The study of heat transfer and stress-strain state of a material, taking into account its fractal structure

: pp. 400–409
Received: May 21, 2020
Accepted: September 09, 2020
Ukrainian National Forestry University
Ukrainian National Forestry University
Lviv Polytechnic National University

In the work, on the basis of the apparatus of fractional integro-differentiation, the mathematical models of heat-and-moisture transfer and of deformation-relaxation processes in the medium with "memory" effects and self-organization are constructed.  Numerical implementation of the mathematical models of heat-transfer and moisture-transfer is based on the adaptation of the predictor-corrector method.  That is why the mathematical models obtained in this work are in a finite-difference form.  For the explicit difference scheme, the stability conditions are determined on the basis of the method of conditionally defined known functions as well as by means of the Fourier integral method.  An integral representation of the deformation and stress of the fractional-differential rheological model is obtained using the Laplace transform method.  Including the numerical and analytical methods of implementation of the constructed models, in this paper, the main results are presented, in particular, identification of fractal parameters for the creep function according to the experimental data.

  1. Sokolovskyy Ya., Levkovych M., Mokrytska O., Kaplunskyy Ya.  Mathematical models of biophysical processes taking into account memory effects and self-similarity.  Informatics and Data-Driven Medicine. 2255, 215–228 (2018).
  2. Sokolovskyy Ya., Levkovych M., Shymanskyi V.  Mathematical Modeling of Non-Isothermal Moisture Transfer and Visco-Elastic Deformation in the Materials with Fractal Structure.  Proceedings of the 2016 IEEE 11th International Scientific and Technical Conference on Computer Sciences and Information Technologies.  91–95 (2016).
  3. Uchajkin V.  Method of fractional derivatives.  Publishing house "Artishok", Ulyanovsk (2008).
  4. Kostrobij P., Markovych B., Viznovych B., Zelinska I., Tokarchuk M.  Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations. Mathematical Modeling and Computing. 6 (1), 58–68 (2019).
  5. Kostrobij P. P., Markovych B. M., Ryzha I. A., Tokarchuk M. V.  Generalized kinetic equation with spatio-temporal nonlocality.  Mathematical Modeling and Computing. 6 (2), 289–296 (2019).
  6. Welch S., Rorrer R., Duren R.  Application of time-based fractional calculus method to viscoelastic creep and stress relaxation of materials.  Mech. Time-Dependent Materials. 3 (3), 279-303 (1999).
  7. Sokolovskyy Ya., Levkovych M., Shymanskyi V., Yarkun V.  Mathematical and software providing of research of deformation and relaxation processes in environments with fractal structure.  Proceedings of the 12th International Scientific and Technical Conference on Computer Sciences and Information Technologies. 24–27 (2017).
  8. Golub V. P., Pavlyuk Ya. V., Fernati P. V.  Determining the parameters of fractional exponential hereditary kernels for nonlinear viscoelastic materials.  International Applied Mechanics. 49 (2), 220–231 (2013).
  9. Zhao B., Zhang Y., Wang G., Zhang H., Zhang J., Jiao F.  Identification of Fractal Scale Parameter of Machined Surface Profile.  Applied Mechanics and Materials. 42, 209–214 (2011).
  10. Pyanilo Ya., Vasyunyk M., Vasyunyk I.  The use of Lagerra polynomials to the spectral method for solving equations of fractional derivatives over time.  Physical and mathematical modeling and information technologies. 17, 163–168 (2013).
  11. Gafiychuk V., Datsko B.  Different types of instabilities and complex dynamics in reaction-diffusion systems with fractional derivatives.  J. Comput. Nonlinear Dynam. 7 (3), 031001 (2012).
  12. Meilanov R. P., Shabanova M. R.  Peculiarities of solutions to the heat conduction equation in fractional derivatives.  Technical Physics. 56, 903–908 (2011).
  13. Kexue L., Jigen P.  Laplace transform and fractional differential equations.  Appl. Math. Lett. 24 (12), 2019–2023 (2011).
  14. Sokolovskyy Ya, Sinkevych O.  Calculation of the drying agent in drying chambers.  Proceedings of the 2017 14th International Conference The Experience of Designing and Application of CAD Systems in Microelectronic. 27–31 (2017).
  15. Yang Q., Liu F., Turner I.  Numerical methods for fractional partial differential equation with Riesz space fractional derivatives.  Applied Mathematical Modelling. 34 (1), 200–218 (2010).
  16. Al-Khaled K.  Numerical solution of time-fractional partial differential equations using Sumudu decomposition method.  Rom. Journ. Phys.  60 (1–2), 1–12 (2015).
  17. Boyko S., Eroshenko A.  Modeling of physico-mechanical properties of modified wood by finite element method.  Technical Sciences and Technologies. 2 (4), 184–188 (2016).
  18. Sokolovskyy Ya., Levkovych M., Mokrytska O., Atamanvuk V.  Mathematical Modeling of Two-Dimensional Deformation-Relaxation Processes in Environments with Fractal Structure.  Proceedings of the 2018 IEEE 2nd International Conference on Data Stream Mining and Processing (DSMP), Lviv, 21–25 August 2018, 375–380 (2018).
  19. Podlubny I.  Fractional Differential Eguations.  Vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA (1999).
  20. Samko S., Kilbas S., Marichev O.  Integrals and derivatives of fractional order and some of their. Science and Technology. Minsk (1987).
  21. Lavrenyuk M.  Models of real deformable solids of inhomogeneous media.  Kyiv National University. Taras Shevchenko, Kyiv (2012).
  22. Birger I., Mavlyutov R.  Materials resistance: textbook.  Nauka, Phys. Math. Met. (1986).
  23. Bodig J., Jayne B.  Mechanics of wood and wood composites.  Krieger Publishing Company (1993).
  24. Liu T.  Creep of wood under a large span of loads in constant and varying environments.  Experimental observations and analysis.  Holz als Roh und Werkstoff. 51, 400-405 (1993).
Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 400–409 (2020)