List of Symbols used

1. Introduction

Surrounding the Antarctic continent are several large ice shelves, many smaller ice shelves and floating ice tongues. It is from these features that most icebergs are generated. With increasing activity in the polar regions within the last decade, it has become necessary to understand better the mechanical behavior of a floating ice mass because:

• (1) scientific bases have been constructed on them,

• (2) shipping lanes in these waters are often governed by the presence of enormous tabular icebergs which calve off from the shelves periodically, and

• (3) certain hypotheses (viz. Reference WilsonWilson, 1964) concerning the dynamics of continental ice sheets feeding such shelves, demand a knowledge of the subsequent behavior of the floating ice masses once they have been formed.

Figure 1 shows three relatively small ice tongues. The Erebus Glacier Tongue (a) on Ross Island is known, from gravimeter measurements (Holdsworth, unpublished data in 1966) to be oscillating as far inland as approximately the section yy′. From altimetry the central ice thickness is estimated as being up to 200 m, assuming a mean ice density of 0.88 Mg m⁻³. The width is 1.5 to 2 km and the length 8.5 km.

Fig. 1. Floating ice tongues in Antarctica.

The Suvorov Ice Tongue (Oates Coast), Figure 1(b), is substantially larger, and unlike the Erebus Glacier Tongue, its length is about equal to its width, assuming a hinge line in the position marked. It is particularly amenable, therefore, to the kind of analysis which follows, under the specific set of assumptions that are made.

In recent years the positions of the ice fronts have been mapped by the U.S. Coast Guard icebreakers, and it has subsequently been deduced that sections, if not all, of a particular ice tongue have broken off at some time within a known year (for example, the Nordenskjöld Ice Tongue (Fig. 1(c)) probably broke off some time in 1964).

2. Deflection of a Floating Ice Mass

Several components are involved in producing the vertical oscillation of a floating ice tongue (or a major ice shelf), namely, oscillations due to:

• (1) wave motion (small amplitude of slab movement in millimeters; short period, generally less than 1 min, e.g. Erebus Glacier Tongue, ≈ 16 s);

• (2) diurnal tidal motion (medium amplitude, 1 to 3 m, period 12 to 24 h, see Thiel and others, 1960);

• (3) astronomical tides of medium to large amplitude, with periods from several weeks to 19 years. Included here are the well known “spring tides” (Reference StewartStewart, [1963]);

• (4) catastrophic impulses, e.g. tsunamis, earthquakes, violent storms, abnormal ocean currents, etc.;

• (5) miscellaneous influences, e.g. differential changes in atmospheric pressure over the shelf surface, differential accumulation or ablation on the surface of the shelf, forced wind oscillations, all of which are probably insignificant compared with the preceding influences.

One method of detecting vertical displacements of an ice shelf at any point, has been described by Reference WeinmanWeinman (1958) and Reference Thiel and OstensoThiel and others (1960).

3. The Problem

From the foregoing discussion and consideration of Figure 2, which is taken from Reference Thiel and OstensoThiel and Ostenso (1961), it is apparent that the system being studied might be represented by a simplified model, to be discussed in detail later. Certain assumptions are made, viz.:

• (1) The width y is considered to be much greater than the thickness 2h of the slab which is of length L measured horizontally from the effective hinge (x = 0). The order of magnitude of L is discussed later. A “long” or semi-infinite slab is considered to be one exceeding about 4 km in length with an average thickness of 200 m. Plane strain conditions are assumed to exist and beam theory is used for simplicity of analysis. However, for very wide tongues (where L is also large), secondary bending in the yz plane may become significant. In the present analysis this will be neglected.

• (2) For large values of x the ice mass may be treated as a freely floating slab of uniform thickness and density, responding to changes in sea-level. If h = h(x) is known then this could be used, but it greatly increases the difficulty of solution of the differential equations. Similarly the assumption of uniform density simplifies the solution.

• (3) The horizontal displacement of the slab is neglected; i.e. the slab is assumed constant in length over the period of time involved in a cycle of flexure.

