Methods Of Analytical Mechanics Of A Growing Elastic Rod Dynamics

Dynamical modeling of growing elastic rod with deformation, movement and growing are discussed with the background of a kind of slender body in nature. Dynamics of the growing elastic rod is studied from the perspective of analytical mechanics.On the premise of some basic assumptions, analyze the deformation, movement and growing geometric rules of a growing elastic rod. The definition of virtual displacement and the constitutive equation of the elastic rod are showed. Based on the definition of virtual displacement on configuration space, Gauss principle of growing elastic rods is given. On one hand, the application of Gauss principle is extended, on the other hand, new ways of growing elastic rods modeling are provided. By defining the inverse of dyad, Gauss principle and Gauss minimum constraint theory in classical mechanics are used to model of growing elastic rod dynamics. Get the two equivalent forms of Gauss variation, which reflects the symmetry between time and arc coordinates in the expression of rod dynamics. Two forms of the Gauss minimum constraint theory are given, which indicates that the actual motion of an elastic rod made the function stationary value, and the minimum value.Conserved quantities of a growing elastic rod dynamics are studied on the basis of Lagrange equation. Several of conserved quantities and their condition of existence are derived, which expanded the concept of conversation from the time to the space, i.e. arc coordinate. Lagrange equation is applied to analyzed energy integral and cyclic integral. First integrals are of practical significance in both theoretical and numerical analysis for a growing elastic rod dynamics.Finally, research results can be extended to the complex model of movement, deformation and growth elastic plate or elastic shell, which provides a new research field for the analysis of classical mechanics.