Iceberg calving and associated processes are in principle a function of fracture formation (crevassing) and propagation in response to applied stresses (van der Veen, 1998b; Benn et al., 2007b). The development of crevasses within the terminal zone of water–terminating glaciers therefore plays a key role in the location, magnitude and timing of calving by providing pre– existing or new lines of weakness along which calving is initiated (Benn et al., 2007b; Mottram and Benn, 2009). As a result, the temporal distribution of calving is related to the development of fractures in the ice, the rate at which they enlarge, how deep they can penetrate, and whether they are preferentially orientated to initiate calving.
Glacier ice is typically represented in models as a visco–elastic medium (Bamber and Payne, 2004; Mottram, 2008). However, once an applied stress (tensile or compressive) exceeds the fracture strength of ice, brittle fracture occurs instead of plastic deformation. In theory, fractures and crevasses develop perpendicular to the principle direction of stress. However, this does not always occur, with crevasses often rotated due to changes in glacier flow and the complex arrangement of forces applied to an ice body (Cuffey and Paterson, 2010).
Three principle modes of fracture formation for ice have been identified (Figure 2.6), based on linear elastic fracture mechanics (Benn et al., 2007b). These are tensile cracking (Mode I), sliding (Mode II), and tearing (Mode III). As the name suggests Mode I fractures occur due to tensile stress causing the walls of the fracture to pull apart. Fracture development along a shear plane, where the walls stay in contact, is Mode II (sliding). Mode III describes the propagation of fracture at right angles to the parallel shearing which initiates fracture development. If a combination of modes occurs to initiate and propagate crevasses, it is called PL[HG PRGH fracture. Historically, only Mode I fracturing has been considered in models of glacier ice deformation (van der Veen, 1998b, a; Benn et al., 2007b).
Once initiated, crevasses can either propagate through an ice body or be closed due to compressive forces. Compressive forces are applied due to glacier flow changing the orientation of the crevasse and the direction of applied stress, or where tensile stresses are matched or offset by compressive forces of the overlying ice body (Benn et al., 2007b; Benn and Evans 2010). Several studies (see Benn et al. (2007b) for overview) have sought to model the process of fracture growth (e.g., Iken, 1977; Kenneally and Hughes, 2002; Pralong and Funk, 2005) and depth (e.g., Nye, 1957; Benn et al., 2007b), as the rate and depth of propagation of crevasses is considered by many authors to be the rate–controlling factor at many calving margins (Kenneally and Hughes, 2002; Pralong and Funk, 2005).
Figure 2.6: Schematic diagram showing the three principal modes of crack propagation from Benn et al. (2007b). Arrows indicate direction of applied force. Mode I is the development of a fracture due to tensile stresses. Mode II occurs when a fracture develops along a shear plan and the fracture walls remain in contact and sliding past each other. Mode III describes the tearing of an ice body, with propagation occurring at right angles to the original parallel shearing. The depth of a crevasse at the terminus, in particular, plays a significant role in calving by isolating blocks of ice to an extent that calving is inevitable. The point at which this occurs is typically assumed to be the point at which the crevasse intercepts with the waterline (Benn et al., 2007b). As a crevasse penetrates through an ice body tensile stress at the tip of the fracture is eventually counteracted by the weight of the ice, limiting crevasse depth (Figure 2.7A). Based on this assumption Nye (1957) derived a formula to calculate crevasse depth (d) for a given strain rate: ݀ = 2 ߩ݃ ൬ ɂሶ A൰ భ (2.1)
where ɏi is ice density, gLVJUDYLWDWLRQDODFFHOHUDWLRQ$DQGQDUHIORZODZSDUDPHWHUVDQGȑLV
longitudinal strain rate. However, several authors (Weertman, 1973; Robin, 1974; van der Veen, 1998b; Alley et al., 2005; Benn et al., 2007b; Hart et al., 2011) have noted that the presence of water within a crevasse can have a significant impact on crevasse depth. This is due to cryostatic compressive forces (weight of ice) being opposed by the pressure of water as well as tensile stresses (Figure 2.7B). As a result, Benn et al. (Benn et al., 2007a; Benn et al., 2007b) have modified Equation (2.1) to incorporate water pressure within a crevasse:
݀ = 2 ߩ݃൬ ߝሶ ܣ൰ భ + (ߩ୵݃݀୵)൩ (2.2) where dw represents the water depth in the crevasse andߩw is water density in the crevasse. As
water pressure acts in the same direction as tensile stresses, crevasses will penetrate deeper with the presence of water. If enough water is present they have the potential for penetrating indefinitely through an ice body. The presence of water within a crevasse is particularly important for calving as water can enter from the proglacial water body. This can cause an
increase in the propagation of crevasses that have developed due to processes acting at the terminus (section 2.5.2).
The orientation of crevasses near the terminus has a significant influence on the geometry of calving margins. In valley glaciers, tensile and shear stresses applied due to glacier flow and lateral drag lead to the development of several common patterns of crevasses (Figure 2.8). They are chevron, transverse and splaying crevasses. Such crevasse patterns can be advected passively to the terminus region or develop due to near–terminus stresses. For example, transverse crevasses commonly develop at the termini of tidewater glaciers due to the simple shear along valley margins and longitudinal extension near the centre, which is often mirrored in the geometry of the calving margin (see Figure 6A in Benn et al. (2007b)).
Figure 2.7: Diagram showing the factors that affect the crevasse development and propagation via Mode I (Figure 2.6) fracture. (A) Tensile and compressive forces are matched limiting the depth of penetration. (B) Water within the crevasse increases pressure at the fracture tip, allowing the crevasse to penetrate to greater depths (Benn and Evans 2010).
Figure 2.8: Common crevasse patterns found in valley glaciers. (A) Chevron crevasses from due to lateral shear. (B) Transverse crevasses develop with a slightly curved form due to a combination of extending glacier flow and lateral shear. (C) Splaying crevasses develop due to lateral shear stress and longitudinal shear stress as a result of compressive flow (after Nye, 1957; Benn and Evans 2010).