  We study elastic snap-through induced by a control parameter that evolves
dynamically. In particular, we study an elastic arch subject to an
end-shortening that evolves linearly with time, i.e. at a constant rate. For
large end-shortening the arch is bistable but, below a critical end-shortening,
the arch becomes monostable. We study when and how the arch transitions between
states and show that the end-shortening at which the fast 'snap' happens
depends on the rate at which the end-shortening is reduced. This lag in
snap-through is a consequence of delayed bifurcation and occurs even in the
perfectly elastic case when viscous (and viscoelastic) effects are negligible.
We present the results of numerical simulations to determine the magnitude of
this lag as the loading rate and the importance of external viscous damping
vary. We also present an asymptotic analysis of the geometrically-nonlinear
problem that reduces the salient dynamics to that of an ordinary differential
equation; the form of this reduced equation is generic for snap-through
instabilities in which the relevant control parameter is ramped linearly in
time. Moreover, this asymptotic reduction allows us to derive analytical
results for the observed lag in snap-through that are in good agreement with
the numerical results of our simulations. Finally, we discuss scaling laws for
the lag that should be expected in other examples of delayed bifurcation in
elastic instabilities.
