  In this paper, we propose a robust low-order stabilization-free virtual
element method on quadrilateral meshes for linear elasticity that is based on
the stress-hybrid principle. We refer to this approach as the Stress-Hybrid
Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner
variational principle is adopted, wherein both the equilibrium equations and
the strain-displacement relations are variationally enforced. We consider
small-strain deformations of linear elastic solids in the compressible and
near-incompressible regimes over quadrilateral (convex and nonconvex) meshes.
Within an element, the displacement field is approximated as a linear
combination of canonical shape functions that are $\textit{virtual}$. The
stress field, similar to the stress-hybrid finite element method of Pian and
Sumihara, is represented using a linear combination of symmetric tensor
polynomials. A 5-parameter expansion of the stress field is used in each
element, with stress transformation equations applied on distorted
quadrilaterals. In the variational statement of the strain-displacement
relations, the divergence theorem is invoked to express the stress coefficients
in terms of the nodal displacements. This results in a formulation with solely
the nodal displacements as unknowns. Numerical results are presented for
several benchmark problems from linear elasticity. We show that SH-VEM is free
of volumetric and shear locking, and it converges optimally in the $L^2$ norm
and energy seminorm of the displacement field, and in the $L^2$ norm of the
hydrostatic stress.
