The steady response of an infinite unbroken floating ice sheet to a moving load is
considered. It is assumed that the ice sheet is supported below by water of finite
uniform depth. For a concentrated line load, earlier studies based on the linearization
of the problem have shown that there are two ‘critical’ load speeds near which
the steady deflection is unbounded. These two speeds are the speed c0 of gravity
waves on shallow water and the minimum phase speed cmin. Since deflections cannot
become infinite as the load speed approaches a critical speed, Nevel (1970) suggested
nonlinear effects, dissipation or inhomogeneity of the ice, as possible explanations.
The present study is restricted to the effects of nonlinearity when the load speed is
close to cmin. A weakly nonlinear analysis, based on dynamical systems theory and
on normal forms, is performed. The difference between the critical speed cmin and
the load speed U is taken as the bifurcation parameter. The resulting normal form
reduces at leading order to a forced nonlinear Schrödinger equation, which can be
integrated exactly. It is shown that the water depth plays a role in the effects of
nonlinearity. For large enough water depths, ice deflections in the form of solitary
waves exist for all speeds up to (and including) cmin. For small enough water depths,
steady bounded deflections exist only for speeds up to U*, with U* < cmin. The weakly
nonlinear results are validated by comparison with numerical results based on the full
governing equations. The model is validated by comparison with experimental results
in Antarctica (deep water) and in a lake in Japan (relatively shallow water). Finally,
nonlinear effects are compared with dissipation effects. Our main conclusion is that
nonlinear effects play a role in the response of a floating ice plate to a load moving
at a speed slightly smaller than cmin. In deep water, they are a possible explanation
for the persistence of bounded ice deflections for load speeds up to cmin. In shallow
water, there seems to be an apparent contradiction, since bounded ice deflections have
been observed for speeds up to cmin while the theoretical results predict bounded ice
deflection only for speeds up to U* < cmin. But in practice the value of U* is so close
to the value of cmin that it is difficult to distinguish between these two values.
The Non-Unicity of the Film Thickness in the Hydrodynamic Lubrication: Novel Approach Generating Equivalent Micro-Grooves and Roughness