Numerical solution techniques for nonlinear elasticity have been an active area of research over the last few decades. Right from the early days, it was recognized that constructing well-performing FE approximations for nonlinear elasticity was a challenging task. Earlier attempts to construct purely displacement based formulations resulted in numerical approximations that often performed poorly. Stabilization methods based on assumed strain and enhanced strain were introduced to circumvent the poor performance of the displacement based FE approximation. These stabilization terms had little to do with the underlying physics and were introduced purely from a numerical standpoint. In this work, we propose a mixed FE approximation for nonlinear elasticity based on a Hu-Washizu (HW) type variational principle discussed in Dhas and Roy [2]. This version of HW variational principle uses kinetics and kinematics written in terms of vector fields and differential forms. Discrete approximations for the differential forms appearing in the HW variational principle are constructed with ideas borrowed from finite element exterior calculus [1]. These approximations are in turn used to construct a discrete approximation to the HW functional. The discrete equations describing mechanical equilibrium, compatibility and constitutive rule, are obtained by seeking an extremum of the discrete functional with respect to appropriate degrees of freedom. The discrete extremum problem is then solved numerically using Newton's method. This mixed FE technique is then applied to benchmark problems wherein conventional displacement based approximations suffer from locking and checker boarding. These studies help establish that our mixed FE approximation is free from these numerical bottlenecks. REFERENCES [1] Arnold, D.N., Falk, R.S., Winter, R., “Finite element exterior calculus, homological techniques, and applications”, Acta numerica, vol-15, page(s)-155 (2006). [2] Dhas, B., Roy, D., “A geometric approach to Hu-Washizu variational principle in nonlinear elasticity”, arXiv preprint arXiv:2009.00275..