• (4) Inertia effects are neglected, as all vertical movements are relatively slow. The problem is treated as one in statics.

• (5) The bending effect produced by the imbalance of ice pressure and hydrostatic pressure at the immediate ice front, as well as at the edges of an unconfined slab, is a local (though very significant) effect (Reference ReehReeh, 1968), but it is the concern of this paper to investigate the bending in the much larger zone from the hinge line to within a few hundred meters from the terminal cliff and lateral margins. Thus the edge boundary condition of hydrostatic pressure is avoided.

Fig. 2. Profile of ice thickness of part of the Ross Ice Shelf; Antarctica, after Reference Thiel and OstensoThiel and Ostenso (1961, p. 824).

According to Reeh’s analysis, icebergs produced by his mechanism would tend to be strongly prismatic, whereas many of the Antarctic icebergs are strongly tabular with both horizontal dimensions measured in kilometers or tens of kilometers, and with thicknesses of 200 to 400 m. Consequently it is evidently necessary to investigate the stresses due to bending along the entire length of the slab. It may also be noted that Reference ReehReeh (1968, p. 225–26) has provided time estimates for his proposed calving mechanism. These indicate that, for typical Antarctic ice masses, a deflection rate at the edge, of about 2 mm per 12 h period is likely to occur (assuming ice to have a constant viscosity). This is seen to be very small compared with the tidal (6 to 12 h) deflections of several meters, with which this analysis is primarily concerned.

The flexural analysis will be considered separately according to elastic-plastic and steady-state creep theories. It will be tacitly assumed that the bending function moment M(x) for a long ice slab has the same form in all cases. Plastic deformation due to bending is presumed to be of the same order of magnitude as a corresponding elastic deformation, because of the cyclical variation from tensile to compressive strains.

Finally, it is to be expected that the shorter the period of oscillation of the slab, the more the response would tend to be an elastic one (see section 2(1) and (2)).

4. Elastic Analysis

It is necessary now to discuss in detail, the model used for the present analysis. For this it is convenient to use the analogue of the system in which a semi-infinite elastic beam, resting on a horizontal elastic foundation, is depressed at its center by a point load. This classical problem has been solved by Reference HetényiHetényi (1946). By inverting this system, eliminating the half to the left of the center of symmetry, and shifting the origin to the neutral axis of the slab in the line of the point load, a model, which will be called the Hetényi model, is formed (Fig. 3), in which for large values of x the deflection tends to a constant value along the slab. The unbent position corresponds to mean sea-level. Consider first a depression of sea-level. It is assumed that at large values of x the ice is freely floating, thus ω = ω _a = ω _s. In accordance with the assumptions made by Hetényi, for the foundation reaction, it will be initially assumed here that the distributed load acting vertically downward on the displaced parts of the slab, is directly proportional to the vertical distance ω _a−ω(x) (Fig. 4). This turns out to be correct because the ice in this region remains above its equilibrium floating position for the new sea-level, corresponding to (b) in Figure 4, and the downward-acting distributed load acting on a strip of width dx at distance x from the origin, is exactly ρ _w g(ω _a−ω(x)) dx.

Fig. 3. Hetényl model (vertical scale exaggerated).

Fig. 4. Model ice tongue (developed from the Hetényi model), distorted vertical scale.

The force P acting upward at the origin may be identified with the “reaction” or vertical shear force at the hinge.

Considering inertial effects of this system to be minimal it is sufficient to treat the problem as one in statics. Starting from the well known equation for a beam in bending,

where M(x) is the bending moment at x, E _b is Young’s modulus in bending, I _a is the moment of inertia of the section per unit width, and μ is Poisson’s ratio, it may be shown that

for the present problem. The solution turns out to he:

where

λ is known as the damping factor. The significant quantity is 1/λ, whose value in the present case is approximately 900 m, taking the following values: ρ _w = 1.028 Mg m⁻³,  μ  = 0.3,  E _b,   = 2.7×10⁴bar (Reference DorseyDorsey, 1940, p. 445; Reference Tabata and OuraTabata, 1967, p. 493), where h = 100 m.

