The Wave Behaviors And Propagation Properties Of Finite Deformation Elastic Rods

In this paper, the general principles are introduced to establish the wave-motion equations by use of two descriptions (Lagrangian and Eulerian descriptions) in the continuum mechanics, when the finite deformation is taken into account. After that the dynamics behaviors and wave properties of finite deformation elastic thin rods are paid more attentions to in Lagrangian coordinate. Using basic equations in continuum mechanics, the wave-motion equation in the geometrical nonlinear elastic rod is derived, and then the characteristic curves and their characteristic relations are deduced by the characteristic line method, and the alterations of the wave profiles are analyzed during propagation. Furthermore using the variation principle in the elasticity, the wave-motion equation is derived in the finite deformation elastic thin rod with viscous and transverse inertia effects, and the characteristic curves and their characteristic relations are obtained by characteristic line method,and the influence of viscous and geometrical-dispersive effects on the propagation of wave is analyzed.In this paper, the multi-scale technology is introduced to study the wave-motion equations with viscous or transverse inertia effects or both of them. Burgers equation, K-dV equation and KdV-Burgers equation are deduced respectively, and correspondingly the steady shock-wave solutions, solitary wave solutions and oscillation solitary wave or shock-wave solutions are obtained. All of these reveal the further properties of the wave propagating in one-dimensional elastic continuum. On phase plane, the qualitative analysis are made, which exhibits four kinds of orbits: the heteroclinic orbit of Burgers equation, the homoclinic orbit of K-dV equation, and saddle-foci heteroclinic orbit or saddle-joint heteroclinic orbit of KdV-Burgers equation.Finally, two-dimensional wave-motion equations of the finite deformation elastic rod are derived by variation principle taking simultaneously geometrical and physical nonlinearity into consideration.