#### 539.38:621.01:004.7 Numerical simulation the post-buckling nonlinear deformation of axisymmetric membranes

##### Podkopaev S. A. (Bauman Moscow State Technical University)

###### NONLINEAR STRAINING, THIN-WALLED AXISYMMETRIC SHELL, MEMBRANE, POST-BUCKLING BEHAVIOR, DISCRETE SWITCHING, CONTINUATION BY PARAMETER, CHANGE OF THE SUBSPACE OF PARAMETERS

doi: 10.18698/2309-3684-2020-1-6487

The theoretical foundations of nonlinear straining of thin-walled axisymmetric shells are considered. The operational characteristics of the membranes in various switching devices, valves and pressure sensors are presented. The types of non-linear behavior of post-buckling behavior of axisymmetric membranes are considered. A mathematical model is presented to describe nonlinear straining of axisymmetric membranes, a discrete continuation by parameter method, and the “changing the subspace of control parameters” technique. Using the hinged spherical shell as an example, a study of post-buckling behavior is performed. A rational mathematical model has been selected to describe nonlinear straining of thin-walled axisymmetric shells. A numerical algorithm for studying the processes of nonlinear straining of multi-parameter systems has been developed and implemented as an author program.

 Alfutov N.A. Osnovy rascheta na ustojchivost' uprugih sistem. Biblioteka raschetchika [Fundamentals of calculation on the stability of elastic systems. Calculator library]. Moscow, Mashinostroenie Publ., 1977, 488 p.
 Andreeva L.E. Uprugie elementy priborov: uchebnoe posobie [Elastic elements of devices: Tutorial]. Moscow, Mashinostroenie Publ., 1982, 456 p.
 Biderman V.L. Mekhanika tonkostennyh konstrukcij. Statika. Biblioteka raschetchika [Mechanics of thin-walled designs. Statics. Calculator library]. Moscow, Mashinostroenie Publ., 1977, 488 p.
 Valishvili N.V. Metody rascheta obolochek vrashcheniya na ECVM [Methods for calculating the shells of rotation on electronic digital computer]. Moscow, Mashinostroenie Publ., 1976, 278 p.
 Volmir A.S. Ustojchivost' deformiruemyh sistem [Resistance of deformable systems]. Moscow, Fizmatgiz Publ., 1967, 984 p.
 Gavrushin S.S. Razrabotka metodov rascheta i proektirovaniya uprugih obolochechnyh konstrukcij pribornyh ustrojstv: dissertaciya [Development of methods for calculating and designing elastic shell structures of instrument devices: dissertation]. Moscow, 1994, 316 p.
 Gavrushin S.S., Baryshnikova O.O., Boriskin O.F. Chislennye metody v dinamike i prochnosti mashin [Numerical methods in dynamics and strength of machines]. Moscow, BMSTU Publ., 2012, 492 p.
 Grigolyuk E.I., Lopanitsyn E.A. Konechnye progiby, ustojchivost' i zakriticheskoe povedenie tonkih pologih obolochek [Finite deflections, stability and post-buckling behavior of thin shallow shells]. Moscow, Moscow State University of Mechanical Engineering Publ., 2004, 162 p.
 Grigolyuk E.I., Kabanov V.V. Ustojchivost' obolochek [The stability of the shells]. Moscow, Nauka Publ., 1978, 359 p.
 Il'gamov M.A. Trudy Matematicheskogo centra im. N.I. Lobachevskogo — Proceedings of the Lobachevsky Mathematical center, 2010, vol. 42, pp. 5–19.
 Podkopaev S.A., Gavrushin S.S., Nikolaeva A.S. Analiz processa nelinejnogo deformirovaniya gofrirovannyh membrane [Analysis of the process of nonlinear straining of corrugated membranes]. Sb. tr. Matematicheskoe modelirovanie i eksperimental'naya mehanika deformiruemogo tverdog [Collection of works. Mathematical modeling and experimental mechanics of a deformable solid], 2017, iss. 1, pp. 31–36.
 Podkopaev S.A., Gavrushin S.S., Nikolaeva A.S., Podkopaeva T.B. Raschet rabochej harakteristiki perspektivnyh konstrukcij mikroaktyuatorov [Calculation of the working characteristics of promising designs microactuators]. Sb. tr. Matematicheskoe modelirovanie i eksperimental'naya mehanika deformiruemogo tverdog [Collection of works. Mathematical modeling and experimental mechanics of a deformable solid], 2017, iss. 1, pp. 45–51.
 Ponomarev S.D., Biderman V.L., Likharev K.K. et al. Raschety na prochnost' v mashinostroenii. T.2. [Strength calculations in mechanical Engineering]. Moscow, Mashgiz Publ., 1958, 975 p.
 Feodosyev V.I. Uprugie elementy tochnogo priborostroeniya [Elastic elements of precision instrument engineering]. Moscow, Oborongiz Publ., 1949, 342 p.
 Feodosyev V.I. Prikl. matematika i mekhanika — Applied Mathematics and Mechanics, 1946, vol. 10, no. 2, pp. 295–306.
 Gavriushin S.S. Matematicheskoe modelirovanie i chislennye metody — Mathematical modeling and Computational Methods, 2014, no. 1, pp. 115–130.
 Belhocine A. Exact analytical solution of boundary value problem in a form of an infinite hypergeometric series. International Journal of Mathematical Sciences and Computing (IJMSC), 2017, vol. 3, no. 1, pp. 28–37. DOI: 10.5815/ijmsc.2017.01.03
 Crisfield M.A. A fast incremental/iterative solution procedure that handles "snapthrought". Cоmput. and Structures, 1981, vol. 13, no. 1, pp. 55–62.
 Chuma F.M., Mwanga G.G. Stability analysis of equilibrium points of newcastle disease model of village chicken in the presence of wild birds reservoir. International Journal of Mathematical Sciences and Computing (IJMSC), 2019, vol. 5, no. 2, pp. 1–18. DOI:10.5815/ijmsc.2019.02.01
 Gupta N.K., Venkatesh. Experimental and numerical studies of dynamic axial compression of thin walled spherical shells. Int. J. of Impact engineering, 2004, vol. 30, pp. 1225–1240. Marguerre, K.: Zur: Theorie der gerkrümmten Platte groβer Formänderung. Proceedings of the Fifth International Congress for Applied Mechanics, 1939, pp. 93–101.
 Moshizi M.M., Bardsiri A.K. The application of metaheuristic agorithms in automatic software test case generation, International Journal of Mathematical Sciences and Computing (IJMSC), 2015, vol.1, no. 3, pp. 1–8. DOI: 10.5815/ijmsc.2015.03.01
 Mescall J. Numerical solution of nonlinear equations for shell of revolution. AIA
 A J., 1966, vol. 4, no. 11, pp. 2041–2043.
 Reissner E. Alisymmetrical deformations of thin shells of revolution. Proc. of Symp. In Appl. Math., Ameг. Math. Soc., 1950, vol. 3, pp. 27–52.
 Riks E. The application of Newton's method to the problem of elastic stability. J. Appl. Mech., 1972, vol. 39, pp. 1060–1065.
 Crisfield M.A. A fast incremental/iterative solution procedure that handles "snapthrought". Cоmput. and Structures, 1981, vol. 13, No 1, pp. 55–62.

Подкопаев С.А. Численное моделирование закритического нелинейного деформирования осесимметричных мембран. Математическое моделирование и численные методы, 2020, № 1, с. 64–87.