A free, square, point-loaded ice sheet: A finite element-discrete element approach

Abstract

Deflection of a free, thin, square, elastic, point-loaded ice sheet is studied. A three-dimensional, combined finite element-discrete element method (FEM-DEM) is applied. Examined is a set of square, self-similar (plan view) ice sheet samples with the side lengths of L=20, 40, 80, and 160 m; thicknesses of h=0.5, 1.0, and 1.5 m; and the average discrete element sizes of l=2h and 3h. The samples are generated and meshed via a centroidal Voronoi tessellation procedure. Each mesh is unstructured and consists of convex polyhedra as the discrete elements and a Delaunay-triangulated, in-plane beam lattice of co-rotational, viscously damped Timoshenko beam finite elements. Two load cases are considered: a sheet sample subjected to i) a vertical, centrally applied point load; and ii) a vertical point load on an edge. In both load cases, a good agreement is found between the deflections computed with the FEM-DEM approach and with FEM. Maximum, non-dimensionalized deflections w¯max (w¯max=q/Fz, where q=q(wmax) is an approximate, resultant reaction force due to the buoyancy, wmax the computed maximum deflection, and Fz the applied point load) scale, in each load case, either by a power-law, w¯max∝hm1(L), or exponentially, w¯max∝10m2(h)L, depending on whether computed over h (for each L) or vice versa. The exponents m1 and m2 are not constants, but functions of L and h, respectively, and approximately equal in either load case. In-plane size estimates are sought for an FEM-DEM sheet sample to represent either a) an infinite, elastic, point-loaded ice sheet; or b) a semi-infinite, elastic, point-loaded (with the load on the free edge) ice sheet with a free edge. A free, square, point-loaded FEM-DEM sheet sample well approximates both a) and b), if L/2>≈5lch, where lch denotes a characteristic length.