Tensile Strength and Fracture Mechanics of Cohesive Dry Snow Related to Slab Avalanches by Christopher P. Borstad B.Sc. Physics, Colorado State University, 2002 M.A.Sc. Civil Engineering, University of British Columbia, 2005 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) The University Of British Columbia (Vancouver) August 2011 c© Christopher P. Borstad, 2011 Abstract Fracture mechanics has been applied for over 30 years to explain the release of slab avalanches, but most studies have focused on the initial shear fracture which governs the loss of slab stability rather than the ultimate tensile fracture which releases the avalanche. The application of continuum fracture mechanics to snow—a porous material near the melting temperature—requires a homogenization scheme which accounts for the characteristic length scales associated with the diffuse nature of cracking in snow. An experimental campaign was conducted to measure the strength, fracture mechanical properties, and length scales in the tensile fracture of cohesive dry snow related to slab avalanches. Over 1000 natural snow samples were fractured in beam bending tests in a cold laboratory. Significant rate and size effects were observed in the experiments, though the loading rates were sufficiently high to justify an effective elastic analysis of the data. Using beam theory, the tensile strength was calculated from hundreds of unnotched bending tests and compared with over 2000 synthesized tensile strength measurements from the literature. From the results of three different types of fracture experiments, the fracture toughness and effective fracture process zone length were calculated using equivalent elastic fracture mechanics, which approximately accounts for the nonlinearity engendered by the distributed nature of microcracking in snow. A thin-blade penetration resis- tance gauge was developed which characterizes structural variations in cohesive snow. The maximum force of penetration was the best index variable for correlating with tensile strength and fracture toughness. A nonlocal damage mechanics model, implemented in a finite element code, was calibrated using the results of ten series of experiments, providing a foundation for future predictive modeling applications related to slab avalanches. The tensile strength and fracture toughness of cohesive snow are now well constrained as functions of the snow density, penetration resistance, grain size, strain rate and sample size. The tensile frac- ture process zone was determined to be about 10-20 times the grain size, a length scale which necessitates the use of nonlinear fracture mechanics in the analysis of all but the very largest slab avalanches. ii Preface Chapter 3 contains published material. The bibliographic citation is: Borstad, C.P. and D.M. McClung (2011), Thin-blade penetration resistance and snow strength, Jour- nal of Glaciology, Vol. 57, No. 202, pp. 325–336. All of the research, analysis and writing of the paper was carried out by myself as first author. Copyright for all material is jointly vested between myself and the International Glaciological Society (IGS). Section 5.3 of Chapter 5 contains published material. The bibliographic citation is: Borstad, C.P. and D.M. McClung (2009), Size effect in dry snow slab tensile fracture, Proceedings of the 12th International Conference on Fracture, Ottawa, Canada, 12-17 July 2009, 10 pp. David McClung, as second author, contributed a statistical analysis of a published data set and approximately 5% of the writing. The remainder of the research, analysis and writing was carried out by myself as first author. Copyright over all material from this publication was retained by the authors. Chapter 6 contains material submitted for publication. All of the research, analysis and writing of the manuscript was carried out by myself as first author. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Description of Snow and Avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Alpine snow from a material science perspective . . . . . . . . . . . . . . . . . . . 3 1.1.2 Classification of avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Requisite components of a slab avalanche . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Historical Analysis of Slab Avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Predicting avalanches using a stability index . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Fracture sequence in slab avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Fracture Mechanics of Snow Slab Release . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Shear fracture and initial slab instability . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Tensile properties of cohesive snow relevant to avalanche release . . . . . . . . . . . 11 1.3.3 Measurement and calculation of fracture properties . . . . . . . . . . . . . . . . . . 13 1.4 Theoretical Framework and Guiding Principles of Thesis . . . . . . . . . . . . . . . . . . . 14 1.4.1 Cohesion threshold for fracture propagation . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Length scales in the analysis of slab avalanches . . . . . . . . . . . . . . . . . . . . 17 1.4.3 Nonlinear quasi-brittle fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.4 Rate effects in the fracture of snow . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.5 Note on temperature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5.1 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 iv 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1 In Situ Snow Stratigraphy Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.1 Study plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.2 Snow pit preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.3 Identification of layering and distinct stratigraphic boundaries . . . . . . . . . . . . 31 2.1.4 Snow crystal identification and classification . . . . . . . . . . . . . . . . . . . . . 32 2.1.5 Density and hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Snow Sample Extraction, Transport and Storage . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Sample extraction and transportation . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Sample storage prior to testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Laboratory Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1 Universal testing machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.2 Bending test apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.3 Horizontal weight compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.4 Vertical weight compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.5 Specimen notching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.6 Density calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.7 Temperature measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.8 Crystal identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.9 Practical limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.10 Post-peak behaviour and apparent softening displacement . . . . . . . . . . . . . . 51 2.3.11 Friction between snow and polycarbonate . . . . . . . . . . . . . . . . . . . . . . . 52 3 Thin-Blade Penetration Resistance and Snow Strength . . . . . . . . . . . . . . . . . . . . . . 56 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Hardness Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 Thin blade hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.2 Hand hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.3 Probe hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.4 Compaction of snow in hardness measures . . . . . . . . . . . . . . . . . . . . . . 61 3.2.5 Hardness and strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Force gauge and blade attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Measurement technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.3 Standard stratigraphic profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.4 Laboratory strength testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.1 Density versus blade hardness index . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.2 Penetration rate effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.3 Blade orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.4 Blade size effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.5 Blade and hand hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.6 Blade hardness index as a proxy for strength . . . . . . . . . . . . . . . . . . . . . 75 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.5.1 Density and hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.5.2 Penetration rate effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 v 3.5.3 Blade orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.4 Blade size effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.5 Hardness and strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.5.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4 Tensile Strength of Dry Alpine Snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1 Review and Analysis of Previous Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1 Centrifugal tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.2 In situ tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1.3 Laboratory tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2 New Tensile Strength Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2.1 Strength calculation from beam theory . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2.2 Pertinent variables and range of values . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.3 Tensile strength versus blade hardness index . . . . . . . . . . . . . . . . . . . . . 117 4.2.4 Tensile strength versus density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.2.5 Influence of grain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.6 Loading rate effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2.7 Specimen size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3 Models of Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.3.1 Density power law models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3.2 Hardness models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5 Determination of Fracture Parameters using Nonlinear Fracture Mechanical Scaling Laws . 157 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.1.1 Linear elastic fracture mechanics (LEFM) . . . . . . . . . . . . . . . . . . . . . . . 159 5.1.2 Sources of deviation from LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.3 Experimental evidence of deviation from LEFM in snow . . . . . . . . . . . . . . . 163 5.1.4 Nonlinear correction using equivalent elastic crack . . . . . . . . . . . . . . . . . . 167 5.2 Notched Size Effect Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3 Unnotched Size Effect Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.4 Zero-Brittleness (Notched-Unnotched) Method . . . . . . . . . . . . . . . . . . . . . . . . 200 5.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.5 Aggregate Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.5.1 Fracture toughness versus density . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.5.2 Fracture toughness versus blade hardness index . . . . . . . . . . . . . . . . . . . . 214 5.5.3 Comparison with previous fracture toughness regression models . . . . . . . . . . . 215 vi 5.5.4 Error terms in size effect law derivations . . . . . . . . . . . . . . . . . . . . . . . . 215 6 Numerical Simulation of Bending Experiments using Nonlocal Damage Mechanics . . . . . . 220 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.1.1 Continuum approximations of heterogeneous materials . . . . . . . . . . . . . . . . 221 6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2.1 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2.2 Nonlocal isotropic damage model . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2.3 Beam theory for experimental load-displacement curves . . . . . . . . . . . . . . . 228 6.2.4 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.2.5 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.3 Experimental Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 6.4.1 Model sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 6.4.2 Simulations of zero-brittleness data sets . . . . . . . . . . . . . . . . . . . . . . . . 241 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.1 Rate Effects and Validity of Effective Elastic Analysis . . . . . . . . . . . . . . . . . . . . . 253 7.1.1 Estimates of bulk creep strain at failure . . . . . . . . . . . . . . . . . . . . . . . . 254 7.1.2 Creep effects within the fracture process zone . . . . . . . . . . . . . . . . . . . . . 257 7.1.3 Failure times in slab avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.2 Relation Between Beam Bending Tests and Tensile Fractures in Slab Avalanches . . . . . . 258 7.2.1 Homogenization of a layered, orthotropic snow slab . . . . . . . . . . . . . . . . . 258 7.2.2 Length scales in avalanches and experiments . . . . . . . . . . . . . . . . . . . . . 259 7.3 Scale-Cohesion Classification Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.3.1 Size requirements for LEFM applicability . . . . . . . . . . . . . . . . . . . . . . . 266 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.4.1 Experimental lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.4.2 Applicability of results to avalanche operations . . . . . . . . . . . . . . . . . . . . 270 7.4.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Appendix A: Fracture Morphology From Bending Tests . . . . . . . . . . . . . . . . . . . . . . . 292 Appendix B: Analysis of Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Appendix C: Viscoelastic Deformation of Snow over Short Timescales . . . . . . . . . . . . . . . 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 vii List of Tables Table 2.1 Dimensions of beam-shaped snow sample cutters . . . . . . . . . . . . . . . . . . . . . 37 Table 3.1 Correlations among strength, hardness and density for data reported by Martinelli (1971) 64 Table 3.2 Correlations among strength, hardness and density for data from the present study . . . . 77 Table 4.1 Sources of published centrifugal tensile strength data . . . . . . . . . . . . . . . . . . . 87 Table 4.2 Rate of acceleration of spin tester in different studies . . . . . . . . . . . . . . . . . . . . 95 Table 4.3 Descriptive statistics of tensile strength test series from the present study . . . . . . . . . 112 Table 4.4 Correlations between tensile strength versus density model residuals and other variables . 138 Table 4.5 Correlations between tensile strength versus blade hardness index model residuals and other variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Table 5.1 Descriptive statistics of notched size effect data . . . . . . . . . . . . . . . . . . . . . . 175 Table 5.2 Fracture parameters determined from the notched size effect . . . . . . . . . . . . . . . . 177 Table 5.3 Descriptive statistics for unnotched size effect data . . . . . . . . . . . . . . . . . . . . . 192 Table 5.4 Calculated and reported values of the Weibull modulus . . . . . . . . . . . . . . . . . . 195 Table 5.5 Fracture parameters determined from the unnotched size effect . . . . . . . . . . . . . . 197 Table 5.6 Descriptive statistics for notched/unnotched test data . . . . . . . . . . . . . . . . . . . . 205 Table 5.7 Fracture parameters determined from notched/unnotched tests . . . . . . . . . . . . . . . 208 Table 6.1 Descriptive statistics of notched-unnotched bending tests used for sensitivity analysis in numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Table 6.2 Numerical model parameter values for a sensitivity analysis of the nonlocal interaction radius using the nonlocal isotropic damage model . . . . . . . . . . . . . . . . . . . . . 237 Table 6.3 Numerical model parameters used in simulations of ten different experimental data sets . 242 Table 7.1 Ratio of creep strain to instantaneous elastic strain for representative times to failure of bending tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 viii List of Figures Figure 1.1 Photograph of the fracture boundaries of a large slab avalanche . . . . . . . . . . . . . . 6 Figure 1.2 Photograph of exposed surface hoar crystals, a common weak layer responsible for slab avalanches once buried . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 1.3 Schematics of initial tensile fracture upward through a slab and then laterally across the slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Figure 1.4 Schematic illustrating the porous and heterogeneous nature of dry seasonal snow . . . . 15 Figure 1.5 Scale-Cohesion classification scheme for snow . . . . . . . . . . . . . . . . . . . . . . 20 Figure 2.1 Overview of Rogers Pass and the Parks Canada compound where the cold lab is located 29 Figure 2.2 Photographs of study plots used for sourcing natural snow samples . . . . . . . . . . . . 30 Figure 2.3 Photograph of technique for preparing the observation corner of a snow pit . . . . . . . 31 Figure 2.4 Photographic showing observation corner of a snowpit with markers indicating strati- graphic boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 2.5 Photographs showing the procedure for observing and classifying snow crystals . . . . . 33 Figure 2.6 Photograph of a snowpit following stratigraphic profiling and snow sample extraction . . 35 Figure 2.7 Thin-shaved snow sample, lit from behind, showing the layered structure of the snow . . 36 Figure 2.8 Schematic showing orientation of notched beams of different sizes taken from the same layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 2.9 Photographs of snow pits showing holes left by snow sample extraction . . . . . . . . . 38 Figure 2.10 Photograph of specimens of different sizes stored in the cold lab prior to testing . . . . . 39 Figure 2.11 Photograph of testing machine with a small sample mounted for a bending test . . . . . 40 Figure 2.12 Photographs of rocker supports used in bending tests and LVDTs used for deflection measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 2.13 Photograph showing testing machine set up for horizontal weight compensation . . . . . 44 Figure 2.14 Photographs of a small sample being prepared for testing in the horizontal configuration. 44 Figure 2.15 Photograph of the testing machine oriented vertically, with a large sample mounted in a weight-compensated fashion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Figure 2.16 Photograph of tool used to cut notches into snow samples . . . . . . . . . . . . . . . . . 46 Figure 2.17 Photograph showing a snow sample being weighed for the calculation of bulk density . . 46 Figure 2.18 Photograph showing the measurement of temperature of a sample after a bending test . . 47 Figure 2.19 Photograph of the author studying snow crystals under a microscope for classification. . 48 Figure 2.20 Photograph showing the largest specimen size used in the present study, mounted for a bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Figure 2.21 Photograph showing the smallest sample size used in the present study, just after failure in a bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ix Figure 2.22 Photograph of a shear failure in a large sample following an attempted unnotched bend- ing test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Figure 2.23 Characteristic post-peak load-displacement curves measured in bending experiments at different loading rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 2.24 Post-peak load-displacement curves measured in flexural tests of thin glass strips . . . . 53 Figure 2.25 Experimental curves used to measure the friction coefficient between snow and polycar- bonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 2.26 Example load-displacement curve showing the possible influence of friction between snow and polycarbonate support table . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Figure 3.1 Schematic of blade penetration force versus penetration distance, showing the definition of the blade hardness index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 3.2 Schematics of different penetrometer tips, to scale . . . . . . . . . . . . . . . . . . . . 62 Figure 3.3 Photograph of blade hardness gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Figure 3.4 Photograph of a blade hardness measurement being carried out . . . . . . . . . . . . . . 67 Figure 3.5 Schematic of paired density and blade hardness measurements in situ . . . . . . . . . . 69 Figure 3.6 Schematic of different techniques of grouping blade hardness measurements in a homo- geneous snow layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 3.7 Schematic of paired tests of flexural strength and blade hardness . . . . . . . . . . . . . 70 Figure 3.8 Scatter plot of density versus blade hardness index from in-sit data . . . . . . . . . . . . 71 Figure 3.9 Box plots showing dependence of blade hardness results on blade orientation relative stratigraphic layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 3.10 Box plot of blade hardness indices for different hand hardness categories . . . . . . . . 76 Figure 4.1 Schematics of centrifugal tensile test specimens . . . . . . . . . . . . . . . . . . . . . . 88 Figure 4.2 Scatter plot of nominal centrifugal tensile strength versus density, grouped by study . . . 90 Figure 4.3 Schematics of stress concentration factor geometries used for correcting centrifugal ten- sile strength data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Figure 4.4 Scatter plot of corrected centrifugal tensile strength versus density, grouped by study . . 94 Figure 4.5 Estimated systematic bias in centrifugal tensile strength values versus spin rate at failure 96 Figure 4.6 Estimated tensile strain rate at failure for centrifugal tensile tests, grouped by study . . . 99 Figure 4.7 Temperature dependence of centrifugal tensile strength . . . . . . . . . . . . . . . . . . 100 Figure 4.8 Nominal centrifugal tensile strength as a function of ram hardness and density. . . . . . 101 Figure 4.9 Scatter plot of in-situ tensile and flexural strength data as a function of density, grouped by study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Figure 4.10 Schematic of in-situ uniaxial tensile strength test geometry . . . . . . . . . . . . . . . . 106 Figure 4.11 Scatter plot of tensile strength versus density, grouped by grain form . . . . . . . . . . . 107 Figure 4.12 Scatter plot of laboratory measurements of tensile strength versus density, grouped by study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure 4.13 Schematic of unnotched bending tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Figure 4.14 Kernel density plots of variables associated with flexural strength tests . . . . . . . . . . 116 Figure 4.15 Scatter plot of tensile strength versus blade hardness index . . . . . . . . . . . . . . . . 117 Figure 4.16 Scatter plot of tensile strength versus blade hardness index, grouped by test series . . . . 118 Figure 4.17 Tensile strength versus density data superimposed on previously measured laboratory strength data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 4.18 Scatter plot of tensile strength versus density, grouped by test series . . . . . . . . . . . 120 x Figure 4.19 Scatter plots of tensile strength versus blade hardness index and density, grouped by grain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 4.20 Strain rate dependence of tensile strength for two test series . . . . . . . . . . . . . . . 122 Figure 4.21 Scatter plots of tensile strength versus beam depth for four different size-effect test series 124 Figure 4.22 Strain rate dependence of tensile strength, grouped by beam depth, for two test series. . . 126 Figure 4.23 Nonlinear regression fits through individual data sets of tensile strength as a function of density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Figure 4.24 Residual plots for assessing the goodness of fit of an initial tensile strength-density model for data from the present study . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Figure 4.25 Residual plots for assessing the goodness of fit of the weighted regression through group means of tensile strength-density data . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Figure 4.26 Residual plots for assessing the goodness of fit of the weighted regression through square-root transformed group means of tensile strength-density data . . . . . . . . . . 136 Figure 4.27 Tensile strength as a function of density, grouped by date of testing, with regression model fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Figure 4.28 Residual plots for assessing the goodness of fit of the transformed and variance modeled tensile strength-density model for the data of Jamieson (1988) . . . . . . . . . . . . . . 140 Figure 4.29 Residual plots for assessing the goodness of fit of the transformed tensile strength- density model for the data of Sigrist (2006) . . . . . . . . . . . . . . . . . . . . . . . . 142 Figure 4.30 Residual plots for assessing the goodness of fit of the transformed tensile strength- density model for the data of Martinelli (1971) . . . . . . . . . . . . . . . . . . . . . . 144 Figure 4.31 Tensile strength versus density for the data of Martinelli (1971) with regression fit . . . 145 Figure 4.32 Residual plots for assessing the goodness of fit of the linear model of tensile strength versus blade hardness index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Figure 4.33 Residual plots for assessing the goodness of fit of the power law model of tensile strength versus blade hardness index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Figure 4.34 Tensile strength versus blade hardness index with regression model fits . . . . . . . . . 152 Figure 4.35 Tensile strength versus ram hardness for data of Martinelli (1971) with regression fits . . 154 Figure 4.36 Scatter plot of tensile strength versus density data from the present study superimposed against shaded regions bounding the limits of previously reported data . . . . . . . . . . 155 Figure 5.1 Schematic of nonlinear crack tip zones for ductile and quasi-brittle materials . . . . . . 161 Figure 5.2 Power law scaling of nominal strength versus specimen size for notched tests . . . . . . 164 Figure 5.3 Load-midspan displacement curves from notched bending tests conducted at three dif- ferent loading rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Figure 5.4 Load-midspan displacement curves from bending tests with different initial notch depths 166 Figure 5.5 Schematic of the homogenization of snow using equivalent elastic crack concept . . . . 168 Figure 5.6 Schematics of notched size effect test series, relatively scaled . . . . . . . . . . . . . . . 176 Figure 5.7 Notched size effect data fit to the size effect law, including a reanalysis of published data 178 Figure 5.8 Notched size effect data fit to the size effect law, showing rate dependence . . . . . . . . 179 Figure 5.9 Density dependence of the fracture toughness determined from the notched size effect, with regression fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Figure 5.10 Density dependence of the non-dimensional critical equivalent crack extension from notched tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Figure 5.11 Rate dependence of the critical equivalent crack extension from notched tests . . . . . . 183 Figure 5.12 Notched size effect transitional size versus density . . . . . . . . . . . . . . . . . . . . 185 xi Figure 5.13 Size dependence of the modulus of rupture . . . . . . . . . . . . . . . . . . . . . . . . 193 Figure 5.14 Unnotched data fit to the size effect law for the modulus of rupture . . . . . . . . . . . . 