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The Curious Case of 2D Isotropic Incompressible Neo-Hookean Composites

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Abstract

The homogenized behavior of a hyperelastic composite material is characterized by an effective stored-energy function that is functionally very different from the stored-energy functions that describe the underlying hyperelastic constituents. Over the past two decades, several analytical and computational results suggest that the case of isotropic incompressible Neo-Hookean composites in 2D may be the exception. This Note conjectures that the homogenized behavior of an isotropic hyperelastic solid made of incompressible Neo-Hookean materials is itself an incompressible Neo-Hookean material. To support this conjecture, earlier results are summarized, a new Reuss lower bound is derived, and a set of computational results is presented for the physically relevant cases of a Neo-Hookean matrix filled with random isotropic distributions of rigid and liquid circular particles of identical size.

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Notes

  1. This is so even in the most specialized case of isotropic incompressible composite materials made of isotropic incompressible constituents, when the resulting effective stored-energy function \(\overline{W}(\overline{{\mathbf{F}}})\), much like the local stored-energy function \(W({\mathbf{X}},{\mathbf{F}})\), admits representations in terms of just \(N-1\) invariants.

  2. Thanks to its objectivity \(\overline{W}({\mathbf{Q}}\overline{{\mathbf{F}}})=\overline{W}( \overline{{\mathbf{F}}})\) \(\forall {\mathbf{Q}}\in Orth^{+}\) and incompressibility \(\overline{W}(\overline{{\mathbf{F}}})=+\infty \) if \(\det \overline{{\mathbf{F}}}\neq 1\), the effective stored-energy function (3) admits representations in terms of two scalar variables, this regardless of its anisotropy. In this work, we found it convenient to use the representation \(\overline{\varPsi}(\lambda ,\varphi ):=\overline{W}(\overline{{\mathbf{F}}}_{ \varphi}(\lambda ))\), where \(\overline{{\mathbf{F}}}\) is given by (15) with \(\lambda \in \mathbb{R}\) and \(\varphi \in [0,\pi /2]\). In our calculations, we discretized the latter range as \(\varphi \in \{0,\pi /8,\pi /4,3\pi /8,\pi /2\}\).

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Acknowledgements

Support for this work by the National Science Foundation through the Grant DMREF–1922371 is gratefully acknowledged. G.A.F. also acknowledges the support of the Simons Foundation. V.L. would like to acknowledge support through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA

    Victor Lefèvre

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Gilles A. Francfort

  3. LAGA, Université Paris-Nord, Avenue J.-B. Clément, 93430, Villetaneuse, France

    Gilles A. Francfort

  4. Department of Civil and Environmental Engineering, University of Illinois, Urbana–Champaign, IL, 61801-2352, USA

    Oscar Lopez-Pamies

  5. Département de Mécanique, École Polytechnique, 91128, Palaiseau, France

    Oscar Lopez-Pamies

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  1. Victor Lefèvre
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  2. Gilles A. Francfort
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  3. Oscar Lopez-Pamies
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Correspondence to Oscar Lopez-Pamies.

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Lefèvre, V., Francfort, G.A. & Lopez-Pamies, O. The Curious Case of 2D Isotropic Incompressible Neo-Hookean Composites. J Elast 151, 177–186 (2022). https://doi.org/10.1007/s10659-022-09907-2

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