Closed circular elastica under external forcing deform into a wide variety of patterned and localized states, with different physical systems producing strikingly similar shapes. To explore this phenomenon, in this dissertation, a closed elastic membrane subject to an imposed pressure difference between its interior and exterior and coupled to either solid or liquid substrate supports is considered, and a simple governing equation is derived. Specifically, a system in which the substrate is a soft, elastic material with a preferred natural extension is considered and a system where the membrane surrounds and is surrounded by fluids of different densities in a rotating Hele-Shaw cell is considered. In the former system, intended to provide a model of a cross-section of biological arteries, the membrane is taken to be inextensible, while in the latter system the membrane is separately considered in the inextensible and highly extensible limits. In both systems, although the physical interpretations and viable parameter values differ, the resulting governing equation is the same or similar.
A natural length scale is identified from system parameters, and the ratio of system scale to this natural length scale is identified as a key parameter found to set the onset order of primary wrinkle branch bifurcations from the unperturbed circle state as well as the secondary bifurcation onset and solution form. The results of this thesis explain how the wrinkle wavelength is selected as a function of the parameters in compressed wrinkling systems and show how localized folds and mixed mode states form in secondary bifurcations from wrinkled states. In the case of a solid substrate, this key parameter expresses the balance of membrane bending modulus, substrate stiffness, and system scale. In the case of a fluid substrate, the key parameter is a function of the difference in fluid densities, the rotation rate, and the system scale.
Steady state solutions are obtained via asymptotic expansions, numerical continuation, and other analytical and numerical approaches, all with good mutual agreement. Numerical continuation is applied to organize solutions in bifurcation diagrams using system-response metrics such as the baseline tension, compression, energy, or maximum curvature, all as a function of the pressure difference. Solutions are found to come in the classes of wrinkling or buckling primary states with a single wavenumber, mixed mode secondary solutions with two wavenumbers, and folding secondary solutions. This discussion is primarily focused on the solution subset of reflection symmetric states but is augmented by a brief study of symmetry-broken chiral states, computed using numerical continuation and shooting and identified as secondary states.
The governing equation is found to be part of a family of exactly solvable equations describing the wrinkling response of a membrane subject to a variety of substrate forces. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The relationship between wrinkling states which have nontrivial onset ordering and buckling states which occur in the absence of a substrate and destabilize monotonically is explored and explained, a previously poorly understood connection.
Finally, the stability of the system is considered from a global energy based perspective and a local temporal linear stability analysis.
