Introduction

Fracture is a process that has received little attention among glaciologists, but it may be a critical process in ice-shelf dynamics. Since both surface and basal crevasses exist in ice shelves, and they can join to fracture the whole ice thickness (Reference BarrettBarrett, 1975; Reference HughesHughes, 1979), a fracture criterion that can be applied to ice shelves should be developed. Figure 1 shows the distribution of Antarctic ice shelves. The CLIMAP ice-sheet disintegration model predicts that the marine West Antarctic ice sheet would collapse were it not buttressed by ice shelves (Reference Stuiver, Denton and HughesStuiver and others, 1981). Disintegration of these buttressing ice shelves would probably be controlled by fracture.

Fig. 1.

Antarctic identification map. Shown are the edge of the continental shelf (broken line), tidewater calving margins (solid line), ice-shelf calving margins (hatchured lines), ice-shelf grounding lines (dotted lines), ice divides (dashed lines), and regions of mountain glaciation (black areas).

Figure 2 shows the kinds of fracture patterns that exist in Antarctic ice shelves. Shear rupture crevasses are shown alongside the floating tongue of Byrd Glacier (1), shear/fatigue rupture crevasses are shown around Crary Ice Rise (2) and in Fashion Lane along the Shirase Coast (3), radial crevasses are shown for bending converging flow around Minna Bluff (4) and for radially diverging flow in the freely-floating tongue of Stancomb-Wills Glacier (6), and transverse crevasses are shown in the Grand Chasms of Filchner Ice Shelf (5). Crevasses disintegrate the ice shelf along its calving front and weaken its links to the ice streams that feed it, to the ice rises that anchor it to the sea floor, and to the sides of the embayment where it is confined.

Fig. 2.

Characteristic crevasse patterns in Antarctic ice shelves. Crevasses associated with weak links include shear rupture alongside the floating tongues of ice streams (1, lateral rifts of Byrd Glacier tongue, long. 161° E., lat. 80.2° S., Reference HughesHughes, 1977), fatigue rupture around ice rises (2, “horst and graben” rifts in the lee of Crary Ice Rise, long. 171° W., lat. 83° S., Reference BarrettBarrett, 1975), shear-fatigue rupture along lateral grounding lines (3, “Fashion Lane” along the Shirase Coast, long. 151° W., lat. 79.5° S., Thiel and Ostenso, 1975). Crevasses associated with the general strain field include shear crevasses from bending converging flow (4, radial crevasses around Minna Bluff, long. 167° E., lat. 78.5° S., see Reference HughesHughes, 1977, fig. 26f), transverse crevasses from straight parallel flow (5, The Grand Chasms on the Filchner Ice Shelf, long. 40° W., lat. 78.7° S., American Geographical Society, 1970), and longitudinal crevasses from radially diverging flow (6, crevasses normal to the calving front of Stancomb–Wills Glacier Tongue (long. 22° W., lat. 75.0° S., American Geographical Society, 1970).

The fracture criterion used in this paper is based on the concept of a critical fracture strain. Strain energy accumulated at grain boundaries in polycrystalline ice is released when a critical strain triggers viscoplastic instability. Strain energy is relieved rapidly by fracture and slowly by recrystallization. Ice moving from the grounding line to the calving front of an ice shelf passes through a strain field that is constantly changing as a result of both general flow and deformation at weak links. Strain energy in the moving ice is minimized if the ice fabric changes continuously in order to be always compatible with the changing strain field, thereby obeying Neumann’s Principle (Reference NyeNye, 1960, p. 20–24, p. 104). An ice fabric that is stable for a given strain field becomes metastable as the strain field changes and is unstable when strain energy is maximized at a critical strain. Further strain causes fracture or recrystallization. Recrystallization reduces strain energy by creating a new ice fabric that is stable in the new strain field. Strain hardening and primary creep occur before recrystallization. Strain softening and tertiary creep occur during recrystallization. The critical strain is the strain of viscoplastic instability. If this strain requires a stress that exceeds the fracture stress, the ice will crack instead of recrystallizing. Crevasses open on ice shelves where strain energy is reduced by fracture instead of by recrystallization. Tertiary creep reflects both recrystallization and crack propagation.

