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Research Papers

# Fractal Pattern Formation at Elastic-Plastic Transition in Heterogeneous Materials

[+] Author and Article Information
J. Li

Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory, and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801junli3@uiuc.edu

M. Ostoja-Starzewski

Department of Mechanical Science and Engineering, Institute for Condensed Matter Theory, and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801martinos@uiuc.edu

J. Appl. Mech 77(2), 021005 (Dec 09, 2009) (7 pages) doi:10.1115/1.3176995 History: Received October 31, 2008; Revised April 03, 2009; Published December 09, 2009; Online December 09, 2009

## Abstract

Fractal patterns are observed in computational mechanics of elastic-plastic transitions in two models of linear elastic/perfectly plastic random heterogeneous materials: (1) a composite made of locally isotropic grains with weak random fluctuations in elastic moduli and/or yield limits and (2) a polycrystal made of randomly oriented anisotropic grains. In each case, the spatial assignment of material randomness is a nonfractal strict-white-noise field on a $256×256$ square lattice of homogeneous square-shaped grains; the flow rule in each grain follows associated plasticity. These lattices are subjected to simple shear loading increasing through either one of three macroscopically uniform boundary conditions (kinematic, mixed-orthogonal, or static) admitted by the Hill–Mandel condition. Upon following the evolution of a set of grains that become plastic, we find that it has a fractal dimension increasing from 0 toward 2 as the material transitions from elastic to perfectly plastic. While the grains possess sharp elastic-plastic stress-strain curves, the overall stress-strain responses are smooth and asymptote toward perfectly plastic flows; these responses and the fractal dimension-strain curves are almost identical for three different loadings. The randomness in elastic moduli in the model with isotropic grains alone is sufficient to generate fractal patterns at the transition but has a weaker effect than the randomness in yield limits. As the random fluctuations vanish (i.e., the composite becomes a homogeneous body), a sharp elastic-plastic transition is recovered.

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## Figures

Figure 1

Plots of equivalent plastic strain on 64×64 domains for models 1 (isotropic grains) and 2 (anisotropic grains) under various BCs: ((a1) and (a2)) kinematic, ((b1) and (b2)) mixed, and ((c1) and (c2)) static

Figure 2

Volume-averaged stress∼strain responses under different BCs for (a) model 1 (isotropic grains) and (b) model 2 (anisotropic grains)

Figure 3

Field images (white: elastic; black: plastic) for model 2 (anisotropic grains) at four consecutive stress levels applied via uniform static BC. The set of black grains is an evolving set with the fractal dimension given in Figs.  4444.

Figure 4

Estimation of the fractal dimension D for Figs.  3333, respectively, using the box-counting method: (a) D=1.667, (b) D=1.901, (c) D=1.975, and (d) D=1.999. The lines correspond to the best linear fitting of ln(Nr) versus ln(r).

Figure 5

Time evolution curves of the fractal dimension under different BCs for (a) model 1 (isotropic grains) and (b) model 2 (anisotropic grains). All loadings are linear in time.

Figure 6

Fractal dimension∼plastic strain curves under different BCs for (a) model 1 (isotropic grains) and (b) model 2 (anisotropic grains)

Figure 7

Comparison by different random variants (RV=5%, 1%, and 0-deterministic case): (a) average stress-strain curves and (b) fractal dimension versus plastic strain

Figure 8

Comparison of the effects of random perturbations in the yield limit and/or elastic moduli: (a) average stress versus average strain and (b) fractal dimension versus plastic strain

Figure 9

Comparison of different material responses: (a) fractal dimension versus plastic strain and (b) fractal dimension versus scaled plastic strain (i.e., scaled by yield strain)

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