  A robust nonconforming mixed finite element method is developed for a strain
gradient elasticity (SGE) model. In two and three dimensional cases, a lower
order $C^0$-continuous $H^2$-nonconforming finite element is constructed for
the displacement field through enriching the quadratic Lagrange element with
bubble functions. This together with the linear Lagrange element is exploited
to discretize a mixed formulation of the SGE model. The robust discrete inf-sup
condition is established. The sharp and uniform error estimates with respect to
both the small size parameter and the Lam\'{e} coefficient are achieved, which
is also verified by numerical results. In addition, the uniform regularity of
the SGE model is derived under two reasonable assumptions.
