Constitutive properties of contacting materials with a finite-sized microstructure

Abstract

Materials consisting of macro-molecules that are set in a structure and interact via contact interactions are investigated. The analysis is analytical, based on the methodology developed for granular materials by Jenkins J.T. and Koenders M.A. “The incremental response of random aggregates of round particles” European Physics Journal E., 13, 2004 113–123. The stiffness tensor is obtained and the results are screened for auxetic properties. Two-dimensional materials are investigated. For regular packings isotropic and anisotropic structures are examined. It is demonstrated that isotropic materials in this category are never auxetic. Heterogeneous materials are also analysed. It is shown that auxetic behaviour is enhanced when structural variation is present. By contrast, auxetic behaviour is suppressed when constitutive heterogeneity is present. The results of the analysis may be employed to scrutinize numerical simulations.

Keywords:

• Auxetic materials
• Isotropy
• Anisotropy
• Heterogeneity
• Elastic-frictional contact interactions

Acknowledgements

Discussions with Professor Peter Foot and Dr Richard Singer have been helpful in clarifying some of the aspects of this paper.

Notes

^¶ J ₀(z) is a Bessel function, see [24].

^‡ In order to carry out the Fourier transforms the medium should—strictly speaking - have infinite dimensions. In practice there is no problem as only locally correlated quantities appear in the end result.

^† The word “distortion” is used rather than the cumbersome “incremental displacement gradient”.