Wiggly strain localizations in peridynamic bars with non-convex potential

We discuss the static states of a peridynamic nonlinear elastic bar of finite length in a hard device, which represents a continuum description of a complex hierarchical structure with interacting long-range crosslinkers, of the type encountered in biological systems. The nonlocal character of the model requires that edge conditions are defined on a boundary layer with the same length of the horizon, affecting the solution in the bulk interior. Assuming that the constituent microscopic ligaments contain bistable units governed by a non-convex potential, we show that the development of coexisting folded-unfolded phases, either synchronized or unsynchronized, induces in the displacement field the formation of undulations at a micro-scale of the length of the horizon, associated with strain localizations triggered at the bar ends. The equilibrium paths, found numerically with a pseudo-arc-length continuation method, become unstable within a certain range of elongation, suggesting the possible occurrence of a negative-stiffness response.