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Review Article

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

[+] Author and Article Information
Saba Saeb

Chair of Applied Mechanics
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

Ali Javili

Department of Mechanical Engineering,
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 68(5), 050801 (Sep 06, 2016) (33 pages) Paper No: AMR-15-1141; doi: 10.1115/1.4034024 History: Received December 03, 2015; Revised June 23, 2016

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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Häfner, S. , Eckardt, S. , Luther, T. , and Könke, C. , 2006, “ Mesoscale Modeling of Concrete: Geometry and Numerics,” Comput. Struct., 84(7), pp. 450–461. [CrossRef]
Musienko, A. , Tatschl, A. , Schmidegg, K. , Kolednik, O. , Pippan, R. , and Cailletaud, G. , 2007, “ Three-Dimensional Finite Element Simulation of a Polycrystalline Copper Specimen,” Acta Mater., 55(12), pp. 4121–4136. [CrossRef]
Galli, M. , Botsis, J. , and Janczak-Rusch, J. , 2008, “ An Elastoplastic Three-Dimensional Homogenization Model for Particle Reinforced Composites,” Comput. Mater. Sci., 41(3), pp. 312–321. [CrossRef]
Lee, K. M. , and Park, J. H. , 2008, “ A Numerical Model for Elastic Modulus of Concrete Considering Interfacial Transition Zone,” Cem. Concr. Res., 38(3), pp. 396–402. [CrossRef]
Reid, A. C. E. , Langer, S. A. , Lua, R. C. , Coffman, V. R. , Haan, S. , and García, R. E. , 2008, “ Image-Based Finite Element Mesh Construction for Material Microstructures,” Comput. Mater. Sci., 43(4), pp. 989–999. [CrossRef]
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He, H. , 2010, “ Computational Modelling of Particle Packing in Concrete,” Ph.D. thesis, TU Delft, Delft University of Technology, Delft, Netherlands.
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Figures

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Fig. 1

Graphical summary of computational homogenization. The macroscopic domain  MB0 is mapped to the spatial configuration  MBt via the nonlinear deformation map  Mϕ. The domain B0 corresponds to a microscopic RVE. The motion ϕ of the RVE is associated with a macroscopic point  MX within the bulk. In view of the first-order strain-driven homogenization, the macroscopic deformation gradient is given, and the macro Piola stress and the macro Piola tangent are sought. These quantities are evaluated through solving boundary value problems at the microscale.

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Fig. 2

Examples of three- and two-dimensional microstructures: cubic microstructure with random distribution of spherical particles (left), rectangular microstructures with random shape and distribution of the inclusions (middle), and pores (right)

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Fig. 3

The inclusions and the matrix deform identically under the Taylor's assumption

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Fig. 4

Taylor's assumption representations via system of parallel springs and multiphase composites

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Fig. 5

The inclusions and the matrix do not necessarily deform identically under DBC

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Fig. 6

Graphical illustration of PBC implementation setting. The boundary of the RVE is decomposed into minus and plus parts. Positions of the boundary nodes are determined through two distinct fields:  MF·X and F̃·X.

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Fig. 7

The entire boundary ∂B0 except point A in both directions and point B in y-direction is prescribed with  MP·N. Point A is fixed in both directions and point B is fixed in y-direction so as to remove rigid body motions. Fixing these points can lead to introduction of spurious forces on the Dirichlet part of the boundary. The spurious tractions are denoted ζ. The dashed line and the solid black line indicate the deformation of the microstructure in the absence and presence of the spurious forces, respectively.

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Fig. 8

Graphical illustration of the TBC implementation setting. We prescribe and update  MP·N and η iteratively until 〈F〉−MF=!0 and ζyB=!0 are satisfied.

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Fig. 9

Graphical illustration of the TBC implementation setting in three-dimensional problems. We prescribe and update  MP·N, ηB, ηC, and ηD iteratively until 〈F〉−MF=!0, ζxB=!0, ζyC=!0, and ζzD=!0 are satisfied. Note that point D is free to move in x- and y-directions.

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Fig. 10

The compatibility of the deformation field is violated under Sachs' assumption

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Fig. 11

The left and right unit elements represent the inclusion and the matrix, respectively

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Fig. 12

Sachs' assumption representation via system of serial springs and multiphase composites

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Fig. 13

Mesh qualities of the three- and two-dimensional samples

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Fig. 14

Two-dimensional microstructure analysis using DBC, PBC, and TBC for r = 0.1 and r = 10. Top: simple-shear deformation in xy-plane. Distribution of the micro Piola stress (xy-component) normalized by its macro counterpart. Bottom: Uniaxial stretch in x-direction. Distribution of the micro Piola stress (xx-component) normalized by its macro counterpart.

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Fig. 15

Three-dimensional microstructure analysis using DBC, PBC, and TBC for r = 0.1 and r = 10. Top: simple-shear deformation in xy-plane. Distribution of the micro Piola stress (xy-component) normalized by its macro counterpart. Bottom: Uniaxial stretch in x-direction: Distribution of the micro Piola stress (xx-component) normalized by its macro counterpart.

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Fig. 16

Evolution of the macro Piola stress due to the increase of simple-shear deformation (top) and uniaxial stretch (bottom) for r = 0.1 (left) and r = 10 (right) for the two-dimensional microstructure. The depicted deformation modes correspond to the results of the PBC for 100% deformation with r = 0.1.

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Fig. 17

Evolution of the macro Piola stress due to the increase of simple-shear deformation (top) and uniaxial stretch (bottom) for r = 0.1 (left) and r = 10 (right) for the three-dimensional microstructure. The depicted deformation modes correspond to the results of the PBC for 100% deformation with r = 0.1.

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Fig. 18

Evolution of macro Piola stress due to the increase of f from 0.0001 to 10,000 when 100% simple-shear deformation (left) and uniaxial stretch (right) is imposed

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Fig. 19

Level n of the random microstructure consists of all the smaller microstructures and the information associated to size n. All the levels have the inclusion volume fraction of 25%.

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Fig. 20

Mesh qualities of periodic and random microstructures

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Fig. 21

Periodic and random microstructures. The inclusion volume fraction in all the microstructures is set to be f = 25%.

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Fig. 22

Evolution of macrostress versus number of inclusions for 1% (top) and 25% (bottom) of uniaxial stretch and r = 0.1. Results of the Taylor's and Sachs' bounds are independent of the distribution pattern of the microstructure and only depend on the volume fraction. Choice of the boundary condition becomes less significant as the number of inclusions inside the RVE increases. Note that the results which are drawn by bolder lines correspond to periodic microstructures.

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Fig. 23

Evolution of macrostress versus number of inclusions for 1% (top) and 25% (bottom) of uniaxial stretch and r = 10. Results of the Taylor's and Sachs' bounds are independent of the distribution pattern of the microstructure and only depend on the volume fraction. Choice of the boundary condition becomes less significant as the number of inclusions inside the RVE increases. Note that the results which are drawn by bolder lines correspond to periodic microstructures.

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Fig. 24

The ratio δF̃/δMF is evaluated through solving linear problems at the converged solution of the microproblem

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Fig. 25

Macro- and microscale samples and the associated finite element discretizations

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Fig. 26

Distribution of the yx-component of the stress within the macrostructure and its microstructures. The microproblem is solved through different boundary conditions and r = 0.1.

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Fig. 27

Distribution of the yx-component of the stress within the macrostructure and its microstructures. The microproblem is solved through different boundary conditions and r = 10.

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