Macroscopic instabilities and domain formation in neo-Hookean laminates

This work is concerned with the characterization of the macroscopic response of soft elastomeric composites following the onset of 'macroscopic' (long wavelength) instabilities. For this purpose, the special case of laminated composites made of two neo-Hookean phases of different stiffnesses subjected to plane strain loading is considered. Indeed, it is known that the macroscopic response of such elastic composites can lose strong ellipticity when the stiff layers are subjected to sufficiently high compressive loads and their volume fraction is large enough. However, their post-bifurcation response is not yet known. Here it will be shown that the laminates can in fact lose (global) strict rank-one convexity, and do so well before losing (local) strong ellipticity. This will be accomplished by first constructing the rank-one convexification of the 'principal' solution for the stored-energy function at finite strains. It will then be shown that the rank-one convexification of the energy is polyconvex, and therefore corresponds to the quasiconvexification or 'relaxation' of the energy. Consequently, the post-bifurcation response corresponds to the formation of lamellar or 'striped' domains, which, in turn, gives rise to 'soft modes' of deformation. One important implication of this work for more general soft elastic composites is that the commonly accepted practice of using loss of strong ellipticity to predict the onset of macroscopic instabilities can significantly overestimate their stability. Instead, the relaxation of the energy should be computed in order to get more accurate (and conservative) estimates for their macroscopic stability, as well as the correct post-bifurcation response.