Dynamical response of hyper-elastic cylindrical shells under periodic load

Abstract

International audienceDynamical responses, such as motion and destruction of hyper-elastic cylindrical shells subject to periodic or suddenly applied constant load on the inner surface, are studied within a framework of finite elasto-dynamics. By numerical computation and dynamic qualitative analysis of the nonlinear differential equation, it is shown that there exists a certain critical value for the internal load describing motion of the inner surface of the shell. Motion of the shell is nonlinear periodic or quasi-periodic oscillation when the average load of the periodic load or the constant load is less than its critical value. However, the shell will be destroyed when the load exceeds the critical value. Solution to the static equilibrium problem is a fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of fixed point is related to the property of the dynamical solution and motion of the shell. The effects of thickness and load parameters on the critical value and oscillation of the shell are discussed

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

“…Through analyzing qualitatively the solutions of (12) that describe the radial motion of the spherical membrane composed of the transversely isotropic incompressible Rivlin-Saunders material (4), we discuss the effects of material parameters on the qualitative properties of solutions of (12), and obtain the following interesting conclusions.…”

“…Without loss of generality, from the normal conditions in (5), we take g(I 2 ) = (I 2 − 3) q , h(I 5 ) = (I 5 − 1) b , where q > 1 2 , b > 1, and substitute them into (12). Let y =ẋ, (12) is equivalent to the following system of first-order ordinary differential equations:…”

“…More recently, the problems of cavity formation and motion of a solid sphere composed of incompressible Valanis-Landel materials under the periodic step loads relating to time have been contributed [11] , and the conditions that control the nonlinear periodic oscillation of the formed cavity are proposed. The dynamical responses such as the motion and destruction of hyperelastic cylindrical shells subjected to periodic or a suddenly applied constant load on the inner surface are studied [12] , and the effects of thickness and load parameters on the critical value and oscillation of the shell are discussed.…”

Stability analysis of radial inflation of incompressible composite rubber tubes

“…Recently, the dynamic stability problems of radial inflation of incompressible rubber tubes have been presented by some references, such as Refs. [6][7][8]. In particular, the finite oscillation problem of a cylindrical tube composed of an incompressible Mooney-Rivlin material model was examined by Knowles [6] , and the conditions of periodic oscillation of the tube and the formulas of oscillation period and oscillation amplitude were presented.…”

“…In particular, the finite oscillation problem of a cylindrical tube composed of an incompressible Mooney-Rivlin material model was examined by Knowles [6] , and the conditions of periodic oscillation of the tube and the formulas of oscillation period and oscillation amplitude were presented. The dynamic response of a hyperelastic cylindrical tube composed of an incompressible neo-Hookean material under periodic loads was studied by Ren [7] . The radial oscillation problem of a cylindrical tube composed of a class of incompressible Ogden materials under periodic step loads was investigated by Yuan et al [8] , and the effects of material parameters, structure parameters, and loading forms on nonlinearly periodic oscillation of the tube were discussed in detail.…”