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In this paper, we present a first-order Stress-Hybrid Virtual Element Method(SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adoptthe Hellinger--Reissner variational principle to construct a weak equilibriumcondition and a stress based projection operator. On applying the divergencetheorem to the weak strain-displacement relations, the stress projectionoperator is expressed in terms of the nodal displacements, which leads to adisplacement-based formulation. This stress-hybrid approach assumes a globallycontinuous displacement field while the stress field is discontinuous acrosseach element. The stress field is initially represented by divergence-freetensor polynomials based on Airy stress functions. However, for flexibility inchoosing basis functions, we also present a formulation that uses a penaltyterm to enforce the element equilibrium conditions. This method is referred toas the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numericalresults are presented for PSH-VEM and SH-VEM, and we compare their convergenceto the composite triangle FEM and B-bar VEM on benchmark problems in linearelasticity. The SH-VEM converges optimally in the $L^2$ norm of thedisplacement, energy seminorm, and the $L^2$ norm of hydrostatic stress.Furthermore, the results reveal that PSH-VEM converges in most cases at afaster rate than the expected optimal rate, but it requires the selection of asuitably chosen penalty parameter.

Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity