Parameter differentiation method in solution of axisymmetric soft shells stationary dynamics nonlinear problems

E. A. Korovaytseva

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 3, pp. 556-570.

Résumé

An algorithm of axisymmetric unbranched soft shells nonlinear dynamic behaviour problems solution is suggested in the work. The algorithm does not impose any restrictions on deformations or displacements range, material properties, conditions of fixing or meridian form of the structure. Mathematical statement of the problem is given in vector-matrix form and includes system of partial differential equations, system of additional algebraic equations, structure segments coupling conditions, initial and boundary conditions. Partial differential equations of motion are reduced to nonlinear ordinary differential equations using method of lines. Obtained equation system is differentiated by calendar parameter. As a result problem solution is reduced to solving two interconnected problems: quasilinear multipoint boundary problem and nonlinear Cauchy problem with right-hand side of a special form. Features of represented algorithm using in application to the problems of soft shells dynamics are revealed at its program realization and are described in the work. Three- and four-point finite difference schemes are used for acceleration approximation. Algorithm testing is carried out for the example of hinged hemisphere of neo-hookean material dynamic inflation. Influence of time step and acceleration approximation scheme choice on solution results is investigated.

Détail

Keywords: soft shell, hyperelastic material, dynamic inflation, method of lines, parameter differentiation method, physical nonlinearity, geometrical nonlinearity.

Bibliographie

[1] Druz' B. I., Druz' I. B., Teoriia miagkikh obolochek [Theory of Soft Shells], Maritime State Univ., Vladivostok, 2003, 381 pp. (In Russian)

[2] Ridel' V. V., Gulin B. V., Dinamika miagkikh obolochek [Dynamics of Soft Shells], Nauka, Moscow, 1990, 204 pp. (In Russian)

[3] Gimadiev R. Sh., Dinamika miagkikh obolochek parashiutnogo tipa [Dynamics of Soft Parachute-Type Shells], Kazan State Energ. Univ., Kazan, 2006, 208 pp. (In Russian)

[4] Libai A., Simmonds J. G., The Nonlinear Theory of Elastic Shells, Cambridge Univ. Press, Cambridge, 1998, 560 pp. | DOI | Zbl

[5] Lyalin V. V., Morozov V. I., Ponomarev A. T., Parashiutnye sistemy. Problemy i metody ikh resheniia [Parachute Systems. Problems and Methods of Their Solution], Fizmatlit, Moscow, 2009, 576 pp. (In Russian)

[6] Knowles J. K., “On a class of oscillations in the finite deformation theory of elasticity”, J. Appl. Mech., 29:2 (1962), 283–286 | DOI | Zbl

[7] Akkas N., “On the dynamic snap-out instability of inflated non-linear spherical membranes”, Int. J. Non-Linear Mechanics, 13:3 (1978), 177–183 | DOI | Zbl

[8] Calderer C., “The dynamical behaviour of nonlinear elastic spherical shells”, J. Elasticity, 13 (1983), 17–47 | DOI | Zbl

[9] Verron E., Khayat R. E., Derdouri A., Peseux B., “Dynamic inflation of hyperelastic spherical membranes”, J. Rheology, 43:5 (1999), 1083–1097 | DOI

[10] Yuan X. G., Zhang R. J., Zhang H. W., “Controllability conditions of finite oscillations of hyperelastic cylindrical tubes composed of a class of Ogden material models”, Comput. Mater. Continua, 7:3 (2008), 155–166

[11] Ren J., “Dynamical response of hyper-elastic cylindrical shells under periodic load”, Appl. Math. Mech., 29:10 (2008), 1319–1327 | DOI | Zbl

[12] Ren J., “Dynamics and destruction of internally pressurized incompressible hyper-elastic spherical shells”, Int. J. Eng. Sci., 47:7–8 (2009), 745–753 | DOI | Zbl

[13] Yong H., He X., Zhou Y., “Dynamics of a thick-walled dielectric elastomer spherical shell”, Int. J. Eng. Sci., 49:8 (2011), 792–800 | DOI

