Старушенко Г. Asymptotic Methods and Models in the Mechanics of Composite Materials

Українська версія

Thesis for the degree of Doctor of Science (DSc)

State registration number

0524U000179

Thesis Registration Form

0524U000179.pdf

Applicant for

Specialization

  • 05.23.17 - Будівельна механіка

14-06-2024

Specialized Academic Board

Д 08.085.02

Prydniprovsk State Academy of Civil Engineering and Architecture, Dnipropetrovs'k, Ukraine

Essay

The research is devoted to the construction of asymptotic models and the development of mathematical methods that make it possible to correctly describe composites of various structures and the physical processes occurring in them, including in the regions of limiting values of their physical and geometric parameters. The following asymptotic models are constructed, mathematically described and physically substantiated in the scientific work: three-phase models of fibrous composites with circular and square inclusions; improved three-phase model; models of lubrication approach; model of two dimensional composite hexagonal structures; models of two-phase fibrous composites for inclusions of various shapes and “inclusion – matrix” contact conditions; models of composites with a thin interlayer at the phase boundary; two-phase models of composites; continuum models of a 1D discrete medium; continuum model with chaotic behaviour. The areas of applicability of the obtained asymptotic models are determined, asymptotic and numeric estimates of their accuracy are given, and a qualitative and quantitative analysis of the reliability of the results is carried out. The problems of determining the effective thermal conductivity of a composite material with periodic cylindrical inclusions of a circular and square cross-section are analysed. Defining mathematical relationships are derived on the basis of a three-phase composite model. The analytical expressions for the effective coefficients are obtained in the zero-order approximation and the corrections in the first-order approximation by the boundary shape perturbation method. This correction allows taking into account the geometry of inclusions, not just their volume fraction. The possibilities of generalizing the three-phase composite model are discussed. The obtained solution with Padé approximants fundamentally expands the applicability limits of the model. A modified three-phase composite model gives reliable results in the whole area of change of both composite parameters – geometric and physical. Solutions for densely packed, high-contrast fibre composites with different structure and shape of inclusions are obtained based on the lubrication approach. The solution of the problem of natural vibrations of a rectangular membrane rigidly clamped along the contour, which is a composite structure with periodically arranged in a hexagonal lattice of circular inclusions, is obtained. The apparatus of asymptotically equivalent functions is used to study the limit states of composite structures. Composite material with periodically distributed cylindrical inclusions of square cross-section is investigated, and analytical interpolation formula of asymptotic expansions for the effective thermal conductivity is obtained. Employing the non-smooth argument substitution for description of local and discretely varying properties of the inhomogeneous structures and using the method of asymptotically equivalent functions, models are constructed, and asymptotic representations are obtained for fibrous composites of various structures. Different conditions of “matrix – inclusion” contact in composite structures are investigated and mathematically described. Asymptotic approximations in models of two-phase fiber composites are analysed; the concept of physical equivalence of composite structures is defined and relations for their effective parameters are given. The asymptotic expressions of the effective parameters of composites of different structures are classified – when the contact conditions at the interface of the composite phases change. Higher approximations of the Maxwell formula are constructed based on the two-phase composite model and alternating Schwarz method. Generalizing relations for the Maxwell formula in the case of circular inclusions are constructed using the Schwarz – Padé expansion. Questions of higher order asymptotic homogenization for dynamic problems are investigated. The solution of the periodic problem of the theory of elasticity for two-phase layered composite massif problem is obtained by application of the technique of non-symmetrical saw-tooth argument transformation method. The non-smooth temporal transformation is applied to construct a periodic solutions of a weakly non-linear system under the parametric impulsive excitation. The various continuum models of a 1D discrete media are considered: classical, intermediate, quasi-continuum, and improved quasi-continuum models. The impact of symmetry change of asymptotic wave behaviour during the transition from discrete to continuous media is analysed for the Lagrange lattice. Continualization procedure for Verhulst-like ordinary differential equations based on Padé approximants is considered. The presence of chaos in a continuous system is proven correctly by calculation the Lyapunov exponents and the Lyapunov dimensions.

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Research papers

1. Andrianov I. V., Starushenko G. A., Danishevs’kyy V. V., Tokarzewski S. Homogenization procedure and Padé approximants for effective heat conductivity of composite materials with cylindrical inclusions having square cross-section. The Royal Society Proceedings: Mathematical, Physical and Engineering Sciences. 1999. Vol. 455. Is. 1989. P. 3401–3413.

