| Popis: |
Many natural phenomena encountered in nature may be understood through the paradigm of buckling instability. Examples include the design of columns in structural engineering, the folding of geological formations, the collapse of blood vessels, and the fragmentation of uncooked spaghetti, to name only a few. The phenomenon of buckling has traditionally been seen only as a nuisance, but more recently it has also proven useful as a potential tool for pattern formation, particularly at small scales. Many studies have focused on the features of static buckling, particularly pattern formation. This thesis focuses on how dynamic buckling affects the spontaneous selection of patterns in a number of canonical problems. We first study the effect of curvature on dynamic buckling in circular geometries. We study the pattern observed when a ring made of either a viscous liquid or an elastic solid is subject to a suddenly applied external pressure. We develop numerical schemes to study the evolution of the ring’s profile and compare those results to linear stability analysis. A weakly nonlinear analysis is performed to understand the behaviour of the observed shape beyond the onset of instability, with results that compare well with both numerical and experimental work. In the late stages of instabilities, buckles merge — the wrinkle pattern coarsens. To understand this coarsening we consider the non inertial problem of a beam sit- ting on a viscous layer. We explain how wrinkle coarsening occurs as the result of wave dispersion and quantify coarsening by presenting an asymptotic analysis of the governing equation. We also study the effect of confinement and show that it has a crucial effect on coarsening at late times. We develop an analytical expression for the effect of confinement, which allows us to better explain previously published experimental data. Finally, our numerical solution reveals a new regime as the wavelength of wrinkles become comparable to the size of the system. In this regime the dynamics is relatively slow with the system remaining for long periods with one particular mode before rapidly switching to another. We study this regime, the so-called temporal cascade, in detail. The problems we are considering have different geometries and different material constitutive relations, but highlight a number of common themes. Mathematically, those systems are described by a Partial Differential Equation with a global constraint whose numerical solution is found by framing it in the language of Differential Algebraic Equations. |
