REVIEW OF WORKS ON THE APPLICATION OF THE METHOD OF R-FUNCTIONS IN THE MECHANICS OF DEFORMABLE SOLIDS

Visnyk LNAU: Architecture and Farm Building 2018 №19: 5-9

REVIEW OF WORKS ON THE APPLICATION OF THE METHOD OF R-FUNCTIONS IN THE MECHANICS OF DEFORMABLE SOLIDS

Yu. Bodnar, Candidate of Technical Sciences
Lviv National Agrarian University

https://doi.org/10.31734/architecture2018.19.005

Annotation

An analytical review of publications on the application of the theory of R-functions in the problems of the mechanics of a deformable solid is carried out. The plane and axisymmetric problems of the theory of elasticity of isotropic and orthotropic bodies, problems of the theory of thermoelasticity and plasticity theo-ry, dynamic problems of elasticity and thermoelasticity are considered. A brief analysis of methods and algorithms for solving these problems based on theories of R-functions, variational and difference methods is given. The results of solu-tions of specific problems for finite-size bodies have been analyzed, in particular, the effect of changes in the elastic properties, voids and rigid inclusions on the stress-deformed state of isotropic and orthogonal bodies, the interaction of period-ic systems of cracks, holes, cutouts in the strip, dynamic thermoelasticity problems with sudden use of force or temperature loads, eigenvalues and the forms of oscillations of elastic bodies with cavities, etc.

The method of R-functions has become widely developed for the problems of the mechanics of a deformable solid. Structures and algorithms of the solution of plane and axisymmetric problems of the elasticity theory of isotropic and orthotropic bodies, problems of plasticity theory, dynamic problems of elasticity and thermoelasticity are constructed. The considered methods and algorithms can also be used for modeling of the elements of buildings and structures at different loads and influences.

Key words

R-functions, elasticity, thermoelasticity, plasticity, isotropic bodies, orthotropic bodies, variational method, difference method, dynamic problems

