Fiber plasticity and loss of ellipticity in soft composites under non-monotonic loading

In this work, we relate fiber plasticity in soft composites to the loss of ellipticity of the governing equations of equilibrium of a composite under non-monotonic uniaxial loading. The loss of ellipticity strongly indicates the emergence of localization phenomena in the composite, reminiscent of the emergence of kinking instabilities in tendon, which occur as a response to tendon “overload” without requiring any macroscopic compressive loading. We examine soft composites where both fibers and matrix can be highly extensible and plastic deformations are present in the fiber phase. We first propose a transversely isotropic constitutive model for the fibers allowing for plastic deformations, taking into account a single slip direction, consistent with the microstructure of hierarchically assembled collagen fibers. Following, we propose a simple hyperelastic model for the matrix and combine the two following the Voigt assumption. We then formulate a general loss of ellipticity criterion for an elastoplastic material subjected to finite deformations. We use this criterion to indicate critical conditions for loss of ellipticity in the soft composite and individually in the fiber phase, under various loading–unloading paths. Results show that plastic deformation of the fiber phase during tensile loading can lead to ellipticity breakdown during elastic unloading while, macroscopically, the material is still in tension, indicating the possible onset of an instability.