Nonlinear dynamics of loaded visco-hyperelastic spherical shells
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ORIGINAL PAPER
Nonlinear dynamics of loaded visco-hyperelastic spherical shells Zhentao Zhao . Datian Niu . Hongwu Zhang . Xuegang Yuan
Received: 23 November 2019 / Accepted: 29 July 2020 Ó Springer Nature B.V. 2020
Abstract In this paper, the nonlinear dynamic behaviors, especially, limit cycles and chaos, are investigated for the spherical shell composed of a class of visco-hyperelastic materials subjected to uniform radial loads at its inner and outer surfaces. To include the thickness effect, a more general model compared with the membrane and thin plate is proposed to investigate the dynamic characteristics of the viscohyperelastic structure. Then, the coupled integrodifferential equations describing the radially symmetric motion of the spherical shell are derived in terms of the variational principle and the finite viscoelasticity theory. Due to both the geometrical and physical nonlinearities, there exists an asymmetric homoclinic orbit for the hyperelastic structure. Particularly, under constant loads, the system converges to a stable equilibrium point, and the convergence position and speed are closely related to both the initial condition and the The graphics program used in the paper ‘‘Nonlinear dynamics of loaded visco-hyperelastic spherical shells’’ is Mathematic V11.3. Z. Zhao H. Zhang State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China D. Niu X. Yuan (&) School of Science, Dalian Minzu University, Dalian 116600, China e-mail: [email protected]
viscosity because of the existence of different basins, while under periodic loads, some complex phenomena, such as the limit cycles and chaos, are found, and the chaotic phenomena are analyzed by the bifurcation diagram and Lyapunov exponent. Moreover, by numerical analyses, parametric studies are carried out to illustrate the effects of viscosity, load amplitude, external frequency and initial condition. Keywords Visco-hyperelastic spherical shell Dynamic load Geometrical and physical nonlinearities Asymmetric homoclinic orbit Limit cycle and chaos
1 Introduction Rubber is a typical hyperelastic material, and it has the features of large deformation, light weight and low cost [1, 2]. In practical applications, this polymeric material simultaneously exhibits elastic and viscous material characteristics [3]. The mechanical behavior is rate-dependent and exhibits hysteresis upon cyclic loading [4]. Besides, as a special visco-hyperelastic material, dielectric elastomers have been well developed in applications such as artificial muscles, Braille displays, lifelike robots, tunable lenses and power generators [5, 6]. Furthermore, the idealized models of an internally pressurized hollow cylinder or sphere are
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instructive as typical models to investigate structures like living cells, saccular aneurysms, blood vessels and aircraft
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