  This paper presents a new hybridizable discontinuous Galerkin (HDG) method
for linear elasticity on general polyhedral meshes, based on a strong symmetric
stress formulation. The key feature of this new HDG method is the use of a
special form of the numerical trace of the stresses, which makes the error
analysis different from the projection-based error analyzes used for most other
HDG methods. For arbitrary polyhedral elements, we approximate the stress by
using polynomials of degree k>=1 and the displacement by using polynomials of
degree k+1. In contrast, to approximate the numerical trace of the displacement
on the faces, we use polynomials of degree k only. This allows for a very
efficient implementation of the method, since the numerical trace of the
displacement is the only globally-coupled unknown, but does not degrade the
convergence properties of the method. Indeed, we prove optimal orders of
convergence for both the stresses and displacements on the elements. In the
almost incompressible case, we show the error of the stress is also optimal in
the standard L2-norm. These optimal results are possible thanks to a special
superconvergence property of the numerical traces of the displacement, and
thanks to the use of a crucial elementwise Korn's inequality. Several numerical
results are presented to support our theoretical findings in the end.