The specified boundary conditions for Equation (4.2) are

ω(x) has the value ω _a when

Between these points the deflection curve follows a flat wave trajectory of decreasing amplitude with distance x.

Differentiating Equation (4.2) twice,

and using Equation (4.0),

M(x) = 0, when

Of greatest interest is the stress σ _xx generated at the hinge-line position (x = 0). The maximum tensile stress produced by a downward deflection +ω _a will obviously occur at the surface z = −h. Using beam theory and Equation (4.6)

σ _xx (−h) = 0 when

Now where the positive sign denotes tension. If the previously a used values of ρ _w, h, I _a and λ are used, this reduces to

For a downward deflection of +50 cm the surface stress is about

At values of maximizes, alternating between tension and compression along the slab surface.

The case where the slab is elevated above the mean position is taken to be the mirror image of the depressed case, assuming that the ice at the hinge does not lift off the bottom, a situation requiring the basal ice to be resting on a sharp rise or rock threshold, or where the thickness of the ice necks down rapidly to the hinge (Fig. 2). Thus, if ω(x→∞) = −ω _a, the signs of the stress at these points are all reversed. The stresses at π/2λ and 3π/2λ are respectively less than ±1 bar and ±0.1 bar for ω _a = ± 50 cm. Equation (4.7) gives values of bending stresses on the surface of the slab produced by various values of ω _a . If fracture is to occur it will first be manifest at the hinge line, then at a distance of approximately 1.4 km from the hinge line.

At this point it would be convenient to investigate the effect of the length of the slab on the bending behavior. So far, only long slabs have been considered, i.e. where L ≫ 5π/4λ, or about 3.5 km. “Intermediate length” slabs, where L ≈ 4 km, could also be treated in a similar way to a first approximation, but for short slabs where L < c. 3.5 km, the problem will have to be treated somewhat differently.

Assume that the deflection curve of the top surface is parallel to the deflection curve of the neutral axis. Suppose that sea-level is depressed an amount ω _s but that the end of the slab deflects only ω _L . The necessary condition is that ω _L ≤ ω _s.

In the absence of prior knowledge about the form of the deflection curve, ultimately to be derived, the approximation is now made that the distributed load due to ice above its equilibrium position varies linearly along the slab, from ρ _w gω _s dx to ρ _w g(ω _s−ω _L ) dx at the end. This is not such a severe assumption if ω _L . is small compared with L which is the condition of the present case. Consequently the bending moment at any point x is given by

Using Equations (4.0) and (4.8),

where

The boundary conditions for Equation (4.9) are

A solution of this equation is

where the constants may be found from the boundary conditions.

Without some simplification this equation turns out to be very ponderous and will not be pursued, except to mention that the initial approximation is justified by it. What is more important to the present problem is the dependence of the hinge-line stress on the length of the slab and the end deflection.

Using Equations (4.0) and (4.9), in which x = 0,  ω(0) = 0 and the relationship σ _0x (−h) = M(0)h/I _a, it may be shown that:

where the limits of L are approximately given by

If the limiting case ω _L = ω _s is taken, Equation (4.10) may be rewritten

where c has the value 0.251 N m⁻⁵.

For ω _s = ±50 cm and L = 3.5 km, the hinge stress is ±15.4 bars, which is of the order of the bending strength of ice at about −20°C with a density above 0.85 Mg m⁻³ (Reference Tabata and OuraTabata, 1967, p. 488–89). For values of L greater than this, but leaving ω _s, unchanged, there is, according to this composite analysis, a reduction of the hinge stress to about of its maximum value.

Due to surface hinge-line stresses of several bars indicated in the case of an elastic slab (eliminating temporarily the horizontal steady-state creep, which is always tensile) it might be supposed that this is the region where plastic yield due to bending only would be likely to occur first. Consequently it is thought appropriate to present an elastic-plastic analysis of the same problem.

