You are using a piecewise continuous function to approximate your displacement field. Infact, for linear elasticity your approximation u(x) \in H^1(\Omega) where H^1(\Omega) is the space of functions which are square integrable and whose derivatives are square integrable. The piecewise continuous functions or say (S^1_0 as described in these notes) which are contained within H^1(\Omega) do not have the requirement of continuity of the derivative \nabla u.
And since the stress depends on the gradient of the displacement field \nabla u, it is at best piecewise discontinuous of order one less than the displacement field. So a piecewise linear displacement should give a piecewise constant stress.