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Stiffening solids with liquid inclusions

A Corrigendum to this article was published on 03 February 2015

This article has been updated

Abstract

From bone and wood to concrete and carbon fibre, composites are ubiquitous natural and synthetic materials. Eshelby’s inclusion theory describes how macroscopic stress fields couple to isolated microscopic inclusions, allowing prediction of a composite’s bulk mechanical properties from a knowledge of its microstructure. It has been extended to describe a wide variety of phenomena from solid fracture to cell adhesion. Here, we show experimentally and theoretically that Eshelby’s theory breaks down for small liquid inclusions in a soft solid. In this limit, an isolated droplet’s deformation is strongly size-dependent, with the smallest droplets mimicking the behaviour of solid inclusions. Furthermore, in opposition to the predictions of conventional composite theory, we find that finite concentrations of small liquid inclusions enhance the stiffness of soft solids. A straightforward extension of Eshelby’s theory, accounting for the surface tension of the solid–liquid interface, explains our experimental observations. The counterintuitive stiffening of solids by fluid inclusions is expected whenever inclusion radii are smaller than an elastocapillary length, given by the ratio of the surface tension to Young’s modulus of the solid matrix. These results suggest that surface tension can be a simple and effective mechanism to cloak the far-field elastic signature of inclusions.

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Figure 1: Stretching droplets embedded in soft solids.
Figure 2: Aspect ratio of stretched ionic-liquid droplets in a soft (E = 1.7 kPa) silicone gel as a function of size and strain.
Figure 3: Young’s modulus of soft composites as a function of liquid content.
Figure 4: Theoretical predictions of composite behaviour.

Change history

  • 08 January 2015

    In the text following equation 7, the expression describing the limit where stiffening occurs was incorrect and should have read: surface tension dominates over elasticity (R « ϒ /E ). This has now been corrected in the online versions of the Article.

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Acknowledgements

We thank J. Fernandez-Garcia for the ionic liquids, and T. Kodger and R. Diebold for advice in preparing silicone. We also thank J. Singer, M. Rooks, F. Spaepen, S. Mukhopadhyay, P. Howell and A. Goriely for helpful conversations. We gratefully acknowledge funding from the National Science Foundation (CBET-1236086) to E.R.D., the Yale University Bateman Interdepartmental Postdoctoral Fellowship to R.W.S., and the John Simon Guggenheim Foundation, the Swedish Research Council and a Royal Society Wolfson Research Merit Award to J.S.W.

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R.W.S., R.B., B.A., K.E.J., H.P.F. and E.R.D. designed experiments. R.W.S., B.A. and R.B. acquired data. R.W.S. and E.R.D. analysed data. R.W.S., J.S.W. and E.R.D. developed the theory. R.W.S., J.S.W. and E.R.D. wrote the paper.

Corresponding author

Correspondence to Eric R. Dufresne.

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The authors declare no competing financial interests.

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Style, R., Boltyanskiy, R., Allen, B. et al. Stiffening solids with liquid inclusions. Nature Phys 11, 82–87 (2015). https://doi.org/10.1038/nphys3181

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