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.
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September 2016
Review Articles
Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound
Saba Saeb,
Saba Saeb
Chair of Applied Mechanics
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: saba.saeb@ltm.uni-erlangen.de
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: saba.saeb@ltm.uni-erlangen.de
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Paul Steinmann,
Paul Steinmann
Chair of Applied Mechanics
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: paul.steinmann@ltm.uni-erlangen.de
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: paul.steinmann@ltm.uni-erlangen.de
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Ali Javili
Ali Javili
Department of Mechanical Engineering,
Bilkent University,
Ankara 06800, Turkey
e-mail: ajavili@bilkent.edu.tr
Bilkent University,
Ankara 06800, Turkey
e-mail: ajavili@bilkent.edu.tr
Search for other works by this author on:
Saba Saeb
Chair of Applied Mechanics
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: saba.saeb@ltm.uni-erlangen.de
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: saba.saeb@ltm.uni-erlangen.de
Paul Steinmann
Chair of Applied Mechanics
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: paul.steinmann@ltm.uni-erlangen.de
University of Erlangen–Nuremberg,
Egerland Str. 5,
Erlangen 91058, Germany
e-mail: paul.steinmann@ltm.uni-erlangen.de
Ali Javili
Department of Mechanical Engineering,
Bilkent University,
Ankara 06800, Turkey
e-mail: ajavili@bilkent.edu.tr
Bilkent University,
Ankara 06800, Turkey
e-mail: ajavili@bilkent.edu.tr
1Corresponding author.
Manuscript received December 3, 2015; final manuscript received June 23, 2016; published online September 6, 2016. Assoc. Editor: Martin Schanz.
Appl. Mech. Rev. Sep 2016, 68(5): 050801 (33 pages)
Published Online: September 6, 2016
Article history
Received:
December 3, 2015
Revised:
June 23, 2016
Citation
Saeb, S., Steinmann, P., and Javili, A. (September 6, 2016). "Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound." ASME. Appl. Mech. Rev. September 2016; 68(5): 050801. https://doi.org/10.1115/1.4034024
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