Research Papers

Scale-Dependent Homogenization of Inelastic Random Polycrystals

[+] Author and Article Information
Shivakumar I. Ranganathan

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801srangan3@uiuc.edu

Martin Ostoja-Starzewski

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801martinos@uiuc.edu

J. Appl. Mech 75(5), 051008 (Jul 15, 2008) (9 pages) doi:10.1115/1.2912999 History: Received September 01, 2007; Revised November 14, 2007; Published July 15, 2008

Rigorous scale-dependent bounds on the constitutive response of random polycrystalline aggregates are obtained by setting up two stochastic boundary value problems (Dirichlet and Neumann type) consistent with the Hill condition. This methodology enables one to estimate the size of the representative volume element (RVE), the cornerstone of the separation of scales in continuum mechanics. The method is illustrated on the single-phase and multiphase aggregates, and, generally, it turns out that the RVE is attained with about eight crystals in a 3D system. From a thermodynamic perspective, one can also estimate the scale dependencies of the dissipation potential in the velocity space and its complementary potential in the force space. The viscoplastic material, being a purely dissipative material, is ideally suited for this purpose.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

The homogenization methodology

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Figure 2

Quartet for the selection of dissipation potential (at macroscale)

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Figure 3

The multiplicative decomposition

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Figure 4

Single crystal compression test: L11=L22=0.5, L33=−1, L12=L13=L21=L23=L31=L32=0: (a) Sachs type and (b) Taylor type

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Figure 5

(a) The polycrystal compression test: L11=L22=0.5, L33=−1, L12=L13=L21=L23=L31=L32=0, (b) texture-Taylor type, and (c) texture-Sachs type

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Figure 6

Single crystal and the polycrystalline aggregate yield surfaces at a strain of 0.5 (in the “π-plane”)

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Figure 7

(a) Geometry of eight grains, (b) mesh, (c) deformed view-Neumann problem, (d) deformed view-Dirichlet problem, (e) von Mises stress-Neumann problem, and (f) von Mises stress-Dirichlet problem

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Figure 8

Bounds on the aggregate response

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Figure 9

(a) Deformed view in the Neumann problem (two-phase material), (b) deformed view in the Dirichlet problem (two-phase material), (c) the von Mises stress in the Neumann problem, and (d) the von Mises stress in the Dirichlet problem

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Figure 10

Bounds on the multiphase aggregate response




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