196 Figure 5.15 Fracture toughness as a function of density and penetration resistance from unnotched data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Figure 5.16 Linear regressions of notched-unnotched test data . . . . . . . . . . . . . . . . . . . . . 207 Figure 5.17 Rate dependence of the non-dimensional critical equivalent crack extension from not- ched-unnotched tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Figure 5.18 Rate dependence of the fracture toughness from notched-unnotched tests . . . . . . . . 210 Figure 5.19 Fracture toughness as a function of density and penetration resistance from notched- unnotched data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Figure 5.20 Dependence of fracture toughness on density and blade hardness index for all data from present study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Figure 5.21 Comparison of existing relations between fracture toughness and density . . . . . . . . 216 Figure 5.22 Range of ratios between critical equivalent crack extension and beam depth. . . . . . . . 217 Figure 6.1 Schematic definition of the minimum Representative Volume Element size for treating snow as a continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Figure 6.2 Finite element mesh for unnotched beam bending tests . . . . . . . . . . . . . . . . . . 231 Figure 6.3 Finite element mesh for notched beam bending tests . . . . . . . . . . . . . . . . . . . 231 Figure 6.4 Loading curves using deflection measured at three different points of the beam in a series of notched-unnotched bending tests . . . . . . . . . . . . . . . . . . . . . . . . . 234 Figure 6.5 Load and displacement data plotted versus time, showing the dynamic loss of elastic stability near peak load in the experiments . . . . . . . . . . . . . . . . . . . . . . . . . 235 Figure 6.6 Sensitivity of numerical simulations to the nonlocal interaction radius . . . . . . . . . . 237 Figure 6.7 Sensitivity of numerical simulations to Poisson’s ratio . . . . . . . . . . . . . . . . . . 238 Figure 6.8 Sensitivity of numerical simulations to the stiffness of the beam supports . . . . . . . . 239 Figure 6.9 Sensitivity of numerical simulations to the fracture energy . . . . . . . . . . . . . . . . 240 Figure 6.10 Load-displacement curves from notched-unnotched experiments with model fits . . . . . 244 Figure 6.11 Load-displacement curves from notched-unnotched experiments conducted at two dif- ferent loading rates, with finite element model fits . . . . . . . . . . . . . . . . . . . . . 246 Figure 6.12 Load-displacement curves from notched-unnotched experiments conducted at three dif- ferent loading rates, with finite element model fits . . . . . . . . . . . . . . . . . . . . . 249 Figure 7.1 Kernel density plots of the time to peak load in 370 bending experiments, covering three orders of magnitude in loading rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Figure 7.2 Schematics showing possible definitions of equivalent structural (beam) size for a given shape of stress and strain gradient in a snow slab . . . . . . . . . . . . . . . . . . . . . 260 Figure 7.3 Scale-Cohesion classification, revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Figure A.1 Broken half of a beam sample following an unnotched bending test . . . . . . . . . . . 293 Figure A.2 Broken halves of a beam sample following a notched bending test . . . . . . . . . . . . 294 Figure A.3 Tensile fracture (crown) surface from a soft slab avalanche . . . . . . . . . . . . . . . . 295 Figure A.4 Before and after images of a notched bending test . . . . . . . . . . . . . . . . . . . . . 297 Figure A.5 Before and after images of a notched bending test . . . . . . . . . . . . . . . . . . . . . 298 Figure A.6 Before and after images of an unnotched bending test . . . . . . . . . . . . . . . . . . . 299 Figure A.7 Schematic indicating field of view of high speed camera for subsequent fracture sequence300 xii Figure A.8 Crack sequence image 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Figure A.9 Crack sequence image 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Figure A.10 Crack sequence image 3. Crack initiation apparent in particle tracking results . . . . . . 303 Figure A.11 Crack sequence image 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Figure A.12 Crack sequence image 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Figure A.13 Crack sequence image 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Figure A.14 Crack sequence image 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Figure A.15 Crack sequence image 8. First visible indication of crack coalescence . . . . . . . . . . 308 Figure A.16 Crack sequence image 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Figure A.17 Crack sequence image 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Figure A.18 Crack sequence image 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Figure A.19 Crack sequence image 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Figure A.20 Crack sequence image 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Figure A.21 Crack sequence image 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Figure A.22 Crack sequence image 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Figure A.23 Crack sequence image 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Figure A.24 Crack sequence image 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Figure A.25 Crack sequence image 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Figure A.26 Crack sequence image 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Figure A.27 Crack sequence image 20. Visible crack has propagated about 10 cm across sample . . . 320 Figure C.1 Viscoelastic elements as combinations of springs and dashpots . . . . . . . . . . . . . . 325 xiii List of Symbols Latin A Area ae Equivalent elastic crack length a◦ Original crack or notch length b Beam width B Blade hardness index B Geometric function in Bažant’s notched size effect law c f Critical equivalent elastic crack extension or effective fracture process zone length D Characteristic specimen dimension, here typically the beam depth Db Length scale characterizing the boundary layer of microcracking prior to crack coalescence D◦ Notched size effect transitional size De Elastic stiffness tensor E Grain size E Young’s modulus Et Post-peak strain softening tangent modulus f Frequency in revolutions per second fr Modulus of rupture (flexural strength) fr∞ Asymptotic large-size limit of the modulus of rupture ft Tensile strength F Force g Dimensionless linear elastic fracture mechanics geometry function (g = k2) g◦ Shorthand notation for g(α◦) GF Fracture energy k Dimensionless linear elastic fracture mechanics geometry function k◦ Shorthand notation for k(α◦) k Scaling factor relating c f between notched and unnotched tests KI Mode I stress intensity factor KIc Mode I fracture toughness KINu Mode I apparent fracture toughness Ktn Net stress concentration factor l Length m Weibull modulus M Bending moment nd Similitude dimension P Applied load xiv r Radius r2 Coefficient of determination for linear regression models rs Spearman’s correlation coefficient R Snow hardness, here denotes hand hardness index unless otherwise specified R Nonlocal interaction radius R2 Coefficient of determination for nonlinear regression models Rram Ram hardness R Radius of cylindrical tensile strength sample S Beam support span t Time T Temperature V Crosshead speed Greek α Nondimensional crack length (a◦/D) β Brittleness number (D/D◦) γ Quasi-brittle multiplier for Irwin’s plastic zone radius estimate ε Strain ε◦ Limit elastic strain under uniaxial tension ε f Post-peak strain softening (ductility) parameter ε̇ Strain rate ε̇N Nominal strain rate predicted by beam theory ε̄ Nonlocal scalar equivalent strain ε̃ Local strain η Multiplier for Db in the limit for D→ 0 in Bažant’s ”universal” size effect law κ Internal history variable representing maximum previous level of equivalent strain ν Poisson’s ratio ρ Density σ Stress σx Tensile stress σN Nominal stress σNu Nominal strength, defined as maximum nominal stress at failure σ̇ Stress rate φ Nondimensional measure of ductility in nonlocal isotropic damage model χ Nonlinear term in the regression analysis of the ”universal” size effect law ω Scalar damage parameter Ω Angular frequency xv Glossary COV Coefficient of Variation FPZ Fracture Process Zone, the spatially distributed zone of softening damage ahead of a crack tip in a quasi-brittle material such as snow. Different in size and behaviour from the hardening crack-tip plastic zone in ductile materials. LCD Liquid Crystal Display LEFM Linear Elastic Fracture Mechanics LVDT Linear Variable Differential Transformer, used for accurate measurements of linear displacement. RVE Representative Volume Element, typically the minimum volume size over which continuum physical relations are applicable for a heterogeneous material. SMP SnowMicroPenetrometer, a motor-driven cone penetrometer that records snow hardness at sub-millimeter resolution. xvi Acknowledgements I would like to thank my supervisor David McClung for supporting me, challenging me, and allowing me the space to develop into an independent scientist. Thanks also to the rest of my committee members, Sid Mindess, Reza Vaziri and Erik Eberhardt for many interesting discussions, insightful comments and sage advice over the years. A special thanks to the Natural Sciences and Engineering Research Council of Canada, the University of British Columbia, and Canadian Mountain Holidays for financial support. This research would not have been possible without the generous in-kind support of Parks Canada and the staff of the Avalanche Control Section at Rogers Pass. Thanks to Bruce McMahon and Jeff Goodrich for defending the cold lab space over the years against cubicle expansion and for allowing researchers to interact with the forecasting operation on a daily basis. The support and front-line contact were invaluable. Thanks to the rest of the forecasting staff and wardens for the engaging discussions, the rides to Mt. Fidelity, and the opportunity to work in such a beautiful and dynamic natural environment. A special thanks to Steve Conger for introducing me to many different facets of the avalanche industry, for assistance with field and lab work, and for many useful discussions and challenging questions over the years. Thanks to Eirik Ainer Sharp and Andrea McLeod for assistance with the field and lab work and for taking great care with the delicate snow samples. To my family, I owe you an undying gratitude for your unwavering love and support as I followed my passions. Thank you Jed for your unconditional love and companionship and for never failing to lift my spirits. To Beth, I cannot summon the words to express my thanks for your patience, support, encouragement, forgiveness and love during this time. I could not have achieved this without you. xvii Chapter 1 Introduction The scientist studying snow and ice is not to be envied. It may be wonderful to work on glaciers and snowy slopes. On the other hand, ice and snow are probably the most complex bodies ever considered in continuum mechanics. –Hans Ziegler Snow avalanches are a hazard to people and structures in most mountainous areas of the world. Avalan- ches have caused human fatalities and captured the imagination of people inhabiting mountainous terrain for thousands of years. In recorded history, the largest losses of life to avalanches have been in great military expeditions in Europe (Bader and Kuriowa, 1962; Voight et al., 1990). During Hannibal’s crossing of the Alps in 218 B.C., thousands of men and a great many of the horses who died were likely consumed by avalanches. During World War I, between 40,000 and 80,000 troops died in avalanches in the mountains of Tyrol, with as many as 10,000 dying in the course of just one to two days on the Austro-Italian front in 1916. In North America, most avalanche accidents prior to the late 1900s were related to mining or railroad operations in mountainous terrain (Voight et al., 1990; Jamieson and Stethem, 2002). In more recent times, most fatalities involve recreationists who were voluntarily exposed to the avalanche hazard (Voight et al., 1990; Jamieson and Stethem, 2002; McClung and Schaerer, 2006). In addition to causing human fatalities, avalanches have direct and indirect economic costs to the con- struction, transportation and tourism sectors every winter in regions with sufficient snowfall and slopes greater than about 25 degrees (McClung and Schaerer, 2006). In Canada, the direct economic costs related 1 to avalanches exceed $10 million per year (McClung and Schaerer, 2006), and in the U.S. the number is probably an order of magnitude greater (Voight et al., 1990). Climate change is likely to influence the global distribution and extent of avalanches, though predicting these effects will be very difficult. Climate change will alter the distribution and variability of temperature and precipitation (IPCC, 2007), both of which influence avalanche activity (Stethem et al., 2003). In Europe, the seasonal timing of the most destructive avalanches may shift toward spring, and the relative proportion of wet slab avalanches compared to dry slab avalanches may increase (Martin et al., 2001). Changing weather and temperature patterns may shift avalanche activity away from regions where it is currently common toward regions where it is currently more rare (Glazovskaya, 1998). The link between climate change and avalanche activity is very tenuous, and more research will be needed to address these questions, but it appears that the avalanche problem will persist well into the future even in a warming climate. Overview An investigation was conducted into the tensile fracture properties of cohesive snow related to slab avalanches. In a slab avalanche, a large volume of cohesive snow is released all at once following the propagation of fractures. The first fracture is beneath a cohesive snow slab in a layer or interface which is weak in shear. This initial fracture propagates widely before a tensile fracture ultimately releases the avalanche (McClung and Schaerer, 2006). Though the shear fracture is the initial instability, this mode of fracture has been studied more widely and was not considered in the present study. However, the shear fracture is only possible if the snow is sufficiently cohesive to support tensile stresses (Mellor, 1968). The tensile fracture properties of the snow slab also determine the distance over which the shear fracture must propagate before the strength or fracture toughness of the slab is overcome. Thus the tensile properties of cohesive snow both enable a slab avalanche and influence its dimensions and destructive potential (McClung and Schweizer, 2006). The majority of the present study involved experimental methods to understand the response of snow to tensile stresses applied at sufficiently high rates to minimize creep effects and cause fast fracture, as in slab avalanches. Natural snow was sampled over a wide range of conditions for the experiments, and the results were correlated with a variety of fundamental snow properties and testing conditions to enable comparison with the state of snow in a slab avalanche and to facilitate the in-situ estimation of fracture 2 properties using index measurements. Equivalent elastic theories, which treated the snow as an elastic continuous material and approximately accounted for the observed nonlinearities, were applied to explain the experimental results and calculate properties such as tensile strength, fracture toughness, and effective fracture process zone size. These calculated properties were used in a continuum damage mechanics model to simulate a variety of the experiments, a first step toward the refined calibration of models that can predict the response of snow to the types of loading relevant to slab avalanches. A general and more thorough description of the avalanche phenomenon is given here first, followed by a review of applications of fracture mechanics to explain avalanche triggering and release. A newly developed conceptual framework is presented to orient and contextualize the research, followed by the guiding principles and hypotheses of the research project. Finally, the chapter structure of the thesis is delineated. 1.1 Description of Snow and Avalanches 1.1.1 Alpine snow from a material science perspective Snow is a particularly difficult material to study and analyze using common experimental methods and physical theories. One of the characteristic properties of alpine snow is its inherent thermodynamic insta- bility (Bader and Kuriowa, 1962), a result of the proximity of snow to its melting temperature. Alpine snow in which avalanches form is usually within 90% of its melting temperature on an absolute scale (McClung and Schaerer, 2006). Other than solid ice, no other common natural or engineering material exists so close to its melting temperature (one exception may be magmatic structures in the asthenosphere). Con- sequently, in some regards, snow and ice may be considered model “high temperature” materials. However, relative to other common engineering materials such as metals, ice has a very low melting point diffusivity, which allows it to fracture right up to the melting temperature (Schulson et al., 1984). In addition to its proximity to the melting temperature, alpine snow is also characterized by a very high porosity, or, equivalently, a low solid volume fraction. For most slab avalanches, the density of the snow slab is between 100–350 kg m−3 (Perla, 1977). Given the density of solid freshwater ice (917 kg m−3), this snow density range corresponds to a volume fraction filled by solids of about 0.1–0.4. Therefore most of the volume of snow is filled by air. The large pore space can allow for grain rearrangement and shearing of 3 grain contact areas without appreciable deformation of the grains themselves (Bader and Kuriowa, 1962), leading to different bulk properties under different modes of loading (e.g. Butkovich, 1956; Mellor, 1975). The material properties of snow have important temperature-dependent and rate-dependent characteris- tics (e.g. Mellor and Smith, 1966; McClung, 1977; Narita, 1980; McClung, 1981; Schweizer, 1998). Creep effects in snow are important for all but the fastest rates of loading (Bader and Kuriowa, 1962), and the same can be said for solid ice (Schulson and Duval, 2009). In fact, snow slabs may not ever respond fully elastically for relevant rates of loading in avalanches. The highly irregular alpine snow cover is built up over the course of a winter by varying environmental conditions and successive storm and wind events, each of which produce snow of different types. The result is a layered and heterogeneous snow structure with spatial variability in properties over a number of length scales relevant to the triggering of avalanches. Spatial variability is the result of environmental processes such as wind and radiation and their interaction with the snow as a function of the local terrain and ground cover (e.g. Schweizer et al., 2008). Metamorphism of snow on the ground, caused by changes in liquid water content, temperature, or a temperature gradient, gives rise to significant changes in the grain size, shape, and internal cohesion of the snow (Colbeck, 1982). The above-mentioned effects combine to complicate the comparison of experimental results from dif- ferent snow studies which typically involve different loading rates, temperatures, specimen geometries and sizes, and snow from different sources with different thermodynamic histories. These effects also make experimental design difficult in the study of avalanches, for the experimental conditions in situ or in a lab may not replicate to the conditions at the point where an avalanche is triggered. In situ snow studies give better indications of the properties of snow in its natural state (Perla, 1969; McClung, 1979a; Jamieson, 1988; Conway and Abrahamson, 1984), though laboratory tests measuring similar properties can give quite different results (Martinelli, 1971; Narita, 1980; Sigrist, 2006). In either case, small-scale measurements are complicated by statistical and deterministic size effects when attempting to relate experimental re- sults to the avalanche scale (Sommerfeld, 1974; Perla and Beck, 1983; Bažant et al., 2003; McClung, 2003; Sigrist et al., 2005b; Sigrist, 2006). 4 1.1.2 Classification of avalanches Snow avalanches can be classified into two types. The first are loose avalanches, which initiate with the failure of a very small volume of cohesionless surface snow. Below the initiation point, loose avalanches fan out and gain mass as they slide downslope. This type of slide more closely resembles the commonly-held notion of an avalanche as “snowball effect” involving a continuously growing mass of loose snow. Loose avalanches are typically small in volume, flow relatively slowly, and have limited destructive effects (e.g. McClung, 2003; McClung and Schaerer, 2006). For these reasons loose avalanches have relatively little practical importance or emphasis in academic studies, and will not be considered here. The second and more important type of avalanche is a slab avalanche, which occurs when the snow is sufficiently cohesive to transmit tensile stresses and thereby permit fracture propagation (Mellor, 1968). Slab avalanches are released by fractures that propagate long distances and isolate a large volume of snow (Figure 1.1) which then flows rapidly downslope with sometimes great destructive effects (Perla and LaChapelle, 1970; Perla, 1977; McClung, 1979b, 1981, 1987, 1996; Bažant et al., 2003; Schweizer et al., 2003). The slope-normal slab thickness is typically less than 1 meter in slab avalanches (Perla, 1977), and the ratios of width-to-depth and length-to-depth of the slab are on the order of 10–103 with a median for both ratios around 100 (McClung, 2009a). The mean slope angle at which slab avalanches are released is 38◦, with nearly all recorded observations falling between 30◦ and 45◦ (Perla, 1977). Most slab avalanches which in- volve humans are caused by humans (e.g. Jamieson and Johnston, 1992), whereas natural slab avalanches, caused primarily by storm snow loading (McClung and Schaerer, 2006) are the primary threat to civil in- frastructure (Schweizer et al., 2003). Slab avalanches can be further classified according to the moisture content of the snow, the hardness of the slab, the location of the sliding surface or weak layer, topographic features of the slope, or triggering fac- tor (Mellor, 1968; Martinelli, 1971). In most alpine areas, most avalanches over the course of a winter occur in dry snow, that is, snow with no liquid water content (McClung and Schaerer, 2006). Slab avalanches in dry snow cause more destruction and fatalities than other types of avalanches in most mountainous regions (McClung and Schaerer, 2006). Moist or wet slab avalanches, which occur in snow with a limited amount of free water (< 3% water content by volume for moist snow, 3–8% for wet snow), are typically of less concern over the course of a winter, other than a punctuated period at the end of the season as temperatures 5 AA A A BB BB C D Figure 1.1: Photograph of the fracture boundaries of a large slab avalanche. The exposed bed surface—the weak stratigraphic layer that was the first to fail—is denoted “A.” The tensile fracture surface at the upslope boundary (the crown), which ultimately released the slab, is labeled “B.” The lateral fracture boundaries (flanks), only one of which is visible in this image (labeled “C”), can fail by a combination of shear and tensile fractures in the slab. The downslope fracture boundary, or stauchwall, is often not visible after the slab overruns it. A secondary step- down tensile fracture surface is visible at “D,” caused by the force of the flowing snow from upslope. increase. Moist or wet slabs also have very different mechanical properties from dry slabs, and it would be difficult to handle and maintain wet snow in a stable state in a cold lab. For these reasons, the focus of the present investigation was limited to dry snow. 1.1.3 Requisite components of a slab avalanche There are three necessary components for a slab avalanche to occur. The first, as mentioned above, is a sufficiently cohesive snow slab to support the propagation of fractures. The second is a weak layer or weak interface beneath the slab. Finally, a slab avalanche requires some kind of trigger to cause the initial unstable shear fracture. The initial cohesion of newly fallen storm snow is usually low, with any cohesive strength due primarily to interlocking of the snow crystals (Fukue, 1977). Instabilities during a storm are often limited to loose avalanches, unless the new snow is loading a pre-existing slab and weak layer which are near critical. Depending on the temperature, humidity, and wind during and after a storm, bond formation (sintering) in the newly fallen snow may proceed rapidly. Sintering occurs as a function of time, temperature, and 6 the temperature gradient within a snow layer (e.g. McClung and Schaerer, 2006). Bond formation and growth during the sintering process promotes bulk cohesion of the snow, and, once a threshold in cohesion is reached, the snow may first begin to transmit tensile stresses. Estimating the timing at which this cohesion threshold is reached—either in slab snow losing cohesion or loose snow gaining cohesion—is important for forecasting avalanches. Weak layers involved in slab avalanches are often snow surfaces that were exposed for long periods of time before burial. Specific types of weak crystals form or metamorphose at or near the surface of the snow pack depending on the environmental conditions. Weak layer crystals which are commonly re- sponsible for slab avalanche activity, such as surface hoar and near-surface facets, form under a large temperature or vapour pressure gradient at or near the snow surface caused by a strong radiation imbalance (Schweizer et al., 2003). Buried layers composed of these crystal forms have been termed “persistent” weak layers, as the crystals have a tendency to resist bond formation and persist for long periods of time in a weak state (Jamieson and Johnston, 1992). Sun crusts, rain crusts, or other weathered surfaces may also be formed during long spells between storms, and these surfaces may also be failure layers once buried (Schweizer et al., 2003; McClung and Schaerer, 2006). Figure 1.2 is a photograph of a layer of surface hoar crystals developing on the snow surface prior to burial. Persistent forms such as surface hoar and facets tend to be anisotropic, behaving much weaker in shear than in slope-normal compression. This anisotropy is an important characteristic in the mechanics of slab avalanches (e.g. McClung, 2003; McClung and Schaerer, 2006; Reiweger and Schweizer, 2010). The third necessary component for a slab avalanche is a trigger that causes the initial unstable frac- ture in the weak layer. Most slab avalanches are triggered naturally by precipitation or wind loading (McClung and Schaerer, 2006). Rapid artificial loading by explosives, snow machines or people can also trigger a slab avalanche. Most avalanches (greater than 80%) in which humans are partially or fully buried or killed were triggered by the victims themselves (McClung and Schaerer, 2006). In the absence of an increase in load, a change in snowpack properties caused by a sudden change in temperature (McClung, 1996; Schweizer et al., 2003) may also render a slab unstable. 