Nonlinear analysis of post-buckling dynamics and higher order instabilities of flexible structures

New methods and algorithms in nonlinear mechanics and dynamics are developed to investigate the secondary and higher order instabilities, and dynamic snapping phenomena of deeply buckled flexible structures. Particular attention is paid to various static and dynamic aspects of the secondary bifurcation and mode jumping of thin plates under generalized (thermal and mechanical) loading. The principal contribution of this work to the field of nonlinear dynamics lies in the development of both analytic methods and numerical algorithms to broaden the investigation of mode jumping from a primarily static approach to include dynamics, and from mechanical loading situations to those of thermal loading, where the plate exhibits much stronger geometric nonlinearity. Innovations from the analytical approach include the development of a compact form of ODES to study the effects of arbitrary combinations of mode components on the secondary bifurcation and the post-buckling dynamics, and the introduction of a linear temperature sweep scheme in a transient analysis to capture the snapping phenomenon dynamically. For the numerical approach, the solution is pursued under more generalized load types and boundary conditions. Secondary instability and the local post-secondary bifurcation behavior are investigated using an asymptotic finite element method. An adaptive non-stationary algorithm is then developed to study the mode jumping phenomenon in a global context. Novel aspects of this approach include: the development of a multi-mode dynamical reduction method to evaluate post-buckling equilibrium branches and determine their stability properties, the design of an effective load-sweeping algorithm to ensure the dynamical transition between two stable branches, and the exploration of the spurious convergence phenomenon of the transient response to an unstable equilibrium using the classical hybrid static-dynamic method. Aspects of mode interactions and their relation with the secondary bifurcation and mode jumping are explored. Typical static and dynamic features are presented, including the local post-secondary bifurcation forms, the change of natural frequencies with respect to the load, the propagation of the buckling patterns of varying complexity, and the breaking of symmetry of both static configurations and vibration mode shapes after the occurrence of mode jumping. Effects of various geometric imperfections are also studied.