Nonlinear hardening/softening dynamic analysis and its application to cables: A geometrical framework

•A hardening/softening analysis is proposed for identifying dominant dynamics with competing mechanisms.•Cable's hardening/softening dynamics is studied along and across curves of iso-Irvine parameter.•Hardening/softening transition dynamics is attacked by a refined asymptotic analysis. Competing mechanisms like quadratic/cubic nonlinearities associated with continuous structures lead to complicated dynamics, and it is meaningful to identify the dominant mechanism in a specific parameter domain. However, when the same parameter arises simultaneously in different competing mechanisms, either hardening (H) or softening (S), it implies that a certain constraint between various parameters and thus leads to difficulties for identifying its dominant dynamics with reference to physics parameters. For example, in a nonlinear cable model, its stiffness α, initial sag f, and Irvine parameter λc are all closely related to H/S behaviors, but they are not independent. A novel nonlinear hardening/softening dynamic analysis procedure is asymptotically developed by geometrically interpreting the underlying constraint between parameters as a curved surface located in an auxiliary parameter space. For cables, it consists of stiffness α, initial sag f, and Irvine parameter λc. Further, iso-λc curves on this curved constraint surface is properly defined, which represents a family of cable models with the same Irvine parameter and turn out to be useful for the proposed hardening/softening analysis. Though currently developed for a cable model, the framework can be meaningfully extended to other nonlinear structures with an initial curvature like shallow arches, buckled beams, or imperfect beams, etc.