Non-monotonic dynamics (mixed hardening/softening) in nonlinear continuous structures: An asymptotic formulation

Non-monotonic dynamics of nonlinear continuous structures, i.e., mixed hardening(H)/softening(S) behavior in the vicinity of H/S transition, is comprehensively investigated by developing a generic asymptotic formulation. Non-monotonic dynamics is due to high-order competition between cubic and quintic mechanisms and thus qualitatively distinct from routine monotonic dynamics (either softening or hardening) associated with Duffing-type cubic mechanism. The general theoretical formulation is applied to both a nonlinear foundation beam model and a nonlinear shallow sagged cable model, with various non-monotonic responses found. In particular, by leveraging frequency response curves (FRCs) and backbone curves (BBCs), reversal of FRCs/BBCs in the mixed softening/hardening dynamics is further connected to zero dispersion phenomenon, with its activation condition also established.