  This paper presents an analytical method to predict the delayed switching
dynamics of nonlinear shallow arches while switching from one state to another
state for different loading cases. We study an elastic arch subject to static
loading and time-dependent loading separately. In particular, we consider a
time-dependent loading that evolves linearly with time at a constant rate. In
both cases, we observed that the switching does not occur abruptly when the
load exceeds the static switching load, rather the time scale of the dynamics
drastically slows down; hence there is a delay in switching. For
time-independent loading, this delay increases as the applied load approach the
static switching load. Whereas for a time-dependent loading, the delay is
proportional to the rate of the applied load. Other than the loading
parameters, the delay switching time also depends on the local curvature of the
force-displacement function at the static switching point and the damping
coefficient of the arch material. The delay switching occurs due to the
flatness of the energy curve at static switching load. Therefore, we linearize
the arch near the static switching point and get a reduced nonlinear ordinary
differential equation to study the switching dynamics of the arch. This reduced
equation allows us to derive analytical expressions for the delay switching
time of the. We further compare the derived analytical results with the
numerical solutions and observed a good agreement between them. Finally, the
derived analytical formulae can be used to design arches for self-offloading
dynamic footwear for diabetics.