5. Elastic-Plastic Analysis

Again, only flexure in the xz plane of the slab is admitted. The stresses considered are the result of the bending moment only. The other component of stress in the x direction is Weertman’s steady-state creep stress (Reference WeertmanWeertman, 1957) which can only be tensile in the upper layers.

All the assumptions of sections 3 and 4 are again made. Further it is assumed that the bending contributes nothing to σ _zz , σ _xy and σ _yz . Consequently the only non-vanishing stresses produced by bending are σ _xx , σ _yy and σ _xz

The stress deviators are

and σ_kk = Σ σ_ij , i = j, hence

But for plane strain σ′ _yy must be zero, therefore:

Using the von Mises yield criterion:

where k is the yield stress in pure shear, the relationship

is obtained. It may be shown (Reference Prager and HodgePrager and Hodge, 1951, p. 44–53) that within the framework of beam theory σ _xz ≪ σ _xx and in the plastic regions σ _xz actually vanishes. Indeed on the upper and lower surface σ _xz = 0, always. Accordingly the yield condition may be sufficiently written:

If the vertical displacement ω(x) ≪ 2h and the usual assumptions of the bending of beams are adopted then to a reasonable approximation

at all points where the material behaves elastically.

If the section in Figure 5 is considered, then the upper plastic region is specified by ξ ≤ z ≤ h, the elastic region by −ξ ≤ z ≤ ξ, and the lower plastic region by −h ≤ z ≤ −ξ, then the stresses in these regions are respectively

Fig. 5. Elastic plastic regions of the slab at the hinge region, for a fully plastic section x = 0.

and σ _xx = −2k when the yield stress has been reached in the outer parts of the section.

The bending moment bearing in mind that the stresses due to bending only are considered.

For an elastic section

For an elastic-plastic section

For an entirely plastic section

Assuming that M(x) takes the same form for all sections as discussed earlier, the form of Equation (4.6) will be used, namely:

The solution of Equation (5.3) has already been worked out in section 4.

For the elastic-plastic section, from Equation (5.1):

Equating Equation (5.4) with Equation (5.6), and using this last expression, the curvature becomes:

The deflection curve is given by

under the boundary conditions ω(0) = ω′(0) = 0.

The solution to Equation (5.8) can be rewritten in the form

where A = 2k/√3E _b h and α = M(0)/2kh ². τ is a dummy variable. This curve is valid only over a distance 0 ≤ x ≤ x ^⋆ where x ^⋆ is a solution of the equation

The value of x ^⋆ ≈π/16λ ≈ 180 m, or approximately the thickness of the slab, so for practical purposes Equation (5.8) or its equivalent is of little use.

Of specific interest is the stress generated at the hinge, which by hypothesis cannot be greater than |2k| or about 2 bar. This corresponds to a bending moment of 2kh ² at an approximate value of This is then equivalent to a full tidal range of about I m, which is about the average for the Ross Sea (Reference ThielThiel and others, 1960, p. 633).

6. Steady-State Creep Analysis

An analysis based on creep bending requires a number of severe assumptions in addition to those already outlined in section 3. These are that:

• (1) Plane sections remain approximately planar after flexure and the strain-rate is proportional to the distance z from the neutral axis, which is assumed to be at mid-thickness of the slab (due to temperature and density variations with depth it is bound to he slightly displaced).

• (2) An average flow law can be assigned to the full section to take account of varying density and temperature of the ice with depth.

• (3) The steady-state creep law holds for both tension and compression induced by the influence of bending.

• (4) The stresses due to bending are cyclical and may be analysed separately from those due to other causes.

