February 3, 2021 /

New publication online: Tangent second-order homogenisation estimates for incompressible hyperelastic composites with fibrous microstructures and anisotropic phases

A new publication of our institute is available online. It deals with homogenisation methods for nonlinear two-phase materials and has been published in the Journal of the Mechanics and Physics of Solids (DOI: https://doi.org/10.1016/j.jmps.2020.104251). The paper is the outcome of a close collaboration between our institute and the University of Pennsylvania (Philadelphia, USA).

This work presents new estimates for the effective mechanical response of composites with fibrous microstructures and hyperelastic, anisotropic, incompressible phases. The composites are two-phase materials consisting of aligned cylindrical fibres of circular cross section that are randomly distributed in the transverse plane of the matrix phase. The proposed homogenisation is based on the tangent second-order method proposed by Avazmohammadi and Ponte Castañeda (2013), which utilises variational principles and a suitably chosen “linear comparison material” to provide estimates for effective strain energies of nonlinear materials. To take into account the incompressibility of the fibre phase, we employed an appropriate asymptotic expansion of the fibre deformation field. By doing so, a constraint pressure is identified as the consistent limit of emerging indeterminate forms in the dilatational part of the fibre stress tensor. Furthermore, we performed an asymptotic analysis to obtain regular expressions of the underlying equations in the incompressible matrix limit. While we addressed the general case of incompressible and generally anisotropic phases, we put particular emphasis on composites with transversely isotropic phases whose symmetry axes are collinear with the fibre direction. For such materials, we derived estimates for the effective strain energies in terms of macroscopic strain invariants. In addition, for composites that are described by isotropic phase energies augmented with J4-invariant-based anisotropic terms, which is the case for many biological materials, we show that the anisotropic part of the energy can be easily homogenised by means of a simple Voigt-type average. Moreover, we present closed-form results for composites with isotropic Neo-Hookean phases augmented by transversely isotropic energy contributions. Finally, it is shown that the new estimates correspond to certain exact results under longitudinal shear deformation and that they properly linearise for small strains by recovering the corresponding linear-elastic estimates. 10.1016/j.jmps.2020.104251

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