Discrete averaging relations for micro to macro transition

TL;DR

This paper proves that Hill's averaging relations hold exactly in finite element discretizations, even with discontinuous stress fields, ensuring accurate micro-macro scale transitions in multiscale finite element methods.

Contribution

It demonstrates the exact validity of Hill's averaging relations under standard finite element discretizations for various boundary conditions, including effects of body forces and inertia.

Findings

A rigorous proof of discrete averaging relations for finite element discretizations.

Numerical verification through finite element simulations of large deformations.

Extension of proofs to include body forces and inertial effects.

Abstract

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.