  The well-known Hill's averaging theorems for stresses and strains as well as
the so-called Hill-Mandel principle of macrohomogeneity are essential
ingredients for the coupling and the consistency between the micro and macro
scales in multiscale finite element procedures (FE$^2$). We show in this paper
that these averaging relations hold exactly under standard finite element
discretizations, even if the stress field is discontinuous across elements and
the standard proofs based on the divergence theorem are no longer suitable. The
discrete averaging results are derived for the three classical types of
boundary conditions (affine displacement, periodic and uniform traction
boundary conditions) using the properties of the shape functions and the weak
form of the microscopic equilibrium equations. The analytical proofs are
further verified numerically through a simple finite element simulation of an
irregular representative volume element undergoing large deformations.
Furthermore, the proofs are extended to include the effects of body forces and
inertia, and the results are consistent with those in the smooth continuum
setting. This work provides a solid foundation to apply Hill's averaging
relations in multiscale finite element methods without introducing an
additional error in the scale transition due to the discretization.
