  In this paper, an enhanced Virtual Element Method (VEM) formulation is
proposed for plane elasticity. It is based on the improvement of the strain
representation within the element, without altering the degree of the
displacement interpolating functions on the element boundary. The idea is to
fully exploit polygonal elements with a high number of sides, a peculiar VEM
feature, characterized by many displacement degrees of freedom on the element
boundary, even if a low interpolation order is assumed over each side. The
proposed approach is framed within a generalization of the classic VEM
formulation, obtained by introducing an energy norm in the projection operator
definition. Although such generalization may mainly appear to have a formal
value, it allows to effectively point out the mechanical meaning of the
quantities involved in the projection operator definition and to drive the
selection of the enhanced representations. Various enhancements are proposed
and tested through several numerical examples. Numerical results successfully
show the capability of the enhanced VEM formulation to (i) considerably
increase accuracy (with respect to standard VEM) while keeping the optimal
convergence rate, (ii) bypass the need of stabilization terms in many practical
cases, (iii) obtain natural serendipity elements in many practical cases, and
(vi) effectively treat also nearly incompressible materials.