Using assumption (1)

Using assumptions (2) and (3) the steady-state creep law may be written

therefore

Assuming small displacements:

For simplicity of analysis the bending moment will be assumed to be a function of the rate of deflection but not of the actual time. Thus if the rate of deflection is assumed roughly constant, the right-hand side of Equation (6.4) can be assumed to be independent of time, so

where n has been put equal to 3 (Reference GlenGlen, 1955, p. 528–29) and

In solving Equation (6.5) it must be realized that although the resulting deflection ω(x) turns out to be an increasing function of time as well as x, it has to have very definite limits which are in this case controlled by sea-level.

The boundary conditions that must be satisfied are:

which enable a complete solution for Equation (6.5) to be written

This equation gives the deflection curve for given values of ω _a and at the far end of the slab.

If is the rate of change of sea-level and is the rate of vertical movement of the slab for large x, then if the movement of the slab would lag behind the sea-level change. On the other hand, if then the deflection rate of the slab can only be exactly equal to . This last case is the one considered to apply to diurnal tidal movements.

where W _S is the full departure of sea-level from mean sea-level it may be shown that:

The value of M(0) is thus dependent on but is rather insensitive to changes in this quantity. Suppose the value of is known, then Equations (6.3) and (6.7) give, at x = 0,

However A = A (ρ _i , temperature, salinity, etc.) is not well known, therefore rewriting Equation (6.8) in terms of strain-rate,

By using a sensitive strain measuring device such as a network of electro-tapes installed at the estimated hinge-line position, a value of might be found, bearing in mind that this value will actually be the change in strain-rate at this point over the appropriate time interval. This is because of the horizontal steady-state creep of the slab (Reference WeertmanWeertman, 1957). The measured value could then be compared with the calculated value in order to test the theory.

Taking a typical value of for the Ross Sea, , which is well above the general range for fracturing of ice (Reference HoldsworthHolds-worth, 1965); therefore at least a tide crack should form, and indeed these are generally visible.

7. Discussion

Before being able to predict any failure criterion, it is necessary to have a knowledge of the bending strength of ice. Reference Tabata and OuraTabata (1967, p. 486) indicates that at stress rates below 0.1 bar s⁻¹ the bending strength increases with decreasing stress rate, but the data are not extended to the stress rates generally dealt with in the present analysis. Although Tabata’s values for the bending strength cannot be used directly here, because a different type of ice was used, it is expected that the same order of bending strengths (9–13 bar) would probably apply in the present case. Additional information comes from tables in Reference MantisMantis (1951) which indicate values of the bending strength of unspecified ice as being up to 244 lb in⁻² (17.5 bar). These miscellaneous model tests suffer from the well-known “notch effect”—so in all probability the bending strength of polar glacier ice is at least as great as 15 bar and probably greater, because if the elastic analysis is valid, ice tongues such as the Erebus Glacier Tongue must have passed through the critical length 5π/4λ without breaking off completely under 15 bar tension.

The map of Antarctica (e.g. that published by the American Geographical Society, 1: 5 000 000, in 1965) discloses the fact that in the Weddell Sea, ice tongues of the type found in the Ross Sea are not found. A reason for this may be provided by the tidal data which show (Reference ThielThiel and others, 1960) a tidal range in the Weddell Sea about three times greater than the Ross Sea. Consequently hinge bending stresses in a similar slab there would be three times as great as in the latter place, so total failure might be expected.

The present discussion has dealt with tidally induced stresses with half periods from 6 to 12 h (Weddell and Ross Seas respectively). Oscillations of the slab due to wave action (or swell) with half periods of less than 30 s should place the system within the limits of elastic theory. However, estimated deflections are generally small. It is quite possible, nevertheless, that violent storms may induce swells of sufficiently large amplitude to generate stresses at the hinge line of the magnitude of tidally produced ones. Moreover these stresses may be generated by moments about the z- as well as they-axis, depending on the direction of the swell.

Acknowledgements

The author is indebted to Dr C. Bull for helpful suggestions and to Dr T. Hughes and Dr J. Weertman for making corrections to parts of the manuscript.

This work was supported by grants GA-205 and GA-532 from the National Science Foundation to the Ohio State University Research Foundation.