In-plane surface modes of an elastic toroidal membrane

In this work, the dynamics of in-plane surface deformation modes of an inflated toroidal membrane has been studied. We have considered both isotropic and anisotropic but homogeneous material properties. The covariant form of the equation of motion assuming a general in-plane small amplitude displacement field has been derived from the variational formulation which clearly shows the effect of curvature on the dynamics. The curvature term in the equation of motion may be interpreted as an effective quadratic potential in the Lagrangian with a coupling proportional to the Ricci curvature scalar of the membrane. The variational problem is discretized, and is subsequently analyzed to obtain the eigenfrequencies and modes of vibration. The effect of geometric and material properties on the modal dynamics has been studied. The effect of anisotropy on the modal dynamics of the torus has also been studied. Certain invariant deformation measures have been defined which are found to characterize the modes in terms of presence or absence of nodal curves/points.