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.
Highlights
Dating back to when [1,2] was written, fractional calculus models have been known to provide good fits to constitutive responses of many viscoelastic materials, both natural and man-made On the other hand, many natural materials display fractal spatial patterns [6], which has motivated some researchers to hypothesize that fractal structures dictate the fractional response
A question arises: Are fractional viscoelastic models dictated by fractalstructures of materials? Put differently, do fractal materials made of non-fractional phases require integer-order or fractional-order viscoelastic models? This issue has been studied for fractal arrangements of rheological elements [7,8,9,10], albeit without a 2D or 3D field theory—such as continuum mechanics, homogenization, and associated boundary value problems—truly taken into account
To further quantify this elastic-viscoelastic transition, we propose the parameter relative viscoelasticity, which is calculated by dividing the reduction in normalized modulus in the composite by the maximum possible reduction, and that is the reduction in a normalized modulus if only a pure viscoelastic phase exists
Summary
Dating back to when [1,2] was written, fractional calculus (i.e., non-integer order derivatives and integrals) models have been known to provide good fits to constitutive responses of many viscoelastic materials, both natural and man-made (see [3,4,5]) On the other hand, many natural materials display fractal spatial patterns [6], which has motivated some researchers to hypothesize that fractal structures dictate the fractional response. The spatial homogeneity and ergodicity of the material statistics allow for the application of uniform kinematic- or traction-controlled boundary conditions, which, for sufficiently large domains, provide nearly macroscopic (i.e., almost scale-independent) responses In other words, this involves the upscaling of a statistical volume element (SVE) (i.e., from a mesoscale level) to a representative volume element (RVE) (i.e., to a macroscale level). This involves the upscaling of a statistical volume element (SVE) (i.e., from a mesoscale level) to a representative volume element (RVE) (i.e., to a macroscale level) Overall, with this methodology and the tools at hand, we study the constitutive response at the Bernoulli percolation point and for the entire range of volume fractions of the viscoelastic phase. We explore the applicability of a fractional calculus model for our random microstructure at critical points of percolation of the elastic phase and the viscoelastic phase
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