Summary: The \(h\)-version of the discontinuous Galerkin finite element method \((h\)-DGFEM) for nearly incompressible linear elasticity problems in polygons is analysed. It is proved that the scheme is robust (locking-free) with respect to volume locking, even in the absence of \(H^2\)-regularity of the solution. Furthermore, it is shown that an appropriate choice of finite element meshes leads to robust and optimal algebraic convergence rates of the DGFEM even if the exact solutions do not belong to \(H^2\).