The fictitious absorption method in solving mixed problems for arbitrary simply-connected areas

Authors

  • Pavlova A.V. Kuban State University, Krasnodar, Российская Федерация
  • Kapustin M.S. Kuban State University, Krasnodar, Российская Федерация
  • Telyatnikov I.S. Southern Research Centre, Russian Academy of Sciences, Rostov-on-Don, Российская Федерация

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-15-3-52-61

Abstract

The work is dedicated to the development of methods for solving integral equations (IE) and systems of IE for mixed dynamic problems in the theory of elasticity, given in simply-connected areas of complex shape. A generalization of the fictitious absorption method to the case of a non-convex in the plane area occupied by a defect or stamp is presented.
The method makes it possible to describe solutions not only inside but also in the neighbourhood of the contact area boundaries and can be used to solve contact problems on the vibration of stamps, cavities or rigid inclusions of an arbitrary shape in the plane. For the areas of complex configuration, it is possible to present them as a union of convex bounded closed domains, possibly with common boundary sets.
We propose a modification of the method in the selection of the basis functions, presented in the solution only under the signs of the operators. As the latter, we choose the derivatives of the delta functions, which simplifies the construction of the solution. The results of solving the integral equation of the axisymmetric problem about steady-state oscillations of a stamp on the surface of an elastic layer with a clamped lower bound are given as an example.

Keywords:

fictitious absorption method, integral equation, oscillating kernel, area of complex configuration, factorization

Acknowledgement

Работа выполнена в рамках ГЗ ЮНЦ РАН, проект № 01201354241 и при частичной поддержке РФФИ (проекты 18-01-00124, 18-05-80008).

Author Infos

Alla V. Pavlova

д-р физ.-мат. наук, профессор кафедры математического моделирования Кубанского государственного университета

e-mail: pavlova@math.kubsu.ru

Mikhail S. Kapustin

канд. физ.-мат. наук, доцент кафедры математического моделирования Кубанского государственного университета

e-mail: kmm@fpm.kubsu.ru

Ilya S. Telyatnikov

канд. физ.-мат. наук, научный сотрудник лаборатории математики и механики Южного научного центра РАН

e-mail: ilux_t@list.ru

References

  1. Vorovich, I.I., Aleksandrov, V.M. (eds.) Mechanics of contact interactions. Fizmatlit, Moscow, 2001. (In Russian)
  2. Babeshko, V.A., Glushkov, E.V., Zinchenko, Zh.F. Dynamics of inhomogeneous linearly elastic media. Nauka, Moscow, 1989. (In Russian)
  3. Babeshko, V.A. Generalized factorization method in spatial dynamic mixed problems of the theory of elasticity. Nauka, Moscow, 1984. (In Russian)
  4. Babeshko, V.A. On the theory of spatial contact problems for anisotropic media. Doklady AN SSSR [Rep. of the USSR Academy of Sciences], 1981, vol. 256, no. 2, pp. 324–328. (In Russian)
  5. Vatul'yan, A.O., Ovsepyan, V.V., Pryahina, O.D. Contact dynamic problem for an orthotropic cylinder. Izvestiya AN ArmSSR. Mekhanika [News of Academy of Sciences of the Armenian SSR], 1983, vol. 36, no. 4, pp. 47–55. (In Russian)
  6. Vorovich, I.I., Babeshko, V.A., Pryahina, O.D. Dynamics of massive bodies and resonant phenomena in deformable media. Nauchnyj mir, Moscow, 1999. (In Russian)
  7. Kalinchuk, V.V., Belyankova, T.I. Dynamic contact problems for prestressed electroelastic bodies. Fizmatlit, Moscow, 2006. (In Russian)
  8. Kalinchuk, V.V., Belyankova, T.I. Dynamics of the surface of inhomogeneous media. Fizmatlit, Moscow, 2009. (In Russian)
  9. Babeshko, V.A. On the nonuniqueness of solutions of dynamical mixed problems for stamping systems. Doklady AN SSSR [Rep. of the USSR Academy of Sciences, 1990, vol. 310, no. 6, pp. 1327–1330. (In Russian)
  10. Kapustin, M.S., Pavlova, A.V., Rubtsov, S.E., Telyatnikov, I.S. On the modeling of the interaction of the foundation with the deformed soil environment. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2015, no. 3, pp. 44–51. (In Russian)
  11. Kapustin, M., Pavlova, A., Rubtsov, S., Telyatnikov, I. Model of foundation-base system under vibration load. Communications in Computer and Information Science (CCIS), 2014, vol. 487, pp. 168–173.
  12. Babeshko, V.A., Buzhan, V.V., Williams, R.T. Solid by an array of rigid planar inclusions. Doklady Physics, 2002, vol. 47, iss. 2, pp. 156–158.
  13. Pryahina, O.D., Smirnova, A.V. Effective method for solving dynamical problems for layered media with discontinuous boundary conditions. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 2004, vol. 68, no. 3, pp. 499–506. (In Russian)
  14. Babeshko, V.A., Pavlova, A.V., Ratner, S.V., Williams, R.T. Problems on the vibration of an elastic half-space containing a system of interior cavities. Doklady Physics, 2002, vol. 47, no. 9, pp. 677–679.
  15. Kardovskii, I.V., Pryakhina, O.D. Apparent absorption method for solving planar problems of interfacial cracking. Doklady Physics, 2006, vol. 51, no. 10, pp. 574–577.

Issue

Section

Mechanics

Pages

52-61

Submitted

2018-09-11

Published

2018-09-29

How to Cite

Pavlova A.V., Kapustin M.S., Telyatnikov I.S. The fictitious absorption method in solving mixed problems for arbitrary simply-connected areas. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 3, pp. 52-61. DOI: https://doi.org/10.31429/vestnik-15-3-52-61 (In Russian)