Modeling the equilibrium configuration of a piecewise orthotropic pneumatic envelope and the phase separation problem in a membrane.

摘要

Large super-light structural systems that for functional reasons require large surfaces are composed at least in part of structural membranes. For efficiency of design, membrane components that will experience low film stresses can be made of lighter material, while those components expected to experience high film stresses can be reinforced with tendons or made from a stronger (albeit heavier) material. The design engineer must balance efficiency of design without compromising structural performance and safety. The under-constrained nature of such structural membranes poses analytical difficulties, and leads to challenging mathematical problems in modeling, analysis, and numerical simulation. In Chapter 1, we present a mathematical model for a tendon-reinforced piecewise-orthotropic thin pressurized membrane motivated by the problem of modeling the shape of a high altitude large scientific balloon. Our model includes contributions due to a position dependent hydrostatic pressure, relaxed film and tendon strain, and film and tendon weight. Using direct methods in the calculus of variations, a variational principle for a quasiconvex Caratheodory Lagrangian is developed and rigorous existence theorems for our model are established. Theorem 1.3.2 is the main result in Chapter 1 of this dissertation. Our mathematical model is implemented into a numerical code which we use to explore equilibrium configurations of a strained pumpkin-shaped balloon at low pressure where the symmetric shape is unstable and the pumpkin-shape is not fully-developed.;Singular perturbation and asymptotic analysis are powerful methods in studies of nonlinear pattern formation problems arising from physical and biological systems. In Chapter 2, we apply some of these techniques to problems that involve objects with non-trivial geometry. We are particularly interested in the role played by the intrinsic geometric properties, such as the Gauss curvature, in determining the shapes and locations of phase domains in a multi-component system. Our model example is a cellular plasma membrane, an archetypal semi-permeable barrier that defines the boundary of living cells. In this work we deal with a two component system where the membrane of a vesicle consists of lipids of two different types. Assuming one type lipids are more numerous than the other type lipids, we observe islands of higher concentration of the minority lipids surrounded by the majority lipids. We want to investigate how the geometry of a membrane determines the location of a small patch. Our work will show that a local maximum point of the Gauss curvature is most likely to attract a small patch. The main theorem in Chapter 2 is stated precisely as Theorem 2.3.3. The case when M has constant Gauss curvature (i.e. M is a sphere) is covered in Theorem 2.3.4.