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Ned Bair US Army Corps of Engineers Cold Regions Research and Engineering Laboratory Earth Research Institute, UC - Santa Barbara AVPRO 9-10AM 2/27/14 Fracture mechanics

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Snow as a material Sigrist, C., 2006: Measurement of fracture mechanical properties of snow and application to dry snow slab avalanche release, doi:10.3929/ethz-a-005282374. 2

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The Weibull modulus of snow Materialm Snow1-2 Freshwater ice3-5 Concrete12 Steel25 Kirchner, H. O., H. Peterlik, and G. Michot, 2004: Size independence of the strength of snow. Phys. Rev. E, 69, 011306. 3

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Snow is weak, full of flaws, and unreliable. The extremely low Weibull modulus means that the failure of snow is highly unpredictable. Jamieson, B. Regional Danger Ratings and the Odds of Triggering a Potentially Fatal Avalanche, The Avalanche Review, Vol 28, No. 1, October 2009, p. 15. “If you are trying to find an avalanche expert, look under the avalanche debris.” – Jim Kanzler, 1949-2011 4

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Snow slope stability (SNOSS) model Strength/stress (stability index) Shear strength fx: Shear stress σ Density ρ A=1.8-2.2×10 4 Pa; b = 1.55-2.07 (Perla et al., 1982; Jamieson, 1995) P=SWE over some time interval t g and θ – gravity and slope angle 5 Perla, R., Beck, T., and Cheng, T.: The shear strength index of alpine snow, Cold Regions Science and Technology, 11-20, 10.1016/0165-232X(82)90040-4, 1982. Jamieson, J. B.: Avalanche prediction for persistent snow slabs, Department of Civil Engineering, University of Calgary, Calgary, AB, Canada, 258 pp., 1995. Conway, H., and Wilbour, C.: Evolution of slope stability during storms, Cold Regions Science and Technology, 30, 67-77, 10.1016/S0165-232X(99)00009-9, 1999.

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Shortcomings of SNOSS Snoss does not account account for fracture mechanical concepts, e.g. crack length and slab stiffness. In a fracture mechanical model, the concept of “strength” depends on these parameters. 6

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What is fracture mechanics? In the 1920’s it was known that glass fractured at stresses one or two orders of magnitude lower than its theoretical strength, computed based on breaking molecular bonds. For his Ph.D. work, A.A. Griffith hypothesized this discrepancy was caused by many flaws (he called them “scratches”) that weakened the material. 7

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Griffith’s Criterion The stress f at fracture decreases with increasing crack length a. C is a constant. 8

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Fundamental properties – stress (Pa), internal forces, often separated into shear ( – strain (%), deformation E – (Effective) Elastic Modulus (MPa), slope of stress:strain curve; a measure of slab stiffness w f - specific fracture energy or critical energy release rate (J m -2 ), energy release rate required for fracture; a measure of failure layer strength 9

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Linear elastic fracture mechanics The simplest form of fracture mechanics Stress is linearly proportional to strain . p. 76 of Sigrist (2006) 5.5 MPa 6.5 MPa 10

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Brittle, ductile, and quasi-brittle p. 21 of Sigrist (2006) 11

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Brittle to ductile transition 12 Narita, H., 1980: Mechanical behavior and structure of snow under uniaxial tensile stress. Journal of Glaciology, 26, 275-282. Snow is ductile at low strain rates ( 10 -3 s -1, Narita 1980) Photo courtesy of John McGrath

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Atomic level fracture 13

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Grain scale fracture 14

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Slope scale fracture 15

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Fractures and cracks Fracture is a process that creates a crack. A crack is a flaw or discontinuity. As long as there is more energy to feed the crack (i.e. bending and collapse of the slab) than resistance (i.e. strength of the failure layer), the crack will propagate. The critical crack size is comparable to the critical length in a PST. Cracks in failure layers travel at 10-45 m/s (22-101 mph, van Herwijnen and Birkeland, 2012). 16 Birkeland, K. W., and A. van Herwijnen, 2012: Using High-speed Video to Better Understand Extended Column Tests. Proceedings of the 2012 International Snow Science Workshop, 98-103.

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Crack arrest G(a) is the driving force (energy release rate) of the crack. R is the material resistance. It can be constant, or it can change. 17 Anderson, T. L.: Fracture Mechanics: Fundamentals and Applications, Taylor and Francis, Boca Raton, FL, 2005, p 38.

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Fracture modes 18 Anderson, T. L.: Fracture Mechanics: Fundamentals and Applications, Taylor and Francis, Boca Raton, FL, 2005, p 43.

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19 1.Simple shear model (McClung, 1979) – Fractures propagate in shear waves – Triggering depends on slope angle – Fractures cannot propagate on flat ground 2.Anticrack model (Heierli et al., 2008) – Fractures propagate as collapse waves – Initiation does not depend on slope angle, but whether a slab slides after collapse does. – Explains “whoompfing” and remote triggering. Fracture models p 96 of The Avalanche Handbook, 3 rd edition Heierli, J., Gumbsch, P., & Zaiser, M. (2008). Anticrack nucleation as triggering mechanism for snow slab avalanches. Science, 321, 240-243, doi: 10.1126/science.1153948. McClung, D.M. (1979). Shear fracture precipitated by strain softening as a mechanism of dry slab avalanche release. Journal of Geophysical Research, 84, 3519-3526, doi: 10.1029/JB084iB07p03519. Heierli et al. (2008)

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Critical lengths for shear and anticrack models McClung (1979) Heierli et al. (2008) 20

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The anticrack model 21 An anticrack is collapse. The model envisions fractures traveling in a mixed-mode collapse wave. Triggering does not depend on slope angle. Whether the slab “whoompfs” or whether it “whoompfs” and releases depends on the slope angle. Heierli, J., 2008: Anticrack model for slab avalanche release. Institut für Zuverlaessigkeit von Bauteilen und Systemen, 113.

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22 Heierli, J., 2008: Anticrack model for slab avalanche release. Institut für Zuverlaessigkeit von Bauteilen und Systemen, 113.

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Crack energy V 1 2 3 4 5 6 r, m 7 23 Heierli, J., 2008: Anticrack model for slab avalanche release. Institut für Zuverlaessigkeit von Bauteilen und Systemen, 113.

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Testing the anticrack model ECT and PST triggering shows slope angle invariance on persistent weak layers Heierli, J., P. Gumbsch, and M. Zaiser, 2008: Anticrack nucleation as triggering mechanism for snow slab avalanches. Science, 321, 240-243, doi:10.1126/science.1153948. Heierli, J., K. W. Birkeland, R. Simenhois, and P. Gumbsch, 2011: Anticrack model for skier triggering of slab avalanches. Cold Regions Science and Technology, 65, 372-381, doi:10.1016/j.coldregions.2010.10.008. 24

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Implications of the anticrack model You can remotely trigger avalanches on all types of failure layers (e.g. precipitation particles, faceted crystals, and surface hoar). You can dig pits in flat terrain, away from hazardous avalanche areas. 25 Z. Guy, Colorado Avalanche Information Center

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Collapse wave speed 26

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Field measurements of the elastic modulus E and specific fracture energy w f van Herwijnen, A., E. H. Bair, B. Reuter, K. W. Birkeland, and J. Heierli, in preparation: Energy-based method for deriving fracture energy and elastic properties of snowpack layers. Journal of Glaciology. 27

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28 van Herwijnen, A., E. H. Bair, B. Reuter, K. W. Birkeland, and J. Heierli, in preparation: Energy-based method for deriving fracture energy and elastic properties of snowpack layers. Journal of Glaciology.

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