Figure 3 shows schematically how strain ε varies with strain energy E, stress σ, and time t during a recrystallization episode as ice moves through the changing strain field of an ice shelf. The flow curve shows the variation of σ with ε, where dσ/dε is positive during strain hardening and negative during strain softening. Strains for which dσ/dε = 0 exist at an upper yield stress where recrystallization begins and a lower yield stress where recrystallization ends. Fracture and crevassing, if present, begin at the upper yield stress. The creep curve shows the variation of with t, where strain-rate

decreases during primary creep and increases during tertiary creep. Secondary creep occurs when is constant during stable and unstable steady-state flow. Unstable flow occurs when strain hardening is exactly cancelled by strain softening at the beginning of recrystallization, and results in slow secondary creep. Stable flow occurs after recrystallization is complete, and results in fast secondary creep. Crevasses that open during slow secondary creep propagate through the ice shelf during tertiary creep.

Fig. 3.

Conditions for recrystallization or fracture in polycrystalline ice. The dependence of strain ε with strain energy E, stress σ, and time t has critical values at the elastic limit strain ε_e when strain hardening and transient creep begin, at the strain of viscoplastic instability ε_v when strain softening and recrystallization begin, and at strain ε_r when strain softening and recrystallization end. Metastable equilibrium exists until ε_e, unstable equilibrium exists at ε_v, and stable equilibrium exists beyond ε_r. Fracture occurs if recrystallization is unable to relieve the strain energy, with brittle fracture occurring instantaneously at ε_v and ductile fracture occurring progressively from ε_v onward.

Three common constitutive equations are used to relate stress σ, strain ε, strain-rate

, and time t to each other in Figure 3. For the flow curve:

where

is kept constant (Reference DieterDieter, 1961, p. 247). For the creep curve:

where σ is kept constant (Reference DieterDieter, 1961, p. 348). For the steady-state or secondary creep:

where

(Reference DieterDieter, 1961, p. 350). Temperature T is kept constant in Equations (1) through (3), σ _s is a strength coefficient, S is a strength exponent, is a strain-rate coefficient, c is a creep exponent, A is a hardness coefficient, and n is a viscoplastic exponent. For elastic strain, σ _s is the elastic modulus and s = 1. For viscoplastic strain, σ _s is a viscoplastic modulus, σ = σ _v is an upper yield stress when viscoplastic instability triggers recrystallization, σ = σ _r is a lower yield stress when recrystallization is complete, s → 0 at these yield stresses, s > 1 during strain hardening, s < 0 during strain softening, 0 < c < 1 during primary creep, c = 1 during both slow and fast secondary creep, and c > 1 during tertiary creep.

Figure 4 shows the viscoplastic spectrum for secondary steady-state creep, and illustrates the basis for two fracture criteria. In the viscoplastic creep spectrum, n = 1 and A = η ₀ is the fluid viscosity for viscous flow and n = ∞ and A = σ ₀ is the yield stress for plastic flow. The effective viscosity for viscoplastic flow is obtained by differentiating Equation (3):

Equation (4) is plotted in Figure 5.

Fig. 4.

The viscoplastic spectrum for steady-state creep and two criteria for viscoplastic yielding. A sharp knee develops in stress–strain rate curves when the viscoplastic exponent n increases, where

is the strain-rate at the maximum shear stress τ_m,

is the strain rate at the plastic yield stress σ₀ and

. Viscoplastic yielding occurs at the knee in the maximum stress-curvature yield criterion and at the stress intercept of the tangent line at

in the critical strain rate yield criterion. For n = 3, σ_v = 0.386σ₀ at the knee and σ_v = 0.667σ₀ at the stress intercept.

Fig. 5.

