Discrete Averaging Relations for Micro to Macro Transition

The well-known Hill's averaging theorems for stresses and strains as well as the so-called Hill–Mandel condition are essential ingredients for the coupling and the consistency between the micro- and macroscales in multiscale finite-element procedures (FE²). We show in this paper that these averaging relations hold exactly under standard finite-element (FE) 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 (BC) (affine displacement, periodic, and uniform traction BC) using the properties of the shape functions and the weak form of the microscopic equilibrium equations without further kinematic constraints. The analytical proofs are further verified numerically through a simple FE simulation of an irregular representative volume element (RVE) 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 FE methods without introducing an additional error in the scale transition due to the discretization.