Homogenization of the static anti-plane shear model for the reinforced composite

E. I. Leonova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 1042-1063

Résumé

The static problem of anti-plane shear of a thermoelastic composite, stitched with reinforcing threads, is considered. The original formulation contains a small positive parameter $\varepsilon$, which characterizes the distance between neighboring threads. It is also assumed that the thermomechanical characteristics of the composite body depend on $\varepsilon$. The asymptotic behavior of solutions as the parameter $\varepsilon$ tends to zero is investigated. The limiting transition as $\varepsilon \rightarrow 0+$ is mathematically rigorously justified and represents a homogenization procedure. This transition is based on the application of the standard Allaire–Nguetseng two-scale convergence method and its version by G. Allaire, A. Damlamian, U. Hornung for homogenization on thin inclusions. The result consists of the construction of a limit averaged model of anti-plane shear of the composite material. Using the newly obtained model, numerical experiments are performed, which show consistency of the theoretical conclusions.

Détail

Keywords: composite material, thin inclusion, anti-plane shear, homogenization, numerical experiment.

Bibliographie

[1] S.A. Sazhenkov, I.V. Fankina, A.I. Furtsev, P.V. Gilev, A.G. Gorynin, O.G. Gorynina, V.M. Karnaev, E.I. Leonova, “Multiscale analysis of a model problem of a thermoelastic body with thin inclusions”, Sib. Èlektron. Mat. Izv., 18:1 (2021), 282–318 | DOI | Zbl

[2] V.V. Zhikov, “Homogenization of elasticity problems on singular structures”, Izv. Math., 66:2 (2002), 299–365 | DOI | Zbl

[3] A.I. Furtsev, E.M. Rudoy, “Variational approach to modeling soft and stiff interfaces in the Kirchhoff-Love theory of plates”, Int. J. Solids Struct., 202 (2020), 562–574 | DOI

[4] E.M. Rudoy, “Asymptotic modelling of bonded plates by a soft thin adhesive layer”, Sib. Èlectron. Mat. Izv., 17 (2020), 615–625 | DOI | Zbl

[5] A.I. Furtsev, I.V. Fankina, A.A. Rodionov, D.A. Ponomarev, “Asymptotic modeling of steady vibrations of thin inclusions in a thermoelastic composite”, Z. Angew. Math. Phys., 74:5 (2023), 195 | DOI | Zbl

[6] S. Dumont, F. Lebon, M.L. Raffa, R. Rizzoni, “Towards nonlinear imperfect interface models including micro-cracks and smooth roughness”, Ann. Solid Struct. Mech., 9 (2017), 13–27 | DOI

[7] S. Dumont, R. Rizzoni, F. Lebon, E. Sacco, “Soft and hard interface models for bonded elements”, Composites Part B: Engineering, 153 (2018), 480–490 | DOI

[8] R. Rizzoni, F. Lebon, “Imperfect interfaces as asymptotic models of thin curved elastic adhesive interphases”, Mech. Res. Commun., 51 (2013), 39–50 | DOI

[9] I.V. Fankina, A.I. Furtsev, E.M. Rudoy, S.A. Sazhenkov, “The homogenized quasi-static model of a thermoelastic composite stitched with reinforcing threads”, J. Comput. Appl. Math., 434 (2023), 115346 | DOI | Zbl

[10] I.V. Fankina, A.I. Furtsev, E.M. Rudoy, S.A. Sazhenkov, “Multiscale analysis of stationary thermoelastic vibrations of a composite material”, Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences, 380:2236 (2022), 20210354 | DOI

[11] V.D. Kupradze, T.G. Gegelia, M.O. Bašeleǐšvili, T.V. Burčuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, Amsterdam etc., 1979 | Zbl

[12] G. Allaire, A. Damlamian, U. Hornung, “Two-scale convergence on periodic surfaces and applications”, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (1995), eds. A. Bourgeat et al., World Scientific Pub., Singapore, 1996, 15–25

[13] G. Allaire, “Homogenization and two-scale convergence”, SIAM J. Math. Anal., 23:6 (1992), 1482–1518 | DOI | Zbl

[14] G. Nguetseng, “A general convergence result for a functional related to the theory of homogenization”, SIAM J. Math. Anal., 20:3 (1989), 608–623 | DOI | Zbl

[15] E. Sanchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Physics, 127, Springer, Berlin etc., 1980 | DOI | Zbl

[16] N.S. Bakhvalov, G.P. Panasenko, Homogenisation: Averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Kluwer, Dordrecht etc., 1989 | DOI | Zbl