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This and the next image are from:
Ilinca Stanciulescu (2005), Nonlinear finite element formulations and bifurcation analysis for structures undergoing large deformations, Ph.D. Dissertation, Department of Civil and Environmental Engineering, Duke University.
ABSTRACT: Nonlinear phenomena are common in structural and solid mechanics; in general, the corresponding system of equations cannot be solved analytically, and numerical techniques are necessary. Furthermore, bifurcations of the solution are frequent; they can appear either at the physical level (the system may have multiple equilibrium configurations), or at the numerical level (related to the algorithm utilized in calculating the solution). Unfortunately no algorithm can solve every nonlinear system, and most of the time the recovery of solutions needs alternative iterative techniques.
This thesis is concerned with finite element formulations and solution techniques for structures undergoing large deformations. The two applications examined are the steady state frictional rolling of tires and the postbuckling analysis of slender structures.
A formulation for steady state rolling calculations is introduced, focusing on the inclusion of frictional sliding conditions between a rolling tire and a flat roadway. Algorithmically, it is seen that traditional return mapping strategies are often ineffective for this problem even when frictional solutions exist; accordingly, an approach utilizing a global stick predictor is proposed to recover solutions to the sliding contact problem. Numerical examples are presented, to demonstrate the effectiveness of the approach advocated. Difficulties associated with enforcing frictional conditions within such a framework are discussed. The interaction of frictional conditions with bifurcation phenomena is also studied in the case of adherent contact conditions. Such phenomena are observed in the context of multiple solutions of the discretized system, and are also manifested in the behavior of the iterative map used to solve the nonlinear algebraic system of equations.
Another example of bifurcation is the buckling of slender structures with direct application to solar sail booms. An interesting aspect in the boom design is that postbuckled configurations are not avoided as is usually the case in structural de- sign; instead, they are sometimes encouraged. In this context, the understanding of the structural behavior after buckling is essential. Various structural systems and loadings appropriate for the boom modeling are examined here. Natural frequencies of vibration about highly–deflected equilibria are extracted, exposing the high sensitivity that these structures have to minor changes in the geometry and loading.
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