Studying the stressed state of elastic medium using the argument functions of a complex variable

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.177514

Keywords:

theory of elasticity, argument functions, Cauchy-Riemann conditions, Laplace equation, boundary conditions

Abstract

Based on the argument function method and the complex variable function method, we have derived the generalizing solutions to a flat problem on the theory of elasticity using the invariant differential ratios capable of closing the result for the set system of equations. The paper reports the approaches whose application defines not the permitting functions themselves but the conditions for their existence. This makes it possible to expand the range of harmonious functions of varying complexity, which satisfy any boundary conditions in the applied problems that are constantly renewed. Two basic functions have been taken into consideration: trigonometric and fundamental, whose arguments are the unknown coordinate dependences. Introducing the argument functions into consideration changes the approaches to determining permitting dependences because the problem is considerably simplified when establishing a differential relationship among them in the form of the Cauchy-Riemann and Laplace ratios. Several analytical solutions of varying complexity have been presented, which are matched with different boundary conditions. Comparison with the results reported by other authors, at the same initial data, leads to the same result; and when considering the test problem on the interaction between a metal and an elastic half space, it leads to the convergence between defining schemes of force influence on the elastic medium.

Thus, a new approach has been proposed to solving a flat problem from the theory of elasticity, which is associated with the use of argument functions, which makes it possible to close the problem via the differential Cauchy-Riemann and Laplace ratios. These generalizations expand the range of harmonious functions that correspond to different boundary conditions for the applied problems.

Author Biographies

Valeriy Chigirinsky, Kremenchug Wheel Plant PJSC Yaroslavsky passage, 8, Kremenchuk, Ukraine, 39611

Doctor of Engineering Sciences, Professor, Associate Director of Scientific and Technical Issues

Olena Naumenko, Dnipro University of Technology Dmytra Yavornytskoho ave., 19, Dnipro, Ukraine, 49005

