Static bifurcations of short FPU softening chains with second-order interaction: The non-degenerate case

Fermi–Pasta–Ulam chains, made of n=2,3,4 masses, taut by an end force, are considered in the static field. Each mass has a second-order interaction with the surrounding masses. Springs are of cubic type, with softening behavior. The equilibrium equations are derived by the total potential energy theorem. The fundamental nonlinear path of each chain, exhibiting a limit point, is evaluated. Then, by controlling the loading process by a displacement, instead of the force, a bifurcation analysis is carried out, to investigate both the ascending and descending branches of the fundamental path. The analysis allows evaluating the exact bifurcation points along this path, which manifest in number of n−1. The study also highlights the existence of a degenerate case in which all the bifurcation points coalesce at the limit point, an occurrence which is excluded here by assuming that all the bifurcation points are well-separated. The n−1 primary bifurcated paths are parametrically described in asymptotic form. Then, a post-bifurcation analysis is carried out along each of them, aimed at discovering further bifurcation points. Stability of all the branches found (first bifurcated and secondary bifurcated branches) is characterized from application of Lagrange–Dirichlet theorem. We show, for the present discrete nonlinear elastic systems, that for some parameters of interest, the first bifurcated branch may be stable, even if it loses stability in a secondary bifurcation scenario. The complex bifurcation scenario is depicted by 3D and 2D bifurcation diagrams, and asymptotic results validated by numerical analyses.