Variations of yield stress and viscosity across the viscoplastic spectrum of steady-state creep. Ratios of viscoplastic yield stress σ_v and plastic yield stress σ₀ increase with the viscoplastic exponent n according to Equation (8) for the maximum stress-curvature yield criterion (broken curve) and according to Equation (10) for the critical strain-rate yield criterion (dashed curve). The ratio of effective viscosity η_v and fluid viscosity η₀ decreases with increasing n (solid curve).

Fracture criteria for ice

Consider principal stresses σ_k where k = 1, 2, 3 in tensor notation. Let σ ₁ be the maximum principal stress and σ ₂ be the minimum principal stress on the surface of an ice shelf. Fracture occurs when the maximum shear stress

reaches the viscoplastic yield stress σ _v. In order to specify σ _v it is useful to rewrite Equation (3) in the form with σ = τ _m:

where σ ₀ is the plastic yield stress where viscoplastic instability, followed by recrystallization or fracture, occurs at a critical strain-rate

If recrystallization occurs, strain softening causes τ _m to drop and fracture is prevented. With fracture, τ _m = σ _v is maintained at the tip of an opening crevasse only if the tip moves into the ice.

Figure 4 illustrates two criteria for specifying σ _v. In the maximum stress-curvature criterion. σ _v is the stress at which τ _m changes most rapidly with respect to

. The radius of stress curvature obtained from Equation (5) is

Setting dR_σ /d(τ _m/σ ₀) = 0 at the maximum stress curvature where τ _m = σ _v gives

In the critical strain-rate criterion, σ _v is the value of τ _m at

that is obtained at the stress intercept of straight lines that are tangent to curves of (τ _m/σ ₀) versus at critical strain-rate . The equation of these straight lines is

Setting τ _m = σ ₀ at

and differentiating Equation (5) to obtain gives

Figure 5 compares Equations (8) and (10) over the same range of n.

The value of σ ₀ is related to the value of A, which depends on the density, fabric, texture, and purity of glacial ice. For polycrystalline ice having maximum density ρ _I and high purity, Reference BakerBaker (1981) has obtained the following relationship between the effective stress τ and the effective strain rate

during steady-state creep:

where B is a constant, d is crystal size. f is fabric intensity, Q is thermal activation energy, T is absolute temperature, and k is Boltzmann’s constant. Density ρ increases substantially with depth through an ice shelf, and hardness increases with density. If A ∞ (ρ/ρ _I) ^κ and κ ≃ 1, Equation (11) gives

Variations of ρ, d, f, and T with depth in Antarctic ice shelves can be obtained from the data published by Reference GowGow (1963).

Laboratory fracture tests can be conducted to determine whether Equation (8) or Equation (10) provides the best fracture criterion for ice. Figure 4 shows that when τ _m = σ _v fracture occurs at

for any value of n when Equation (10) is employed. The value of σ ₀ is therefore obtained from Equation (6) for measured at the moment of fracture and A computed from Equation (12) for given values of d, f, and T, where ρ = ρ _I for bubble-free ice. The state of stress also allows to be determined at the moment of the fracture. Since n = 3 for ice, Equation (10) applies if (σ _v/σ ₀) = 0.667, as determined by fracture experiments. If this test fails, Equation (8) must apply.

Fracture for shelf flow

Flow in an ice shelf is determined by the geometry of its confining embayment and by the number and location of the ice streams that feed it and the ice rises where it is anchored to the sea floor. Fracture in the ice shelf occurs when its flow pattern results in principal surface stresses such that

. In tensor notation, subscripts i, j, k denote rectilinear axes x, y, z in succession so that

where σ_ij and

are components of the stress and the stress deviator, δ_ij is the Kronecker delta, and P = σ_kk /3 is hydrostatic pressure. The tensor form of Equation (3) is (Reference GlenGlen, 1958)

where

is the effective stress. Since P → 0 at the surface of an ice shelf, from Equations (13) and (14). Consequently, σ _v can be determined if A, τ, and the principal surface strain-rates are known when a crevasse opens.

