Sir
I agree with the conclusion of Robin’s letter. Another way to reach his conclusion is as follows. Consider material that contains an infinite row of parallel cracks. Let the spacing d between the cracks be much smaller than the length L of the cracks. The stress analysis of this problem when the outer boundary of the material is subjected to a uniform stress is given in Reference Sneddon and LowengrubSneddon and Lowengrub (1969 p. 57–62), in Reference IsidaIsida (1973), and in Reference TadaTada and others (1973, p. 12.1). The result of this analysis is that the stress singularity that exists at the crack tips is approximately the same as that for an isolated crack whose length is equal to the spacing d between the parallel cracks. Thus if an applied tensile stress T is applied (in a direction normal to the plane of the cracks) at the outer boundary, the stress singularity is such that the tensile stress at distance r ahead of a crack tip is of the order of T(d/r)½ where 0 ≤ r ≪d. The stress singularity at the tips of the set of parallel cracks of length L and spacing d is approximately the same as that for a set of parallel cracks of length d and spacing d (or of larger spacing).
For a set of closely spaced crevasses of spacing d and length L (d ≪ L) the stress singularity at the crevasse tips, by analogy with the results just mentioned, is a tensile stress ahead of the crack that is approximately equal to (T + p − ρ igL)(d/r)½ where ρ i is the ice density and g is the gravitational acceleration so that ρ igL is the ice overburden pressure at the tips of the crevasses, T is the longitudinal tensile stress present in the glacier, and p is the water pressure at the bottom of the crevasses. If no water is present (p = o), a tensile stress singularity is present if L is smaller than T/ρ ig. Thus because of the presence of a high tensile stress at the crevasse tips, closely-spaced water-free crevasses can penetrate a glacier until their depth reaches Nye’s value of L = T/ρ ig. Deeper penetration is not possible in this case because a compressive stress singularity then would come into existence. For completely water-tilled crevasses, P = ρ wgL where ρ w is the density of water. Since ρ wρ i, a tensile stress singularity exists if T ≥ o for all values of L. Thus, as concluded by Robin, a set of closely spaced water-filled crevasses should be able to penetrate to the bottom of a glacier.
8 January 1974
