Modeling the Equilibrium Configuration of a Piecewise Orthotropic Pneumatic Envelope and the Phase Separation Problem in a Membrane
Open Accessn 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 hydro-static pressure, relaxed film and tendon strain, and film and tendon weight. Using directmethods in the calculus of variations, a variational principle for a quasiconvex CarathéodoryLagrangian 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 isimplemented into a numerical code which we use to explore equilibrium configurations ofa strained pumpkin-shaped balloon at low pressure where the symmetric shape is unstableand the pumpkin-shape is not fully-developed. Singular perturbation and asymptotic analysis are powerful methods in studies of non-linear pattern formation problems arising from physical and biological systems. In Chapter2, 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 amulti-component system. 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.
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