7 Figure 1.2: Photograph of exposed surface hoar crystals, a common weak layer responsible for slab avalanches once buried. Grid units are in mm. 1.2 Historical Analysis of Slab Avalanches 1.2.1 Predicting avalanches using a stability index An early and still common method for analyzing the stability of a snow slab and thereby predicting avalan- ches is to calculate a stability index, or ratio of stress to strength (or vice-versa) of the snow. Bucher (1948) proposed as a stability index the ratio of shear stress to shear strength within a weak layer. Bradley (1966) proposed a ratio of compressive strength of a weak basal layer (such as depth hoar) to the normal load of the entire snowpack. Mellor (1968) assumed that slab avalanches initiate when shear stress exceeds shear strength “over a significant area of the snow cover.” Applying these ratios implicitly assumes that strength is the governing factor for slab stability, a classic strength of materials approach. Calculating the stress component of a simple stability index is relatively straightforward; the mean slab density, slope-normal slab thickness, and slope angle are the only requirements. These terms are known or can be estimated with relative confidence in most cases. The more uncertain term is the snow strength, which is typically calculated from the results of in situ testing that directly measures the layer of interest (most commonly in shear). However, due to the spatial variability of snow properties, approximate nature of the strength calculation, and small volume of snow sampled relative to the avalanche scale, these strength values likely contain most of the overall uncertainty in a calculated stability index. Sommerfeld (1969) emphasized a problem with this type of stability factor analysis, namely that a great 8 many large avalanches involve slabs with stability indices (ratios of strength to stress) greater than 1. Perla (1977) reported a mean stability index, calculated after the fact for 80 avalanches, of 1.7± 1. This value suggested that the index would not have, on average, predicted the observed avalanches. The large variability of the calculated indices also cast doubt on the applicability of such a technique for predictive purposes. A number of subsequent refinements to simple stability indices were introduced that attempted to ac- count for size effects, the load produced by a skier, or other contributing factors (e.g. Sommerfeld and King, 1979; Conway and Abrahamson, 1984; Föhn, 1987; Jamieson, 1995). These modified indices still neglected the effects of slab temperature and hardness, which hampers their applicability and accuracy in forecasting avalanches (McClung and Schweizer, 1999). Stability indices do not (and for all practical purposes cannot) take into account the size and distribution of imperfections within the weak layer, which are commonly be- lieved to be the fundamental source of slab avalanche instability (McClung and Schaerer, 2006). A fracture mechanical slope stability index would take the form of a ratio between the shear stress intensity factor and the shear fracture toughness of a weak layer (McClung, 2003), though the shear fracture toughness of the weakest portion of the weak layer, that which governs the loss of stability, can never be measured in the field prior to avalanche release because its size and location are unknown. Though some field studies have found direct evidence for such “shear deficit zones” (e.g. Conway and Abrahamson, 1988), many extensive stud- ies have not (Schweizer, 1999). For these reasons, and for relative simplicity, stability indices as a function of stress and strength have remained in favor in some applications despite their drawbacks. 1.2.2 Fracture sequence in slab avalanches Observations of the geometry and inclination of the fracture surfaces in slab avalanches provided an early benchmark for models of avalanche release. In slab avalanches, the uppermost tensile fracture surface through the slab, or crown (labeled “B” in Figure 1.1), is nearly always oriented perpendicular to the bed surface, plus or minus ten degrees (Perla and LaChapelle, 1970). This indicates that the maximum principal stress in the slab is oriented parallel to the weak layer when it fails, an important observation that informed an early debate over the fracture sequence in slab avalanches. Many investigators assumed that shear failure of the weak layer was the primary failure which governed the instability of a slab (Bucher, 1948; Jaccard, 1966). Others held that the tensile fracture through the slab was the initial failure which led to slab avalanche release (Haefeli, 1963; Roch, 1966; Sommerfeld, 1969; 9 Perla, 1975). Following this initial tensile failure of the slab, it was postulated that the weak layer beneath the slab would be overstressed and subsequently fail, releasing the avalanche. Many of the early tensile failure models for slab release were incompatible with the observed angle of the crown surface with respect to the weak layer. A static tensile failure model for an inclined slab, independent of the action of an underlying weak layer, would have a slope angle dependence in the orientation of the principal tensile stress. The discrepancy between observed and predicted angles of intersection of the crown and bed surface was explained by some as a result of slope geometry. The influence of a weak basal layer undergoing large-scale shear slip was later postulated as an explanation for the rotation of principal stresses, even if the tensile fracture was still thought to be the initial instability (Perla and LaChapelle, 1970). A more consistent explanation of the observed crown fracture angles followed from the introduction of fracture mechanics to explain slab avalanche release. McClung (1979b, 1981, 1987) was the first to apply principles of fracture mechanics to the problem of slab avalanche triggering. Based on experimental evidence of strain softening in snow under simple shear (McClung, 1977), the pioneering slip surface model of Palmer and Rice (1973) was applied by McClung (1979b) to explain the release of a slab avalanche following the growth of a strain-softening shear band in a weak layer beneath a snow slab. This model predicted principal tensile stresses in the slab nearly in alignment with the weak layer. McClung (1981) was the first to introduce the shear stress intensity factor KII and shear fracture tough- ness KIIc as fundamental parameters governing the stability of a snow slab. McClung (1987) outlined further detail on shear fracture propagation conditions and discussed effects such as layered slab stratigraphy, dy- namic effects and weak layer anisotropy that would favor tensile fractures oriented perpendicular to the weak layer following an initial shear fracture. The current consensus opinion is that the perpendicularity of the crown fracture surface to the weak layer is the result of an initial failure between the slab and the substratum which propagates widely before tensile failure through the slab releases the avalanche (e.g. Schweizer et al., 2003; McClung and Schaerer, 2006). 10 1.3 Fracture Mechanics of Snow Slab Release 1.3.1 Shear fracture and initial slab instability The initial shear fracture which governs the loss of slab stability is typically assumed to nucleate from an imperfection or especially weak region within the weak layer (Bažant et al., 2003; Schweizer et al., 2003). The length scale associated with this initial imperfection is expected to be on the order of at least 10 cm (McClung, 2005, 2009b). The strain-softening shear fracture process zone in the weak layer is expected to be large, perhaps on the order of the slab depth (Bažant et al., 2003). The nonlinear effects caused by a large fracture process zone can be approximately accounted for using equivalent elastic fracture mechanics, whereby an infinitely-sharp crack tip which obeys linear elastic fracture mechanics (LEFM) is extended into the fracture process zone of the actual crack (Bažant et al., 2003). A similar procedure was applied for the tensile fractures in the present study. Since the fracture in the weak layer beneath the slab is not within the scope of the present research, it will not be discussed further here. A review of slab release models which focus on the initial shear instability can be found in Schweizer et al. (2003). 1.3.2 Tensile properties of cohesive snow relevant to avalanche release The tensile properties of a cohesive snow slab play two fundamentally important roles in slab avalanches. First, the cohesion of the snow supports the transmission of tensile stresses through the slab and thus enables the initial shear fracture to propagate beneath the slab. This particular role of the snow slab is often taken for granted. For example, no common or standard test or index property exists which distinguishes cohesionless snow from snow with sufficient cohesion to support the propagation of fractures. Experienced observers can often make this distinction (McClung and Schaerer, 2006), one that is typically based on some measure of snow hardness (penetration resistance). However, prior to this study, an objective and quantifiable classi- fication for slab snow as distinct from loose snow had not been addressed since an early study by Fukue (1977). Previous investigators have considered the cohesive strength of snow as synonymous with tensile stren- gth (Bader and Kuriowa, 1962; Mellor, 1968). The tensile strength of cohesive dry snow has been mea- sured using a diversity of field and lab techniques (e.g. de Quervain, 1951; Bader et al., 1951; Roch, 1966; Sommerfeld and Wolfe, 1972; Sommerfeld, 1974; McClung, 1979a; Narita, 1980; Jamieson, 1988). The 11 most common index variable for the tensile strength is the snow density, though reported values of strength vary by nearly two orders of magnitude at a given density. This is the result of different specimen sizes, loading rates, testing techniques, and important variations in snow structure at a given density. In other words, cohesion is a function of much more than density. Provided that a snow layer is sufficiently cohesive to support fracture propagation, the second fundamen- tal role that the slab plays is in governing the release dimensions, and therefore indirectly the destructive potential, of a slab avalanche (McClung and Schweizer, 2006). The tensile fracture toughness (or fracture energy) governs the distance that the underlying shear fracture will propagate before the slab fails (Figure 1.3a). The tensile fracture is assumed to initiate in a boundary layer at the base of the slab without requiring a stress concentration or initial flaw in the slab (McClung and Schweizer, 2006). This boundary layer is characterized by a gradient in stress and strain at the base of the slab, which is caused by the propagating shear fracture beneath the slab and the typical increase in density and hardness of the slab as a function of depth. (a) (b) Figure 1.3: Modes of tensile fracture through a slab. Following shear fracture propagation be- neath the slab, the initial tensile fracture is assumed to propagate from the bottom to the top of the slab after coalescence of a tensile crack in a highly stressed boundary layer at the bottom of the slab (a). Once the initial tensile fracture has reached the surface of the slab, the tensile fracture may propagate laterally across the slope as the shear fracture continues to propagate be- neath the slab (b). The characteristic length scale for the fracture mechanics of slab avalanches is the slope-normal slab thickness D. From thousands of observations, the median half-width of a released slab is around 50D (McClung, 2009a). 12 Once the initial tensile fracture coalesces and propagates to the surface of the slab, the fracture may, in part, proceed laterally across the slope while the weak layer continues to fail in shear or anti-plane shear (Figure 1.3b). This latter mode of tensile fracture, represented in a plane of symmetry in Figure 1.3b, may be conceptualized as a center-cracked panel on a frictional bed. However, the role of side friction in combination with the downslope motion of the slab may combine to produce a curved fracture trajectory (McClung, 2009a). The ratio of fracture energy or fracture toughness in tension to that in shear is important in determining the overall release dimensions of a slab avalanche (McClung and Schweizer, 2006). The mean tensile frac- ture energy of the slab is about 10 times greater than the shear fracture energy of the weak layer (McClung, 2007b). However, the surface area of the perimeter fractures that fail in tension is approximately 30 times smaller than the basal shear fracture area (McClung, 2009a). Therefore, the total fracture energy consumed in tension is roughly comparable to that in shear, based on median measurements of avalanche dimensions (McClung, 2009a). 1.3.3 Measurement and calculation of fracture properties The tensile fracture toughness of cohesive snow was first calculated from the results of notched cantilever beam tests (Kirchner et al., 2000, 2001, 2002a; Schweizer et al., 2004). These studies used the framework of Linear Elastic Fracture Mechanics (LEFM) to calculate the critical stress intensity factor at which the beam samples failed. In using LEFM, these investigators implicitly assumed that any inelastic nonlinear zone ahead of the notch tip had a negligible size compared to all other specimen dimensions in the experiments (Bažant and Planas, 1998). However, for a heterogeneous and porous material such as snow, the specimen size requirements may not have been met for this assumption to be valid. A further assumption for the use of LEFM or any other continuum mechanical theory for a highly porous material such as snow is that the specimen dimensions are sufficiently large compared to the scale of hetero- geneity (in the case of snow, the grain size) to ensure that the specimen is, in bulk, homogeneous. This is a requisite for the approximation of snow as a continuum. For solid ice, the homogeneous limit is on the order of 10–200 times the grain size (Dempsey et al., 1999b; Schulson and Duval, 2009). An analogous relation for snow does not exist, but might take into account, in addition to the grain size, the mean grain spacing or pore space in the snow. 13 The first nonlinear fracture theory applied to tensile fracture data was by Sigrist et al. (2005b); Sigrist (2006) for the analysis of both notched cantilever beam tests and notched three-point bending tests conducted in a cold laboratory. These studies applied the equivalent elastic crack approach (e.g. Bažant and Kazemi, 1990a,b; Bažant and Planas, 1998) to account for the presence of a large and distributed fracture process zone ahead of the notch tip. Using the same theoretical approach, McClung and Schweizer (2006) calculated both the shear and tensile fracture toughness of dry snow slabs using data from other sources. They estimated the length of the effective fracture process zone in tension on the order of 1–10 cm, though the uncertainty was large and possibly important rate effects were not tested for in the data. However, their results strongly support the assertion that nonlinear fracture mechanics is necessary, and that LEFM is inapplicable, for most length scales of interest in slab avalanche applications. A further critical implication is that the fracture parameters determined from laboratory-scale tests will be inapplicable for full-scale analysis of avalanches unless a proper nonlinear size scaling correction is applied. As with most cohesive snow properties, the published fracture toughness data has been primarily indexed against the snow density. The reported values of fracture toughness vary by nearly an order of magnitude at a given density (Kirchner et al., 2000, 2002a,b; Schweizer et al., 2004; Sigrist et al., 2005b; Sigrist, 2006; McClung and Schweizer, 2006), though Schweizer et al. (2004) binned data into “hard” and “soft” snow categories (from hand hardness index values) prior to fitting models of tensile strength as a function of density. The variability between studies is due to a number of factors, from variations in the microstructure and grain size of snow at a given density, to rate, size and geometry effects, to the assumption about which fracture theory (linear or nonlinear) on which to base the analysis. 1.4 Theoretical Framework and Guiding Principles of Thesis Many open questions remain with regard to the tensile properties of snow slabs related to avalanches, and this study addressed many of these. Outlined below is a conceptual classification that was devised for the purpose of framing much of the analysis and discussion in the present study. This classification has two primary divisions: the spatial scale of interest and the amount of internal cohesion of the snow. Both should be addressed with the porous and heterogeneous nature of snow in mind, as shown schematically in Figure 1.4. 14 Figure 1.4: Schematic illustrating the porous and heterogeneous nature of dry seasonal snow. The layering of the snow structure and the densification with depth is represented. Characterization of snow using continuum mechanics is only valid for volume elements much greater than the grain size. 1.4.1 Cohesion threshold for fracture propagation As discussed above, slab snow and loose snow are delimited by a cohesion threshold. Above this threshold, the snow is able to support the transmission of tensile stresses and therefore fracture propagation. The ability to quantify this cohesive threshold in-situ would have practical benefits since forecasting the onset of slab avalanche activity in newly fallen snow can be difficult. Similarly, the loss of cohesion in snow undergoing destructive metamorphism or warming to the melting temperature can cause loose avalanches which can also be difficult to predict. The cohesion threshold also defines the domain for which the response of snow is adequately character- ized by material properties, for loose snow, versus structural properties for slab snow. Material properties may be taken as the bulk snow density, grain size and grain shape as well as inherited properties of the parent material (ice). Structural properties in some way measure or account for the manner in which snow is bonded into a coherent slab structure which gives it specific bulk behaviour. Examples of structural prop- erties include tensile strength, Young’s modulus and fracture toughness. These parameters depend on the 15 number and area of bonds per grain or unit volume of the snow and therefore only make sense to define for cohesive snow. The most well-defined index properties which are sensitive to snow structure are the various measures of penetration resistance or snow hardness (e.g. Shapiro et al., 1997). The most common hardness test is a subjective hand penetration test, but the results from this index test vary across observers and are difficult to use quantitatively. Fukue (1977) studied the penetration of a thin blade into snow and related penetration resistance to cohesive strength, though this promising study appears to have been largely overlooked in the last 30 years. Currently, no hardness test or structural index measurement exists which is objective, quantifiable, and widely adopted. Consequently, most studies reporting strength or fracture properties have not reported any hardness measurements or correlations with properties other than density. Density is thus the most commonly used variable to index cohesive snow properties, even though it is commonly agreed and often stated that density is an inadequate measure of cohesion or snow structure (Bader and Kuriowa, 1962; Ballard and Feldt, 1966; Ballard and McGaw, 1966; Mellor and Smith, 1966; Shapiro et al., 1997). Density is, however, a reasonable first approximation for snow structure in the ab- sence of an alternative index property which is more sensitive to structure. Density also has the advantage of being easy to measure, relatively objective, and easily comparable across data sets. However, snow properties such as strength often display large scatter when expressed as a function of the snow density (or porosity), and most of this scatter can be attributed to variations in snow structure at a given density (e.g. Mellor and Smith, 1966; Shapiro et al., 1997; Schweizer et al., 2003). For example, the uniaxial ten- sile strength of snow slabs with rounded grains is about twice that of slabs with angular or faceted grains at the same density (Jamieson, 1988; Jamieson and Johnston, 1990), and the same is true of other cohe- sive properties. In most studies, however, the lack of a superior, repeatable and objective index measure for snow structure has left most investigators with no better alternative variable than density for correlating with strength and fracture properties. This need was addressed in the present study by the development of a new penetration resistance gauge, the results of which were consistently correlated with measured tensile properties and compared to density for predictive merit. 16 1.4.2 Length scales in the analysis of slab avalanches In the majority of slab avalanches, the slope-normal slab thickness is in the range of 0.1–1 m, with a mean of about 0.7 m (Perla, 1977). The slab thickness is the characteristic length scale for analysis of slab avalanches using fracture mechanics. Other important length scales include the continuum limit and the size of the fracture process zone during crack initiation and propagation in snow. These length scales are more uncertain than the slab thickness, which can be directly measured, but are important for scaling analyses of slab strength and fracture properties with changing slab thickness and for relating lab-scale measurements to the slope scale. Continuum limit The continuum limit defines the length scale above which a mass of snow may be considered, in bulk, to be homogeneous. This distinction allows the use of continuum mechanics for the analysis of the bulk response of a snow slab or specimen (provided that the snow is also above the cohesive threshold). For a material such as snow, which has a highly porous and heterogeneous microstructure, continuum equations do not apply for arbitrarily small volume elements (see e.g. Figure 1.4). For most applications related to avalanches, however, the volume element of interest is much greater than the grain size and likely sufficient for the application of continuum mechanics (Salm, 1971). Below the continuum limit, discrete models are necessary to describe the mechanical response of a heterogeneous material (Bažant and Jirásek, 2002). The continuum limit is an important characteristic length scale in the analysis of slab avalanches, one that can be defined in a number of ways. It can be considered as the size of the Representative Volume Element (RVE), the minimum volume for which continuum relations are applicable for the material (e.g. Bažant and Pang, 2006). As with polycrystalline ice, the continuum or homogeneity limit may be expressed as a multiple of the grain size. For freshwater and sea ice, this limit has been expressed as a requirement that the initial crack length as well as the unbroken ligament in a fracture test be greater than about 10–100 times the grain size (Dempsey et al., 1999a,b; Mulmule and Dempsey, 2000; Schulson and Duval, 2009). The homogeneity requirement for snow may be of a similar order of magnitude in terms of a grain scale multiple. 17 Fracture process zone Provided that the continuum limit is satisfied, another critical length scale in the fracture of snow is the length of the fracture process zone (FPZ). The FPZ is a zone of softening damage ahead of an existing or coalescing crack in a heterogeneous material (e.g. Cotterell and Mai, 1996; Bažant and Planas, 1998). For a material such as snow, this inelastic zone is likely characterized by distributed bond breakage ahead of a traction-free crack or notch. The open structure of the ice matrix in porous cohesive snow will necessarily force an advancing fracture to have a distributed or diffuse nature, making an unambiguous definition of a “crack” in snow difficult. The FPZ may also be defined as the minimum length scale, dependent on the material microstruc- ture, over which strain can localize (Bažant and Pijaudier-Cabot, 1988). In this sense, the FPZ may be physically related to the size of the RVE at the continuum limit. Initial estimates of the effective process zone length in both shear and tension are on the order of 50-100 times the grain size (Bažant et al., 2003; McClung and Schweizer, 2006; Sigrist, 2006). The length scale of the FPZ is fundamental for any scaling relation for the fracture mechanics of slab avalanches and determines, for example, the structural scale (if any) at which Linear Elastic Fracture Me- chanics (LEFM) is applicable for analysis of slab avalanches. In order for LEFM to be applicable, the char- acteristic length scale(s) in the fracture problem need to be at least an order of magnitude greater than the effective process zone size or more, depending on geometry (Bažant and Planas, 1998). Therefore, careful consideration of several length scales is necessary in the analysis of slab avalanches. No fracture mechanical study to date has addressed explicitly the continuum limit, though the application of continuum mechanics implicitly assumes that this limit is satisfied. If the RVE is of similar size as the FPZ, and if these length scales are on the order of about 100 times the grain size, then specimen sizes on the order of 5-10 cm are necessary for homogeneity and the applicability of continuum mechanics. Specimen sizes on the order of 1 m would be necessary for the direct applicability of LEFM, though sampling and testing natural snow specimens of this size would be impractical, if not impossible. However, the grain size multiple that defines the continuum limit and process zone size are highly uncertain given the sparsity of data currently and the large scatter in the data on which these estimates are based. The cohesion threshold for fracture propagation and the length scales discussed above are represented 18 schematically in a Cartesian classification scheme in Figure 1.5. The cohesion is represented on the ordinate (y-axis) and the spatial scale on the abscissa (x-axis). The domain of applicability of fracture mechanics for analyzing avalanches is represented as the region above both the continuum limit length scale and the cohesive threshold. The LEFM limit length scale defines the regions for which quasi-brittle (nonlinear) frac- ture mechanics versus LEFM apply. Below the cohesive threshold, loose snow is adequately characterized by material properties. Above the cohesive threshold, slab snow takes on structural properties that may not be adequately characterized by material properties such as density. The present study constrained the quantitative definition of the cohesive threshold, the LEFM limit, and the effective process zone length. 19 Increasing cohesion Increasing spatial scale Loose snow Slab snow Grain scale Macroscale Quasi-brittle Fracture Mechanics Soft slab Hard slab Brittle Fracture Mechanics Fracture propagation possible Fracture propagation not possible Domain of applicability of fracture mechanics Heterogeneity is influential Homogeneous approximation justifiable Cohesive threshold Continuum Limit: Some multiple of the grain size LEFM Limit: ? Figure 1.5: Scale-Cohesion classification of snow for the purpose of analyzing avalanches. The cohesive threshold is defined as the limit at which snow has sufficient internal cohesion (tensile strength) to support the propagation of fractures. Continuum mechanics only applies to snow at length scales above the continuum limit, a length scale which has not been rigorously defined but may be on the order of 10–100 times the grain size. The size of the fracture process zone in slab avalanche fractures may also be on the same order of magnitude. The domain of applicability of fracture mechanics is defined by both the cohesive threshold and the continuum limit length scale. Linear elastic fracture mechanics (LEFM) is only valid above a length scale for which the size of the fracture process zone is negligible. This limit is ill-constrained for snow. 20 1.4.3 Nonlinear quasi-brittle fracture mechanics Provided that the continuum limit is satisfied, and that a large fracture process zone is present (a safe as- sumption), a nonlinear fracture theory is necessary for analysis of fractures in cohesive snow. One approach to accounting for nonlinearity is by defining an elastically equivalent crack system that gives the same global response, such as energy dissipation or the stress-displacement curve, as in the actual specimen (e.g. Cotterell and Mai, 1996; Bažant and Planas, 1998). This type of approach smears out all of the toughening mechanisms in the process zone using a single parameter that represents the difference in length between the actual and equivalent cracks (Bažant and Gettu, 1992). The framework of LEFM is then applied to the equivalent specimen, which is an advantage of this approach since the theoretical foundation of LEFM is well developed. Furthermore, the quasi-brittle fracture mechanical scaling laws of Bažant and Planas (1998); Bažant (2005) contain LEFM as an asymptotic limit for vanishing process zone size or increasing specimen size, so the safest and most general assumption for the analysis of fractures in heterogeneous materials is that a large fracture process zone is present. The equivalent elastic crack technique is not designed to explore the micromechanical details of the process zone itself (Mindess, 1991). If this was the objective, a variety of direct and indirect techniques are available: acoustic emissions, scanning electron microscopy, stereo imaging, and interferometry, to name just a few (e.g. Mindess, 1991; Cotterell and Mai, 1996). Alternatively, if the full softening-displacement curve is known or measured for the actual specimen, the elastically-equivalent process zone length can be related to the length of the actual process zone (e.g. Planas and Elices, 1989; Bažant and Kazemi, 1990b). For situations in which the nonlinearity is too great to use the equivalent elastic crack approach, a variety of alternative nonlinear techniques are available. Examples include the crack band model, the cohesive crack model, the J-integral, multiple-parameter models, or numerical techniques (e.g. Cotterell and Mai, 1996; Bažant and Planas, 1998; Bažant, 2005). 1.4.4 Rate effects in the fracture of snow The proximity of snow to its melting temperature and the resulting thermodynamic instability gives rise to important rate effects for most rates of loading (e.g. Bader and Kuriowa, 1962; Mellor and Smith, 1966; Narita, 1980; Schweizer, 1998). In solid ice, creep effects are also important in all but the fastest fracture 21 tests (e.g. Dempsey et al., 1999b; Schulson and Duval, 2009). Rate-dependent results in experimental data for snow and ice should be expected as a general rule. Accordingly, systematic and thorough testing for rate effects should be part of nearly any experimental study of the mechanical properties of snow. For laboratory tests on snow, Bader and Kuriowa (1962) suggested that loading times to failure of less than 10 seconds are necessary to avoid inelastic effects. Rate effects are present in snow at least up to strain rates of 10−3 s−1 in uniaxial tension (Narita, 1983) and 10−2 s−1 in unconfined compression (Mellor and Smith, 1966). In both studies, the beginning of what appeared to be an asymptotic limit strength for higher strain rates was observed, suggesting an approach to fully elastic response. A ductile-to-brittle transition has been observed for snow at a critical strain rate of 2.5× 10−3 s−1 in unconfined compression (Mellor and Smith, 1966), 10−4 s−1 in uniaxial tension (Narita, 1980, 1983), and about 10−3 s−1 in shear (Schweizer, 1998). A similar transition in solid ice has been observed as a function of strain rate (Schulson and Duval, 2009). This transition can be explained as a balance between competing effects of creep and fracture in the material; below the transition, creep leads to crack blunting, while above the transition crack propagation dominates (Mellor and Smith, 1966; Schulson and Duval, 2009). As such, this transition might be more appropriately labeled a “creep-to-fracture” transition. This terminology was adopted for the present study to avoid confusion between the terms “brittle” and “quasi-brittle” as applied to the linear and nonlinear fracture theories, respectively, discussed above. The creep-to-fracture transition marks the strain rate at which maximum strength values have been measured as a function of strain rate for a given type of snow (Mellor and Smith, 1966; Narita, 1980, 1983; Schweizer, 1998). The critical transition shifts toward higher strain rates as the snow temperature approaches the melting point (Narita, 1983), a similar trend as in solid ice (Schulson and Duval, 2009). These transition strain rates do not define the limits of applicability of elasticity theory, rather the point at which viscous and elastic effects are critically balanced. Strain rates much higher than the critical transition are likely necessary before snow responds fully elastically. However, no consistent and systematic approach for defining a fully elastic strain rate for different types of snow under various loading scenarios has been developed. That said, an equivalent elastic fracture analysis as an approximation to a fully viscoelastic solution is acceptable as long as the creep strain at failure is not too large (Bažant and Gettu, 1992). In this type of elastic-viscoelastic correspondence, the elastic modulus is replaced by an appropriate effective 22 modulus, such as the creep compliance for the time to failure or the secant modulus at peak load (e.g. Schapery, 1997; Dempsey and Palmer, 1999). The effective size of the fracture process zone in snow is expected to have a rate dependence, as in concrete (Bažant and Gettu, 1992). Even above the creep-to-fracture transition, creep effects within the fracture process zone and bulk relaxation away from the FPZ are likely to diminish the effective size of the FPZ (see e.g. Figure 7.2, Bažant, 2005). Thus, separate creep effects may need to be considered in the bulk of the material and within the strain-softening fracture process zone (Cotterell and Mai, 1996; Bažant and Li, 1997). Though rate dependence in tensile strength measurements has been demonstrated (e.g. Narita, 1980, 1983), no studies to date have investigated the sensitivity of fracture parameters such as the fracture toughness or process zone length to loading rate. When speaking of rate effects, distinction should also be made between the rates of loading (or unload- ing) relevant for an avalanche and analogous rates in lab-scale or in situ tests. For the avalanche case, the dynamic unloading of the slab as the weak layer fractures should occur at a high enough rate that the slab behaviour can be considered mostly elastic (Bažant et al., 2003). Viscous effects may still play a role in some avalanche cases, especially for post-control releases or for cases where progressive strain softening in the weak layer rather than rapid fracture is responsible for avalanche triggering (McClung, 1981). Further- more, humans, snow machines and explosives apply loads to the snow surface at different rates, for which the response of the slab is expected to differ from the perspective of avalanche triggering. 1.4.5 Note on temperature effects Given the proximity of snow to the melting temperature, one would expect the material and structural prop- erties of snow to display strong temperature dependence. In some cases, this is true. The creep and creep rate of snow are more sensitive to temperature than any other properties (e.g. Bader and Kuriowa, 1962). Creep parameters for ice are also highly sensitive to temperature (Schulson and Duval, 2009). However, elastic properties of snow and ice are only weakly dependent on temperature. The strength of snow weakly increases with decreasing temperature (Roch, 1966; Narita, 1983). The stiffness, or the initial tangent modulus, may be more sensitive to temperature than strength (McClung, 1996). For solid ice, the elastic stiffness increases by only 5% as the temperature decreases from 0◦C to−50◦C (Schulson and Duval, 2009). The fracture toughness of solid ice is only weakly dependent, if at all, on temperature over the range 23 −2◦C to −50◦C, and the tensile strength follows a similar trend over a temperature range from 0◦C to −30◦C (Schulson and Duval, 2009). Therefore, strong temperature effects in experimental data are indirect evidence of the presence of creep effects. The present study aimed to measure properties that were mostly elastic in accordance with the expectation of slab behavior in avalanches. In order to achieve this, the loading rates in experiments were chosen to produce nominal strain rates well above the creep-to-fracture (ductile-to-brittle or viscous-to- elastic) transition. Thus, temperature effects were expected to be second-order at most. 1.5 Summary To summarize and emphasize many of the points raised above, several key principles and hypotheses which guided the present study are reviewed here. 1. The tensile fracture properties of cohesive snow have yet to be systematically correlated with any index properties which represent the structure of the snow. The tensile strength has been shown in a few studies to correlate well with penetration resistance, thus parameters such as fracture toughness are likely to be well explained by some quantifiable measure of penetration resistance or other index property which is sensitive to snow structure. Such an index is also likely to help explain some of the observed scatter in properties when expressed as a function of density. 2. If a continuum property such as strength or toughness is to be correlated with a hardness measure or other index measure for snow structure, the measure should characterize the snow over a continuum length scale, that is a length scale for which a continuum approximation of snow is justified. 3. Important length scales in the fracture of snow, specifically the length of the fracture process zone or length defining the limit of applicability of linear elastic fracture mechanics, remain to be constrained. These length scales are likely to be influenced by rate effects. 4. Systematic testing for rate effects needs to be carried out for any fracture experiments on snow. A significant difference in properties calculated from tests at different rates should be expected in most cases. Rate effects and temperature effects may be difficult to separate and easy to confuse. 24 5. The most appropriate starting assumption for an analysis of snow fracture, which follows from phys- ical reasoning related to the structure of snow, is that a large fracture process zone engenders non- linearity in fracture. This nonlinearity may be accounted for approximately using equivalent elastic fracture mechanics, which in some cases requires only the measurement of peak loads in fracture ex- periments and allows the general framework of LEFM to be used, but leads to scaling laws which are nonlinear. 6. A size effect will be important for relating the results of lab-scale strength or fracture experiments to the avalanche scale. Careful measurement of the size-dependence of test results, over as wide a size range as possible, should to be conducted for fitting results to fracture mechanical scaling relations and for calculating relevant properties for full-scale analysis of avalanches. 1.5.1 Chapter outline Chapter 2 contains a description of the experimental methods used for the present study. Some methods were adopted from previous snow studies or analogous studies of other heterogeneous materials. Most of the techniques for characterizing the in situ properties and stratigraphy of snow came from industry-adopted standards. Beam bending tests were conducted in a cold lab to measure the tensile (flexural) properties of cohesive snow. The nature of snow compared to other engineering materials necessitated the development of several new tools and techniques for the laboratory testing. Many of the newly-developed experimental methods represent a significant original contribution of the present study. In Chapter 3, a new thin-blade penetration resistance gauge is introduced that was developed for charac- terizing snow structure over a length scale (on the order of 10–100 grain contacts) relevant for a continuum description of the fracture of slab avalanches. The blade hardness index, defined as the maximum resistance to penetration, was a highly repeatable measure across observers compared to the common and subjective hand hardness test. This new tool is small, inexpensive, easy to use and is being adopted by several practi- tioners in the avalanche industry. The blade hardness index was a better variable than density, or any other variable, for correlating with the tensile strength and fracture toughness data from the present study. Chapter 4 contains a review of the extensive literature on the tensile strength of snow, a re-analysis of much of this data and a contribution of hundreds of new measurements. The literature review synthesized 25 around 2000 measurements from 20 sources, mostly expressed as a function of density. Much of the data was re-analyzed to account for neglected geometric stress concentrations in the experiments. The data from the present study included 245 unnotched beam bending experiments in the cold lab which were used to calculate the tensile (or flexural) strength. The new data was systematically correlated with and analyzed for dependence on density, hardness, grain size, loading rate, and specimen size. General agreement was found between the existing and new strength data when expressed as a function of snow density, though with large scatter. The results of the current study challenge the existing norm of indexing snow properties only against the density, in light of the wide scatter in strength values at a given density and the better correlation of strength with penetration resistance. This chapter now represents the largest collection of data on the tensile strength of dry alpine snow in the literature. Chapter 5 contains an analysis of 23 different test series, covering nearly 300 experiments, that measured the nominal strength of beam samples in three and four point bending over different specimen sizes and relative notch depths. The data were analyzed using Bažant’s equivalent elastic crack (quasi-brittle) theories for the size effect. Fitting of these theories through the experimental data led to a collection of fracture parameters such as the fracture toughness, effective process zone length, transitional size bridging plasticity and LEFM in notched tests, and boundary layer thickness over which cracks initiate in unnotched tests. These properties were related to density, penetration resistance and loading rate. As with the tensile strength data, the fracture toughness was better correlated with thin-blade penetration resistance than density. The fracture toughness and effective fracture process zone length both showed rate dependence. Best estimates for the length of the fracture process zone in snow slab tensile fractures, at rates sufficiently high to minimize creep strains, are given as a function of the grain size. A numerical modeling approach based on continuum damage mechanics and some results of simulations of the laboratory experiments are given in Chapter 6. The nonlocal isotropic damage model was applied for the first time to simulate the initiation and propagation of tensile fractures in snow. The length scale over which nonlocal averaging was conducted in the simulations was related to the critical equivalent crack extension from the experimental data. A sensitivity analysis was conducted to explore the dependence of the numerical results on the most uncertain model parameters. Subsequently, a prescriptive algorithm was developed for determining numerical parameters based on the results of experimental bending tests. This 26 algorithm was tested against the results of 10 different experimental series, and generally good agreement was found between the simulations and experiments. The model was able to reproduce key features of the experimental load-displacement curves, such as the mean peak load, rounding of the curve near peak load, and strain softening following peak load, that are consistent with a quasi-brittle fracture mechanical interpretation of the failure process. Since no additional tuning of numerical parameters was conducted to improve the fits, these results are promising for future predictive applications of the numerical model to simulate full-scale avalanche fractures for which no experimental data is (or ever will be) available. Chapter 7 contains overall conclusions of the present study and a discussion of the results with respect to the hypotheses, guiding principles and general themes laid out here in the introduction. A number of recommendations for future research are discussed, building on and refining the methods and results of the present study. The implications of the present study and their link with the larger field of snow and avalanche mechanics are given. Three appendices are include for reference. Appendix A contains images and analysis of the fracture morphology from the bending experiments, demonstrating that the experiments indeed failed by the prop- agation of single tensile fractures. Appendix B contains background related to the goodness of fit of the nonlinear regression models used to fit much of the experimental data. Appendix C contains equations for the order-of-magnitude calculations of the creep strains in the experiments given the observed times to failure. 27 Chapter 2 Methods The experiments in the present study were carried out at Rogers Pass in the Selkirk range of the Columbia mountains of British Columbia, Canada. The TransCanada highway travels through Rogers Pass, which lies within Glacier National Park of Canada. The pass sits at an elevation of 1320 m.a.s.l., and the cold lab and primary study plot are located in a compound (Figure 2.1) of several buildings operated by Parks Canada as part of the service of road maintenance, avalanche forecasting and control, and parks operations. The natural setting of the Rogers Pass area is a great location for snow and avalanche research. It was possible to sample natural snow of a variety of types from multiple elevations. Natural snow samples were used entirely for the present research. Avalanche activity is frequent along the highway through Rogers Pass in the winter, which allows researchers to gain a close (though hopefully not too close) appreciation for the phenomena at hand. Living quarters were provided by Parks Canada for me and field assistant(s) for each of the three winters of research conducted for this study. This was a tremendous benefit, as the cold lab and primary study plot were both within walking distance from the apartment building. It would not have been possible to carry out such a large number of experiments without this kind of institutional support for avalanche research. This chapter contains descriptions of the principal experimental methods of the present study. Only one previous experimental study of a similar kind had been conducted on which to build (Sigrist, 2006), but much of the detail related to experimental methods was omitted. Some similar methods were adopted, however, as will become apparent. The chapter sections are organized around the primary locations where each 28 Figure 2.1: Overview of Rogers Pass and the Parks Canada compound where the cold lab is lo- cated. technique was utilized, starting with in situ characterization of snow properties, followed by the methods of snow sample extraction, transport and storage in the cold lab. Finally, detailed summaries are given of cold lab testing equipment, methods, and a few results related to the unique experimental design considerations for a material such as snow. 2.1 In Situ Snow Stratigraphy Characterization At the start of each field day, a standard snow profile observation was conducted, following avalanche in- dustry guidelines (Canadian Avalanche Association (CAA), 2007). This profile included the observation and description of the stratigraphic layering of the snowpack and the demarcation of the snow into a discrete set of layers, each with approximately homogeneous properties. For each identified layer, measurements or descriptions of the hand hardness index, size and form of the snow crystals (grains), density, temperature, and the wetness of the snow (only dry snow was considered in the present study). Additionally, meteorolog- ical and site characteristics were recorded such as the slope angle, aspect, elevation, sky cover, wind speed and direction, air temperature, type and rate of precipitation, and depth of foot penetration into undisturbed 29 snow. Detailed descriptions of the observations and measurements most important to this study are given below. 2.1.1 Study plots Snow was extracted from two primary sources in the present study. The most common source of snow was from a sheltered study plot within the Rogers Pass compound, a short walk from the cold lab at an elevation of 1320 m (Figure 2.2a). This plot was on mostly flat ground, surrounded on three sides by trees and on the fourth by two apartment buildings. The plot was therefore mostly sheltered from wind effects. The area of the study plot was approximately 10 m by 50 m. All snow sampled from this study plot was taken a distance of, at minimum, 1.5–2 m away from any previous snow pit location. This was to ensure that previously exposed snow surfaces did not alter the natural state of the sampled snow, at least no more than would be detectable above other environmental drivers of snow metamorphism. (a) (b) Figure 2.2: Primary study plot at Rogers Pass, showing the remnant holes of previous snow pits (a); secondary study plot at Mt. Fidelity research station, accessed by snow machines (b). Occasionally, when snowmobile or snowcat transportation was available, snow was sampled from a study plot near treeline at an elevation of 1900 m on Mt. Fidelity (Figure 2.2b). This mountain is a short drive west from Rogers Pass and is the location of a permanent research station used by the Avalanche Control Section of Parks Canada for monitoring snow and weather conditions. Snow was typically sampled from flat terrain near the research station. Snow samples were packed into insulated boxes for transportation back to the cold lab. The samples were first transported on a snow cat back down to the highway and were 30 then driven to Rogers Pass and carried into the cold lab. 2.1.2 Snow pit preparation Once a study site was chosen for a particular day, a snow pit was dug in an area free of disturbances with as uniform a shape as possible. The dimensions of the pit were typically around 2 m by 4 m, with a depth that depended on the particular layer of interest that day. The pit was typically dug to a depth of at least 50 cm below the layer of interest to facilitate more ergonomic sample extraction. Figure 2.3 shows a sharp corner being prepared in a new snow pit. One such sharp corner, the principal location for making stratigraphic observations, was prepared in every pit. The walls of the snowpit were shaved such that the 90◦ corner was fully shaded to prevent solar radiation from changing the exposed snow crystals before or during the observations. The axis of the corner was oriented vertically regardless of slope angle. All standard stratigraphic observations prior to sample extraction were made from these corners. Figure 2.3: Preparing the observation corner of a snow pit. Photograph by Elisabeth Hicks. 2.1.3 Identification of layering and distinct stratigraphic boundaries Once the snowpit was finished, the major stratigraphic boundaries were identified. The primary indicator of a major stratigraphic boundary was a distinct change in the snow hardness, as measured using the force required to penetrate the snow by hand or using a thin plate or card. Many interfaces could also be identified visually from a change in the size or type of snow crystals. Often, one or more layers within the snowpack could be identified by the presence of dirt or dust which was deposited on the snow surface earlier in the 31 season and then buried by subsequent snowfall. These “dirty” layers could be dated and used to more easily locate and reference other layers within the snowpack. A common paint brush was typically used to brush out layering detail on an exposed pit wall. This technique often helped to locate thin and weak layers within the snowpack. Figure 2.4: Photograph showing observation corner of a snowpit, with the author and field as- sistant discussing a weak layer. Markers next to the meter stick in the corner indicate major stratigraphic boundaries within the snowpack. Photograph by Elisabeth Hicks. Layer boundaries were identified using markers placed next to a meter stick in the observation corner, as in Figure 2.4. Note also the presence of two digital thermometers in the observation corner. The shovel on the snow surface was used to shade the measurement of the snow surface temperature and the temperature within the first 30 cm below the surface. The stratigraphic boundaries were recorded using the distance of the boundary from either the snow surface or the ground. 2.1.4 Snow crystal identification and classification Once the stratigraphic boundaries were located and recorded, the type and size of snow crystals (the terms snow crystals, crystals and grains are used synonymously throughout this text) were determined and recor- ded. Crystals were sampled from the exposed snowpit wall by lightly scraping the wall with a metal or plastic card. The crystals were then gently disaggregated by tapping the card, allowing individual crystals or small 32 clusters of crystals to spread out on the card. The surface of the card had several painted grids, varying in size from typically 0.5 mm to 2 mm or more in spacing. This grid aided in the determination of the grain size, defined as the average maximum linear extension of the grains (Canadian Avalanche Association (CAA), 2007). Note that this procedure and definition of grain size is highly subjective. Crystals were viewed using a hand lens with a magnification of typically 8-12 times (Figure 2.5). Care was taken to ensure that the snow crystals were not exposed to sunlight or heat from hands or gloves. Often, several samples of crystals were taken from within a layer to add confidence to the observations. (a) (b) Figure 2.5: Photographs showing the procedure for observing and classifying snow crystals. Pho- tographs by Steve Conger (a) and Elisabeth Hicks (b). The crystals within each layer were classified and recorded using the International Classification for Seasonal Snow on the Ground (Colbeck et al., 1990; Fierz et al., 2009). Note that during the course of this investigation the old standards (Colbeck et al., 1990) were updated, and several changes to the grain form classification were made in the updated standards (Fierz et al., 2009). For reporting the grain forms in this text, all grain form classes have been converted, where possible, to the new (2009) standard. 