  A four-field mixed variational principle is proposed for large deformation
analysis of Kirchhoff rods with a $C^0$ mixed FE approximations. The core idea
behind the approach is to introduce a one-parameter family of points (the
centerline) and a separate one-parameter family of orthonormal frames (the
Cartan moving frame) that are specified independently. The curvature and
torsion of the curve are related to the relative rotation of neighboring
frames. The relationship between the frame and the centerline is then enforced
at the solution step using a Lagrange multiplier (which plays the role of
section force). Well known frames like the Frenet-Serret are defined only using
the centerline, which demands higher-order smoothness for the centerline
approximation. Decoupling the frame from the position vector of the base curve
leads to a description of torsion and curvature that is independent of the
position information, thus allowing for simpler interpolations. This approach
is then cast in the language of exterior calculus. In this framework, the
strain energy may be interpreted as a differential form leading to the natural
force-displacement (velocity) pairing. The four-field mixed variational
principle we propose has frame, differential forms, and position vector as
input arguments. While the position vector is interpolated linearly, the frames
are interpolated as piecewise geodesics on the rotation group. Similarly,
consistent interpolation schemes are also adopted to obtain finite dimensional
approximations for other differential forms aswell. Using these discrete
approximations, a discrete mixed variational principle is laid out which is
then numerically extremized. The discrete approximation is then applied to
benchmark problems, our numerical studies reveal an impressive performance of
the proposed method without numerical instabilities or locking.
