MATHEMATICAL MODEL OF ICE SHEET DEFORMATION CAUSED BY SUBMARINE MOTION V. M. Kozin, V. L. Zemlak, S. D. Chizhiumov Shipbuilding Department, State Technical University Komsomolsk-na-Amure, Russia ABSTRACT A model for the stress-strain state of ice cover under hydrodynamic loads caused by the motion of a submarine is constructed. Numerical calculations using this model are carried out. The deflections and stresses in the ice cover predicted by the model are computed using a finite element method (FEM) in combination with a boundary element method (BEM). The diffraction of gravity waves in ice cover that has longitudinal cracks or bands of open water are computed. The results for various open-water widths are presented together with the corresponding data from a model test basin. ISOPE-2010, Beijing, China

MATHEMATICAL MODEL Ice sheet deformation by the field of hydrodynamic pressures. The water - ideal incompressible liquid; the ice - viscoelastic plate. The equation of the uniform plate motion: Hydrodynamic load: (1) (2) where f o is the hydrodynamic pressure to the ice surface as if to a hard plate as a result of the submarine motion; f st is the hydrostatic response of the liquid to the plate deflection; f d is the intensity of the wave damping force: f w is the intensity of the liquid inertia force caused by the plate deflection.

ISOPE-2010, Beijing, China To define the hydrodynamic load f o, let us consider the submarine motion with the given velocity v (t). The field of the liquid velocities at each moment of time is found by solving the boundary-value problem for the Laplace equation:

ISOPE-2010, Beijing, China To find the hydrodynamic load f w, let us consider the ice sheet deflections at the velocity. The field of the liquid velocities at each moment of time is determined by the solution of the boundary-value problem for Laplace equation:

ISOPE-2010, Beijing, China Differential boundary-value problems are transformed to boundary integral equation, which is calculated by a boundary elements method (BEM): NUMERICAL MODEL

ISOPE-2010, Beijing, China

where

ISOPE-2010, Beijing, China FEM modelling

ISOPE-2010, Beijing, China Direct integrating of the motion equation:

ISOPE-2010, Beijing, China

EXAMPLE OF NUMERICAL RESULTS

ISOPE-2010, Beijing, China EXPERIMENTAL EQUIPMENT

ISOPE-2010, Beijing, China

Relative deflection depending on width of open water

ISOPE-2010, Beijing, China Normal stress vs. ship speed and width of free-water patches Corresponds to thickness of an ice plate 0.5 m and motion of a submarine with relative depth h0/L of 0.1.

ISOPE-2010, Beijing, China continuous ice

ISOPE-2010, Beijing, China b = 60 mm

ISOPE-2010, Beijing, China b = 120 mm

CONCLUSIONS The comparison of the surface displacement and surface normal stresses predicted by the theoretical model constructed in the article to those measured in a physical model of the motion of a vessel under an ice sheet shows the feasibility of using the theoretical model to predict the stress/strain state of an ice sheet containing cracks and free-water patches when subjected to hydrodynamic loads caused by the motion of a submarine. This numerical model provides a tool that can be used to make recommendations as to how inhomogeneous surface conditions such as cracks and free-water patches can be used to facilitate ice breakup with the aim of improving the effectiveness of the resonance method of breakup of ice cover by a submarine in the event of a requirement to surface in case of emergency. ISOPE-2010, Beijing, China