  A geometrically exact membrane formulation is presented that is based on
curvilinear coordinates and isogeometric finite elements, and is suitable for
both solid and liquid membranes. The curvilinear coordinate system is used to
describe both the theory and the finite element equations of the membrane. In
the latter case this avoids the use of local cartesian coordinates at the
element level. Consequently, no transformation of derivatives is required. The
formulation considers a split of the in-plane and out-of-plane membrane
contributions, which allows the construction of a stable formulation for liquid
membranes with constant surface tension. The proposed membrane formulation is
general, and accounts for dead and live loading, as well as enclosed volume,
area, and contact constraints. The new formulation is illustrated by several
challenging examples, considering linear and quadratic Lagrange elements, as
well as isogeometric elements based on quadratic NURBS and cubic T-splines. It
is seen that the isogeometric elements are much more accurate than standard
Lagrange elements. The gain is especially large for the liquid membrane
formulation since it depends explicitly on the surface curvature.
