MODELS OF LARGE-SCALE CRUSHING AND SPALLING RELATED TO HIGH-PRESSURE ZONES

Transcription

1 Ice in the Environment: Proceedings of the 16th IAHR International Symposium on Ice Dunedin, New Zealand, 2nd 6th December 2002 International Association of Hydraulic Engineering and Research MODELS OF LARGE-SCALE CRUSHING AND SPALLING RELATED TO HIGH-PRESSURE ZONES Andrew Palmer 1 and John Dempsey 2 ABSTRACT There is much field, laboratory and theoretical evidence that the contact force between ice and a structure during a crushing or spalling event is very far from uniformly distributed, but that most of it is concentrated in localised high-pressure zones ( hot spots ). This behavior is expected for brittle materials, and is confirmed by the JOIA field-scale experiments using tactile sensors, and by Sodhi s laboratory results. It has important implications for the structural design of Arctic offshore structures, and for the understanding of pressure/area relationships. The paper presents new work on high-pressure zones and how they link to total forces in contact events with wide structures, and relates them to alternative modes such as indentation spalling and buckling. The results make it possible to develop mode maps consistent with dimensional analysis and based on fundamental parameters such as fracture toughness. INDENTATION AND HIGH PRESSURE ZONES A stationary structure indents the edge of an ice sheet that moves into contact with it. Edge indentation is the classical and central problem in ice mechanics. There have been many experiments at a laboratory scale, a smaller number of measurements at field scale, and many theoretical models, adopting widely different material idealisations. The notion that ice can usefully be idealised as a plastic material characterised by a strength has shown itself to be fundamentally flawed and at variance with observation. A more rewarding approach is to focus on fracture. If the ice moves very slowly, the force builds up relatively slowly, and contact develops over the full thickness of the ice. Near-horizontal cleavage cracks parallel to the ice surface at first spread stably outward from the contact zone, but later become unstable and grow much longer, though they remain parallel to the surface (Hirayama et al., 1974, Kry, 1981). Ultimately a region on one side of the crack becomes unstable, the crack turns towards a free surface, and a spall or flake breaks away. Any relative vertical movement 1 Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK acp24@eng.cam.ac.uk 2 Department of Civil and Environmental Engineering, Clarkson University, Potsdam, New York USA john@clarkson.edu

2 (related, for example, to tide or to large-scale flexural buckling) makes spalling more likely. Repeated spalling leads to a sequence of wedge-shaped profiles. If the ice moves rather faster, the distribution of contact force alters. The largest contact pressures occur close to mid-thickness, along a line-like high-pressure zone, shown schematically in Figure 1(a). The zone is composed of myriad small contact regions, high-pressure zones (HPZs), which continually form and fade away, each contact region centered on a local asperity left by an earlier fracture. An individual HPZ may only carry a large force for a very short time, perhaps hundredths of a second. Together, however, the HPZs transmit a force that is almost uniformly distributed across the contact breadth (though not through the contact thickness). At any instant, the distribution of contact force looks like a rather smooth range of rounded hills. The ice literature calls this simultaneous contact, and calls the failure mode ductile crushing. If the edge indentation speed surpasses some critical speed, the transition velocity, the local contact pressure is much less uniform, and the ice pressure (averaged over the nominal contact area, the ice thickness multiplied by the contact breadth) is distinctly lower. Although the HPZs are still forming and fading rapidly across the breadth, they are only weakly correlated in space and time. At any instant, the distribution of contact force looks like a jagged mountain range, roughly centered on the mid-thickness of the ice, with many sharp peaks. Figure 1(b) is an idealisation. The ice literature describes this scenario as non-simultaneous contact, and the mode as brittle crushing. In contacts with compliant structures, there is an intermediate indentation speed range in which the mode alternates cyclically between a load-up phase (associated with simultaneous contact and a reduced relative indentation speed) and a briefer extrusion phase that follows failure of the ice. In the extrusion phase the structure rebounds and the relative indentation speed is much higher. A plot of force against time looks like the teeth of a ripsaw. This is intermittent crushing. MODELS OF HIGH-PRESSURE ZONES In order to try to understand the behavior of high-pressure zones, we consider models that are deliberately - and without apology - highly idealised to have very simple geometries. Some of the analysis has been described in more detail in a separate paper (Dempsey et al., 2001). h h a) b) Figure 1: a) Line-like and b) localized distribution of high-pressure zones.

