Aspects of computational homogenization at finite deformations: A unifying review from Reuss' to Voigt's Bound

Saeb, S., Steinmann, P. and Javili, A. (2016) Aspects of computational homogenization at finite deformations: A unifying review from Reuss' to Voigt's Bound. Applied Mechanics Reviews, 68(5), 050801. (doi: 10.1115/1.4034024)

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 FE2 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.

• Item Type: Articles
• Additional Information: Acknowledgment The support of this work by the Cluster of Excellence “Engineering of Advanced Materials” at the University of Erlangen–Nuremberg, which is funded by the DFG within the framework of its “Excellence Initiative,” is greatly appreciated.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Steinmann, Professor Paul
• Authors: Saeb, S., Steinmann, P., and Javili, A.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Applied Mechanics Reviews
• Publisher: American Society of Mechanical Engineers (ASME)
• ISSN: 0003-6900
• ISSN (Online): 1088-8535
• Published Online: 06 September 2016