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Technical Briefs

# On a Mass Conservation Criterion in Micro-to-Macro Transitions

[+] Author and Article Information
İ. Temizer, P. Wriggers

Institute of Mechanics and Computational Mechanics, Leibniz University of Hannover, Appelstrasse 9a, 30167 Hannover, Germany

$⟨Q⟩Ψ=def1∕∣Ψ∣∫ΨQdΨ$ denotes the volume average of the quantity $Q$ with respect to the domain $Ψ$.

The notation (●) will be used to denote the macroscopic counterpart of a microscopic quantity (●).

In this reference, only UT-BCs and LD-BCs are considered. However, it is a straightforward task to extend the same line of discussion to PR-BCs.

The enforcement procedure for the BCs employs tractions only (see Appendix). Here, kinematic admissibility is used in the sense that the proposed displacements match the solution displacements on the boundary.

In Ref. 7, the fact that $det(F)=⟨J⟩V0$ for $F$-LD-BCs has been used to derive a key identity.

If the tests are $P$-controlled, then $⟨F⟩V0$ is different for each realization of the random case and for each sample size and therefore it would not make sense to compare the results.

For periodic microstructures, one simply takes a unit cell and subjects it to PR-BCs.

For $F$-PR-BCs and $F$-UT-BCs, translational degrees of freedom should additionally be constrained to avoid rigid body motion.

J. Appl. Mech 75(5), 054503 (Jul 17, 2008) (4 pages) doi:10.1115/1.2913042 History: Received September 27, 2007; Revised January 10, 2008; Published July 17, 2008

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## Figures

Figure 1

The original and the homogenized micromechanical problems

Figure 2

The periodic (LEFT) and a random realization (RIGHT) of a micromechanical sample size with 64 particles. The volume fraction of the disklike particles in the reference configuration is set to 0.4

Figure 3

The inconsistency associated with the micro-macro mass balance is plotted for periodic (LEFT) and random (RIGHT) samples employing F-UT-BCs on a particulate hyperlastic microstructure depicted in Fig. 2. F has components F11=F22=1.2 and F12=F21=0.4.

Figure 4

The variation of the stress component P¯11 and its ensemble average (arithmetic mean) is monitored for F-LD/PR/UT-BCs for the random microstructure that was considered in Fig. 3

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