Since

in Equation (14), the ratio R of surface strain-rates is

Solving Equation (15) for σ ₂ gives

Substituting Equation (16) into the expression for τ in terms of principal stresses gives

Substituting Equation (16) into the expression for

in terms of principal stresses gives

Substituting Equations (17) and (18) into Equation (14) gives the flow law for a horizontal ice shelf in terms of R and its principal stresses σ _I and σ ₃:

Terms containing R in Equation (19) can be collected to form a constant R′ defined as

The principal strain-rates for an ice shelf are then:

where Equation (21c) expresses the first invariant of strain-rate for conservation of volume

.

Since hydrostatic pressure increases linearly with depth for an ice shelf having thickness h_I and density ρ _I,

where g is gravity acceleration and z = 0 at the base of the ice shelf. Substituting Equation (22) into Equation (21a) and solving for σ _I, gives

The base of the ice shelf is below sea-level at depth h _w in water of density ρ _w. Balancing hydrostatic pressure in a given ice column by the hydrostatic pressure of water in the column if the ice melted:

Note that the effect of

is accounted for by measuring to compute R′. Substituting Equation (23) for σ _I, integrating, and solving for gives:

where Equation (20) is substituted for R′, buoyancy requires that h _w = (ρ _I/ρ _w)h _I, and the average value of A is:

Equation (12) incorporates vertical density variations into A in Equation (26). Reference SandersonSanderson (1979) presents an alternative procedure.

Principal strain rates

are obtained from strain rates using the Mohr circle construction:

where ϕ is the angle between coordinates x, y and 1, 2.

Most Antarctic ice shelves occupy embayments, so ice entering such an ice shelf crosses a grounding line perimeter that is substantially longer than the calving perimeter crossed by ice leaving the ice shelf. Consequently, a typical flowband is bent around the z direction and converges in the x direction. Figure 6 shows bending parallel flow in a flowband at distance r from the bending axis and having constant width Δr. Bending through angle θ changes the are length an amount Δs across the flowband, which has an average arc length along its centerline of x′ − x″. If ice converges along this distance, the flowband width decreases uniformly from y′ to y″ and the flowband velocity increases uniformly from u′ to u″. The strain and strain-rates for simple shear in the flowband are:

where the pure shear strain-rate is

, the rigid body rotation rate is , the velocity along x is u, the velocity along y is v, and dv/dx = 0. Consequently, . For flow that bends around z and converges along x, the average horizontal strain-rate components are those when axes x, y correspond to cylindrical coordinates θ, r:

Fig. 6.

Bending parallel flow on an ice shelf. In a flowband at distance r from the bending axis and having width Δr, simple shear deformation Δs increases with the angle of bending θ about the bending axis and the angle of shear θ′ in the flowband, where θ = Δs/Δr = tan θ′. A crevasse (lens-shaped opening) that initially opens at angle ϕ = 45° to the bending radius rotates such that ϕ decreases as θ increases, causing the shear crevasse to become a transverse crevasse, where θ = 90° − 2ϕ. Simple shear in bending parallel flow is analogous to the shear between pages of a book when the book is bent. Axes x, y move with the flow-band. Axes θ, r are fixed.

Equations (27) and (29) can be combined to give an expression relating strain ε_yy to u, ω, and θ.

If no creep thinning occurs,

and bending converging flow is plane strain. By volume conservation, and:

Combining Equations (30) and (31), no creep thinning gives:

Figure 7 shows bending converging flow on the Ross Ice Shelf related to Equation (30). Figure 8 plots ϕ versus θ for various y″/y′ ratios in Equation (32).

Fig. 7.

