  We develop a lowest-order nonconforming virtual element method for planar
linear elasticity, which can be viewed as an extension of the idea in Falk
(1991) to the virtual element method (VEM), with the family of polygonal meshes
satisfying a very general geometric assumption. The method is shown to be
uniformly convergent for the nearly incompressible case with optimal rates of
convergence. The crucial step is to establish the discrete Korn's inequality,
yielding the coercivity of the discrete bilinear form. We also provide a
unified locking-free scheme both for the conforming and nonconforming VEMs in
the lowest order case. Numerical results validate the feasibility and
effectiveness of the proposed numerical algorithms.
