The Researches On Propagation Of Nonlinear Waves In Hyperelastic Axisymmetric Structures

Axisymmetric structures composed of hyperelastic materials,such as cylindrical rods,cylindrical shells and cylinders,have widespread applications in aerospace,transportation,mechanical manufacturing and other fields due to their excellent mechanical properties of vibration damping,energy absorption and structural stability.In particular,these structures are usually encountered collision or impact loads in applications,which may cause deformation,instability,and damage.In engineering science,many problems need to be analyzed and solved from the perspective of waves,such as non-destructive testing,determination of material parameters,and structural stability analyses.Therefore,the researches on the waves in hyperelastic structures have important theoretical and practical significance.Based on the theory of nonlinear elastodynamics and bifurcation theory of dynamical system,the propagation of nonlinear waves in axisymmetric structures composed of hyperelastic materials is studied,and some new conclusions are obtained,as follows.1.The radial propagation of steady-state waves in an infinite cylindrical shell composed of isotropic compressible neo-Hookean materials is studied.Firstly,the cylindrical shell undergoes radial finite deformation under the radial uniform load.This problem is divided into two stages:the first stage is to consider the radial finite deformation of the cylindrical shell while the radial uniform load is relatively smaller;the second stage is to consider the radial movement of the shell while the load is relatively larger.Based on the cylindrical shell,a radial displacement disturbance is superimposed by the theory of small deformations superposed on large elastic deformations,and the existence conditions of steady-state waves are given.Combining with numerical examples,the effects of loads,material parameters and structural parameters on steady-state waves are discussed.The results indicate that the frequency of steady-state waves increases but the amplitude decreases with the load increases.The material parameters play a significant role on the frequency and amplitude.Moreover,for some special structure parameters,the frequency will increase discontinuously.2.The radial and axial propagation of strongly nonlinear travelling waves is examined in a semi-infinite rod,which is composed of a class of transversely isotropic incompressible hyperelastic materials about the radial direction.According to the theory of nonlinear elastodynamics,a mathematical model describing the axisymmetric motion of the rod is established.The model is simplified by using incompressible conditions and end conditions.The first integral of the equation is obtained and the implicit integral solutions of travelling waves are given.Based on the bifurcation theory of dynamical system,under different parameters,dynamic behaviors of the orbits,which contained in the phase diagram of the system are analyzed,and different types of bounded travelling waves existing in the rod are obtained.Particularly,for the transversely isotropic incompressible neo-Hookean material model about the radial direction,some bounded travelling waves describing the radial motion of the rod are obtained,including solitary waves with the peak form and periodic waves.For the transversely isotropic incompressible Mooney-Rivlin material model about the radial direction,solitary cusp waves and periodic cusp waves with the peak form,solitary waves with the peak form and periodic waves can be generated in the rod.Moreover,the effects of physical parameters on the quantitative properties(such as period and amplitude)of two types of periodic waves,which are generated in different positions of the rod are analyzed.3.The radial and axial propagation of strongly nonlinear travelling waves in an infinite cylindrical shell composed of transversely isotropic compressible neo-Hookean materials about the radial direction is studied.Coupled nonlinear evolution equations describing the radially and axially symmetric motion of the shell are obtained by Hamiltons principle,and the motion equations are decoupled by the relationship between the radial deformation function and the axial stretch.The travelling wave equation for radial direction is derived by the travelling wave transform,and the implicit integral solutions of travelling waves are given.According to the qualitative theory of differential equations,the qualitative properties of equilibrium points are discussed,and the existence conditions of bounded travelling waves are given.Based on the bifurcation theory of dynamic systems,solitary waves with the peak form,periodic waves describing the radial motion and solitary waves with the valley form,periodic waves describing the axial motion are obtained.Quantitative analyses of different waveforms show that the anisotropy parameter plays a significant role on the quantitative properties(such as period and amplitude)of bounded travelling waves.4.A class of generalized hyperelastic rod equations(KdV equation with variable coefficients),which can be used to describe the propagation of nonlinear waves in a non-uniform(with variable cross-sections and variable material densities)hyperelastic rod,are investigated.In terms of Lou's direct method,two sets of symmetric transformations are obtained,and the analytical solutions of soliton-like waves corresponding to the two sets of symmetric transformations are given.In particular,while the variable coefficients are taken as some constants,it can be used to describe the propagation of longitudinal strain waves in a rod composed of general incompressible materials,by using the same method,two sets of symmetric transformations and analytical solutions of soliton-like waves are given.