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Hyperelastic fiber composites - homogenization and macroscopic stability

Gal DeBotton (Ben-Gurion Univ.), Gal Shmuel (CalTech), Stephan Rudykh (Massachusetts Institute of Technology), Tal Oren (Ben-Gurion Univ.)

SES Medal Symposium in honor of D.J. Steigmann

Tue 2:40 - 4:00

MacMillan 115

We characterize the macroscopic response of hyperelastic fiber composites in terms of the behaviors of the constituting phases. To this end we make use of a unique representation of the deformation gradient in terms of a set of transversely isotropic invariants. Respectively, these invariants correspond to extension along the fibers, transverse dilatation, out-of-plane shear along the fibers, in-plane shear in the transverse plane and the coupling between the shear modes. With the aid of this representation, it is demonstrated that under a combination of out-of-plane shear and extension along the fibers there is a class of nonlinear materials for which the exact expression for the macroscopic behavior of a composite cylinder assemblage can be determined. The macroscopic response of the composite to shear in the transverse plane is approximated with the aid of an exact result for sequentially laminated composites. Assuming no coupling between the shear modes, these results allow to construct a closed-form homogenized model for the macroscopic response of a fiber composite with neo-Hookean phases. A new variational estimate allows to extend these results to more general classes of materials. The resulting explicit estimates for the macroscopic stresses developing in composites with one and two families of neo-Hookean and Gent-like phases are compared with corresponding finite element simulations of periodic composites with a hexagonal unit cell. Estimates for the critical stretch ratios at which the composites loose stability at the macroscopic level are compared with the corresponding numerical results too. It is demonstrated that both the primary stress-strain curves and the critical stretch ratios are in fine agreement with the corresponding numerical results.