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Internal friction and the Stieltjes analytic representation of the effective properties of two-dimensional viscoelastic composites

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Abstract

Mathematical model for viscous internal friction is suggested. The model relates the viscous phenomena on the grain boundaries in a polycrystalline composite to the spectral measure in the analytic representation of the effective viscoelastic properties. This spectral measure contains all information about the geometry of a finely structured material. The spectral measure can be recovered from the measurements of viscoelastic effective properties over a range of frequencies. The Stieltjes analytic representation of the effective modulus is derived for the two-dimensional viscoelastic problem. It is shown that the spectral function in this representation determines the internal memory variables and the viscous internal friction.

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  1. Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT, 84112-0090, USA

    Elena Cherkaev

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  1. Elena Cherkaev
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Correspondence to Elena Cherkaev.

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This paper is dedicated to my longtime colleague and dear friend Prof. Konstantin Lurie.

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This work was supported by the US National Science Foundation through Grants DMS-0940249 and DMS-1413454.

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Cherkaev, E. Internal friction and the Stieltjes analytic representation of the effective properties of two-dimensional viscoelastic composites. Arch Appl Mech 89, 591–607 (2019). https://doi.org/10.1007/s00419-019-01514-3

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