2.1.5 Density and hardness The density and hardness of the snow were measured in a wall of the snow pit adjacent to the observation corner, parallel to the snow layering. The density was measured by extracting a sample of snow using a stainless steel rectangular cutter and then weighing the snow sample. The most common density sampler had a volume of 100 cm3. Since the density sampler was 3 cm thick, the density was typically recorded 33 every 3 cm of depth from the surface of the snow to the bottom of the snowpit (often the ground). This produced a stepwise continuous profile of density as a function of depth. The hand hardness index of each layer was recorded by pushing a gloved hand into the snow. The objective of the hand test is to record the object (gloved hand in various cross-sectional shapes, pencil or knife) that can be pushed into the snow using the given 10–15 N force. For example, if the snow was too hard to insert a gloved finger using no more than 10–15 N of force, then the blunt end of a pencil or a knife was used to penetrate the snow. The hardness was recorded as the object which most closely required the given (10–15 N) force to penetrate. Variations in hardness were recorded using “+” and “-” qualifiers, representing approximately one-third level deviations above and below the given index value, respectively. A new thin-blade penetration resistance gauge was developed for measuring an alternative and more objective value of hardness for each snow layer (Borstad and McClung, 2011). A 10 cm wide blade with a 0.6 mm blunt leading edge was attached to a digital push-pull gauge. The blade was inserted into the snow at a penetration rate slightly faster than the hand hardness test, and the maximum force of penetration was recorded as the blade hardness index. Chapter 3 is devoted to the design, use and analysis of data obtained using this new gauge. Figure 2.6 shows an overview of a snowpit after the stratigraphic profiling and sample extraction were completed. The meter stick is still present in the observation corner. The small round holes to the left of the meter stick are finger holes left by the hand hardness test. Further to the left, the regular array of holes are remnants of the density sampling. Next to each density sample, the blade hardness index was measured. Repeated measurements of the blade hardness index were also conducted within the layer from which samples were extracted. 34 Figure 2.6: Photograph of a snowpit following stratigraphic profiling and snow sample extraction. 2.2 Snow Sample Extraction, Transport and Storage All snow samples in the present study were extracted from layers of natural snow which were approximately homogeneous in snow properties with depth. Following the standard stratigraphic profile, a specific layer was selected from which to extract samples for lab testing. This selection was based on a number of criteria. First, the layer needed to be reasonably homogeneous in properties such as density and hardness with depth. The standard stratigraphic profiling technique, which separated the snowpack as best as possible into indi- vidual layers with approximately homogeneous properties, ensured that this criterion was met. However, even snow layers which were considered homogeneous in the field typically had a layered structure, even if properties such as density and hardness did not change appreciably (Figure 2.7). Second, the snow layer needed to be at least 10 cm thick to allow the insertion of the beam-shaped 35 10 cm Figure 2.7: Thin-shaved snow sample, lit from behind, showing the layered structure of the snow. This sample was taken from a layer judged to be homogeneous in the field. Common hand tests for determining stratigraphic layer boundaries often miss subtle changes in layering such as this. For this reason, fracture specimens were oriented such that tensile fracture propagated parallel to the layering (into or out of the page) rather than across the layering (up or down through the layering). sample cutters, all of which had a width of 10 cm. The width of the beam was always oriented normal to the slope, which allowed beams of different depth and span to be extracted from any layer at least 10 cm thick. Layers of 15 cm or more thickness were preferred in order to guarantee that the sample cutters did not deviate from the layer during insertion. If multiple layers were available that met the criteria of homogeneity and thickness, then selection was based on the particular type of test series that was being conducted on that day. As much as possible, a wide variety in types of natural snow were sought for study. If a particular test series had been conducted previously using high-density or high-hardness snow, then lower density and hardness snow would be given preference if available. 2.2.1 Sample extraction and transportation The samples were excised using stainless steel rectangular boxes, open on both ends. Table 2.1 shows the dimensions of the five different sample cutters used in this study. Note that the free ends of the beam-shaped samples often needed to be trimmed off to ensure consistency across all samples within a data set, so the values of L represent the maximum possible length of the samples. In the lab, the actual length of the prepared sample prior to testing was recorded in place of the value of L in Table 2.1. 36 The width W of the cutters was always oriented slope-normal during extraction. This forces the tensile fracture in the laboratory bending test to be oriented parallel to the stratigraphic layering of the snow, as in Figure 2.8 (also Figure 1.3b). This orientation provides an average measure of the fracture properties of the layer, as the tensile crack initiates across all layers simultaneously and propagates parallel to the layer- ing. The chosen orientation was judged to allow the most consistent sampling technique for repeatability. The alternative would be to orient the samples and resulting tensile fracture perpendicular to the layering. Although this orientation would correspond to the initial tensile fracture in a slab avalanche (Figure 1.3a), homogeneous layers that are 20 cm thick or more, which would be necessary to test for size effects, are rare in an alpine snowpack. Even 10 cm-thick layers can be difficult to find at times over the course of a winter. W D L Figure 2.8: Schematic showing orientation of notched beams of different sizes taken from the same layer (not to scale). The width of the beams was oriented normal to the slope so that the fracture propagated parallel to the layering and across all layers simultaneously. D (cm) L (cm) W (cm) 2.5 12.5 10 5 25 10 10 50 10 15 75 10 20 100 10 Table 2.1: Dimensions of beam-shaped snow sample cutters. Prior to sample extraction, either the top or bottom of the snow layer of interest was marked along the snowpit wall adjacent to the observation corner (Figure 2.9). All but approximately 30 cm of snow above the layer of interest was removed in order to facilitate cutting the back of the samples once the cutter was inserted. The saw used to cut the back of the samples is just sticking out of the snow in the right of Figure 2.6. Figure 2.9a shows a layer from which all samples extracted had the same dimensions (10 cm by 50 cm 37 by 10 cm), whereas figure 2.9b shows a layer from which samples of a variety of sizes were extracted. (a) (b) Figure 2.9: Photographs of snow pits showing holes left by snow sample extraction from the study plot at Rogers Pass (a) and at Mt. Fidelity (b). Note the coloured markers placed along the boundary of the extraction layer for reference. The cutter for the most common sample size used in the present study (with D = 10 cm) is shown in the lower right of Figure 2.6. Next to the cutter is a plunger used to gently push the sample out of the cutter. Samples were pushed onto a large, thick styrofoam sheet on the snow surface. Once the sheet was full of samples, it was either carried directly to the cold lab (if using the study plot near the lab) or, if using the Mt. Fidelity study plot, packed into a large insulated box for transport back to the lab (this box is just visible to the right of the person in Figure 2.9b). 2.2.2 Sample storage prior to testing Once in the cold lab, the samples were stored in the open (Figure 2.10) for as short a period of time as possible prior to testing. This was to ensure that the snow changed as little as possible, compared to the state it was in at the time of extraction, due to the different thermodynamic environment of the lab. Though the temperature of the lab was usually set to mimic the temperature of the snow layer from which the samples originated, the humidity of the lab was lower than that of the pore air in the snowpack, which is typically at 100% (McClung and Schaerer, 2006). Small changes in the snow crystals were observed within just a couple hours after arrival in the lab. As a result, testing of the samples commenced within typically 2 hours after all samples were in the lab. Most test series in the lab lasted 2-4 hours, so the maximum time that a 38 sample was exposed in the lab prior to testing was about 6 hours. Figure 2.10: Photograph of specimens of different sizes stored in the cold lab prior to testing. 2.3 Laboratory Testing The experiments were carried out in a cold laboratory located in the Parks Canada compound at Rogers Pass. The floor plan of the lab is 3.3 m by 3.3 m with a ceiling height of 2.2 m. This small size limits the space available for storage of samples amid the other testing equipment and space for two people to move about and work. The temperature of the lab can be controlled down to about -20◦C. The air temperature within the lab fluctuates within about 2 degrees of the set temperature. The lab was always kept colder than about -5◦C to avoid any melting of the snow samples during preparation or handling, as the heat generated by some of the instruments in the lab melted snow crystals at higher ambient air temperatures. Humidity control was unavailable in the lab. The lab contained a microscope, a digital scale, table top space for sample storage and preparation, miscellaneous tools and instruments related to sample preparation and handling, and a universal testing machine. Details and photographs of key experimental procedures are given below. 2.3.1 Universal testing machine A bench-top universal testing machine supplied by Adelaide Testing Machines (model SO-200) was used for the bending experiments (Figure 2.11). A number of custom modifications were made to the machine to ensure reliable and repeatable operation at sub-freezing temperatures. Most importantly, the motor and 39 electrical housing of the machine was heated by a custom-built internal heater with a programmable ther- mostat, which allowed the internal temperature of the machine to remain above freezing. Additionally, the ball-screw on which the actuator travelled was lubricated with a silicone-based lubricant for better operation at low temperatures. The entire base of the machine was insulated with styrofoam sheets. Finally, legs were constructed on the back of the machine to allow it to be placed on its back for horizontally-oriented tests. This manner of testing will be discussed further below. Figure 2.11: Photograph of bench-top testing machine set up for a bending test in the cold lab. The computer which controlled the machine is to the right. The testing machine was operated by a PC running Windows 98 software. The CPU of the computer was insulated with thick styrofoam sheets to buffer the electronics from temperature changes which might promote condensation on the internal circuitry. Surprisingly, no computer problems related to operation at low temperatures were encountered. The control software allowed only basic open-loop displacement control of the machine crosshead. No cyclical or load-control experiments were possible, nor was closed- loop servo control. This software limitation was related to the design specification for as high a crosshead speed as possible. Fast loading rates to minimize viscous effects were one of the most important design considerations in the present study, and this required a simplified software routine given the chosen testing machine and budget constraints. The control software was only capable of recording 1200 data points per test. It was necessary to adjust two control parameters to achieve the desired sampling frequency without going over this limit. One parameter determined the time delay between recorded points relative to the processor speed of the CPU. 40 This parameter was therefore a sort of sampling period. The second parameter determined an averaging interval for curve smoothing. In nearly every test in the present study the average interval was set to 1, that is no averaging was performed and the raw curve was recorded. The sampling period was empirically adjusted in order to maximize the sampling frequency without hitting the 1200 point limit in the duration of a test. If this limit was reached, the test was aborted. In tests performed at the fastest crosshead speed possible, this limit was never reached. However, tests at slower crosshead speed were sometimes recorded with low sampling frequencies if the sampling period parameter was not set correctly. The crosshead displacement and load indicated by the load cell were recorded for every test. An encoder on the DC servo motor drive shaft was used for the crosshead displacement and speed calculations and control. The sampling frequency was adjustable using the control software, as described above. For the fastest loading rates (1.25 cm s−1), the sampling frequency was in the range of 500–1400 Hz, depending on the averaging interval. The actuator (crosshead) of the testing machine was mounted with an HBM RSC load cell. During the winter of 2006–2007, a load cell with 250 N capacity was used. This initial load cell was selected based on estimated values of nominal strength calculated using beam theory compared to published values of the ten- sile strength of snow (Jamieson and Johnston, 1990). However, many tests during this first season had peak loads which exceeded 200 N, the chosen safety cutoff load at which the test was aborted to avoid damage to the load cell. Subsequently, a 1000 N load cell with a 0.5 N resolution (model RSC-200, calibrated with dead weights) was purchased for the following two winters of research. The data obtained from this load cell comprises the majority of the data analyzed in this study. Two Sentech Linear Variable Differential Transformers (LVDTs) were attached to the testing machine for the measurement of beam deflection at various points. These LVDTs were calibrated to an accuracy of ±0.025 mm using digital calipers. The most common deflection measurements were made at the midspan below the beam. In some experiments, the second LVDT was mounted on top of the beam directly above one of the rocker supports. Deflections measured by this LVDT indicated the level of deformation or crushing of snow at the supports. Overall, the testing machine performed in a very satisfactory and consistent manner given the long periods of operation at sub-freezing temperatures. 41 2.3.2 Bending test apparatus The testing machine was outfitted with a set of rails upon which rocker supports were mounted. These rails are visible in Figure 2.11 extending laterally below the snow sample and supports. Figure 2.12a shows a close-up of a rocker support, constructed of thick pieces of polycarbonate. The top support plate was interchangeable with plates of different width, such that wider plates could be used for softer snow to prevent excessive crushing at the supports during testing. (a) (b) Figure 2.12: Photograph of rocker support, constructed of thick pieces of polycarbonate (a) and photograph of testing machine and half of a broken sample following a bending test, showing both rocker supports and two LVDTs for measuring deflection (b). The rails were designed with the objective of allowing low-friction lateral deflection of the rocker sup- ports during testing, such that the boundary conditions during testing approximated a bending beam on rollers. However, operationally the friction of the rail-support system was too large to permit free lateral sliding during testing. For this reason, the supports were locked into place for each test series to ensure the support span was exactly the same for each test (given the same specimen size). As a result, some frictional sliding between the snow sample and the rocker support during testing was unavoidable. Figure 2.12b shows half of a broken sample after a bending test. At the midspan below the sample, one of the LVDTs is mounted for measuring the midspan deflection. The second LVDT is mounted on top of the sample above the right support. The deflection measured by this second LVDT indicated the amount of crushing at the supports. The arms of the LVDTs had flat, round faces that were lightly pressed against the snow using plastic springs. The force applied by these springs at their typical level of deflection was on 42 the order of 1 N, and was judged small enough not to affect the bending test results for all but a few tests involving the weakest snow in the present study. The LVDTs were not used during the first winter of research (2006–2007). Additionally, some of the smallest bending specimens tested did not permit the mounting of LVDTs for lack of physical space between the supports. For three-point bending tests (central loading), a polycarbonate loading plate of the same size and shape as the support plates (Figure 2.12) was bolted to the load cell along with a wide aluminum stiffener. Four- point bending tests (third-point loading) were also conducted. These were achieved by first mounting an aluminum plate to the load cell. Onto this plate two rocker supports (of the same type as in Figure 2.12a) were mounted for load application on the top of the sample. 2.3.3 Horizontal weight compensation An important aspect of the bending tests was weight compensation. The weak nature of snow and the propensity for viscous effects could contribute to a large amount of experimental scatter if the samples were mounted such that a gravitational bending moment contributed significantly to the fracture. In uncompen- sated bending tests on snow, Sigrist (2006) found that the effects of self weight could account for more than 50% of the bending moment required to fail the sample. The first technique devised to eliminate gravitational effects was to orient the testing machine horizon- tally. The snow samples were then supported from below on a smooth piece of polycarbonate, which itself was supported on sturdy polycarbonate tables that sat on either side of the crosshead housing. Figure 2.13 shows the testing machine set up for this type of testing. Clear pieces of polycarbonate, as shown, were cut for each specimen size. The indentation in the clear polycarbonate closest to the crosshead was to allow a sufficient amount of crosshead displacement to fracture the sample before contact was made between the loading plate and the polycarbonate. In some early tests, the clear piece of polycarbonate shown was not used, and the samples were simply cantilevered over the gap between the two tables. In all cases, the poly- carbonate surfaces were sprayed with a silicone lubricant which reduced the friction between the snow and the polycarbonate. Figure 2.14 shows a small sample being prepared for testing in this horizontal configuration. Figure 2.14a shows the manner in which samples were manipulated into place using styrofoam. Small pieces of styrofoam were always used to grip and move snow specimens so that contact with the snow was never made 43 Figure 2.13: Photograph showing testing machine set up for horizontal weight compensation. The load cell and polycarbonate loading plate are not mounted to the crosshead in this photo. directly using a gloved hand. Figure 2.14b shows the same small sample (D = 5 cm) ready for testing, with the load cell/load plate lightly pressed against the sample. The typical pre-load value used to hold samples in place prior to testing was around 1–2 N. (a) (b) Figure 2.14: Photograph of a small sample being moved into place for testing in the horizontal configuration (a); the sample mounted in place and ready for the test signal from the PC (b). Photographs by Steve Conger. 2.3.4 Vertical weight compensation The second technique for achieving weight compensation was placing the testing machine in the common bench-top vertical orientation and moving the rocker supports to the quarter points of the beam. This place- 44 ment of the supports cancels the bending moment due to self weight in the central cross section where the failure occurs. Figure 2.15 shows the testing machine oriented in this manner, with a large sample (D = 20 cm) mounted and ready for testing. Figure 2.15: Photograph of the testing machine oriented vertically, with a large sample mounted in a weight-compensated fashion. The styrofoam piece in the foreground of Figure 2.15 was used to carry the snow samples from the prep bench to the testing machine. The three notches in the styrofoam are spaced to fit around the rocker supports and the midspan LVDT. The snow sample, sitting atop the styrofoam, would be carefully lifted into place such that the supports and LVDT fit between the notches. The styrofoam was then withdrawn from below and the sample then sat upon the supports, ready for testing. 2.3.5 Specimen notching For many tests, the snow specimens were notched at the bottom of the central cross section. The notch was cut using a paint scraper blade, the same kind of blade as used in the blade hardness gauge. Lines were painted on the blade at 1 cm intervals relative to the leading edge (Figure 2.16). The blade was mounted to a right-angle device which kept the leading edge of the blade vertical and square to the sample. The blade was carefully pushed into the snow specimen by hand to the desired depth, with the notch tool pressed against a framing square which itself was aligned against the bottom of the sample. Given the rough nature of the notching technique, the uncertainty in the notch depth was judged to be around ±2 mm. 45 Figure 2.16: Photograph of tool used to cut notches into snow samples. 2.3.6 Density calculation For small and medium sized specimens, the entire specimen was weighed prior to testing for the calculation of the bulk snow density. Figure 2.17 shows a sample on the digital scale prior to notching and testing. The dimensions of every sample were also measured and recorded prior to testing. For larger specimens that could not be weighed on the scale, 1000 cm3 samples were cut from the specimens following the bending test and weighed. Figure 2.17: Photograph showing a snow sample being weighed for the calculation of bulk den- sity. 46 2.3.7 Temperature measurement Following each test, a thermometer was inserted into one of the broken halves of the specimen to record the temperature as close to the time of testing as possible. When using a dial-stem thermometer, as in Figure 2.18, the temperature was recorded to the nearest 0.5◦C. More often, digital thermometers were used, and the temperature using these gauges was recorded to the nearest 0.1◦C. Periodically, all thermometers were calibrated in a slush bath. Figure 2.18: Photograph showing the measurement of temperature of a sample after a bending test. 2.3.8 Crystal identification The snow crystals were classified in the lab by observing them under a microscope at a magnification of up to 50×. This was typically done only a couple times during the course of a test series in the lab, as time did not permit the sampling and studying of crystals after each test. Typically, the classification made using the microscope agreed with that made in the field using a hand lens. However, observation of crystals under greater magnification in the lab led to a tendency to classify the crystals as having more angular or faceted forms than visible under low magnification. McClung and Schaerer (2006) mention this tendency to focus too much on small-scale detail when observing crystals under high magnification, and for this reason lower magnification was preferred when making the initial classification. 47 Figure 2.19: Photograph of the author studying snow crystals under a microscope for classifica- tion. 2.3.9 Practical limitations Given the limitations of time, area available within a given snow pit, and storage area in the lab, the number of samples that could be successfully tested in a day was limited. For test series using specimens of different sizes, the maximum number of possible tests was in the range of about 10-20. Many of the test series used specimens of all the same moderate size, and for these series up to 30 tests in a day could be conducted. More tests would have been possible if samples had been stored in the lab for extended periods of time, but this was deemed undesirable given the metamorphic change of the snow that would take place. The largest samples that were successfully extracted from the natural snowpack, transported to the lab, and successfully tested had a beam depth D = 20 cm. Figure 2.20 shows a sample of this size, with length L = 80 cm and support span S = 40 cm, mounted for an unnotched bending test. It was not deemed possible or practical to attempt the extraction of any larger sizes. Only about one in four of the largest specimens which were extracted successfully resulted in successful tests in the lab. Most large samples broke in some stage of removing the sample from the sample cutter, transportation to the lab, or manipulation in the lab prior to testing. Additionally, the largest samples that were successfully tested may have undergone some 48 weakening or damage prior to testing, but to a degree that was not sufficient to cause a failure or be noticed. For this reason medium sized samples were preferred as standard for many of the test series in this study. Figure 2.20: Photograph showing largest specimen size used in the present study, mounted for a bending test. The smallest samples successfully tested (D = 2.5 cm) were equally difficult to extract, handle and test. The most difficult component was sample extraction in situ. The stainless steel sample cutters created enough grain-scale disturbance during insertion into the snow that the small samples typically came out irregular and unfit for testing. This practical limitation may be considered as related to a minimum length scale over which cohesive snow behaves as a continuum (recall the continuum limit in the scale-cohesion classification introduced in Figure 1.5). Figure 2.21 shows the smallest sized specimen tested. Wooden dowels were used as roller supports and for the central loading device attached to the load cell. The different boundary conditions that were required for testing with these small specimens complicated the comparison of results with those from larger specimens. Very little data from samples of this size were conducted or included in the analysis in this study. The large and flat support and loading plates used to prevent localized crushing in the bending tests occasionally led to adverse effects, especially for unnotched bending tests. In Figure 2.22, a shear failure between one edge of the large loading plate and the adjacent support plate is evident. In this particular case, the loading plate was too wide. Based on experience, the loading and support plates were initially chosen to have a width of 0.25D or less, and only increased in width if the snow was soft enough that excessive 49 Figure 2.21: Photograph showing the smallest sample size used in the present study, just after failure in a bending test. crushing was observed. Note that the shear failure in Figure 2.22 was the exception rather than the norm, and is shown here simply to reflect one of the experimental challenges in working with a material such as snow. See Appendix 7.5 for a description and images of the tensile fracture morphology that was the norm in this study. Figure 2.22: Photograph of a shear failure in a large sample following an attempted unnotched bending test. 50 2.3.10 Post-peak behaviour and apparent softening displacement The measurement of strain-softening displacement was initially desired in the experimental design. This proved to be beyond the capability of the testing machine and bending test apparatus, however. This was in part related to the compliance of the testing machine, which is known to be related to the stability of fracture experiments (Bažant and Becq-Giraudon, 1999). Loading apparatus compliance is known to complicate the measurement of post-peak behaviour in ice (Dempsey et al., 1999b). Though no direct measurements of the compliance of the overall loading apparatus were made in the present study, it was estimated that the combined compliance of the testing machine (rated 2 kN frame) plus the bending apparatus (mostly polycarbonate) was high enough to affect experimental stability. The bending experiments may have been inherently unstable themselves. In a series of fracture ex- periments on Antarctic shelf ice, Rist et al. (1999) could not achieve stable crack growth using three point bending tests. The absence of stable crack growth complicates the measurement of post-peak behaviour in solid ice (Dempsey et al., 1999b). Stable crack growth was never observed in the experiments in the present study; crack growth initiation always appeared unstable. This may have been in part a consequence of the use of open-loop displacement control in the experiments. The instability observed in the experiments was also linked to the rapid loading rates, which were se- lected to minimize viscous effects. The high rates of crosshead displacement led to an apparent post-peak deflection curve which was mostly due to the elastic rebound of the load cell and the continued crosshead travel after receiving the signal to stop when the post-peak load dropped below a threshold value (typically about 5 N). Figure 2.23 shows the post-peak curves from a series of three point bending experiments conducted in the horizontal orientation. The beam depth was 10 cm, the loading span was 20 cm, and all samples were notched to a relative depth of 0.3. Only the crosshead speed was varied between tests. The apparent softening curves are conspicuously consistent for a given crosshead speed. The highest crosshead speed of 1 cm s−1 led to a post-peak displacement of around 0.5 mm. Only when the crosshead speed was reduced by an order of magnitude or more did the softening curves begin to converge. However, at these lower speeds viscous effects during loading would be more significant. Therefore, even though the lower speeds may have suggested a more physically realistic softening displacement (at least for the initial post-peak tangent slope, 51 which is important in governing the fracture energy in Bažant’s size effect laws), the desire to minimize viscous effects was deemed more important than eliminating this spurious post-peak deflection. Figure 2.23: Post-peak curves measured in bending experiments at different loading rates. A further investigation of the post-peak rebound behaviour of the load cell was conducted by breaking thin strips of glass in three point bending at different rates. These tests were assumed to produce fully brittle behaviour, but again an apparent softening displacement was measured as a function of loading rate (Figure 2.24). The excellent fit of the regression through the data in Figure 2.24 is confirmation that the observed, apparent softening displacement was due to the consistent elastic rebound of the load cell. Consequently, due to the combined factors of initial crack growth instability, testing machine compliance, and elastic load cell rebound, no reliable post-peak behaviour was measured for the bending experiments in the present study. 