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Fractal Fract 2018, 2(1), 12; https://doi.org/10.3390/fractalfract2010012

Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

1
Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
2
Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
3
Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
Author to whom correspondence should be addressed.
Received: 8 January 2018 / Revised: 30 January 2018 / Accepted: 7 February 2018 / Published: 13 February 2018
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Abstract

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity. View Full-Text
Keywords: fractal; fractional model; random media; viscoelasticity; homogenization fractal; fractional model; random media; viscoelasticity; homogenization
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Ostoja-Starzewski, M.; Zhang, J. Does a Fractal Microstructure Require a Fractional Viscoelastic Model? Fractal Fract 2018, 2, 12.

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