Bending converging flow on the Ross Ice Shelf. The flow-band from the Siple Coast has a centerline radius r, outer radius r′, and inner radius r″ which do not coincide because flow converges. Bending flow causes velocities along a given radial transect, such as KL, MN, or OP, to be relatively constant. Velocities that increase along a radial transect are localized near grounding lines and form crevasses at 90° to those in Figure 6, as is seen in Figure 2–4 for bending converging flow around Minna Bluff. Flow in Figure 7 differs from flow in Figure 6 in that convergence causes ice to thicken and accelerate, according to Equation (30). Ice velocities are from Reference Bentley and JezekBentley and Jezek (1981).

Fig. 8.

The effect of longitudinal convergence and divergence on bending flow in an ice shelf with no creep thickening. Crevasses initially open at angles ϕ = 45° to the bending radius for all bending angles θ when convergence and divergence are absent, and the subsequent rotation of these crevasses is shown in Figure 6. Converging and diverging flow require that, initially, ϕ < 45° and ϕ > 45°. The value of ϕ is determined by the change in flowband width from y′ to y″ as flow bends through angle θ. Equation (32) is plotted for ratios y″/y′ from 0.3 to 1.8, where 1.0 is bending parallel flow.

Table 1 lists the principal strain-rates obtained from Equations (21), (25), and (30) for the special cases of straight parallel flow

, radially diverging flow , simple shear flow , bending diverging flow , and bending converging flow . Surface and basal crevasses tend to open along directions perpendicular to principal tensile strain-rates. The orientation of crevasses changes along curving flowlines of the ice shelf, where axis x follows flowlines.

Table I.

Principal strain-rates for selected ice-shelf flow conditions

Fracture alongside ice streams

When ice streams merge with ice shelves, fracture by shear rupture may occur alongside the floating tongue of an ice stream if it is moving at a surge velocity. Ice streams have a rather rectangular cross-section (Reference RobinRobin and others, 1970). If the thickness h _I of an ice stream changes an amount Δh _I in horizontal distance

, then the downstream hydrostatic force F _H in an ice stream of width w is

provided that the surface slope greatly exceeds the bed slope. This hydrostatic force is resisted by an up-stream shear force F _s given by

where τ₀ is the basal shear stress, τ _s is the side shear stress, and the ice stream has constant width w and average height

. Balancing up-stream forces against down-stream forces and solving for Δh _I gives

where P ₀ = ρ _I gh _I is the basal hydrostatic pressure. In ice streams,

, τ ₀ decreases steadily from a maximum at the head to zero at the flotation/grounding line, and τ _s is relatively constant along its sides. The deviator longitudinal stress gradient is important within 3 km of the grounding line (Reference SandersonSanderson, 1979), so Δx should exceed this distance. Equation (35) simplifies to give the basal shear stress for flow converging at the head of an ice stream, where τ _s → 0:

and the lateral shear stress alongside the floating tongue of an ice stream imbedded in an ice shelf, where τ ₀ = 0:

Reference SandersonSanderson (1979) has analyzed thickness gradients Δh _I/Δx in floating ice.

The effective stress for an ice stream is

where

, and R_yz ≈ R_zx ≈ 0. From Equation (14):

where R_yy expresses divergence of the floating ice tongue and

The floating tongue of an ice stream imbedded in an ice shelf should have a broad central width w _c where longitudinal strains are important and narrow side widths w _s where simple shear strains dominate. The ice hardness coefficients in these regions are A _c and A _s, respectively, where

due to thermal and strain softening in the lateral shear zones. Taking the x-axis along the center-line of the ice stream, u _c as ice velocity at the center-line, u _s as maximum ice velocity in the lateral shear zones, and a linear variation of with y so that , the variation of longitudinal velocity u_x with y across w _c when y ≤ w _c/2 is

Since u _x = u _s at y = w _c/2,

Subtracting Equation (41) from Equation (42) for y ≤ w _c/2 gives

A simple expression for u _s is obtained if a constant yield stress

exists in lateral shear bands. Simple shear requires that for constant w _s and across w _s, so that R_yy = 0 and gives R″ ≃ 1 in Equation (40). With constant shear stress, the variation of u_x across w _s when y ≤ w _c/2 is

If the lateral shear zones are not rifted. u_x → 0 at

and

If rifting occurs, w _s = τ _s = 0 and u _s is a lateral sliding velocity equal to the ice-stream velocity.

