Mathematical modeling of panels and arches stress-strain state for the case of large displacements and strains at designing thin-walled aircraft structures of hyperelastic materials

DOI: 10.34759/trd-2023-131-08

Аuthors

Dmitriev V. G.^*, Popova A. R.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: vgd2105@mail.ru
**e-mail: Popova.ar@1566.ru

Abstract

The presented work is aimed at mathematical models and computational algorithms developing for studying specifics of deformation processes of the panel and arc structures from hyper-elastic materials at arbitrary displacements and deformations. The initial continual problem discretization by the spatial variables is being accomplished by the finite difference method with approximation of differential operators by the finite differences of the second-order accuracy. Quasi-dynamic form of the settling method with plotting an explicit two-layer difference scheme by the time of second order accuracy is being employed for building the nonlinear boundary problem solving computational algorithm. The stress-strain state specifics were studied and critical loads values were determined for the locally loaded arc structure from the hyper-elastic material while using the relationships of the Mooney-Rivlin model neo- Hookean model for both fastened and hinge-fixed edges.

Keywords:

arches, panels, hyperelastic materials, finite differences, nonlinear problems, stabilization method, approximation, boundary conditions

References

1. Grigolyuk E.I., Shalashilin V.I. Problemy nelineinogo deformirovaniya: Metod prodolzheniya resheniya po parametru v nelineinykh zadachakh mekhaniki tverdogo deformiruemogo tela (Problems of Nonlinear Deforming: Parameter Continuation Method in Nonlinear Problems of Mechanics of Deformed Solid), Moscow, Nauka, 1988, 232 p.
2. Birger I.A. Sterzhni, plastinki, obolochki (Rods, plates, shells), Moscow, Fizmatlit, 1992, 392 p.
3. Grigolyuk E.I., Mamai V.I. Nelineinoe deformirovanie tonkostennykh konstruktsii (Nonlinear deformation of thin-walled structures), Moscow, Nauka. Fizmatlit, 1997, 272 p.
4. Korovaitsev A.V. Raschet uprugikh obolochek vrashcheniya pri bol’shikh osesimmetrichnykh peremeshcheniyakh. Raschet na prochnost’, zhestkost’, ustoichivost’ i kolebaniya (Calculation of elastic shells of rotation with large axisymmetric displacements), Moscow, Mashinostroenie, 1983, pp. 290-295.
5. Evkin A.Y., Kalamkarov A.L. Analysis of large deflection equilibrium states of composite shells of revolution. Part.1. General model and singular perturbation analysis, International Journal of Solids and Structures, 2001, vol. 38, no. 50-51, pp. 8961-8974. DOI: 10.1016/S0020-7683(01)00184-6
6. Evkin A.Y., Kalamkarov A.L. Analysis of large deflection equilibrium states of composite shells of revolution. Part. 2. Applications and numerical results, International Journal of Solids and Structures, 2001, vol. 38, no. 50-51, pp. 8975-8987. DOI: 10.1016/S0020-7683(01)00185-8
7. Korovaitseva E.A. Trudy MAI, 2019, no. 108. URL: http://trudymai.ru/eng/published.php?ID=109235. DOI: 10.34759/trd-2019-108-1
8. Korovaitseva E.A. Trudy MAI, 2020, no. 114. URL: http://www.trudymai.ru/eng/published.php?ID=118881. DOI: 10.34759/trd-2020-114-04
9. Dmitriev V.G. Uchenye zapiski TsAGI, 2023, no. 1, pp. 76-88.
10. Dmitriev V.G., Danilin A.N., Popova A.R., Pshenichnova N.V. Numerical Analysis of Deformation Characteristics of Elastic Inhomogeneous Rotational Shells at Arbitrary Displacements and Rotation Angles, Computation, 2022, no. 10 (10), pp. 184. DOI: 10.3390/computation10100184
11. Korovaitseva E.A. Problemy prochnosti i plastichnosti, 2021, vol. 83, no. 4, pp. 5-13. DOI: https://doi.org/10.32326/1814-9146-2021-83-4-415-423
12. Adamov A.A. Vestnik Permskogo gosudarstvennogo tekhnicheskogo universiteta. Matematicheskoe modelirovanie sistem i protsessov, 2001, no. 9, pp. 6-9.
13. Bondar’ V.D. Prikladnaya mekhanika i tekhnicheskaya fizika, 1990, no. 2, pp. 155-164.
14. Treolar L.R.G. The physics of rubber elasticity, Oxford, Clarendon Press, 1975, 310 p.
15. Rodríguez—Martínez J.A., Fernández—Sáez J., Zaera R. The role of constitutive relation in the stability of hyper-elastic spherical membranes subjected to dynamic inflation, International Journal of Engineering Science, 2015, vol. 93, pp. 31–45. DOI: 10.1016/j.ijengsci.2015.04.004
16. Rivlin R.S. Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philosophical Transactions of the Royal Society of London, 1948, vol. 241, pp. 379-397. DOI: 10.1098/rsta.1948.0024
17. Samarskii A.A. Teoriya raznostnykh skhem (Theory of difference schemes), Moscow, Nauka, 1989, 616 p.
18. Bakhvalov N.S., Zhidkov N.P., Kobel’kov G.M. Chislennye metody (Numerical methods), Moscow, Fizmatlit. Laboratoriya Bazovykh Znanii, 2001, 632 p.
19. Dmitriev V. Applied Mathematic Technologies in Nonlinear Mechanics of Thin-Walled Constructions. Chapter 4. Book Mathematics Applied to Engineering and Management Sciences. Edited by Mangey Ram and S. B. Singh. CRC Press Taylor & Francis Group. BocaRaton, 2019, pp. 71-116. DOI: 10.1201/9781351123303-4
20. Dmitriev V.G., Preobrazhenskii I.N. Izvestiya AN SSSR. Mekhanika tverdogo tela, 1988, no. 1, pp. 177 — 184.
21. Turvey G.J., Der Avanessian N.G.V. Elastic large deflection of circular plates using graded finite-differences, Somruters & Structures, 1986, vol. 23, no. 6, pp. 763-774. DOI: 10.1016/0045-7949(86)90244-0

Download