  Biomembranes consisting of two opposing phospholipid monolayers, which
comprise the so-called lipid bilayer, are largely responsible for the dual
solid-fluid behavior of individual cells and viruses. Quantifying the
mechanical characteristics of biomembrane, including the dynamics of their
in-plane fluidity, can provide insight not only into active or passive cell
behaviors but also into vesicle design for drug delivery systems. Despite
numerous studies on the mechanics of biomembranes, their dynamical viscoelastic
properties have not yet been fully described. We thus quantify their
viscoelasticity based on a two-dimensional (2D) fluid membrane model, and
investigate this viscoelasticity under small amplitude oscillatory loadings in
micron-scale membrane area. We use hydrodynamic equations of bilayer membranes,
obtained by Onsager's variational principle, wherein the fluid membrane is
assumed to be an almost planar bilayer membrane. Simulations are performed for
a wide range of oscillatory frequencies $f$ and membrane tensions. Our
numerical results show that as frequencies increase, membrane characteristics
shift from an elastic-dominant to viscous-dominant state. Furthermore, such
state transitions obtained with a 1-$\mu$m-wide loading profile appear with
frequencies between $O(f) = 10^1-10^2$ Hz, and almost independently of surface
tensions. We discuss the formation mechanism of the viscous- or
elastic-dominant transition based on relaxation rates that correspond to the
eigenvalues of the dynamical matrix in the governing equations.