Rifting occurs when the maximum shear stress τ _m reaches the viscoplastic yield stress σ _v. For simple shear alongside the floating tongue of an ice stream, lateral rifts open when, from the Mohr circle,

An ice stream punching into an ice shelf experiences compressive flow until it can punch through the ice shelf. Compressive flow causes the floating ice tongue to diverge laterally, so that R_yy is negative in Equation (40) and u _s decreases in Equation (45). The lateral rifts remain open until u _s decreases enough so that τ _s < σ _v, where τ _s is given by Equation (37) and σ _v is given by either Equation (8) or Equation (10). In Antarctica, an estimate of σ _v can be made at the down-stream end of the lateral rifts created where Byrd Glacier punches into the Ross Ice Shelf lateral divergence of the floating ice tongue of amount R_yy ≈ −2.2 allows ice thinning of Δh _I ≈ 200 m over rifted length Δx ≈ 40 km and average width w ≈ 35 km (Reference HughesHughes, 1977). Entering these values into Equation (37) gives σ _v ≈ 8 bars for strain-softened fracture at the ends of rifts. As shown in Figure 9, these rifts can allow ice streams to punch through their confining ice shelf.

Fig. 9.

Ice streams and ice shelves on the Amundsen Sea flank of the West Antarctic ice sheet. Thwaites Glacier and Pine Island Glacier have punched through an ice shelf in Pine Island Bay, presumably because the length of their lateral rifts is comparable to the distance from the calving front (hatchured lines) to the grounding line (dotted lines) of the ice shelf. A comparison of 1947 trimetrogon photography with 1972 Landsat imagery shows that Thwaites Glacier tongue has buckled laterally in mode N = 1, and is now rotating about a probable pinning point about 200 km from its grounding line. This rotation was first noted by Robert J. Allen (personal communication, May 1978).

Fracture along grounding lines

An ice shelf cannot effectively resist the punch of a surging ice stream unless the ice shelf occupies a confined embayment and is pinned to bedrock at ice rises in the embayment. However, the links between the ice shelf and the ice rises are weak, as is the link to bedrock along grounding lines of the embayment. These links are weakened primarily by repeated tidal flexure (Reference SwithinbankSwithinbank, 1955; Reference RobinRobin, 1958), but also by shear rupture where the ice shelf moves parallel to grounding lines in the embayment and alongside ice rises (Reference Thiel and OstensoThiel and Ostenso, 1961; Reference BarrettBarrett, 1975). Shear rupture alongside these grounding lines can be analyzed in much the same way as shear rupture alongside the floating tongues of ice streams imbedded in the ice shelf. Our analysis of fracture along grounding lines, therefore, will focus on fatigue rupture caused by cyclic tidal flexure.

Crevasses open normal to the largest tensile principal stress, and a depression along ice shelf grounding lines (Reference SwithinbankSwithinbank, 1955) suggests necking associated with the maximum tensile stress caused by tidal bending. In a tensile test, the applied force F does not change at the upper yield stress σ _v when the strain of viscoplastic instability ε _v is inhomogeneous. This causes localized recrystallization and necking, which terminates in a cup-and-cone fracture for a circular cross-section. An ice shelf has a rectangular cross-section. Let x be the horizontal distance normal to its grounding line, y be the horizontal distance along its grounding line, and z be the vertical distance upward, with the origin of coordinates x, y, z at the neutral axis, taken as midway through the ice shelf. Actually, the vertical density gradient in ice shelves displaces the neutral axis toward the base (Reference GowGow, 1963). Necking along the grounding line is caused by the maximum surface and basal tensile stresses σ _m during tidal bending. If tidal bending force F _x stretches length L _x and reduces cross-sectional area A _x normal to x, where volume L _x A _x is conserved, the necking condition for viscoplastic yielding requires that