2.3.11 Friction between snow and polycarbonate Approximate values of the friction coefficient between snow and polycarbonate were calculated from a series of experiments which involved simply pushing snow samples along the polycarbonate support tables using the crosshead. A total of ten experiments were conducted with the same sample of snow, each time pushed slightly further along the polycarbonate table at the same constant crosshead speed (1 cm s−1). Figure 2.25 shows the results of two of these experiments, with the static and kinetic friction coefficients calculated using a simple ratio between normal gravitational force and tangential applied force. 52 Figure 2.24: Post-peak load-displacement curves measured in flexural tests of thin glass strips. The mean and median static friction coefficient values in this series of experiments was 0.4, with indi- vidual values ranging from 0.14 to 0.6. The kinetic friction values were somewhat more repeatable, typically falling between 0.1 and 0.25. These values are higher than reported by Mellor (1975) for kinetic friction between snow and polycarbonate, though adhesion may have played a role in the present experiments in addition to simple friction. This adhesion may have even been enhanced by the silicone lubricant which was sprayed on the polycarbonate and wiped to a thin film prior to testing (and repeated periodically for all horizontally-oriented tests). Mellor (1975) reported the difficulty in separating the effects of friction and adhesion in experiments. In the friction experiments, the initial peak and then drop in force occurred over the first 1 mm of displacement (Figure 2.25). Most bending tests required at least this amount of crosshead displacement to fracture the sample. Figure 2.26 shows a typical load-displacement curve for a horizontally-oriented test. The circled region represents a commonly-observed shape in the early part of the loading curve for this type of test. This shape is consistent with an interpretation of the snow overcoming the initial peak static friction value at a displacement of about 0.5 mm, dropping thereafter to the kinetic value for the remainder of the test (Figure 2.25). This behaviour was consistent enough to give confidence in the following frictional- correction to the loading curves for horizontally-oriented tests: when processing the load-displacement 53 (a) (b) Figure 2.25: Experimental curves used to measure the friction coefficient between snow and poly- carbonate curves for analysis, a constant force equivalent to the kinetic friction coefficient times the normal force (weight of the sample) was subtracted from the measured force. This typically amounted to a correction of around a few percent to the peak load. 54 Figure 2.26: Example load-displacement curve showing the possible influence of friction between snow and polycarbonate support table. From the shape of the loading curves measured in fric- tion experiments (Figure 2.25), the circled region was interpreted as the snow sample over- coming the initial high static friction (plus perhaps adhesion), thereafter dropping to the kinetic friction value. The characteristic shape of the load-displacement curve in the circled region was common enough in the horizontally-oriented tests to give confidence to this physical in- terpretation. Summary Much of the first of three seasons of research in the cold lab was devoted to development and adaptation of tools and techniques for the unique challenges posed by testing a material such as snow. This was a significant investment in time, resources and patience to get to a point where consistent and confident results could be obtained to meet the objectives of this study. In some cases, special adaptation of existing experimental fracture test methods was possible, and in others the development of entirely new approaches was required. After the first season of development, two full seasons of productive field and laboratory research were conducted using the methods described above. The methods were described here in as much detail as was deemed appropriate to facilitate any future research along the same lines. As challenging a material as snow is to work with, and as challenging as the slab avalanche problem is to analyze, much more data in future studies will surely be welcome. 55 Chapter 3 Thin-Blade Penetration Resistance and Snow Strength 3.1 Introduction Snow hardness is defined as the resistance to penetration of an object into snow (Fierz et al., 2009) and is measured using penetrating devices of various shapes and sizes. The resisting force in any hardness measure comes from a combination of bending and rupture of grain bonds and grain structures, compaction of loose grains and friction between snow and the penetrating object. The relative contribution of each resistance component to the total penetration force is unknown. However, bonding is the critical variable in determining the mechanical properties of snow such as strength (Shapiro et al., 1997). Despite the recognition of the relationship between hardness and the strength or bonding in snow, the bulk density of snow continues to be the most commonly used index variable for mechanical properties of snow. Examples of properties represented as functions of density include strength and Young’s mod- ulus (e.g. Nakamura et al., 2010; Marshall and Johnson, 2009; Shapiro et al., 1997), fracture toughness (Sigrist et al., 2005b), fracture speeds and fracture energy (McClung, 2007a,b), and viscoelastic proper- ties (Camponovo and Schweizer, 2001). The scatter in properties at a given density is typically attributed to differences in snow microstructure (Schweizer et al., 2003), differences which might be captured by a This chapter contains material published as Borstad and McClung (2011). Additional information on this publication is de- scribed in the Preface. Minor modifications were made here for clarity and flow within the overall structure of the dissertation. 56 hardness measure. Several factors explain the widespread use of density in these contexts, when a hardness measure or other parameter representing bonding is theoretically more appropriate. Density is easy to measure and relatively objective, though different density samplers can give rise to inconsistent results with different errors (Conger and McClung, 2009). More importantly, no objective standard for hardness has been adopted to supplement or replace the density as a proxy variable in snow mechanics. A thin blade snow hardness gauge was developed to establish an objective index measure of hardness for direct comparison with strength and other mechanical properties of snow. In order to minimize compaction and displacement of snow ahead of the penetrating tip, the thickness of the blade (0.6 mm at the leading edge) was chosen to be comparable to the grain sizes commonly encountered in alpine snow. The width of the blade (10 cm) was chosen so that around 10–100 grains would be simultaneously in contact with the blade, resulting in an average resistance measure over a length scale that corresponds to the structural scale of interest in most avalanche applications. Examples of relevant length scales in fracture of snow include the critical length of weak layer fractures (often called sweet spots or hot spots) prior to unstable propagation, which are on the order of the slab depth (Bažant et al., 2003) and the scale of distributed damage prior to tensile crack coalescence, which is on the order of 10–100 times the grain size (Borstad and McClung, 2009). The blade hardness gauge consists of an adapted stainless steel paint scraper blade attached to a hand held push-pull gauge. The peak resistance to penetration of the blade into layers or samples of snow was defined as the blade hardness index and was the single quantitative output. Blade hardness measurements were made horizontally in the walls of excavated snow pits. The results were compared against hundreds of density and hand hardness tests. The effects of penetration rate, blade orientation and blade width were explored. The blade hardness index was a consistent measure across observers, overcoming a drawback of the common hand hardness test. The tensile strength of snow samples was measured in a cold lab and the strength correlated better with the blade hardness index than with the density. A threshold in penetration resistance was identified that separated cohesive from cohesionless snow, confirming previous results using a thin blade gauge (Fukue, 1977). This chapter is limited to an exploration of the blade hardness gauge with respect to its response in dif- 57 ferent types of snow and environmental conditions, sensitivity to various testing conditions and usefulness in providing a single quantitative output for correlation with strength and other mechanical properties of snow related to avalanches. Correlation and comparison with common instability evaluation tests in avalanche work was beyond the scope of the analysis but will be an important area of future research. We begin with a brief review of relevant snow hardness literature. Emphasis is given to results related to thin blade penetration, compaction of snow in hardness measures and comparisons between direct mea- surements of strength and hardness. Details on the design and use of the thin blade gauge follow. Tests involving the gauge in excavated snow pits and in the cold lab are described, followed by the results of these tests and discussion. Potential applications of the gauge and limitations of the present study are discussed and conclusions drawn. 3.2 Hardness Measures 3.2.1 Thin blade hardness Bradley (1966) developed a resistograph that recorded hardness using two blades mounted on either side of a probe. The probe was inserted to the base of the snow, rotated by 90◦, and then withdrawn. The resisting force met by the blades during withdrawal was transferred via a spring in the shaft to a scribe that recorded the force on a spool of paper. This design was later modified so that the resistance was met by two upward- pointing cones rather than blades (Bradley, 1968). This change may have been the result of difficulty in turning the blades prior to withdrawal in some types of snow (Floyer, 2008). Bradley’s resistograph was never widely adopted. Fukue (1977) carried out thin blade penetration measurements that met four primary objectives. Namely, the measure was simple to carry out (1), minimized sensitivity to penetration rate (2), minimized densifi- cation of snow around the penetrating object (3) and minimized changes in intergranular bonding between adjacent snow grains (4). The third point has been emphasized as a drawback of many common hardness measures (Shapiro et al., 1997). The last two points may be especially important in any hardness measure with a large compaction zone since ice grains form bonds which gain strength within a fraction of a second after contact (Szabo and Schneebeli, 2007). The blade used by Fukue (1977) was 12 mm wide by 0.6 mm thick with a blunt leading edge. It was 58 mounted to an actuator and driven into snow samples in a cold lab and the penetrating force was measured using a transducer. The individual peaks in the force-depth signal were roughly constant within the first 3 cm of penetration depth and slightly increased with further penetration due to friction between the sides of the blade and the snow grains. A ductile-to-brittle transition in penetration speed of 0.2 mm s−1 was observed. At penetration speeds below this transition the response of the snow was ductile, characterized by penetrating force which increased without bound. At speeds above this transition, brittle bond failures were evident from the spiked shape of the force displacement signal (Figure 3.1). A slight rate dependence in the brittle range, with decreasing peak penetration force with increasing penetration speed, was observed between 0.2-0.6 mm s−1. Above 0.6 mm s−1 the peak force was independent of penetration speed. Penetration depth Pe ne tra tio n f orc e B Figure 3.1: Conceptual schematic (not to scale) of blade penetration force versus penetration dis- tance for brittle penetration rates, based on measurements made by Fukue (1977). In Fukue’s study, the minima following individual peaks in force were located at less than half of the peak force. The wider blade in this study should lead to higher minima with respect to individual peaks due to more structural elements in contact with the blade. The blade hardness index (B) is represented by the dashed line. Similar trends in the vicinity of the ductile-to-brittle transition were also observed in the uniaxial tensile strength tests, expressed as a function of strain rate, reported by Narita (1980). The maximum blade pen- etration force in Fukue’s data and the maximum tensile strength in Narita’s data, both as functions of rate (penetration rate and strain rate, respectively), were observed at the ductile-to-brittle transition. This sug- gests that Fukue’s ductile-to-brittle transition at a penetration speed of 0.2 mm s−1 corresponds to a nominal 59 bond-scale strain rate on the order of 10−4 s−1. 3.2.2 Hand hardness The hand hardness test (de Quervain, 1951) is perhaps the most common hardness test in avalanche fore- casting work (McClung and Schaerer, 2006) and is, consequently, commonly cited in analysis of avalanche and snow stability data (e.g. Schweizer and Jamieson, 2001, 2007). The basic premise of the hand hard- ness test is to penetrate the snow using a standard force. Achieving this standard force requires selecting penetrating objects of different cross sectional area. Five hand hardness categories (excluding solid ice) are defined corresponding to different cross sectional areas that can be driven into the snow layer without exceeding the given force (Fierz et al., 2009). Operationally, half-scale or ± qualifiers are often appended to the categorical result to refine the coarse scale. The current international standard penetration force in the hand hardness test is 10–15 N (Fierz et al., 2009). The previous version of the hardness standard (Colbeck et al., 1990) specified a force of 50 N. However, the North American standard has been 10–15 N for many years (McClung and Schaerer, 2006). When comparing different hand hardness indices from different observers, countries or years, therefore, the difference in the applied force may vary by up to a factor of five. This makes any quantitative analysis using the hand hardness index difficult. Höller and Fromm (2010) measured actual force values associated with the hand hardness test using a push-pull gauge and flat plates with standard cross sectional areas. Their results showed high scatter and overlap between penetration resistance values for adjacent hand hardness categories. Overall, the maximum force values agreed better with the old 50 N force standard (Colbeck et al., 1990), though the force gauge was unable to record values below 10 N. These results illustrate the limitations to quantitative analysis using hand hardness data. 3.2.3 Probe hardness The Swiss rammsonde or ram hardness test (Haefeli (1939), translated in Bader et al. (1954)) is a cone penetration test (60◦ cone tip angle and 40 mm base diameter) adapted from the soil sciences. The ram resistance is defined as the measured amount of force required to drive the rod a given depth into the snow. The large base area and weight of the instrument limit the vertical resolution of the ram hardness to the 60 centimeter scale (Pielmeier and Schneebeli, 2003). The SnowMicroPen (SMP) is a motor-driven cone penetrometer that records hardness at sub-millimeter resolution (Schneebeli and Johnson, 1998; Johnson and Schneebeli, 1999). The cone angle is the same as the Swiss rammsonde but the cone diameter (5 mm) is much smaller. Pielmeier and Schneebeli (2003) compared SMP hardness profiles to hand hardness and ram hardness and found that the SMP most ef- fectively resolved small scale stratigraphy when compared against planar sections of snow layers. No standard algorithm exists for interpreting and processing the SMP resistance signal, making comparison of results from different studies difficult (Marshall and Johnson, 2009). The SABRE probe penetrometer (Mackenzie and Payten, 2002) is another probe hardness gauge, with a 12 mm diameter rounded tip, that has seen limited use (Floyer, 2008). 3.2.4 Compaction of snow in hardness measures Floyer (2008) attached tips of different shape and size to the SABRE probe, filmed the penetration pattern around each, and analyzed the films using particle tracking velocimetry. Floyer and Jamieson (2010) ex- amined in more detail the compaction around the round probe tip specifically. These experiments provide insight into the assumption that compaction around a probe tip can be neglected in the interpretation of the force signal (e.g. Johnson and Schneebeli, 1999; Marshall and Johnson, 2009). This compaction can be separated into horizontal (or normal to the direction of penetration) and forward (ahead of the probe tip) components. The most important qualitative conclusion that can be drawn from the work of Floyer (2008) from the perspective of this study was that the tapered blade tip led to a much smaller zone of horizontal and forward compaction than either of the larger conical or rounded probe tips. The relative size and shape of the blade tip in the current study (and also that of Fukue (1977)) is shown in Figure 3.2a. Since the leading edge of the blade is blunt rather than tapered, it might be more appropriate to consider the full thickness of the blade as the scaling length L rather than half the thickness. However, in either case the length scale is comparable to or smaller than many common grain sizes encountered in seasonal snow (Fierz et al., 2009). This introduces the grain size (or a relationship between the grain size and available pore space for densification) as the dominant scaling parameter for the horizontal deformation around the tip. The tip of the SMP (Figure 3.2b), for comparison, has a base radius of 2.5 mm and a shallower cone 61 xy 𝜽 𝜽 L = 2.5 mm L = 6 mm L = 0.3 mma b c Figure 3.2: Scaled representation of penetrometer tips, each in a plane of symmetry. Thin blade (a) used by Fukue (1977), with the same leading edge dimensions as the blade in the present study; SnowMicroPen (SMP) dimensions (b), with θ = 30◦; tips used by Floyer (2008) (c), with rounded tip (unshaded) and conical tip (light gray, θ = 45◦) of the same radius. The blade tip (dark gray) had L = 1 mm, θ ≈ 45◦. half-angle (30◦) than the conical tip used by Floyer (2008). This should lead to a relatively smaller zone of compaction around the SMP compared to the SABRE probe. The relative difference between the scaling length L for the SMP and the thin blade in the present study is about the same as the relative size difference between the tapered blade and conical tip used by Floyer (2008), though the cone and blade tip angles are different. Though the precise relative shape and size of the compaction zones for the SMP and the thin blade in this study cannot (and need not) be determined, it can be argued based on the results of Floyer (2008) and from simple dimensional scaling arguments that the thin blade will horizontally compact less snow as it 62 penetrates compared to any other common hardness measure considered here. Since the blade tip in the present study is both blunt and thin, the forward compaction may scale dispro- portionately with L. Whiteley and Dexter (1981) found that a 1 mm diameter probe required around 50% more pressure than a 2 mm diameter probe to penetrate sandy soils. The explanation may lie in the devel- opment of a passive nose cone being pushed ahead of the probe, similar qualitatively to that observed by Floyer (2008). The shape and size of this nose cone does not appear to have a simple scaling relationship with the penetrometer shape and size, especially when the probe tip size is comparable to the grain size. Therefore, comparison of the forward compaction for blunt-tipped thin blades versus other penetrometers is more difficult and uncertain. 3.2.5 Hardness and strength Bradley (1968) underlined the importance of direct strength measurements for comparison against hardness tests, though few studies have systematically done this. Comparing resistograph measurements to the com- pressive strength of snow columns containing weak basal layers, Bradley (1968) found that the minimum resisting stress from the resistograph roughly correlated with the compressive strength of the basal layer. Martinelli (1971) reported data relating both ram hardness and density to centrifugal tensile strength. This data was analyzed here to compare the two different proxies for strength. Both the ram hardness and density show very high (and nearly equal) correlations with the nominal centrifugal tensile strength (Table 3.1). The ram hardness and density are also highly correlated with each other. From this data the ram hardness appears no better (nor worse) than density for correlating with strength. Other studies attempting to relate ram hardness to strength have been largely unsuccessful (Shapiro et al., 1997). Fukue (1977) empirically correlated blade penetration force with cohesive strength in several ways. First, artificial snow samples were allowed to sinter over time at a temperature conducive to bond growth. The unconfined compressive strength of the samples increased with age and therefore bond strength. Thin blade penetration tests were then paired with unconfined compressive strength tests on similar snow samples undergoing sintering. The maximum blade penetration force strongly correlated with unconfined compres- sive strength, suggesting a link between bonding and blade penetration. Second, confined compression tests were performed at rates both below and above an identified ductile-to-brittle transition in compression rate. Following each test, a thin blade penetration measurement was performed on the sample. Samples that 63 Table 3.1: Correlation matrix for data reported by Martinelli (1971). The upper diagonal elements contain Spearman’s rank correlation coefficients, rs*, and the lower diagonal elements are the p- values. Bold face indicates statistically significant correlations at the α = 0.05 level. The lower part of the table shows the range of each variable (n=98). ft ρ Rram T E ft 0.919 0.928 0.023 0.011 ρ <0.001 0.940 0.013 -0.045 Rram <0.001 <0.001 -0.054 -0.075 T 0.822 0.899 0.594 0.078 E 0.918 0.662 0.465 0.448 Nominal tensile strength ft 0.4 < ft < 246.0 kPa Density ρ 68 < ρ < 491 kg m−3 Ram hardness Rram 9.81 < Rram < 2207 N Temperature T -22.4 < T < -1.4 ◦C Grain size E 0.2 < E < 1.5 mm *Spearman’s rank correlations are shown rather than Pearson’s product-moment correlations (r) for several reasons. First, Pearson’s r is based on the assumption of linear dependence between the two variables. However, associations among the mechanical properties of snow are often nonlinear. Pearson’s r also contains the assumption that the underlying parent distributions of the two variables are normal (though violations of this assumption are not severe if the sample size is large). Finally, Pearson’s r is much more sensitive to outliers. Spearman’s rs is a non-parametric alternative which tests for any monotonic relationship when the assumptions for using Pearson’s r are not met. had been compressed at rates above the ductile-to-brittle transition, and therefore had broken bonds, had systematically lower penetration resistance than samples that had been slowly compressed in the ductile range. Schneebeli and Johnson (1998) directly compared centrifugal tensile strength and average penetration resistance using an early version of the SMP with a cone half-angle of 45◦ and a 5 mm diameter cone tip. The tensile strength measurements showed a high amount of scatter when expressed either as a function of density or average penetration resistance. However, the uncertain repeatability of the centrifugal tensile tests may have contributed to the large scatter (Schneebeli and Johnson, 1998). No other examples in the literature were found which directly measured snow strength and compared strength with components of SMP resistance signals. Recent studies have analyzed or derived parameters from SMP signals and compared them to either the results of avalanche instability tests, which can provide 64 indices of strength, or to published strength values. Birkeland et al. (2004) found that the maximum re- sistance recorded by the SMP in a weak surface hoar layer increased as a shear strength index increased. Neither the mean nor median resistance in the layer were significantly correlated with the shear strength in- crease. Lutz et al. (2009) used resistance values and drop frequencies from the SMP to calculate a grain-scale strength index and observed changes in this index with artificial load changes in three compression tests. Marshall and Johnson (2009) calculated theoretical values of strength from SMP signals and compared these values against tensile, compressive, and shear strength values, expressed as a function of density, from the literature. Around half of the SMP-derived strength values were higher than any published values. This might be explained by the assumption in the calculations that all of the resisting force (less a small amount of friction) was due to the elastic deflection and rupture of bonds (Marshall and Johnson, 2009). Account- ing for a compaction component in the resistance signal may have brought the derived strength values into better agreement with measurements. 