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We develop a lowest-order nonconforming virtual element method for planarlinear 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 meshessatisfying a very general geometric assumption. The method is shown to beuniformly convergent for the nearly incompressible case with optimal rates ofconvergence. The crucial step is to establish the discrete Korn's inequality,yielding the coercivity of the discrete bilinear form. We also provide aunified locking-free scheme both for the conforming and nonconforming VEMs inthe lowest order case. Numerical results validate the feasibility andeffectiveness of the proposed numerical algorithms.

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems