  Biological membranes are able to exhibit various morphology due to the
fluidity of the lipid molecules within the monolayers. The shape transformation
of membranes has been well described by the classical Helfrich theory, which
consists only a few phenomenological parameters, including the mean and the
Gaussian curvature modulus. Though various methods have been proposed to
measure the mean curvature modulus, determination of the Gaussian curvature
modulus remains difficult both in experiments and in simulation. In this paper
we study the buckling process of a rectangular membrane and a circular membrane
subject to compressive stresses and under different boundary conditions. We
find that the buckling of a rectangular membrane takes place continuously,
while the buckling of a circular membrane can be discontinous depending on the
boundary conditions. Furthermore, our results show that the stress-strain
relationship of a buckled circular membrane can be used to effectively
determine the Gaussian curvature modulus.