3 Figure 2a depicts a line-like HPZ on one plane face of a uniform ice sheet. It carries a local contact pressure p across a zone breadth 2a, small by comparison with the sheet thickness h. Figure 2b shows a model that represents conditions close to the HPZ, a cylinder centered at the mid-height of the HPZ. The ice is idealised as a uniform linearelastic/brittle material characterised by Young s modulus E, Poisson s ratio ν and critical strain energy release rate G c. Position is referred to cylindrical coordinates r,θ,z, radial displacement is denoted u, and the stress components are σ rr, σ θθ and σ zz. The outer boundary of the cylinder at b is loaded by a pressure q. The cylinder in Figure 2b is divided into three cylindrical regions, as follows: In region I, 0 < r < a, σ rr = σ θθ = p, In region II, a < r < ρ, n equally-spaced radial cracks extend to a radius ρ and release the circumferential stress, so that σ θθ is zero, In region III, ρ < r < b, the ice remains intact and elastic. Under these conditions, it is straightforward to show that the radial displacements at the inner and outer boundaries are u(a) = pa ( ln ρ E a + Cbρ 2b2 1 2ν ) (1) b 2 ρ 2 1 ν u(b) = pa bρ Cb2 (2 + C ) (2) E b 2 ρ 2 1 ν where C = qb/pa, E = E/(1 ν 2 ), and in addition to derive the energy release rate G = dπ/da, the derivative of the potential energy Π with respect to the area A = n(ρ a) of the cracks, which is G = p2 a 2 π (1 2Cξ + ξ 2 ) 2 (3) be n ξ(1 ξ 2 ) 2 where ξ = ρ/b. For small ξ, G is inversely proportional to ρ. Whether the growth of the cracks is stable or unstable depends on the stability index dg/dρ, given by b dg G dρ = ξ4 6Cξ 3 + 8ξ 2 2Cξ 1 ξ(1 ξ 2 )(1 2Cξ + ξ 2 ) (a/b < ξ < 1) (4) Figure 2: a) High-pressure zone on the contact face of an ice sheet; b) radially cracked cylinder or sphere.

4 Dempsey et al. (2001) plot the relationships between G, dg/dρ, ξ and C. The ratio C expresses the effect of the inward-acting pressure q on the outer boundary of the cylinder, and therefore reflects the influence of external constraint. If C is zero, the outer boundary is free, and the cracks become unstable when ξ is ( 17 4) = Increasing C makes the cracks more stable, and when C is unity (or greater than unity) the cracks are stable for all values of ξ. If the outer boundary is clamped, so that u(b) is zero, equation (2) determines C, and the corresponding energy release rate is G = p2 a 2 be π n ( ) 1 2ν ξ 2 2 (5) 1 2ν + ξ 2 Figure 3 plots G and dg/dρ for the clamped case: for ξ small the difference from the case with prescribed traction on the outer boundary is small. Now think of half the cylinder in Figure 2b transferred to the edge-loaded sheet in Figure 2a, as in Figure 4. Equilibrium of the half-cylinder requires that pa and qb are equal, and therefore that C is 1. The assumption that q is uniformly distributed around the halfcircumference is of course a severe idealisation, and in addition σ θθ must vanish on the vertical boundaries. At points close to L and N there is little radial constraint and conditions approximate to those for C = 0, whereas close to M the elastic hinterland of the rest of the ice sheet induces much more constraint and conditions approximate to C equal to 1 or greater (since overall equilibrium still has to be maintained). This is consistent with the observation that inclined cracks that propagate towards the upper or lower surface soon =1/2 =1/2 Figure 3: With a clamped outer boundary at r = b, the normalized energy release rate G and crack growth stability quantity (b/g)(dg/dρ) are portrayed for a) a line-like and b) a localized distribution of high-pressure zones.

