  Elastic membranes composed of paramagnetic beads offer the possibility of
assembling versatile actuators operated autonomously by external magnetic
fields. Here we develop a theoretical framework to study shapes of such
paramagnetic membranes under the influence of a fast precessing magnetic field.
Their conformations are determined by the competition of the elastic and
magnetic energies, arising as a result of their bending and the induced dipolar
interactions between nearest neighbors beads. In the harmonic approximation,
the elastic energy is quadratic in the surface curvatures. To account for the
magnetic energy we introduce a continuum limit energy, quadratic in the
projections of the surface tangents onto the precession axis. We derive the
Euler-Lagrange equation governing the equilibria of these membranes, as well as
the corresponding stresses. We apply this framework to examine paramagnetic
membranes with quasiplanar, cylindrical, and helicoidal geometries. In all
cases we found that their shape, energy, and stresses can be modified by means
of the parameters of the magnetic field, mainly by the angle of precession.
