Size Effect in Penetration of Sea Ice Plate

A thorough examination of the quasi-static penetration of a floating elastic-brittle plate via a fracture mechanics approach has been presented by Bažant and Kim. Bažant and Kim reach the conclusion that there is a size effect (in terms of the plate thickness, h). A few of the assumptions made by these authors will be examined in this discussion. The formulation presented by Bažant and Kim assumes both that a radial system of part-through cracks is formed and that the appearance of these radial cracks is accompanied by stable crack growth. The analysis proceeds by subdividing each partthrough crack into narrow vertical strips (the ith strip being of length bi, with ligament h 2 bi). In each strip, the crack is assumed to propagate vertically, independently of the crack propagation in the adjacent strips. A simplified form of a cohesive crack model is adopted, with the crack initially growing as a plastic crack. The assumed stable development of the part-through radial cracks does not match experimental observations, especially for thin to moderately thick ice sheets (h < 0.5 m). The initiation of cracks in ice almost always leads to unstable crack growth (DeFranco and Dempsey 1994). The radial cracking that occurs prior to the formation of circumferential cracks and subsequent penetration is understood to occur suddenly and to be through-the-thickness. In other words, a system of throughthe-thickness radial cracks occurs, with rapid radial and through-the-thickness crack propagation. Even though these radial cracks are subjected to the dome or arching effect, crack growth instability in ice is sufficient to allow through-thethickness cracks to form (in thick ice sheets, it is plausible to assume that the through-the-thickness cracking would be prevented by the arching effect). Dempsey et al. (1995) studied radial cracking with closure for the case of a clamped plate subjected to a concentrated lateral load. By assuming that the closure width was a function of the radial crack length only, Dempsey et al. (1995) obtained an analytical solution that facilitated a thorough examination of the dependencies of the closure width, the nucleated radial crack lengths, the energy release rate, and the penetration load. In particular, the latter analysis made it clear that radial crack growth instability would accompany the nucleation of any radial crack system. A finite-element study of a radially cracked floating plate by Sodhi (1996) confirmed the broad applicability of the conclusions reached by Dempsey et al. (1995). An implicit requirement underlying the size effect analysis presented by Bažant and Kim is the stable formation of process zones (contiguous to each traction-free crack front) that scale self-similarly with the ice sheet thickness. However, if sudden and unstable radial crack formation takes place, with full through-the-thickness crack-face separation and subsequent compressive closure (unilateral contact, in other words), there is no logical way in which one can simultaneously assume the stable formation of process zones; there are, in fact, no ligaments subjected to bending, but instead pairs of completely separated crack faces subjected to ever-increasing pres-