5 become unstable, whereas horizontal cleavage cracks propagate much further before they lose stability. INDENTATION AND SCALING: LINE-LIKE HPZ In order to obtain indications of how forces can be expected to scale, and to secure orderof-magnitude estimates, we now 1 assert that the cracks extend when G equals the critical energy release rate Gc scale at the appropriate scale (since there is evidence that G c may not be scale-independent), 2 identify the outer radius b with the ice half-thickness h/2, 3 incorporate a free-surface correction multiplier Ψ, 4 take C as 1, then from (3) G scale c = p2 a 2 (h/2)e The ice force F per unit breadth is 2pa, and so π n (1 ξ) 2 ξ(1 + ξ) 2 (6) F = 2pa = Ψh 1/2 K scale 2n (1 + ξ) ξ π 1 ξ (7) where K scale EGc scale is the apparent fracture toughness; E can be taken as the short time modulus if the loading is rapid, or as the secant modulus for slower loading. The has a twofold implication. First, it serves to emphasize the probable lack of validity of the one parameter Griffith linear elastic fracture mechanics (K Ic ) for ice, given the associated requisite well defined crack tip and scale invariance characteristics. Second, one is reminded that the fracture energy is closely tied to the degree of stable crack growth encountered in the application at hand. In this paper, the degree of stable crack growth may be assumed to scale with the thickness of the ice sheet; use of the notation K scale Figure 4: Deformation within a high-pressure zone

6 it follows therefore that K scale scales as h β with 0 < β < 1/2 (Mulmule and Dempsey, 2000). The ice force per unit breadth correspondingly scales as h 1/2+β for geometrically similar problems. The ice contact pressure, force per unit nominal contact area (contact breadth times ice thickness) scales as h 1/2+β. If we arbitrarily guess that the terms other than h 1/2 K scale in equation (7) are together of the order of 1, and take the apparent fracture toughness for sea ice as 100 kpa m, the nominal pressure that can be exerted on a fixed structure by an ice sheet 1 m thick is 0.1 MPa. Under the same assumptions - and particularly assuming that K scale does not change - the nominal pressure exerted by an ice sheet 0.04 m thick is 0.5 MPa. These are reasonable values, but a better understanding of the role of ξ and n obviously needs to be developed. The proposed scaling avoids a well-known difficulty in applying the Sanderson pressure/area diagram to wide structures. If contact area rather than ice thickness is the governing parameter, the local force developed at one end of the contact area must somehow be influenced by what is happening at the other end of the contact area: loosely, the ice would have to know what the contact breadth is, which is inherently implausible for wide structures that induce either continuous or intermittent brittle crushing. However, it is not implausible for other modes such as radial cracking or creep buckling. Masterson and Spencer (2001) catalog design ice pressures based on contact area and aspect ratio. They point out that...aspect ratio and area effects are not mutually exclusive and cannot be considered in isolation." and go on to remark that...measurements of forces on structures with large aspect ratios have shown little relationship between the actual aspect ratio and ice crushing pressure, but have shown that the pressure decreases as the ice thickness increases" For wide structures having aspect ratios between 10 and 80, they quote a design nominal ice pressure of 1.5h MPa, where h is in m based in part on data from the Molikpaq. INDENTATION AND SCALING: LOCALISED HPZ Localised HPZs can be analyzed by a similar argument. Return to Figure 2b, and now interpret it as a section through a spherically symmetric system, described by spherical polar coordinates. Again there is a region I in which the three principal stresses are equal, surrounded by a region II in which radial cracks release σ θθ, in turn surrounded by a region III which remains untracked and elastic. The radial load per unit area on the outer boundary is q. The energy release rate is G = p2 a 4 b 3 ξ 3 E (1 ν) 2n (1 3Sξ 2 + 2ξ 3 ) 2 (1 ξ 3 ) 2 (8)

