Kang, Min Kyoo
16 November 2010
A hydrogel consists of a cross-linked polymer network and solvent molecules, capable of large, reversible deformation in response to a variety of external stimuli. In particular, diverse instability patterns have been observed experimentally in swelling hydrogels under mechanical constraints. The present study develops a general theoretical framework based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions for swelling deformation of hydrogels, along with proper boundary conditions. A specific material model is employed for analytical and numerical studies, for which the nonlinear constitutive behavior of the hydrogel is derived from a free energy function combining rubber elasticity with a polymer solution theory. A finite element method is then developed and implemented as a user-defined material (UMAT) in the commercial package, ABAQUS. By numerical simulations, the effect of constraint on inhomogeneous swelling of substrate-attached hydrogel lines is elucidated. It is found that crease-like surface instability occurs when the width-to-height aspect ratio of the hydrogel line exceeds a critical value. Next, by considering a hydrogel layer on a rigid substrate, swell-induced surface instability is studied in details. A linear perturbation analysis is performed to predict the critical condition for onset of the surface instability. In contrast to previously suggested critical conditions, the present study predicts a range of critical swelling ratios, from about 2.5 to 3.4, depending on the material properties of the hydrogel system. A stability diagram is constructed with two distinct regions for stable and unstable hydrogels with respect to two dimensionless material parameters. Numerical simulations are presented to show the swelling process, with evolution of initial surface perturbations followed by formation of crease-like surface patterns. Furthermore, with combined swelling and mechanical compression, the stability analysis is extended to predict a general critical condition that unifies the swell-induced surface instability of hydrogels with mechanically induced surface instability of rubbers. The effect of surface tension is found to be critical in suppressing short-wavelength modes of surface instability, while the substrate confinement suppresses long-wavelength modes. With both surface tension and substrate confinement, an intermediate wavelength is selected at a critical swelling ratio for onset of surface instability. Both the critical swelling ratio and the characteristic wavelength depend on the initial thickness of the hydrogel layer as well as other material properties of the hydrogel. It is found that the hydrogel layer becomes increasingly stable as the initial layer thickness decreases. A critical thickness is predicted, below which the hydrogel layer swells homogeneously and remains stable at the equilibrium state. Finally, three-dimensional finite element models are developed to simulate swelling deformation of hydrogel lines. Depending on the aspect ratio of the cross section as well as the material properties of the hydrogel, two types of swell-induced instability patterns are envisaged, i.e., localized surface instability versus global buckling. / text
Geophysical granular flows, such as snow avalanches, pyroclastic density currents, mudslides and debris flows, can be extremely hazardous to local populations, and understanding their complex behaviour remains an important challenge. This project aims to provide insight into these events by exploring different aspects in isolation, using a combination of mathematical theory, numerical simulations and small-scale experiments. Firstly, the effect of lateral confinement is examined by studying granular material moving in an inclined chute. This can have applications to natural releases flowing down confined valleys or conduits, and the relative simplicity of the geometry also provides a useful test case for new theoretical models. One such model is the recent depth-averaged μ(I)-rheology, which, because of the viscous terms introduced into the depth-averaged momentum balance, may be described as an intermediate approach between full constitutive laws and classical shallow-water-type equations for dense granular flows. Here, a generalisation of the new system to two spatial dimensions is described, and the resulting viscous equations are able to capture the cross-slope curvature of the downslope velocity profiles in steady uniform chute flows. This may be regarded as major progress compared to traditional hyperbolic models, which only admit constant velocity solutions. Particle size-segregation in geophysical granular flows is then investigated, which can cause important feedback on the overall bulk properties as it can lead to the development of regions with different frictional properties. A particularly striking example is segregation-induced 'finger' formation, where large particles are segregated to the flow surface and sheared to form a resistive coarse-rich front, which is unstable and spontaneously breaks down into a series of lobate structures. These travel both faster and further than one might anticipate. To model such segregation-mobility feedback effects, the depth-averaged μ(I)-rheology is extended to bidisperse flows by coupling with a depth-integrated model for size-segregation. The system of equations remains mathematically well-posed and is able to qualitatively capture finger formation, with the newly-introduced viscous terms controlling the characteristics of the leveed channels that develop. A more subtle segregation effect is studied in bidisperse roll waves, which form as small irregularities merge and coarsen as they move downslope, eventually growing into destructive large amplitude pulses. Experimental measurements show lateral, as well as vertical, segregation profiles, with the coarser grains accumulating at the fastest moving wave crests. The disturbances that form in mixtures with higher proportions of large particles grow more slowly, leading to smaller amplitude waves that travel at slower speeds, and the new coupled model predicts qualitatively similar behaviour. Finally, the influence of complex topography is investigated. A smooth two-dimensional bump is placed across the width of a chute, which, depending on the initial conditions, can lead to the formation of an airborne jet or granular shock at steady state. A simple depth-averaged model in a curvilinear coordinate system following the topography accurately captures both regimes, and represents a significant improvement on using an aligned Cartesian approach.
Charakterisierung von Kavitationsblasenpopulationen / Characterization of cavitation bubble populationsThiemann, Andrea 09 June 2011 (has links)
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