Nonlinear effects in the response of a floating ice plate to a moving load

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 <e1>c</e1><inf>0</inf> of gravity waves on shallow water and the minimum phase speed <e1>c</e1><inf>min</inf>. 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 <e1>c</e1><inf>min</inf>. A weakly nonlinear analysis, based on dynamical systems theory and on normal forms, is performed. The difference between the critical speed <e1>c</e1><inf>min</inf> and the load speed <e1>U</e1> 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) <e1>c</e1><inf>min</inf>. For small enough water depths, steady bounded deflections exist only for speeds up to <e1>U</e1>*, with <e1>U</e1>* &lt; <e1>c</e1><inf>min</inf>. 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 <e1>c</e1><inf>min</inf>. In deep water, they are a possible explanation for the persistence of bounded ice deflections for load speeds up to <e1>c</e1><inf>min</inf>. In shallow water, there seems to be an apparent contradiction, since bounded ice deflections have been observed for speeds up to <e1>c</e1><inf>min</inf> while the theoretical results predict bounded ice deflection only for speeds up to <e1>U</e1>* &lt; <e1>c</e1><inf>min</inf>. But in practice the value of <e1>U</e1>* is so close to the value of <e1>c</e1><inf>min</inf> that it is difficult to distinguish between these two values.