7 where S = qb 2 /pa 2 and ξ = ρ/b, and the stability index dg/dρ is given by b dg G dρ = 5Sξ 5 + 5ξ 3 Sξ ξ6 ξ(1 ξ 3 )(1 3Sξ 2 + 2ξ 3 ) (a/b < ξ < 1) (9) Dempsey et al. (2001) plot the behavior of these functions. If instead the outer boundary is clamped, the energy release rate is G = p2 a 4 b 3 ξ 3 E 2(1 ν) n ( ) 1 2ν (1 + ν)ξ 3 2 (10) 2 4ν + (1 + ν)ξ 3 Continuing to Figure 4, we now interpret it as a section through half the spherical region from Figure 2b embedded at the edge of an ice sheet of thickness h. If the half-sphere is in equilibrium S equals unity, and the applied force is πpa 2. Making the same assumptions as before and taking S as 1, from (8) G = p 2 a 4 (h/2) 3 ξ 3 E (1 ν) 2n ( ) 2 (1 + 2ξ)(1 ξ) (11) 1 + ξ + ξ 2 and so the corresponding applied force P is P = πpa 2 = 2Ωh 3/2 K scale π n (1 + ξ + ξ2 )ξ 3/2 (1 + 2ξ)(1 ξ) (12) now calling the free-surface correction function Ω instead Ψ. Under similar conditions, the force per HPZ therefore scales as h 3/2. If HPZs are on average equally spaced at separation λh across the breadth of the contact (perpendicular to the plane of Figure 4), the force per unit breadth is F = P/λh = 2Ωh 1/2 K scale λ 1 π n (1 + ξ + ξ2 )ξ 3/2 (1 + 2ξ)(1 ξ) (13) This has the same form as equation (7), which was derived from a model of line-like HPZs, but has different multiplying terms in ξ. The nominal ice contact pressure again scales as h 1/2+β. It is instructive to consider what happens if there is more than one HPZ within the thickness of the ice. Suppose that there are N equally sized zones through the thickness, and that the breadthwise spacing becomes λh/n. The effect is to replace h/2 in equation (11) by h/2n. Continuing to denote the force on a single HPZ by P, the force per unit breadth is NP/(λh/N), and the force per unit breadth F is multiplied by N 1/2. It follows that the force per unit breadth associated with more than one HPZ through the ice thickness is always larger than the force per unit breadth associated with a single HPZ within the thickness. This is consistent with the observation that at velocities above critical the HPZ s form an irregular single ridge of high pressure close to the mid-thickness, and that two or more HPZs do not form at the same breadthwise location. ACKNOWLEDGMENT This research has been supported in part by the US National Science Foundation under grant OPP , in part by the US Army under grant DAAD , in part

8 by the EU Marine Science and Technology Mast IV STRICE project, and in part by the Jafar Foundation. REFERENCES Dempsey, J.P., Palmer, A.C. and Sodhi, D.S. High-pressure zone formation during compressive ice failure. Engineering Fracture Mechanics 68: (2001). Hirayama, K.I., Schwarz, J. and Wu, H-C. Model technique for the investigation of ice forces on structures. In Proceedings of the 2nd International Conference on Port and Ocean Engineering under Arctic Conditions (1973) Kry, R.H. Scale effects in continuous crushing of ice. In International Association of Hydraulic Research Ice Symposium II (1973) Masterson, D.M. and Spencer, P.A. Ice force calculation for large and small aspect ratios. In Scaling Laws in Ice Mechanics and Ice Dynamics, J.P. Dempsey and H.H. Shen, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands (2001) Mulmule, S.V. and Dempsey, J.P. LEFM size requirements for the fracture testing of sea ice. International Journal of Fracture 102: (2000).