The main objective of this work is to shed light on the effect of fiber plasticity on the macroscopic response and domain formation in soft biological composites. This goal is pursued by analyzing the plane-strain response of two-phase laminates. In the context of this problem, the effect of fiber plasticity is accounted for by allowing the elastically stiffer layers (“fiber” phase) to also yield plastically and by taking the soft layers (“matrix” phase) to be purely elastic solids. The analysis is carried out at finite elastic and plastic strains, but it is restricted to unidirectional, non-monotonic loading paths, applied by initially increasing the macroscopic stretch along the direction of the layers up to a prescribed maximum value and then decreasing the same stretch down to a minimum value. A simple expression is derived for the critical conditions at which the homogenized behavior of the laminate loses strong ellipticity for the first time along the loading path. The relevance of this result stems from the fact that the loss of macroscopic ellipticity of these composites is known to coincide with the onset of bifurcations of the long-wavelength type. It follows from this result that, just like hyperelastic laminates, elastoplastic laminates may lose macroscopic ellipticity whenever their incremental strength in shear perpendicular to the layers vanishes for the first time. It is shown by means of specific numerical calculations that fiber plasticity has a softening effect on the critical shear strength of the laminate under monotonically increasing loading and a hardening effect under monotonically decreasing loading, whereas local elasticity has the opposite effects. Thus, the effects of local elasticity and fiber plasticity compete with each other at every stage of the deformation where both mechanisms are active and whether or not a specific laminate will lose its macroscopic ellipticity along a given loading path depends crucially on the relative strengths of these effects. In this regard, the influence of the relevant loading and material parameters is investigated in detail. For situations in which loss of macroscopic ellipticity does take place, a corresponding post-bifurcation solution for the homogenized behavior of the laminate is computed. The deformed state of the material described by this solution is characterized by twin lamellar domains that are formed at a length scale much larger than the width of the original, microscopic layers, but still much smaller than the overall dimensions of the macroscopic specimen under consideration. The macroscopic response of these composites is found to be much softer along the bifurcated equilibrium path than along the corresponding principal path. Nevertheless, no macroscopic unloading has been observed in the post-bifurcation regime, indicating accordingly that the macroscopic, post-bifurcation behavior of these materials is stable.
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