3.3 Methods The blade hardness measurements were carried out in two settings. The first was in excavated snow pits, alongside standard stratigraphic snow profiling techniques used in avalanche work (Fierz et al., 2009; Canadian Avalanche Association (CAA), 2007). The second was in a cold laboratory, where the blade hard- ness measurements were paired with tensile strength tests on samples extracted from the natural snow cover. The force gauge, blade attachment and measurement technique are detailed first. 3.3.1 Force gauge and blade attachment The force gauge used was a Chatillon DFE series with a full bridge strain gauge load cell. Figure 3.3 shows the gauge and blade attachment. The load cell capacity was 250 N with a resolution of 0.1 N. The gauge accuracy was certified to within±0.25% full scale (±0.6 N). The gauge was periodically tested for accuracy by hanging dead weights from a hook attached to the gauge. These tests confirmed the accuracy of the gauge in the approximate range of 5–95% of the full scale. The operating temperature range of the gauge was specified as -1 to +49◦C, but the typical testing temperature was in the range -10 to 0◦C. When used in the field, the gauge was kept in an insulated container and only brought out just before use. Most tests involving the gauge lasted only a matter of minutes. This 65 1 cm Figure 3.3: Blade hardness gauge. likely limited the actual temperature drop of the load cell in the interior of the gauge relative to the ambient temperature. In the cold lab, however, the gauge was exposed to cold temperatures for longer periods of time and temperature effects might have been more significant. The specified temperature effect on zero load level was 0.09 N/◦C relative to the calibration temperature. The median ambient temperature in the lab was about -5◦C and the calibration temperature was 23◦C, suggesting a possible shift in the zero point of the load cell of up to 2.5 N. However, the actual internal temperature of the load cell was probably slow to change relative to the ambient temperature, buffered by the thick housing of the gauge and the internal circuitry (the gauge manufacturer used 30 minute stabilization times when testing the load cell performance within the range of operating temperatures). The LCD and battery life of the load cell were not affected down to air temperatures as low as -20◦C. The data sampling rate of the force gauge was 5000 Hz. It was possible to record a continuous signal at this rate, which could be sent via a cable to a datalogger or a computer, or simply to record the peak force in tension or compression. For this study only the peak resisting force was recorded. For future studies, high resolution measurements of penetration resistance could be obtained. In particular, the variance of penetration resistance over the length scale of interest (10 cm) could be useful in addition to the peak force. However, this would require a more accurate and sensitive force gauge than used in the present study. The thin blade used was a 10 cm wide by 0.6 mm thick paint scraper blade with a blunt leading edge. Only the leading 2 mm of the blade was 0.6 mm thick. Behind the leading edge the blade tapered to 0.5 mm thickness. This thickness profile was related to a hardening finish at the tip of the blade. The handle was removed from the original paint scraper, and two bolts were used to clamp the blade to an aluminum turnbuckle (Figure 3.3). One of the bolts also clamped a nut that secured the end of the threaded rod 66 extending from the force gauge. The primary cost of the apparatus was the force gauge, as the paint scraper cost around $10 (USD 2007). The digital force gauge cost around $1250 (USD 2007). The leading edge of the blade extended about 30 cm from the front of the force gauge. This distance could have been reduced by about 12 cm by using a shorter threaded rod extending from the load cell. The combined weight of the threaded rod and blade assembly was about 100 g. The effect of this cantilevered weight did not cause a load response in the load cell. The assembled blade apparatus was also rigid torsion- ally, and the load cell was not sensitive to manual twisting of the blade. During penetration tests, twisting, bending or other deflection of the blade was never sensed. 3.3.2 Measurement technique Blade hardness measurements were carried out by pushing the blade 3–5 cm into the surface of the snow, either into an exposed pit wall or into a snow sample in a cold lab, at an estimated penetration speed of around 10 cm s−1 (Figure 3.4). The blade was then withdrawn and the maximum force of penetration was recorded as the blade hardness index. Figure 3.4: Carrying out a blade hardness measurement with the blade oriented parallel to the stratigraphic layering. The adopted notation for recording the blade hardness index was using the symbol ’B’ (Figure 3.1). This distinguished the blade hardness from the commonly used ’R’ for the hand hardness or the ram hardness (Fierz et al., 2009). Unless otherwise noted, the orientation of the blade was parallel to the stratigraphic layering of the snow cover and the blade width was 10 cm. Variations on this notation will be explained as they are introduced below. 67 The penetration speed was high to ensure that the peak force fell into the rate-independent portion of the brittle range identified by Fukue (1977). It was hypothesized that this would maximize the consistency across observers. At this penetration speed, the 5000 Hz data sampling rate of the force gauge records hundreds of samples per centimeter of penetration. This gives high confidence that during rapid penetration of the blade the true peak load was accurately captured. 3.3.3 Standard stratigraphic profiling In an excavated snow pit, the standard profile measurements included hand hardness of identified strati- graphic layers, temperature measurements every 10 cm of depth from the surface to the ground, grain size and form classification by sampling snow crystals from each layer and examining them on a gridded screen under 10 × magnification, density measurements, and water content characterization using a hand test. De- tails of these standard methods can be found in Fierz et al. (2009); Canadian Avalanche Association (CAA) (2007) and in Chapter 2. The characterization of hand hardness in this study was consistently done using the 10–15 N force standard. Plus and minus qualifiers were used for finer scale distinctions. For example, a hand hardness index of “2+” was recorded as 2.3 and a “3-” was recorded as 2.7. An approximate uncertainty was then added to reflect the imprecision and subjectivity of the test and to facilitate comparison with results from the old 50 N force standard. It was estimated that the 50 N force standard would result in a hand hardness index of one level lower for characterizing the same snow. For example, if a snow layer was characterized as having hardness index 3, the approximate range for comparison was reported as 2–3. A factor of 5 force difference, representing the maximum difference between the old and new standards, may lead to an even greater difference in reported hand hardness indices, however. Blade hardness measurements added only a few minutes to standard stratigraphic profiles. Density measurements were typically paired with single blade hardness measurements as a function of depth (Figure 3.5). Often groups of 10 blade hardness measurements were made in manually identified homogeneous snow layers for characterizing the variability of the blade hardness index (Figure 3.6, top). The effects of the blade orientation, width and penetration rate were also investigated in snow pits adjacent to standard profiles. 68 Figure 3.5: Schematic of paired density and blade hardness measurements, looking at the face of an exposed snow pit wall. The central strip represents a meter stick, the open rectangles are the holes left by the density sampling and the solid black lines are the blade hardness measurements. 3.3.4 Laboratory strength testing The blade hardness gauge was also used in a cold laboratory containing a universal testing machine for mea- suring the tensile (flexural) strength of cohesive snow samples. A blade hardness measurement was paired with each strength test (Figure 3.7). Dry cohesive snow samples were first extracted from homogeneous layers of at least 10 cm thickness in the natural snow cover. The samples were cut out using a stainless steel rectangular cutter with a sharpened leading edge. The most common specimen size had dimensions 50 cm long by 10 cm deep by 10 cm wide. Once extracted, the samples were transported to a nearby cold laboratory for testing the same day. The samples were weighed in the lab for the calculation of the bulk density. The samples were fractured in unnotched and weight compensated three (or four) point bending tests. The peak force recorded in the test was used to calculate the nominal tensile strength using Timoshenko beam theory (Timoshenko, 1940). Immediately after a strength test a blade hardness measurement of the sample was taken, along with the temperature, grain size and grain form classification. The grain size and form were determined by examining a sample of snow crystals under a microscope on a similar crystal screen as that used in the field. 69 Figure 3.6: Schematic of blade hardness measurement technique in two different layers, looking at the face of an exposed pit wall. Homogeneous layers are identified manually in a snow pit for this type of test grouping. In the top layer ten measurements are shown, distributed evenly over the thickness of the layer. In the bottom layer an equal number of slope-parallel and slope-normal measurements are shown. Figure 3.7: Schematic of paired tests of flexural strength and blade hardness. The snow sample was first broken in three or four point bending (three point bending shown). Next, a blade hard- ness measurement was taken (upper right of figure) using a part of the sample that experienced low stress during the strength test. 70 3.4 Results 3.4.1 Density versus blade hardness index For a given snow layer, which was typically characterized by a single density, there was wide scatter in blade hardness indices (Figure 3.8). The COV of repeated tests (usually 10 tests) within a layer decreased with increasing layer density, and the slope was statistically significant in a linear regression (p < 0.001). Due to the density-hardness correlation, the COV also decreased with increasing mean blade hardness index, and this slope was also statistically significant (p = 0.02). Cohesionless snow, hereafter defined as snow with B = 0 N, had no clear relation with density. Cohesionless snow was observed with densities ranging from about 30–250 kg m−3. Blade hardness index 'B' (N) D en si ty ( k g m − 3 ) 100 200 300 400 0 10 20 30 40 50 l l l l l l l l ll l ll ll l lll l l l l l l l l ll l l l l ll l l l l l l l l l ll l l l ll l l l l l l l ll l l l l ll l l l l ll lll ll ll l l ll l l l l l l ll l l l l l l l l l l l l l ll l l l ll l l l ll l l l ll l l l l l l l l l l l l l l ll ll l ll l l l l l l l l l l l l l l l l l l ll l l l l l l l lll l l l l l l ll l l l ll l l l l l l l l l l l l l l l l l l ll l l l l l l lll ll l l l ll l l l l l l l l l l l l ll l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l ll l ll l l l l l l l l l l l l l l l lll lll l l l ll l l l l l l l l l l ll l ll l l l ll l l l l ll l l ll l l l l l l l l l l l l l l l lll l l l l l l l l l l ll l l l l l l ll l l ll l l l Figure 3.8: Density versus blade hardness index (B) from 24 snowpit profiles carried out over the winters of 2007/2008 and 2008/2009. The Spearman’s rank correlation coefficient is 0.89, p-value <0.001. (n=628) In the data set of 628 in situ test pairs, not a single blade hardness index was registered between 0.0 and 1.7 N, indicating a gauge sensitivity problem. More than 90% of the blade hardness index values were less than 20 N, indicating only a small range of the full capacity (250 N) of the load cell was used. A total of 99 values of B = 0 N were recorded, many of which were interpreted as legitimate negligible resistance values 71 in cohesionless snow. The true resisting force for some of these tests was likely nonzero, however, but too low to be accurately resolved with the load cell. 3.4.2 Penetration rate effects Penetration rate effects were considered to be the primary source of possible variability for results obtained with different operators. In one test series, pairs of fast and slow blade hardness measurements were carried out side by side. The same gauge operator was used for all tests in order to isolate the rate effect. The penetration rates were subjectively judged, with the standard 10 cm s−1 rate considered fast. For the slow tests, the penetration rate was around a few cm s−1, or approximately one order of magnitude slower. These penetration speeds correspond to bond-scale strain rates on the order of 10−3 s−1 for the slow tests and 10−2 s−1 for the fast tests. These rates are both in the rate-independent portion of the brittle range identified by Fukue (1977) (as a function of penetration rate) and Narita (1983) (as a function of strain rate). A total of 40 pairs of tests were carried out side by side comparing one fast and one slow measurement at the same depth and within the same layer. All tests were done in a single location, with pairs of tests conducted every 3 cm of depth from near the surface of the snowpack to the ground. This ensured that at least one test pair was conducted within every manually identified layer. The blade was oriented slope- parallel for all tests. In this test series, there were 8 test pairs in snow of hand hardness index 1–2 in which one of the two tests (fast or slow) had a blade hardness index of 0 N. Given the gauge sensitivity problem near 0 N these test pairs were thrown out of the following analysis. In the 32 pairs of tests for which both the fast and slow results were nonzero, the ratio of fast to slow hardness (B f ast/Bslow) had a mean and median of 1.1. The ratio of fast to slow hardness did not consistently correlate with any other measured snow property. A Wilcoxon signed rank test was performed on the logarithm of the ratio (the logarithm symmetrizes the ratio about zero) to test whether the fast and slow results were different. The test indicated that the ratio was not significantly different from 1 at the α = 0.05 level (p = 0.07). A second test series was carried out within a single homogeneous layer (hand hardness index 3–4). A total of 20 tests were carried out, 10 fast and 10 slow. For consistency of speed, the fast measurements were carried out first in a spatial cluster, followed by the slow measurements adjacent to the fast cluster. The blade was again oriented parallel to the layering. 72 For this test series, the mean hardness was 18.6 N (range 15.0–22.7) for the fast tests and 15.6 N (range 12.5–18.8) for the slow tests. The coefficients of variation of the fast and slow tests were the same at 0.15. Welch’s t-test indicated significance in the difference between the means at the α = 0.05 level (p = 0.02). 3.4.3 Blade orientation Paired groups of tests were conducted to explore the effect of the orientation of the blade (Figure 3.6, bottom) on the mean hardness and variability. For these tests, snow layers were first sought that were homogeneous and least 10 cm thick to allow blade penetration perpendicular to the layering. In such layers, 10 tests were carried out in each of two orientations. The first 10 tests were carried out with the width of the blade parallel to the layering (B‖) and the next 10 were perpendicular to the layering (B⊥). The groups of penetration tests were carried out immediately adjacent to one another to avoid, as much as possible, encountering horizontal changes in layer properties. The mean blade hardness index of each group of 10 tests (B̄‖ and B̄⊥) as well as the range and standard deviation were recorded. In total, 12 groups of such orientation tests were carried out in different layers. In order to compare the tests across layers with different properties, the ratios of normal to parallel mean blade hardness index (B̄⊥/B̄‖) and coefficient of variation (CoV⊥/CoV ‖) were calculated for each layer tested. The mean hardness appeared to be independent of blade orientation (Figure 3.9, top). The bulk of the mean hardness ratios (B̄⊥/B̄‖) clustered close to 1, other than two outliers. The majority of layers tested showed lower variability in slope-normal than slope-parallel tests, however. In these cases the COV ratios (CoV⊥/CoV ‖) were less than 1. The individual values of the COV ranged from 0.06 to 2.25 for slope-normal tests (mean 0.42, median 0.21) and from 0.13 to 2.0 for the slope-parallel tests (mean 0.35, median 0.16). A Wilcoxon signed rank test indicated that the COV ratio was different from 1 at the α = 0.05 level (p = 0.02). 3.4.4 Blade size effect A wider blade was attached to the gauge in an attempt to bring out more detail than the 10 cm blade, especially in very soft and soft snow (hand hardness indices 1 and 2, respectively). A 20 cm blade with a thickness of 0.48 mm and a blunt leading edge (another off-the-shelf paint scraper blade) was used for comparison. The large blade apparatus weighed about 280 g (compared to 100 g for the 10 cm blade attachment) and had a cantilever length about 3 cm shorter than the 10 cm blade. The additional cantilever 73 lll CO V ra tio M ea n ha rd ne ss ra tio 0.5 1.0 1.5 2.0 Figure 3.9: Box plot showing the ratio of mean blade hardness index normal to the layering to mean hardness parallel to the layering (B̄⊥/B̄‖, top) and mean normal to parallel COV (CoV⊥/CoV ‖ , bottom). The boxes contain the inner quartile range, the whiskers extend to data points within 1.5 times the inner quartile range from the median, outliers are drawn as individual points, and the thick black line is the median. Each box plot represents 12 group means, with each group containing 10 tests in each orientation. The dashed vertical line is drawn to indicate no difference between orientations. weight did not induce an axial load in the load cell. A total of 55 paired tests were carried out with the 10 cm and 20 cm wide blades. Each test pair was conducted side by side within the same layer. As with the variable penetration rate tests, test pairs were carried out every 3 cm of depth from the surface to the ground. In the first 10 pairs of size effect tests in very soft snow (hand hardness index 1) near the surface, both blades registered B = 0 N. As the hardness increased with increasing depth, however, the 20 cm blade was the first to record values of B > 0 N. This was the case in 6 pairs of tests in snow that was transitioning from hand hardness index 2 to 3. In this snow, the 20 cm blade gave nonzero hardness values whereas the 10 cm blade did not register. Overall, 39 of the 55 pairs of tests had nonzero hardness values for both blades. Among these pairs, the ratio of blade hardness between the 20 and 10 cm blades, normalized by the cross sectional area of the 74 blade tip (0.96 cm2 and 0.6 cm2, respectively) ranged from 0.5 to 1.6 with a mean and median of 0.9 and a standard deviation of 0.2. 3.4.5 Blade and hand hardness The lowest values of the blade hardness index (B < 5 N) typically correlated with weakly cohesive snow of hand hardness index 1.7–3.7 (Figure 3.10). There is considerable overlap between blade hardness indices for neighbouring hand hardness categories. The blade hardness data in Figure 3.10 are the group means from 52 different snow layers in which typically 10 blade hardness measurements (parallel to the strati- graphic layering) were taken in each layer. The plotted hand hardness categories are those that correspond to cohesive snow as measured by B > 0 N. The blade hardness in layers of hand hardness index 0.7–1.3 was always 0 N. In the context of this study such snow was considered cohesionless. In layers of hand hardness index 1.7–3 some values of B = 0 N were recorded, but the mean of repeated tests was always greater than zero. The variability of repeated blade hardness tests decreased with increasing hand hardness. The COV was highest in any snow that still contained decomposing and fragmented forms, which were most often found in young snow that was in the process of bond formation (and thus had low hand hardness). The next highest COV was found in faceted crystals. The lowest COV’s in repeated measures were from rounded grains and mixed rounded and faceted grains. 3.4.6 Blade hardness index as a proxy for strength Recently deposited snow layers were monitored and sampled for laboratory testing as soon as the snow was cohesive enough to extract, handle and transport. It was not possible to extract any samples characterized by hand hardness index 1 because the snow was too weak. When snow was just cohesive enough to extract, the blade hardness of the snow would, in nearly every case, register above 0 N. Only 9 of 238 strength tests in the lab were paired with a blade hardness index of 0 N. These nine samples were from the softest and most fragile snow layer that was successfully tested in the lab. The true value of blade hardness index for many (if not all) of these samples was likely between 0 N and 1.7 N. Lack of gauge sensitivity rather than lack of bond strength prevented quantifying the blade hardness index of these samples. 75 l l 1.7−2.7 2−3 2.3−3.3 2.7−3.7 3−4 3.3−4.3 3.7−4.7 4−5 H an d ha rd ne ss in de x 0 5 10 15 20 25 30 Blade hardness index 'B' (N) Figure 3.10: Box plot of hand hardness index versus blade hardness index for cohesive snow. The boxes contain the central 50% of the data points, the whiskers extend to points within 1.5 times the inner quartile range from the median. Outliers are plotted as individual points. Overlap between hand hardness indices is related to the assumed uncertainty and imprecision of the hand test (n=520). In our data the tensile strength correlated much better with the blade hardness index than with the density (Table 3.2). The blade hardness index had a nearly equal correlation coefficient with the density as with the tensile strength. This suggests that the blade hardness index could be used to predict both density and strength, equally well, with a single measurement. A subset of the laboratory strength tests (n = 143) also had precise deflection measurements at the bottom of the beam. These measurements allowed for the calculation of the flexural modulus, analogous to an elastic (or linear viscoelastic) modulus. For this subset, the Spearman correlation coefficient between the blade hardness and the flexural modulus was 0.68 and was highly significant (p < 0.001). 76 Table 3.2: Spearman’s rank correlation coefficients (upper diagonal) and p-values (lower diago- nal) for the laboratory tensile strength data. Bold face indicates statistical significance at the α = 0.05 level. The lower part of the table shows the range of each variable (n=238). ft ρ B T E ft 0.644 0.844 -0.066 -0.325 ρ <0.001 0.819 0.002 0.085 B <0.001 <0.001 -0.046 -0.118 T 0.311 0.972 0.475 -0.099 E <0.001 0.192 0.070 0.126 Tensile strength ft 3.4 < ft < 66.7 kPa Density ρ 149 < ρ < 364 kg m−3 Blade hardness index B 0.0 < B < 17.4 N Temperature T -12.0 < T < 0.0 ◦C Grain size E 0.5 < E < 1.0 mm 3.5 Discussion 3.5.1 Density and hardness Tables 3.1 and 3.2 have similar variables which are significantly correlated. The lower correlation between the blade hardness index and density, compared to that between ram hardness and density, may be due to several factors. The ram hardness, which deflects and compacts more snow as the cone tip penetrates, might be expected to correlate better with density than a thin blade measure which minimizes compaction during penetration. The current data set also contained snow with mixed rounded and faceted forms compared to that of Martinelli (1971) who did not sample any coarse-grained lower layers in which faceted forms may have been present. At equal densities, snow with faceted forms is weaker than rounded forms (Jamieson, 1988). The smaller range of densities tested in the present data may also be a factor in the lower correlation. The wide variability in mechanical properties of snow at a given density is well known. Takeuchi et al. (1998) and Höller and Fromm (2010) observed high scatter between density and flat plate hardness mea- sures. Martinelli (1971) and Keeler and Weeks (1968) reported increasing scatter in ram hardness with increasing density, which is consistent with the blade hardness-density data (Figure 3.8). 77 3.5.2 Penetration rate effects Given the high variability in snow properties it was not surprising that no statistically significant rate effect was observed when pairing single blade hardness tests at different penetration rates. The typical COV of repeated measures in the same layer was on the order of 0.1–1. This high level of variability, inherent in snow properties, makes in-situ testing for systematic rate dependence difficult. In the grouped test series with 10 fast and 10 slow measurements within the same layer, the statistically significant rate effect observed was the opposite of what was expected. The slow tests were weaker than the fast tests, which conflicts with the precise laboratory results of Fukue (1977) and Narita (1980). The strain rate of the fast blade hardness tests, on the order of 10−2 s−1, is higher than any of the previous laboratory results, however. Horizontal spatial variability cannot be ruled out as a factor in these results, as the fast and slow tests were separated by up to 50 cm within the same layer. Takeuchi et al. (1998) and Höller and Fromm (2010) demonstrated horizontal variability using push-pull hardness measures at similar length scales. Therefore it cannot be confirmed that the snow properties were the same for the two spatially separated test series. This point could have been addressed by spatially pairing fast and slow tests, alternately. The rate effects could also be influenced by a rate dependence in the development of a nose cone of grains being pushed ahead of the blade tip. Therefore there is still some uncertainty regarding the dependence of the blade hardness index on pen- etration rate. Rate effects could be further investigated using a universal testing machine to drive the blade into snow samples at precise speeds. The force gauge used in this study has a mounting backplate which would facilitate integration with a testing machine. The testing machine used for the strength tests in this study had a maximum crosshead speed of 1.25 cm s−1, so it could be used to investigate the slower push speeds. Recording the penetration resistance at 5000 Hz in this sort of testing, rather than just the peak force, would address many of these questions. Such tests would indicate if the schematic interpretation of the blade hardness measure (Figure 3.1), based on the slower penetration speeds of Fukue (1977), is appropriate for the fast push speeds used in this study. When comparing the blade hardness results obtained by different operators, the results varied no more than would be expected given the observed variability in repeated measures using the same operator. The 78 data set was not consistently divided by users to permit formal statistical analysis to confirm this point, however. Three people (the first author and two field assistants) were the primary operators of the gauge for the data contained in this paper and the results from each operator were taken as interchangeable. Additional tests are necessary using different people pushing the blade hardness gauge into the same layer in order to more conclusively address the consistency across operators. However, given the commonly observed COV of repeated measures (on the order of 0.1–1) in homogeneous snow from the same operator (assuming consistent push speeds for a given operator), it is doubtful that a statistically significant difference in operator results would be found. This result is unique when compared to, for example, the hand hardness test which requires a subjective judgement about penetration force which can vary across observers. 