Symplectic Methods In Dynamic Buckling Of Elastic Long Cylindrical Shells

The stability of structures is an important research subject in modern solid mechanics, and much attention has been focused on the problem in engineering. Since the cylindrical shell is a conventional structure, its dynamic buckling problems play an important role in the structure stability. Though many researches analyzed the stability of cylindrical shells with various material in different circumstance and different load types, there are still many urgent problems not solved. For the local dynamic stability of structure, the propagation of stress wave is an important factor. Stress wave theory is very active frontier in solid mechanics and is also important basis and method in modern acoustics, geophysics, explosion mechanics and performance research of material mechanics. The further research of stress wave will develop from one dimensional to two dimensional and three dimensional, from single load form to complex stress wave loading conditions. Usually, the radial displacement is considered and the axial displacement and circumferential displacement are neglected in the dynamic stability problem of cylindrical shells. For a shorter cylindrical shell, the axial displacement and circumferential displacement have little effect to the global buckling, but for a longer shell, they will influence the global deformation of the shell.In this paper, the dynamic buckling problem of elastic long cylindrical shells subjected to axial impact load, torsional impact load, coupled impact load and axial pulse impact load, is considered. By introducing a Hamilton function, the Hamiltonian system for the problem is established. It means that the transformation from the Euclidean space to the Symplectic space is finished and the dual equations in the Hamiltonian system can be obtained. In the symplectic space, the critical buckling loads and buckling modes of the problem are substituted by eigenvalues and eigensolutions. Eigenvalues include zero eigenvalue which is corresponding to axisymmetric buckling and non-zero eigenvalues which are corresponding to non-axisymmetric buckling.The dynamic loads discussed in the paper are mainly step impact load and pulse impact load, and the cylindrical shell expresses different buckling modes for the two kinds of loads. The long cylindrical shell occurs local buckling, global buckling and coupling formation of these two buckling under step impact load; and the center of long cylindrical shell occurs "bamboo" buckling unde pulse impact load.The influence of the stress wave on the buckling of long cylindrical shells is researched, and the critical load curves and correspongding buckling modes in various boundary conditions are analyzed. The results show that the boundary conditions have great effects on the impacted cylindrical shell, particularly for the free boundary conditions, the "bell" shape of the buckling mode will occure unde the axial impact load. The basic factors of influence buckling include the impact load, impact time, the physical and geometrical parameters. The results also reveal that the first order buckling modes of long cylindrical shells have rather special forms under impacted loads. When the stress wave goes to a long stage, the critical load curves can appear another type curves different from other order's. Corresponding buckling modes are similar to bending or distorted buckling of beams. The first order buckling of long cylindrical shells contains global buckling and local buckling, and buckling modes will occure separately or simultaneously.