MODELING THE EQUILIBRIUM CONFIGURATION OF A PIECEWISE-ORTHOTROPIC PNEUMATIC ENVELOPE WITH APPLICATIONS TO PUMPKIN-SHAPED BALLOONS

Large superlight structural systems that, for functional reasons, require large surfaces are composed at least in part of structural membranes. For efficiency of design, components that experience low stress can be made of lighter material, while those expected to experience high stress can be reinforced with tendons or made from a stronger, albeit heavier, material. The design engineer seeks an efficient design without compromising structural performance and safety. The underconstrained nature of such structural membranes poses analytical difficulties and leads to challenging mathematical problems in modeling, analysis, and numerical simulation. Motivated by the problem of modeling the shape of a high-altitude large scientific balloon, we present a model for a tendonreinforced piecewise-orthotropic thin pressurized membrane. Using direct methods in the calculus of variations, a variational principle for a quasi-convex Carathéodory Lagrangian is developed, and rigorous existence theorems are established. Our 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 becomes unstable.