[14] Ju Y., Niu D., “On a class of differential equations of motion of hyperelastic spherical membranes”, Appl. Math. Sci., 6:83 (2012), 4133–4136 | Zbl

[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”, Int. J. Eng. Sci., 93 (2015), 31–45 | DOI

[16] Zhao Zh., Zhang W., Zhang H., Yuan X., “Some interesting nonlinear dynamic behaviors of hyperelastic spherical membranes subjected to dynamic loads”, Acta Mechanica, 230:8 (2019), 3003–3018 | DOI | Zbl

[17] Shahinpoor M., Balakrishnan R., “Large amplitude oscillations of thick hyperelastic cylindrical shells”, Int. J. Non-Linear Mechanics, 13:5–6 (1978), 295–301 | DOI | Zbl

[18] Wang A. S. D., Ertepinar A., “Stability and vibrations of elastic thick-walled cylindrical and spherical shells subjected to pressure”, Int. J. Non-Linear Mechanics, 7:5 (1972), 539–555 | DOI | Zbl

[19] Akyüz U., Ertepinar A., “Stability and asymmetric vibrations of pressurized compressible hyperelastic cylindrical shells”, Int. J. Non-Linear Mechanics, 34:3 (1999), 391–404 | DOI | Zbl

[20] Zhu Y., Luo X. Y., Wang H. M., Ogden R. W., Berry C., “Three-dimensional non-linear buckling of thick-walled elastic tubes under pressure”, Int. J. Non-Linear Mechanics, 48:1 (2013), 1–14 | DOI

[21] Soares R. M., Gonc̨alves P. B., “Large-amplitude nonlinear vibrations of a Mooney–Rivlin rectangular membrane”, J. Sound Vibration, 333:13 (2014), 2920–2935 | DOI

[22] Gorissen B., Melancon D., Vasios N., Torbati M., Bertoldi K., “Inflatable soft jumper inspired by shell snapping”, Science Robotics, 5:42 (2020), eabb1967 | DOI

[23] Rogers C., Saccomandi G., Vergori L., “Helmholtz-type solitary wave solutions in nonlinear elastodynamics”, Ricerche Mat., 69:1 (2019), 327–341 | DOI

[24] Pucci E., Saccomandi G., Vergori L., “Linearly polarized waves of finite amplitude in pre-strained elastic materials”, Proc. R. Soc. Lond., Ser. A, 475:2226 (2019) | DOI

[25] Johnson D. E., Greif R., “Dynamic response of a cylindrical shell. Two numerical methods”, AIAA Journal, 4:3 (1965), 486–494 | DOI

[26] Korovaytseva E. A., “On some features of soft shells of revolution static problems solution at large deformations”, Trudy MAI, 114 (2020), 34 pp. (In Russian) | DOI

[27] Shapovalov L. A., “Elasticity equations for a thin shell at nonaxisymmetric deformation”, Izv. Akad. Nauk SSSR. MTT, 3 (1976), 62–72 (In Russian)

[28] Houbolt J. C., “A recurrence matrix solution for the dynamic response of elastic aircraft”, J. Aeronaut. Sci., 17:9 (1950), 540–550 | DOI

[29] Levy S., Kroll W. D., “Errors introduced by finite space and time increments in dynamic response computation”, J. Res. Natl. Bur. Stand., 51:1 (1953), 57–68 | DOI

[30] Korovaytseva E. A., “Combined equations of theory of soft shells”, Trudy MAI, 108 (2019), 17 pp. (In Russian) | DOI

[31] Amabili M., Nonlinear Mechanics of Shells and Plates in Composite, Soft and Biological Materials, Cambridge Univ. Press, Cambridge, 2018, xvi+568 pp. | DOI | Zbl

[32] Grigolyuk E. I., Shalashilin V. I., Problems of Nonlinear Deformation. The Continuation Method Applied to Nonlinear Problems in Solid Mechanics, Springer, Berlin, 1991, viii+262 pp. | DOI