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

A novel four-field mixed variational approach to Kirchhoff rods implemented with finite element exterior calculus