  This article offers a new perspective for the mechanics of solids using
moving Cartan's frame, specifically discussing a mixed variational principle in
non-linear elasticity. We treat quantities defined on the co-tangent bundles of
reference and deformed configurations as additional unknowns. Such a treatment
invites compatibility of the fields with base-space (configurations of the
body), so that the configuration can be realised as a subset of the Euclidean
space. We first rewrite the metric and connection using differential forms,
which are further utilised to write the deformation gradient and Cauchy-Green
deformation tensor in terms of frame and co-frame fields. The geometric
understanding of stress as a co-vector valued 2-form fits squarely within our
overall program. We show that, for a hyperelastic solid, an equation similar to
the Doyle-Erciksen formula may be written for the co-vector part of stress.
Using these, we write a mixed energy functional in terms of differential forms,
whose extremum leads to the compatibility of deformation, constitutive rules
and equations of equilibrium. Finite element exterior calculus is then utilised
to construct a finite dimensional approximation for the differential forms
appearing in the variational principle. These approximations are then used to
construct a discrete functional which is then numerically extremised. This
discertization leads to a mixed method as it uses independent approximations
for differential forms related to stress and deformation gradient. The mixed
variational principle is then specialized for 2D case, whose discrete
approximation is applied to problems in nonlinear elasticity. An important
feature of our FE technique is the lack of additional stabilization. From the
numerical study, it is found that the present discretization also does not
suffer form locking and related convergence issues.
