  In this paper, we construct new finite element methods for the approximation
of the equations of linear elasticity in three space dimensions that produce
direct approximations to both stresses and displacements. The methods are based
on a modified form of the Hellinger--Reissner variational principle that only
weakly imposes the symmetry condition on the stresses. Although this approach
has been previously used by a number of authors, a key new ingredient here is a
constructive derivation of the elasticity complex starting from the de Rham
complex. By mimicking this construction in the discrete case, we derive new
mixed finite elements for elasticity in a systematic manner from known
discretizations of the de Rham complex. These elements appear to be simpler
than the ones previously derived. For example, we construct stable
discretizations which use only piecewise linear elements to approximate the
stress field and piecewise constant functions to approximate the displacement
field.
