Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Abstract

The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

“…This domain—known as a RVE—should be large enough to statistically represent the heterogeneities present at the microscale but small enough to be considered as an infinitesimal volume element . Defining the appropriate RVE for a given heterogeneous material is a topic that has undergone much investigation (see, for example, the review of Saeb et al). Classical homogenization theories rely on the concept of scale separation, which is the assumption that the scale of microscopic field fluctuations should be smaller than the RVE size, which should, in turn, be much smaller than the characteristic length of macroscopic field fluctuations .…”

“…An alternative approach, which is known as stress‐driven homogenization, assumes that the macroscopic first Piola‐Kirchhoff stress tensor is known and then calculates the corresponding macroscopic deformation gradient. This can be performed numerically by treating the macroscopic deformation gradient components as additional unknowns to be solved for . The additional set of equations that must be considered to account for the new unknowns is where are prescribed macroscopic stress values, whereas is the macroscopic stress calculated using Equation .…”

Fuzzy‐stochastic FEM–based homogenization framework for materials with polymorphic uncertainties in the microstructure

“…The use of periodic boundary conditions for the statistically similar (substitute) model is motivated by a number of studies demonstrating that the periodic boundary conditions are the most reliable and converge faster than Dirichlet and Neumann boundary conditions . They are often used even if the model is not periodic, because Dirichlet and Neumann boundary conditions always give an overestimation and an underestimation for the stress in computational homogenization. Moreover, the simple comparison of the homogenized stresses in periodic composites modeled with only one inclusion in the unit cell and composites with a random microstructure, performed in the work of Zabihyan et al, demonstrated close homogenized stress values in both models.…”

“…Some examples of analytical homogenization of deterministic media can be found in the works of Andrianov et al Interesting results in computational homogenization of deterministic media are presented in the works of Chatzigeorgiou et al, Javili et al, Kästner et al, Legrain et al, Moës et al, and Spieler et al For an overview of existing deterministic homogenization techniques, we refer to the work of Saeb et al…”

The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method

“…The FE 2 method has its origins in solid mechanics, [31], [46], [47], [48], [28], [27], [55], and has found considerable interest in academia and industry; as a versatile method FE 2 has been used in non-linear problems of elasticity and inelasticity. For recent, comprehensive overviews of the FE 2 method we refer to [30], [61] and [56]. In order to account for size-dependency observed in materials science, Kouznetsova et al [41], [42] have introduced a second-order homogenization into FE 2 .…”