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Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites

Jun Zhang, Martin Ostoja-Starzewski
Published 27 April 2016.DOI: 10.1098/rspa.2015.0801
Jun Zhang
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
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Martin Ostoja-Starzewski
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USAInstitute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USABeckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
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Abstract

This paper investigates the scaling from a statistical volume element (SVE; i.e. mesoscale level) to representative volume element (RVE; i.e. macroscale level) of spatially random linear viscoelastic materials, focusing on the quasi-static properties in the frequency domain. Requiring the material statistics to be spatially homogeneous and ergodic, the mesoscale bounds on the RVE response are developed from the Hill–Mandel homogenization condition adapted to viscoelastic materials. The bounds are obtained from two stochastic initial-boundary value problems set up, respectively, under uniform kinematic and traction boundary conditions. The frequency and scale dependencies of mesoscale bounds are obtained through computational mechanics for composites with planar random chessboard microstructures. In general, the frequency-dependent scaling to RVE can be described through a complex-valued scaling function, which generalizes the concept originally developed for linear elastic random composites. This scaling function is shown to apply for all different phase combinations on random chessboards and, essentially, is only a function of the microstructure and mesoscale.

  • Received November 18, 2015.
  • Accepted March 22, 2016.
  • © 2016 The Author(s)
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April 2016
Volume 472
, issue 2188
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science: 472 (2188)
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Keywords

random composite
viscoelasticity
representative volume element
mesoscale bounds
scaling function
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Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites
Jun Zhang, Martin Ostoja-Starzewski
Proc. R. Soc. A 2016 472 20150801; DOI: 10.1098/rspa.2015.0801. Published 27 April 2016
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Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites

Jun Zhang, Martin Ostoja-Starzewski
Proc. R. Soc. A 2016 472 20150801; DOI: 10.1098/rspa.2015.0801. Published 27 April 2016

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  • Article
    • Abstract
    • 1. Introduction
    • 2. Problem formulation
    • 3. Scaling to representative volume element: theoretical results
    • 4. Scaling to representative volume element: numerical results
    • 5. Conclusion
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