2. Pilipchuk V. N., Volkova S. A., Starushenko G. A. Study of a non-linear oscillator under parametric impulsive excitation using a non-smooth temporal transformation. Journal of Sound and Vibration. 1999. Vol. 222. Is. 2. P. 307–328.

3. Starushenko G., Krulik N., Tokarzewski S. Employment of non-symmetrical saw-tooth argument transformation method in the elasticity theory for layered composites. International Journal of Heat and Mass Transfer. 2002. Vol. 45. Is. 14. P. 3055–3060.

4. Andrianov I. V., Starushenko G. A., Weicher D. Numerical investigation of 1D continuum dynamical models of discrete chain. ZAMM – Journal of Applied Mathematics and Mechanics. 2012. Vol. 92. Is. 11–12. P. 945–954.

5. Andrianov I. V., Kalamkarov A. L., Starushenko G. A. Three-phase model for a fiber-reinforced composite material. Composite Structures. 2013. Vol. 95. P. 95–104.

6. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Application of an improved three-phase model to calculate effective characteristics for a composite with cylindrical inclusions. Latin American Journal of Solids and Structures. 2013. Vol. 10. No. 1. P. 197–222.

7. Andrianov I. V., Kalamkarov A. L., Starushenko G. A. Analytical expressions for effective thermal conductivity of composite materials with inclusions of square cross-section. Composites: Part B. 2013. Vol. 50. P. 44–53.

8. Kalamkarov A. L., Andrianov I. V., Starushenko G. A. Three-phase model for a composite material with cylindrical circular inclusions. Part I: Application of the boundary shape perturbation method. International Journal of Engineering Science. 2014. Vol. 78. P. 154–177.

9. Kalamkarov A. L., Andrianov I. V., Starushenko G. A. Three-phase model for a composite material with cylindrical circular inclusions. Part II: Application of Padé approximants. International Journal of Engineering Science. 2014. Vol. 78. P. 178–191.

10. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Asymptotic analysis of the Maxwell Garnett formula using the two-phase composite model. International Journal of Applied Mechanics. 2015. Vol. 7. No. 02. P. 1550025-1–1550025-27.

11. Andrianov I. V., Awrejcewicz J., Markert B., Starushenko G. A. Analytical homogenization for dynamic analysis of composite membranes with circular inclusions in hexagonal lattice structures. International Journal of Structural Stability and Dynamics. 2017. Vol. 17. Is. 5. P. 1740015-1–1740015-14.

12. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Asymptotic models and transport properties of densely packed, high-contrast fibre composites. Part I: Square lattice of circular inclusions. Composite Structures. 2017. Vol. 179. P. 617–627.

13. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Asymptotic models for transport properties of densely packed, high-contrast fibre composites. Part II: Square lattices of rhombic inclusions and hexagonal lattices of circular inclusions. Composite Structures. 2017. Vol. 180. P. 351–359.

14. Andrianov I., Starushenko G., Kvitka S., Khajiyeva L. The Verhulst-like equations: integrable OΔE and ODE with chaotic behavior. Symmetry. 2019. Vol. 11. Is. 12. P. 1446.

15. Andrianov I. I., Awrejcewicz J., Starushenko G. A., Gabrinets V. A. Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method. Acta Mechanica. 2020. Vol. 231. Is. 12. P. 4971–4990.

16. Andrianov I. I., Awrejcewicz J., Starushenko G. A., Gabrinets V. A. Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part II: using Padé approximants. Acta Mechanica. 2020. Vol. 231. Is. 12. P. 5145–5157.

17. Andrianov I., Koblik S., Starushenko G. Transition from discrete to continuous media: the impact of symmetry changes on asymptotic behavior of waves. Symmetry. 2021. Vol. 13. Is. 6. P. 1008.

18. Andrianov I. I., Andrianov I. V., Starushenko G. A., Borodin E. I. Higher order asymptotic homogenization for dynamical problems. Mathematics and Mechanics of Solids. 2022. Vol. 27. Is. 9. P. 1672–1687.

19. Andrianov I. V., Koblik S. G., Starushenko G. A., Kudaibergenov A. K. On aspects of gradient elasticity: Green’s functions and concentrated forces. Symmetry. 2022. Vol. 14. Is. 2. P. 188.

20. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Non-stationary heat transfer in composite membrane with circular inclusions in hexagonal lattice structures. Acta Mechanica. 2022. Vol. 233. Is. 4. P. 1339–1350.

21. Andrianov I. V., Awrejcewicz J., Starushenko G. A., Kvitka S. A. Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution. ZAMM – Journal of Applied Mathematics and Mechanics. 2022. Vol. 102. Is. 11. P. e202200216.

22. Andrianov I. V., Awrejcewicz J., Starushenko G. A. Approximate models of mechanics of composites: An asymptotic approach : monograph. CRC Press, Taylor & Francis Group, 2023. 367 p.

23. Старушенко Г. А. Асимптотические методы и модели в теории композитных материалов : монография. LAP LAMBERT Academic Publishing, 2021. 696 c.

24. Pilipchuk V. N., Starushenko G. A. A version of non-smooth transformations for one-dimensional elastic systems with a periodic structure. Journal of Applied Mathematics and Mechanics. 1997. Vol. 61. Is. 2. P. 265–274.

25. Andrianov I. V., Starushenko G. A., Tokarzewski S. Homogenization procedure and Padé approximations in the theory of composite materials with parallelepiped inclusions. International Journal of Heat and Mass Transfer. 1998. Vol. 41. Is. 1. P. 175–181.

26. Starushenko G., Krulik N. Saw-tooth argument transformation method in the theory of composite materials // Progress and Trends in Rheology V : Proceedings of the Fifth European Rheology Conference (Portorož, September 6-11, 1998) / ed. I. Emri. Heidelberg, 1998. P. 120–121.

27. Andrianov I. V., Starushenko G. A., Danishevskyy V. V. Asymptotic determination of the effective thermal conductivity of a pile field. Soil Mechanics and Foundation Engineering. 1999. Vol. 36. Is. 1. P. 31–36.

28. Tokarzewski S., Andrianov I., Danishevsky V., Starushenko G. Analytical continuation of asymptotic expansions of effective transport coefficients by Pade approximants. Nonlinear Analysis: Theory, Methods & Applications. 2001. Vol. 47. Is. 4. P. 2283–2292.

29. Andrianov I. V., Starushenko G. A., Weichert D. Asymptotic analysis of thin interface in composite materials with coated boundary. Technische Mechanik. 2011. Vol. 31. No. 1. P. 33–41.

30. Gluzman S., Mityushev V., Nawalaniec W., Starushenko G. Effective conductivity and critical properties of a hexagonal array of superconducting cylinders // Contributions in Mathematics and Engineering. In Honor of Constantin Carathéodory / eds. P. M. Pardalos, T. M. Rassias. Switzerland : Springer International Publishing, 2016. P. 255–297.

31. Kalamkarov A. L., Andrianov I. V., Pacheco P. M. C. L., Savi M. A., Starushenko G. A. Asymptotic analysis of fiber-reinforced composites of hexagonal structure. Journal of Multiscale Modelling. 2016. Vol. 07. No. 03. P. 1650006-1–1650006-32.

32. Andrianov I. V., Starushenko G. A., Gabrinets V. A. Percolation threshold for elastic problems: self-consistent approach and Padé approximants // Advances in mechanics of microstructured media and structures / eds. F. dell’Isola, V. Eremeyev, A. V. Porubov. Springer International Publishing AG, part of Springer Nature, 2018. Vol. 87. P. 35–42.

33. Kalamkarov A., Andrianov I., Starushenko G. Refinement of the Maxwell formula for fiber-reinforced composites. Journal of Multiscale Modelling. 2020. Vol. 11. No. 01. P. 1950001-1–1950001-33.

34. Andrianov I. V., Starushenko G. A., Kvitka S. A. Calculation of effective characteristics of a 2D composite with rhombic voids using an inhomogeneous cell model. Symmetry. 2023. Vol. 15. Is. 3. P. 646.

35. Andrianov I. V., Awrejcewicz J., Koblik S. G., Starushenko G. A. Nonlinear oscillation of a microbeam due to an electric actuation – comparison of approximate models. ZAMM – Journal of Applied Mathematics and Mechanics. 2023. Vol. 104. Is. 2. P. e202300091.

36. Andrianov I. V., Koblik S. G., Starushenko G. A. Investigation of electrically actuated geometrically nonlinear clamped circular nanoplate. Acta Mechanica. 2024. Vol. 235. Is. 2. P. 1015–1026.

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