Separate expressions for the change in strain at the viscoplastic yield stress σ _v are obtained from Equations (1) and (47), putting σ = σ _v and ε = ε _v in Equation (1),

Equation (48) reduces to the following relationship between σ _s and σ _v:

By comparing Equations (1) and (49) it is clear that ε _v = 1/S. As seen in Figure 3, however, s → 0 at ε = ε _v and this requires that σ _s → 0 in Equation (49). Consequently, a fracture analysis for tidal flexure must consider conditions of homogeneous strain that exist just prior to necking. These would be conditions of parabolic strain-hardening for which s = 2 is observed.

Bending stress σ_xx caused by tidal flexure varies linearly with distance h ₀ from the neutral axis of the ice shelf, which is taken at the mid-point of an ice shelf having thickness h _I. The bending strain ε_xx at distance h ₀ from the neutral axis is, for elastic bending:

where σ _e is the elastic modulus and R _ε is the radius of strain curvature for elastic bending given by

Pinned boundary conditions along the grounding line require that z = dz/dx = 0 at x = 0, and z = z _m as x → ∞ is the maximum vertical tidal displacement of the ice shelf. These boundary conditions require that the ice shelf is not in hydrostatic equilibrium except at mean tide. With departures from mean tide, a vertical shear stress σ _zx is induced by shearing force F _z, where:

By definition, F _z = dM/dx where M is the bending moment.

During tidal flexure resulting in elastic strain and strain-hardening, the longitudinal bending strain ε _xx is related to the longitudinal bending stress σ _xx by writing Equation (1) for the tidal flexure application in the form

where σ _s = σ _e and s = 1 for elastic bending, and σ _e > σ _s > σ _v and s = 2 for parabolic strain-hardening (Reference HughesHughes, 1977). Using Equations (43) and (51), the bending moment is:

Combining Equations (44) and (52) to get the bending curvature

Differentiating twice more

The general solution for elastic strain is obtained by setting s = 1 and integrating. The elastic displacement, first solved by Reference RobinRobin (1958), is

where

is the elastic damping factor. The general solution for parabolic strain-hardening is obtained by setting s = 2 and integrating. The viscoplastic displacement is

where λ _v is the viscoplastic damping factor and

Since elastic and viscoplastic strains are additive, the total bending displacement is the sum of Equations (57) and (59):

where C _I = C ₂ = C ₅ = 0 since z is finite, C ₆ = 0 since z = z _m as x → ∞,

since z = 0 at x = 0, and since dz/dx = 0 at x = 0. Equation (59) results from an attempt to involve strain-hardening in tidal flexure (Reference HughesHughes, 1977; Reference LingleLingle, unpublished; Reference LingleLingle and others, 1981).

Reference LingleLingle and others (1981) studied the tidal flexure of Jacobshavn Isbræ in Greenland. By setting λ _v = λ _e and

in Equation (61), a good fit to flexure data was obtained. Equation (61) then reduces to:

Reference LingleLingle and others (1981) computed σ _m ≈ 5 bars at the side grounding line and σ _m ≈ 1 bar at the second stress maximum for ice in which fatigue fracture has given an uncrevassed ice thickness of 160 ± 48 m in floating ice 750 m thick, where σ _e/σ _s = 247 ± 37 for strain-hardened fracture.