Senior Lecturer

Department of Structural, Theoretical and Applied Mechanics

References

  1. Chigurinski, V. (1999). The study of stressed and deformed metal state under condition of no uniform plastic medium flow. Metalurgija, 38 (1), 31–37.
  2. Chygyryns'kyy, V. V. (2004). Analysis of the state of stress of a medium under conditions of inhomogeneous plastic flow. Metalurgija, 43 (2), 87–93.
  3. Chigirinskiy, V. V. (2009). Metod resheniya zadach teorii plastichnosti s ispol'zovaniem garmonicheskih funktsiy. Izv. vuzov. CHernaya metallurgiya, 5, 11–16.
  4. Chigirinsky, V., Putnoki, A. (2017). Development of a dynamic model of transients in mechanical systems using argument-functions. Eastern-European Journal of Enterprise Technologies, 3 (7 (87)), 11–22. doi: https://doi.org/10.15587/1729-4061.2017.101282
  5. Lur'e, A. I. (1979). Teoriya uprugosti. Moscow: Nauka, 940.
  6. Mushelishvili, N. I. (1966). Nekotorye osnovnye zadachi matematicheskoy teorii uprugosti. Moscow: Nauka, 709.
  7. Timoshenko, S. P. (1934). Teoriya uprugosti. Leningrad: ONTI, 451.
  8. Pozharskii, D. A. (2017). Contact problem for an orthotropic half-space. Mechanics of Solids, 52 (3), 315–322. doi: https://doi.org/10.3103/s0025654417030086
  9. Georgievskii, D. V., Tlyustangelov, G. S. (2017). Exponential estimates of perturbations of rigid-plastic spreading-sink of an annulus. Mechanics of Solids, 52 (4), 465–472. doi: https://doi.org/10.3103/s0025654417040148
  10. Vasil’ev, V. V., Lurie, S. A. (2017). New Solution of Axisymmetric Contact Problem of Elasticity. Mechanics of Solids, 52 (5), 479–487. doi: https://doi.org/10.3103/s0025654417050028
  11. Georgievskii, D. V. (2014). Compatibility equations in systems based on generalized Cauchy kinematic relations. Mechanics of Solids, 49 (1), 99–103. doi: https://doi.org/10.3103/s0025654414010117
  12. Sneddon, I. N., Berri, D. S.; Grigolyuka, E. I. (Ed.) (1961). Klassicheskaya teoriya uprugosti. Moscow: Gos.izd-vo fiz.-mat. lit-ry, 219.
  13. Chygyryns'ky, V. V., Mamuzić, I., Vodopivec, F., Gordienko, I. V. (2006). The influence of the temperature factor on deformability of the plastic medium. Metalurgija, 45 (2), 115–118.
  14. Maciolka, P., Jedrzejewski, J. (2017). Evaluation of different approaches to 3d numerical model development of the contact zone between elementary asperities and flat surface. Journal of Machine Engineering, 4 (17), 40–53. doi: https://doi.org/10.5604/01.3001.0010.7004
  15. Qian, H., Li, H., Song, G., Guo, W. (2013). A Constitutive Model for Superelastic Shape Memory Alloys Considering the Influence of Strain Rate. Mathematical Problems in Engineering, 2013, 1–8. doi: https://doi.org/10.1155/2013/248671
  16. El-Naaman, S. A., Nielsen, K. L., Niordson, C. F. (2019). An investigation of back stress formulations under cyclic loading. Mechanics of Materials, 130, 76–87. doi: https://doi.org/10.1016/j.mechmat.2019.01.005
  17. Lopez-Crespo, P., Camas, D., Antunes, F. V., Yates, J. R. (2018). A study of the evolution of crack tip plasticity along a crack front. Theoretical and Applied Fracture Mechanics, 98, 59–66. doi: https://doi.org/10.1016/j.tafmec.2018.09.012
  18. Li, J., Zhang, Z., Li, C. (2017). Elastic-plastic stress-strain calculation at notch root under monotonic, uniaxial and multiaxial loadings. Theoretical and Applied Fracture Mechanics, 92, 33–46. doi: https://doi.org/10.1016/j.tafmec.2017.05.005
  19. Pathak, H. (2017). Three-dimensional quasi-static fatigue crack growth analysis in functionally graded materials (FGMs) using coupled FE-XEFG approach. Theoretical and Applied Fracture Mechanics, 92, 59–75. doi: https://doi.org/10.1016/j.tafmec.2017.05.010
  20. Correia, J. A. F. O., Huffman, P. J., De Jesus, A. M. P., Cicero, S., Fernández-Canteli, A., Berto, F., Glinka, G. (2017). Unified two-stage fatigue methodology based on a probabilistic damage model applied to structural details. Theoretical and Applied Fracture Mechanics, 92, 252–265. doi: https://doi.org/10.1016/j.tafmec.2017.09.004
  21. Tihonov, A. N., Samarskiy, A. A. (1999). Uravneniya matematicheskoy fiziki. Moscow: Izd-vo MGU, 799.
  22. Malinin, N. N. (1975). Prikladnaya teoriya plastichnosti i polzuchesti. Moscow: «Mashinostroenie», 400.
  23. Nikiforov, S. N. (1955). Teoriya uprugosti i plastichnosti. Moscow: GILSI, 284.
  24. Saint-Venant, B. D. (1855). Mémoire sur la torsion des prismes. Mémoires des Savants étrangers, 14, 233–560.
  25. Sedov, L. I. (2004). Mehanika sploshnoy sredy. Sankt-Peterburg: Lan', 560.
  26. Shtaerman, I. Ya. (1949). Kontaktnaya zadacha teorii uprugosti. Moscow: Gostehizdat, 270.

Downloads

Published

2019-09-05

How to Cite

Chigirinsky, V., & Naumenko, O. (2019). Studying the stressed state of elastic medium using the argument functions of a complex variable. Eastern-European Journal of Enterprise Technologies, 5(7 (101), 27–35. https://doi.org/10.15587/1729-4061.2019.177514

Issue

Vol. 5 No. 7 (101) (2019): Applied mechanics

Section

Applied mechanics

License

Copyright (c) 2019 Valeriy Chigirinsky, Olena Naumenko

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.