Size effect in penetration of sea ice plate with part-through cracks. I: Theory

The paper analyzes the vertical penetration of a small object through a floating sea ice plate. The analysis takes into account the fact that the bending cracks reach only through part of the ice plate thickness and have a variable depth profile. The cracks are modeled according to the Rice-Levy nonlinear softening line spring model. The plate-crack interaction is characterized in terms of the compliance functions for the bending moments and normal forces in the crack plane, which are computed by an energy-based variational finite-difference method. The radial crack is divided into vertical strips, and a numerical algorithm with step-by-step loading is developed to calculate the vertical growth of the crack in each strip for a prescribed radial crack length increment. The initiation of crack strips from the surface of the plate is decided on the basis of a yield strength criterion with a fracture based flow rule. Systems of up to 300 nonlinear equations are solved by the Levenberg-Marquardt optimization algorithm. The maximum load is reached when the circumferential cracks begin to form. Numerical calculations, comparison of the results with test data, and a study of scaling laws are relegated to the companion paper, which follows in this issue. Numerical calculations show a typical quasi brittle size effect such that the plot of log σ_N versus log h (where σ_N = nominal stress at maximum load and h = plate thickness) is a descending curve whose slope is negligible only for h < 0.2 m and then gets gradually steeper, asymptotically approaching -1/2. The calculated size effect agrees with the existing test data, and contradicts previous plasticity solutions.