Buckling up-stream from ice rises

An ice rise most effectively pins an ice shelf by resisting up-stream flow, rather than lateral or down-stream flow. Since an ice shelf is thin compared to its width and length, pushing against an ice rise may cause the ice shelf to buckle instead of thickening uniformly. If so, the compressive force should be analyzed in terms of buckling in a thin sheet. Consider an ice rise having radius r and resisting an up-stream longitudinal force F_x from the ice shelf. Buckling begins with a small vertical displacement z, possibly due to tidal changes. Bending stress σ_xx varies linearly with distance h ₀. The bending moment M, where σ_xx = σ _m at the surface and the base, is

From Equations (50) and (51) the maximum bending strain is

Substituting for σ _m from Equation (63),

where the compressive stress is σ _c = F_x /2rh _I and

The solution of Equation (65) is

where L is the length of the ice shelf which buckles, N is the number of bends that occur in that length, K = Nπ/L from the boundary condition that z = 0 at x = L and C ₂ = 0 from the boundary condition that z = 0 at x = 0. The compressive stress needed to cause buckling is

Vertical buckling must overcome a body force due to gravity, and vertical displacement z is reduced if N is large. However, increasing N increases σ _c. Lateral buckling is not retarded by the body force, so it occurs for N = 1 and σ _c is minimized. However, lateral buckling is possible only if the ice rise pins the floating tongue of an ice stream that is not imbedded in a confined ice shelf. In Antarctica, the floating tongue of Thwaites Glacier seems to have punched through an ice shelf and it has buckled laterally in the mode N = 1, seen in Figure 9, and the Brunt Ice Shelf has buckled vertically in the mode N ≈ 10 up-stream from an ice rise (Reference ThomasThomas, 1973, plate IIIa). Taking σ _e ≈ 9.7 × 10⁴ bars in Equation (68), L ≈ 200 km and h _i ≈ 500 m gives σ _c ≈ 0.23 bar for Thwaites Glacier, and L ≈ 70 km and h _I ≈ 170 m gives σ _c ≈ 23.5 bars for the Brunt Ice Shelf. Converging flow in an ice shelf confined in an embayment may also cause transverse buckling that would create longitudinal undulations (Reference HughesHughes, 1972, p. 53–55).

Discussion

The reason for studying the role of fracture in ice-shelf dynamics is better to understand the stability of Antarctic ice shelves, particularly those that buttress the marine West Antarctic ice sheet, which is believed to be inherently unstable (Reference HughesHughes, 1972; Reference WeertmanWeertman, 1974). An ice shelf is probably metastable; it can survive small temporary perturbations but not large prolonged ones. For a given surface and basal mass balance, it exists so long as the supply of ice crossing its grounding lines is able to replace ice lost along its calving front.

Ice rises and islands typically pin an ice shelf along its calving front, and actually determine the position of the calving front along a line across which the ice discharge velocity matches the iceberg calving rate (Reference SwithinbankSwithinbank, 1955). Any process, such as rising sea-level or surface and basal melting, that floats the ice shelf free from its pinning points reduces the ice discharge velocity by increasing the calving perimeter. A calving bay will then carve away the ice shelf until other ice rises establish a new calving front where ice discharge again matches iceberg calving.

If the new calving front is too close to the grounding line of the ice shelf, ice streams can punch through the ice shelf and surge (see Figures 2–6 and 9). This has two consequences. First, deprived of ice input from these surging ice streams, the mass balance of the ice shelf will turn strongly negative so that the discharge velocity at the calving front falls behind the iceberg calving rate. This allows the calving bay to migrate past the array of ice rises and continue to carve away the ice shelf. Second, being no longer buttressed by the ice shelf, the ice streams will not only be able to surge, the surges can be more vigorous and prolonged so that more interior ice will be drawn down into the ice streams. This compels the grounding line to retreat into the ice sheet at the same time that the calving bay is advancing into the ice shelf. Survival of the ice shelf depends upon which retreat rate is greatest.

The consequences of fracture on the stability of an ice shelf, and ultimately on the stability of the West Antarctic ice sheet, may even now be unfolding. Most of the ice draining from the northern flank of the West Antarctic ice sheet is drawn down into Pine Island Bay through Thwaites and Pine Island Glaciers, two huge ice streams that have apparently punched through a confined and pinned ice shelf and are now surging. The disintegration scenario outlined here for ice shelves may have already been played out in Pine Island Bay (Reference Stuiver, Denton and HughesStuiver and others, 1981; Reference HughesHughes, 1981).