Title: The Stability at the Solid-Solid and Liquid-Solid Interfaces
Description: In this thesis, we studied three small subjects in the realm of continuum mechanics: imbibition in fluid mechanics, beam and rod buckling in solid mechanics and shell buckling at the solid-liquid interface. In chapter 2, we examined the radial imbibition into a homogenous semi-infinite porous media from a point source with infinite liquid supply. We proved that in the absence of gravity (or in the regime while gravity is negligible compared to surface tension), the shape of the wet area is a hemisphere, and the radius of the wet area evolves as a function with respect to time. This new law with respect to time has been verified by Finite Element Method simulation in software COMSOL and a series of experiments using packed glass microsphere as the porous media. We also found that even though the imbibition slows down, the flow rate through the point source remains constant. This new result for three dimensional radial imbibition complements the classic Lucas-Washburn law in one dimension and two dimensional radial imbibition in one plane.

In chapter 3, we studied the elastic beam/rod buckling under lateral constraints in two dimension as well as in three dimension. For the two dimensional case with unique boundary conditions at both ends, the buckled beam can be divided into segments with alternate curved section and straight section. The curved section can be solved by the Euler beam equation. The straight sections, however, are key to the transition between different buckling modes, and the redistributed length of straight sections sets the upper limit and lower limit for the transition. We compared our theoretical model of varying straight sections with Finite Element Method simulation in software ABAQUS, and good agreements are found. We then attempted to employ this model as an explanation with qualitative feasibility for the crawling snake in horizontal plane between parallel walls, which shows unique shape like square wave. For the three dimensional buckling beam/rod confined in cylindrical constraints, three stages are found for the buckling and post buckling processes: initial two dimensional shape, three dimensional spiral/helix shape and final foldup/alpha shape.

We characterized the shape at each stage, and then we calculated the transition points between the three stages using geometrical arguments for energy arguments. The theoretical analysis for three dimensional beam/rod are also complemented with Finite Element Method simulations from ABAQUS. In chapter 4, we investigated the buckling shape of solid shell filled with liquid core in two dimension and three dimension. A material model for liquid is first described that can be readily incorporated in the framework of solid mechanics. We then applied this material model in two dimensional and three dimensional Finite Element Method simulation using software ABAQUS. For the two dimensional liquid core solid shell model, a linear analysis is first performed to identify that ellipse corresponds to lowest order of buckling with smallest elastic energy. Finite Element Method simulation is then performed to determine the nonlinear post-buckling process. We discovered that two dimensional liquid core solid shell structures converge to peanut shape eventually while the evolution process is determined by two dimensionless parameters Kτ/μ and ρR^2/μτ.

Amorphous shape exists before final peanut shape for certain models with specific Kτ/μ and ρR^2/μτ. The two dimensional peanut shape is also verified with Lattice Boltzmann simulations. For the three dimensional liquid core solid shell model, the post buckling shape is studied from Finite Element Method simulations in ABAQUS. Depending on the strain loading rate, the deformations show distinctive patterns. Large loading rate induces herringbone pattern on the surface of solid shell which resembles solid core solid shell structure, while small loading rate induces major concave pattern which resemble empty solid shell structure. For both two dimensional and three dimensional liquid core system, small scale ordered deformation pattern can be generated by increasing the shear stress in liquid core. In the final chapter, we summarized the discoveries in the dissertation with highlights on the role that geometry plays in all of the three subjects. Recommendations for future studies are also discussed.