Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization

Abstract

SUMMARYFormulation of the scale transition equations coupling the microscopic and macroscopic variables in the second-order computational homogenization of heterogeneous materials and the enforcement of generalized boundary conditions for the representative volume element (RVE) are considered. The proposed formulation builds on current approaches by allowing any type of RVE boundary conditions (e.g. displacement, traction, periodic) and arbitrary shapes of RVE to be applied in a unified manner. The formulation offers a useful geometric interpretation for the assumptions associated with the microstructural displacement fluctuation field within the RVE, which is here extended to second-order computational homogenization. A unified approach to the enforcement of the boundary conditions has been undertaken using multiple constraint projection matrices. The results of an illustrative shear layer model problem indicate that the displacement and traction RVE boundary conditions provide the upper and lower bounds of the response determined via second-order computational homogenization, and the solution associated with the periodic RVE boundary conditions lies between them.

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

“…[215][216][217][218][219][220][221][222][223][224], have shown that in pure mechanical linear and nonlinear problems, the effective behavior derived under periodic boundary conditions is bounded by linear displacement boundary conditions from above and constant traction boundary conditions from below for a finite size of the RVE. Kaczmarczyk et al [225] made similar conclusions in the context of second-order computational homogenization. However, this does not imply that the results obtained under periodic boundary conditions are always the closest ones to the exact solutions as clearly stated by Terada et al [221] "there is no guarantee that periodic boundary conditions are the best among a class of possible boundary conditions.…”

Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

“…The microscopic boundary conditions are related to the macroscopic kinematic quantities, which are the macroscopic deformation gradientF and its gradientḠ. These boundary conditions are given in [8] and are formulated in terms of the microscopic displacement fluctuation field ω as…”

“…An effective remedy, which is known as the computational homogenization, has been developed to link up straightforwardly the responses of the large scale problems, also called the macroscopic problems, to the behavior of the smaller scale problems, also called the microscopic problems, where the presence of heterogeneities is considered. The basic ideas of the computational homogenization approach have been presented in papers by Michel et al [1], Terada et al [2], Miehe et al [3,4], Kouznetsova et al [5,6,7], Kaczmarczyk et al [8], Peric et al [9], Geers et al [10] and references therein, as a non-exhaustive list. By this technique, two boundary value problems are defined at two separate scales, Figure 1: Illustration of first-order and second-order multiscale computational homogenization schemes.…”

“…For the multiscale problems, the generalized continuum can potentially be used at both the macroscopic and the microscopic scales. Recent extensions of the FMCH scheme to the second-order continuum, as for the so-called second-order multiscale computational homogenization (SMCH) [6,7,8], provide a systematic way to couple the strain-gradient continuum at the macro-scale with the classical continuum at the micro-scale, see Fig. 1.…”

Relaxed loading conditions for higher order homogenisation approaches

“…The first one goes back to the seminal work of Mindlin [14] and introduces higher gradients of the overall displacement field as additional and independent degrees of freedom. The application of the second gradient continuum as a substitute medium for heterogeneous micro structures has been discussed in literature, e. g. [10,11]. The second extension bases on the micromorphic continuum theory initially proposed by Eringen [3].…”