Full text

pdf

Link

  1. Rvachev V. L. Geometric applications of the algebra of logic. Kiev: Technique, 1967. 212 p.
  2. Rvachev V. L. Theory of R-functions and some of its applications. Kiev: Naukova dumka, 1982. 552 p.
  3. Rvachev V. L., Sinekop N. S. Method of R-functions in problems of the theory of elasticity and plasticity. Kiev: Naukova dumka, 1990. 212 p.
  4. Protsenko V. S., Protsenko V. G., Sinekop N. S. On the solution of some contact problems for an elastic rectangle by the structural method. International Applied Mechanics, 1974. Vol. 10, № 9. P. 60-64.
  5. Rvachev V. L., Protsenko V. S. Contact problems of the theory of elasticity for nonclassical domains. Kiev: Naukova dumka, 1977. 235 p.
  6. Rvachev V. L., Sinekop N. S. Formulas of convolution in the method of R-functions and their application to the construction of the structure of solutions of boundary value problems. Reports of the Academy of Sciences of the USSR. 1980. 225. No. 1. P. 80-83.
  7. Rvachev V. L., Sinekop N. S., Erivantseva L. S. Structures of solutions of some problems in the theory of elasticity for a regular n-gon loaded with concentrated forces. Reports of the Academy of Sciences of the Ukrainian SSR. Ser. A. 1982. P. 33-36.
  8. Sinekop N. S. On the structure of the solution of the contact problem for a piecewise-homogeneous body of finite dimensions. Reports of the Academy of Sciences of the Ukrainian SSR. Ser. A. 1981. № 8. P. 59-62.
  9. Rvachev V. L., Sinekop N. S., Kravchenko L. K. Axially symmetric problem of elasticity theory for an inhomogeneous cylinder. International Applied Mechanics. January 1986. Vol. 22. P. 11-16. (Translated from Prikladnaya Mekhanika. January, 1986. Vol. 22, No. 1. P. 18-23).
  10. Rvachev V. L., Sinekop N. S., Kravchenko L. K. Calculation of a non-uniform cylinder with a rigid inclusion. Mathematical methods of the phisico-mechanical fields. 1987. Is. 26. P. 72-78.
  11. Rvachev V. L., Sinekop N. I. Approximate solution of the plane elasticity problem for an orthotropic body by the R-function method. Reports of the Academy of Sciences of the Ukrainian SSR. Ser. A. 1981. № 10. P. 61-64.
  12. Rvachev V. L., Sinekop N. S., Sinekop N. I. Study of the stress-strain state of hollow curvilinearly orthotropic cylinders. Strength of Materials, October 1983. Vol. 15, Is. 10. P. 1410-1415. (Translated from Problemy Prochnosti. October, 1983. No. 10. P. 49-53). doi: https://doi.org/10.1007/BF01523319.
  13. Rvachev V. L., Sinekop N. S., Sinekop N. I. R-functions in problems of the theory of elasticity of an orthotropic body with concentrators. Abstracts of the Republican Symposium on Stress Concentration (Donetsk, May 31 - July 2, 1983). Donetsk, 1983. P. 98-99.
  14. Rvachev V. L., Sinekop N. S., Sinekop N. I. Two-dimensional problem of the theory of elasticity for an orthotropic body of finite dimensions. International Applied Mechanics. April, 1984. Vol. 20, Is. 4. P. 333-341. (Translated from Prikladnaya Mekhanika. 1984. Vol. 20, № 4. P. 40-49).
  15. 15. Rvachev V. L., Sinekop N. S., Sinekop N. I. Calculation of the state of stress and strain of rotating curvilinearly orthotropic cylinders. Strength of Materials. July 1987. Vol. 19, Is. 7. P. 963-971. (Translated from Problemy Prochnosti. July, 1987. No. 7. P. 76–81). doi: https://doi.org/10.1007/BF01523538.
  16. Rvachev V. L., Sinekop N. S. Kravchenko L. K. On the application of the method of R-functions in problems of the theory of small elastoplastic deformations. Reports of the Academy of Sciences of the Ukrainian SSR. Ser. A. 1983. № 1. P. 49-53.
  17. Rvachev V. L., Sinekop N. S., Kravchenko L. K. Elastoplastic stressed state of infinitely long solids of revolution. Strength of Materials. May, 1984. Vol. 16. P. 721–726. (Translated from Problemy Prochnosti. May, 1984. No. 5. P. 94-97). doi: https://doi.org/10.1007/BF01529057.
  18. Bodnar Yu. I., Sinekop N. S. Approximate solution of the problems of the theory of cracks by the method of R-functions. Reports of the National Academy of Sciences of Ukraine. 1994. № 4. P. 45-48.
  19. Bodnar Yu. I. Аpplication of R-functions to the investigation of the mutual influence of periodic systems of edge cracks and of notches when the band is stretched. Bulletin of the State University "Lviv Polytechnic". No. 277: Differential equations and their applications. Lviv, 1994. P. 8-10.
  20. Bodnar Yu. I. Stretch of a continuous-non-uniform band with a periodic system of bilateral cracks. Bulletin of the Lviv State Agrarian University: Architecture and agricultural-building construction. 2005. № 6. P. 49-54.
  21. Bodnar Yu. I. Investigation of the mutual influence of periodic systems of edge cracks in the strip and holes on its axis. Bulletin of the Lviv State Agrarian University: Architecture and agricultural-building construction. 2008. № 9. P. 79-84.
  22. Bodnar Yu. I. Stress-strain state when stretching strips with periodic system of cracks originating from triangular concentrators. Visnyk LNAU: Architecture and agricultural-building construction. 2009. № 10. P. 10-15.
  23. Rvachev V. L., Sinekop N. S., Molotkov I. P. Quasistatic contact problem of thermoelasticity for bodies of finite dimensions. Reports of the National Academy of Sciences of Ukraine. 1994. № 9. P. 84-87.
  24. Lobanova L. S., Sinekop N. S. An oscillometric dynamic contact problem for a body of finite-length rotation. Reports of the National Academy of Sciences of Ukraine. 1994. № 10. P. 51-53.
  25. Sinekop N. S., Lobanova L. S. Nonstationary deformation of a cylinder of finite length. Problems of mechanical engineering. 1998. Vol. 1, № 2. P. 41-47.
  26. Rvachev V. L., Lobanova L. S., Sinekop N. S. Solution by the method of R-functions of the dynamic problem of the theory of elasticity for an orthotropic body of finite dimensions. Reports of the National Academy of Sciences of Ukraine. 1999. № 4. P. 68-71.
  27. Rvachev V. L., Lobanova L. S., Sinekop N. S. The method of R-functions in dynamic problems of thermoelasticity for bodies of finite dimensions. Reports of the National Academy of Sciences of Ukraine. 1999. № 5. P. 64-68.
  28. Lobanova L. S. Structural method in the problem of unsteady oscillations of a cylinder in contact with stamps. Problems of mechanical engineering. 2000. Vol. 3, № 3-4. P. 99-106.
  29. Lobanova L. S., Sinekop N. S. Nonstationary dynamic problems of thermoelaticity in two-dimensional domains. Problems of mechanical engineering. 2001. Vol. 4, № 1-2. P. 108-120.
  30. Sinеkop M. S., Parkhomenko L. O. Calculation of own oscillations of cylinders. Progressive engineering and technology of food production enterprises, catering business and trade. 2009. 1. P. 295-303.
  31. Sinekop N. S. , Parkhomenko L. O. Investigation of the influence of the type of boundary conditions on its own forms of cylinder oscillations. Progressive engineering and technology of food production enterprises, catering business and trade. 2009. Is. 2. P. 276-283.
  32. Sinekop N. S., Lobanova L. S., Parkhomenko L. A. The method of R-functions in the dynamic problems of the theory of elasticity: monograph. Kharkiv: Kharkiv State University of Food Technology and Trade, 2015. 95 p.
  33. Bodnar Yu., Sinekop N. Approximate solution of the generalized dynamic thermoelasticity problem by the R-functions method. Theory and practice of rational designing, manufacturing and exploitation of machine-building constructions: 3rd International sci.-tech. conf. Lviv, 2012. P. 17-18.
  34. Bodnar Yu., Sinekop N. Application of R-functions to heat conduction problems taking into account relaxation heat flow. 13-th International Symposium of Ukrainian Mechanical Engineers in Lviv. Lviv, 2013. P. 24-25.
  35. Bodnar Yu. R-functions method in unsteady heat conduction problems generalized thermomechanics. Visnyk LNAU: Architecture and agricultural-building construction. 2014. No. 15. P. 21-25.
  36. Bodnar Yu., Bar V. Structure of the solution generalized Neumann problem for the heat equation. 13-th International Symposium of Ukrainian Mechanical Engineers in Lviv. Lviv, 2017. P. 22-23.