determining the effect of the dynamic process on the nonlinear flexural vibrations and the strength of compressed shaft.  Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 5, 69–76 (2017).
  10. Pukach P. Ya., Kuzio I. V., Nytrebych Z. M., Ilkiv V. S.  Asymptotic method for investigating resonant regimes of nonlinear bending vibrations of elastic shaft.  Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 1, 68–73 (2018).
  11. Labibov R. R., Chernyakov Y. A., Sheveleva A. E., Shevchenko A. G.  Strips of localization of plastic deformation.  Archive of Applied Mechanics. 88 (12), 2221–2230 (2018).
  12. Sheveleva A., Lapusta Y., Loboda V.  Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading.  Mechanics Research Communications. 63, 6–12 (2015).
  13. Pukach P. Ya.  Investigation of bending vibrations in Voigt–Kelvin bars with regard for nonlinear resistance forces.  J. Math. Sci. 215 (1), 71–78 (2016).
  14. Pukach P. Ya.  Qualitative Methods for the Investigation of a Mathematical Model of Nonlinear Vibrations of a Conveyer Belt.  J. Math. Sci. 198 (1), 31–38 (2014).
  15. Polozhiy G. N.  Teoriya i primeneniye $p$-analiticheskikh i $(p,q)$-analiticheskikh funktsiy. Moscow, Nauka (1973), (in Russian).
  16. Burak Ya. I., Pabyrivskyy V. V.  Pobudova rozviazkiv prostorovykh zadach teorii pruzhnosti z vykorystanniam metodu holomorfnykh funktsii dvokh kompleksnykh zminnykh.  Lviv, Rastr-7 (2014), (in Ukrainian).
  17. Bondarenko V. I., Samusya V. I., Smolanov S. N.  Mobile lifting units for wrecking works in pit shafts.  Gornyi Zhurnal. 5, 99–100 (2005).
  18. Pabyrivskyi V. V., Kuzio I. V., Pabyrivska N. V., Pukach P. Ya.  Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring.  Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 1, 46–51 (2020).