3.5.3 Blade orientation The lack of dependence of mean hardness on blade orientation in homogeneous layers is likely the result of the careful selection of layers that did not contain thin hard or soft sublayers or noticeable gradients in hardness or other properties from top to bottom. In the presence of stratigraphic changes in layers that were not sensed manually, conditions which were likely present in some cases, the lack of dependence on orientation probably stems from the depth averaging of the slope-parallel measurements. In many cases this technique will capture small hard or thin sublayers. In most practical in-situ applications, it made the most sense to orient the blade parallel to the layering. The observation that tests conducted normal to the layering had lower variability than parallel oriented tests is important, however. In scenarios where only a single measurement was or could be taken, such as in the lab on snow samples that did not permit multiple measurements, the blade was typically oriented perpendicular to the layering. This is effectively equivalent to saying that, given the choice between sampling from two populations with equal means but different variances, preference was given to sampling from the lower- variance population. 3.5.4 Blade size effect When normalized by the cross-sectional area of the blade tip, the 20 cm blade gave slightly (but not signifi- cantly) lower values of penetration resistance. This could be related to a slightly smaller zone of compaction around the 20 cm blade due to its smaller thickness. The dependence of the results on the width of the blade 79 could further be explored, though the 10 cm length scale was motivated by considerations from the fracture mechanics of slab avalanches. This length scale was also convenient from the perspective of ease of use, especially when compared to the 20 cm blade which was heavier and more awkward to align with the snow. The original motivation for using a wider blade was to attempt to capture the transition between cohe- sionless and cohesive snow. It turned out that the force gauge sensitivity problem near zero was the limiting factor in soft snow rather than the blade width, however. The operating range of the gauge was very low compared to the full scale capacity of the gauge. Rather than changing the blade width, a gauge with a lower capacity and higher sensitivity near zero would better identify the threshold penetration resistance that separates cohesive from cohesionless snow. 3.5.5 Hardness and strength The blade hardness index characterizes an averaged measure of penetration resistance over a length scale of about 100 grains. The index compares favourably side-by-side with tensile strength and flexural modulus measurements in the lab. The tensile strength correlated higher with the blade hardness index than with any other variable in the present data (Table 3.2). Other investigations (e.g. Martinelli, 1971) have shown similar correlations with different measures of hardness and tensile strength. The reason that density continues to be used as the primary index variable for strength and other mechanical properties of snow is related to the lack of standardization and adoption of a hardness measure across disciplines interested in snow mechanics and avalanches. The blade hardness gauge in this study is easy to use, inexpensive, and appears promising as a tool for addressing this issue. The observation that the softest snow that could be physically handled and transported to the laboratory had the lowest (0-2 N) values of the blade hardness index is an independent confirmation that thin blade penetration resistance indicates sufficient bonding between snow crystals to give strength to macroscopic volumes of snow. This is the same conclusion with regard to blade hardness and unconfined compressive strength found by Fukue (1977). It is concluded that the blade hardness can be used to classify snow as cohesionless for B ≈ 0 N and cohesive for higher values of B. Future research will aim to more precisely quantify this threshold. 80 3.5.6 Applications The blade hardness gauge developed in this study can be easily adopted by avalanche forecasting and control operations that still rely heavily on snow pit observations. Many operations also cannot afford the cost of a probe penetrometer and are disinclined to adopt technology that requires postprocessing, a steep learn- ing curve, or any subjective judgements. The blade hardness gauge was designed to complement existing observation techniques rather than attempt to eliminate the need to dig a snow pit. The blade hardness is an intuitive measure, analogous to the hand hardness test which is common in avalanche operations. As a research tool, the blade hardness measure shows promise as an objective proxy for macroscopic properties of interest in avalanche applications and snow mechanics generally. The blade hardness gauge could be used to characterize the strength of thick persistent weak layers that are commonly related to slab avalanches (McClung and Schaerer, 2006). For example, the gauge could track the relative hardness of a newly buried weak layer, and the storm snow overlying it, as they both evolve and gain (or lose) strength. The gauge could also be used to track the loss of cohesion in snow during facet formation or as it approaches the melting temperature. The blade hardness may also be useful for characterizing the strength of snow at higher densities, such as in firn snow. A smaller blade could be used in such a scenario because the rationale for the 10 cm length scale related to avalanches would not apply. This would reduce the potential for blade bending or twisting in stiffer snow. A higher capacity force gauge would be necessary, though, and there would likely be a limiting density beyond which a blade could no longer be pushed into the snow. 3.5.7 Limitations In principle it would be desirable to have larger sample sizes for many of the hypothesis tests and other comparisons made in this study. Given the destructive sampling technique and the size of the blade, however, this was often not possible. The area that would be taken up by increasing the number of tests would increase the dependence of the results on the spatial variability of snow properties, making conclusions more difficult to draw even if the hypothesis test results appeared more robust. Moreover, part of the motivation in the development of this gauge was to provide a fast, supplemental piece of information related to or dependent on snow microstructure rather than to investigate in detail the microstructure itself. 81 The types of snow investigated in this study were limited to what was available in the natural snow cover. The range of most properties (Table 3.2) is appropriate for avalanche applications. Most of the tests in this study were done in dry snow. A limited number of tests in moist snow were carried out, but not enough to test for any significant differences with dry snow. Further testing would need to be done to determine how the penetration resistance changes in moist to wet snow. Additional work also needs to be done to relate the cohesion threshold identified in this study to the cohesion threshold at which slab avalanches first begin to occur in storm snow. Snow avalanches are reported in the hand hardness index range of 1–2 (Schweizer and Jamieson, 2001), though a large uncertainty exists in these values. The lowest hand hardness values for samples that could be handled in this study were in the range 1.7–2.7, which suggests that some slab avalanches may occur in snow that is weakly cohesive but too weak to be handled for testing. There is a potential boundary condition effect associated with measuring hardness in the wall of an excavated snow pit and using the results to characterize the properties of snow in situ where the stress state is different. The observations in the present study, with the COV on the order of 0.1–1 for closely-spaced clusters of resistance values in homogeneous snow, suggest that measuring the influence of internal stress amid the spatial variability of natural snow would be difficult. Moreover, the relative hardness of adjacent layers is often as important a piece of information as actual hardness scores in stability evaluation (e.g. Schweizer and Jamieson, 2007). The capacity of the digital force gauge used in this study did not match the operating range, which likely contributed to the observed sensitivity problems at the bottom 1% of the scale. We did not conduct any calibration tests covering the bottom 5% of the scale, so we can only speculate as to the origin of the observed 1.7 N threshold penetration resistance. Temperature effects on the load cell also likely played a role. Errors introduced by temperature effects were likely larger in the laboratory results than the in situ results. As most of the data was obtained at ambient temperatures between 0 and -10◦C, the relative shift in the zero point of the load cell across the data is small (about 0.2 N). A load cell with a capacity in the range 30–50 N with at least 0.1 N resolution and better than 1% accuracy would be more appropriate for future investigations with a 10 cm blade. Additional calibration procedures should be conducted to more precisely characterize the function of the gauge at low temperature 82 and low load. Comparing the results in the present paper against those obtained with a more appropriate load cell will be the subject of future work. 3.5.8 Conclusions A thin blade hardness gauge was developed that characterizes an average penetration resistance over a length scale appropriate for the continuum characterization of snow properties relevant to avalanches. Horizontal compaction of snow around the blade is minimized relative to all other common hardness measures. The gauge is an inexpensive, small and lightweight tool that can be used in the field or lab with results that are objective and consistent across observers. The measurement technique is simple and adds little time to other experimental methods. Compared to other standard measurements such as density, temperature and grain size, the blade hardness index was the best variable for correlating with the tensile strength of snow, one of the most important properties in the triggering and release of slab avalanches. 83 Chapter 4 Tensile Strength of Dry Alpine Snow The tensile strength of snow has long been viewed as a fundamental property related to the release of slab avalanches. Whether early investigators believed that the initial fracture which triggered an avalanche was in shear beneath the slab or tension through the slab, the coherent properties of slab snow have been viewed as important. This recognition is reflected in the numerous studies of the tensile strength of cohesive alpine snow, beginning as early as the 1930’s and continuing through to the present. In many ways, the measurement of tensile strength has been easier than the characterization of the structural factors that influence strength. However, many different types of tests have been conceived and conducted for measuring the strength of snow in tension. These tests span a wide variety of sample volumes, loading geometries, environmental conditions, and strain rates. In most tests only a small portion of the total sample volume is highly stressed and responsible for the failure of the sample. This is typically due to the presence of induced stress concentrations associated with gripping the sample or localizing the failure. The variability in testing conditions associated with the existing strength data, compounded with the microstruc- tural variability both within and across data sets, leads to much difficulty when comparing or synthesizing data from different sources or selecting representative data for a given application. The first section of this chapter contains a review of published data on the tensile strength of seasonal snow. The most widely used variable which is common across data sets, and often the only reported variable available to address the numerous factors which influence strength, is the density. The data come from uniaxial and bending tests performed in situ and in cold labs. Around 2000 tensile strength tests from 20 84 sources are synthesized, primarily as a function of density. Where additional information is available, the dependence of strength on hardness, loading rate, sample size and temperature are reviewed. In the second section of this chapter, new data from the current study is introduced. The data come from three and four point bending tests on unnotched beam samples. A total of 245 tests were conducted over the course of 20 days in the cold lab in the winters of 2007-2008 and 2008-2009. The derivation of the equations for calculating the tensile strength is first presented, followed by an exploratory analysis of the dependence of the strength on the sample density, blade hardness index, grain size, specimen size, loading rate, and beam slenderness. The third and last section contains univariate models of tensile strength fit through the data from the present study and several other representative studies. A common power-law formulation for the strength as a function of density is used in each case, and several models of strength as a function of hardness are explored where appropriate. Numerous graphical and statistical diagnostics of the model residuals are explored in detail to assess the assumptions inherent in these models, judge the goodness of fit of each model, and choose the best univariate model for representing the data. 4.1 Review and Analysis of Previous Data As early as the 1930’s investigators in Switzerland began to measure the strength properties of snow, adopt- ing many experimental techniques from the soil sciences. The first laboratory uniaxial tensile tests were reported by Haefeli (1939) (translated in Bader et al., 1954). Shortly thereafter, also in Switzerland, the centrifugal tensile testing method was developed (Bucher, 1948) which became the predominant method for measuring tensile strength for the next 30 years. Starting in the late 1960’s, in situ methods were de- veloped for testing the properties of undisturbed natural snow. These methods allowed for larger specimen sizes which were believed to be more appropriate for relating to avalanche activity than the small specimens typically used in lab tests. Section 4.1 here is organized first around a discussion of the extensive centrifugal strength data (Section 4.1.1). Next, in Section 4.1.2, the results of in-situ tests are reviewed, followed by strength data from laboratory measurements in Section 4.1.3. 85 4.1.1 Centrifugal tests Test description The 12 data sources reviewed in this section are listed in Table 4.1, which indicates the variables and proper- ties that were reported for each study. The only property which was reported in every study was the strength itself. All but one study reported the density of the snow for each test. Beyond these two properties, one a structural and the other a material property, the types of variables which were reported varied widely. The general procedure for the centrifugal tensile testing was as follows. Snow samples were first ex- tracted from the snowpack using a cylindrical tube. The primary axis of the tube was typically oriented parallel to the slope (rather than toward the ground). This ensured that the sample was from one distinct stratigraphic layer and did not contain weak layers or interfaces between layers of different properties. The tube was weighed prior to testing in order to calculate the snow density. The sample was then slid into the tester and gripped about the center using clips, as illustrated in Figure 4.1a. This clip system reduced the central cross section of the sample, introducing a volumetric and geometric stress concentration. Once gripped, the samples were spun about an axis normal to the axis of the cylinder. The spin rate was increased until the sample failed in tension. The location of failure was generally reported as being in the middle of the sample between the clips (due to the stress concentration). Some low density samples were reported to fail at outer points along the cylinder in low density snow, but these points were discarded (e.g. Keeler, 1969; Martinelli, 1971). In data sets where it was not explicitly stated, I assumed that all samples failed on a plane in the central cross section. It should be emphasized, however, that in most studies no information was given about the location of failure or the exclusion of data points based any criterion. For calculating the nominal strength, the rotational speed at failure was recorded in some fashion. Early versions of centrifugal testers required the operator to observe a dial and manually record the spin rate at the time of sample failure. An improved tester design (Sommerfeld and Wolfe, 1972; Upadhyay et al., 2007) automatically recorded the spin rate at failure. Other investigators may have devised methods to automatically record the spin rate at failure, but very few details of the experimental setup (e.g. acceleration of spin tester, sample storage times, shape of geometric stress concentration) were reported in the sources reviewed here. Typically little more than the calculated nominal tensile strength and the snow density were published (Table 4.1). 86 Source σNu ρ R T E Loading Rate Permeability Geometry n Bucher (1948) X X X X 71 de Quervain (1951) X X X1 X X 12 Bader et al. (1951) X X X X 9 Butkovich (1956) X X X X4 6 Roch (1966) X X X 64 Keeler (1969) X X X X4 150 Keeler and Weeks (1968) X X X X4 183 Martinelli (1971) X X X2 X X X X X4 104 Sommerfeld (1974) X X X5 158 Gubler (1978) X X 20 Schneebeli and Johnson (1998) X X X3 X4 45 Upadhyay et al. (2007) X X X X X 153 Table 4.1: Sources of data and variables reported in published centrifugal tensile experiments on cohesive snow. The nominal tensile strength is given by σNu , the density by ρ , the hardness by R, the temperature by T , the grain size by E. 1Three types of hardness measurements (rammsonde, a small spring-loaded conical tester and the flat plate Canadian hardness gauge) were reported. 2Ram hardness. 3Average SnowMicroPen resistance. 4Only the dimensions of the cylindrical cutter tube or reference to the geometry-specific nomogram developed by Bader et al. (1951) were given. 5Geometry of samples described in Sommerfeld and Wolfe (1972). 87 (a) (b) (c) Figure 4.1: General geometry of centrifugal tensile test specimen with two-pronged clip in the center (a), top view of standard test specimen (b) and relative size and shape of specimens used by Sommerfeld (1974) (c). Strength calculation For a cylindrical sample of radius R spun about an axis normal to the cylinder axis, the differential centrifu- gal force acting on a differential disc of mass dm is given by dF = dmΩ2r = ( ρpiR2dr ) Ω2r (4.1) where Ω is the angular frequency (rad s−1) and r is the radial distance of the disc from the center of the cylinder. This relation is integrated over the half-length l/2 of the cylinder to get the total centrifugal force acting on the central cross section of the cylinder: F = ∫ r=l/2 r=0 ρpiR2Ω2rdr = 1 8 ρpiR2Ω2l2 = 1 2 ρpi3R2 f 2l2 (4.2) where angular frequency Ω has been expressed as 2pi f where f is the number of revolutions per second. The result was often reported in the literature in terms of the number of revolutions per minute N in relation to the angular frequency Ω via Ω = 2piN/60. The nominal tensile strength σNu is found by dividing the 88 maximum force in Equation 4.2 (from the value of f at failure) by the effective cross sectional area, which is reduced from the gross area due to the two-pronged clip that holds the sample in the center (Equation 4.1). Expressing the result in terms of frequency f leads to σNu = F Ae = 1 2Ae ρpi3R2 f 2l2. (4.3) Originally reported data Figure 4.2 shows a summary of nominal centrifugal tensile strength data from the 11 sources in Table 4.1 that reported the snow density (Gubler (1978) is the study that did not report density). For a given density, the scatter in strength covers up to two orders of magnitude. This in large part reflects changes in loading rate, snow hardness, temperature, and snow microstructure between and within data sets. The lowest mean strength values came from Sommerfeld (1974), who also had specimen sizes a factor of four larger than the standard sample adopted by the rest of the studies. The data of Keeler and Weeks (1968) are systematically higher, by a factor of 2-3, than any other source. This difference could be attributable to differences in hardness or microstructure at the same density compared to the previous studies at different field sites. It could also be due to the presence of some moist or wet snow samples which were subsequently refrozen prior to testing. Keeler and Weeks reported that not all of their tests were done before the isothermal transition at the end of the winter season, so some of the samples may have been moist or wet when extracted and then refrozen by the time the tensile test was carried out. A further explanation could be related to a different stress concentration in the central cross section resulting from a differently shaped pronged clip that held the samples into place. The nomogram developed by Bader et al. (1951) for relating the spin rate at failure and the mass of the sample to the nominal strength was referenced in the strength calculations of Keeler and Weeks (1968), but if any geometric factors were different from those which went into the derivation of that particular nomogram, the results would have been systematically biased. No mention was made of any change in experimental technique, however. A final factor may have been sample storage for long periods of time prior to testing, which would likely promote sintering and an increase in strength at constant density. However, no indication was given on whether samples were stored or, if so, for how long. 89 0 100 200 300 400 500 Density [kg/m3 ] 10-1 100 101 102 103 N o m i n a l t e n s i l e s t r e n g t h [ k P a ] Bucher (1948) deQuervain (1950) Bader (1951) Butkovitch (1956) Roch (1966) Keeler & Weeks (1968) Keeler (1969) Martinelli (1971) Sommerfeld (1974) Schneebeli & Johnson (1998) Upadhyay (2007) Figure 4.2: Originally reported centrifugal (nominal) tensile strength as a function of snow density. These data represent the majority of the most commonly cited tensile strength data in avalanche applications. 90 Stress concentration in notched samples The nominal strength values as originally reported in each study were defined using the nominal stress at failure (Equation 4.3). As mentioned above, the notched cross section caused by the two-pronged clip that held and spun the samples in the centrifugal tests introduced a geometric stress concentration. This stress concentration was mentioned by Sommerfeld and Wolfe (1972), but was never accounted for in the strength calculations of any of the studies reviewed here. If the tensile strength was to be equated with the maximum tensile stress at failure, a more consistent definition than using the nominal stress, the data as originally reported in the literature need to be corrected. Otherwise the nominal strength values from tests with different geometries, and therefore different stress concentrations, could not be directly compared. Relations were thus sought for a stress concentration factor to relate the nominal to maximum tensile stress for the geometry of the centrifugal tests. The stress concentration factor Ktn relates the nominal tensile stress σN to the maximum stress σmax via Ktn = σmax σN . (4.4) The subscript ’n’ in the Ktn term indicates that the stress concentration factor uses the net cross sectional area of the specimen for the nominal stress calculation (the gross section can alternately be used). Stress concentration factors were calculated or estimated for the data sources represented in Table 4.1, and the tensile strength ft was equated with the maximum stress at failure σmax. The key assumption associated with this argument, then, is that the tensile strength ft should correspond with the maximum stress at failure rather than the nominal stress, which can vary widely depend on the presence and geometry of stress concentrators. Stress concentration factors for the net cross section were taken from Pilkey (1997). The numerical values used to correct the published data were taken as the average of Ktn for the flat tension specimen with a U-shaped groove (Figure 4.3a and c) and the round bar with a circumferential groove (Figure 4.3b and d) since the geometry of the actual snow specimens (round sample with straight notches/grooves) was not available. This averaging led to Ktn = 2.25 for the “standard” data using the sample and notch geometry reported by Bader et al. (1951). Sommerfeld (1974) improved upon the standard design of the centrifugal testing machine to reduce the stress concentration and increase the size of the sample, leading to Ktn = 1.28. 91 Of all the studies listed in Table 4.1, only Bader et al. (1951) and Sommerfeld (1974) reported the spe- cific shape and dimensions of the notched central cross section. Most other studies in Table 4.1 reported only the diameter and length of the cylindrical sampling tube, neglecting to specify the shape and size of the notch induced by the pronged sample holder and the resulting effective cross sectional area. Five of the sources in Table 4.1, comprising nearly 500 tests, referred to the nominal strength equation reported by Bader et al. (1951), which took the form of Equation 4.3 with numerical values for Ae, R, and l already plugged in (as reported: σNu = 1.166× 10−9MN2, where M is the sample mass, N is the number of rev- olutions per minute at failure and σNu is in kg/cm2). For the present analysis, any literature sources that referenced this equation were given the benefit of the doubt that they indeed used the exact same geometry to justify the use of the published geometry-dependent equation. However, not enough information was typically published to critically evaluate this assumption. For example, it may not be safe to assume that each study used pronged clips of the exact same size and shape to notch and hold the cylindrical samples during testing. There is therefore some uncertainty in the actual value of Ktn appropriate for most studies. (a) Ktn=2.434 (b) Ktn=2.132 (c) Ktn=1.321 (d) Ktn=1.244 Figure 4.3: Standard geometries used to approximate the stress concentration factor Ktn. Geome- tries in (c) and (d) are for the samples tested by Sommerfeld (1974), (a) and (b) are for all others (not drawn to scale). A further complication arises from the nonuniform tensile stress as a function of radius about the center of rotation. The common stress concentration factors referenced from Pilkey (1997) are calculated based 92 on the assumption of a uniform, remotely applied load. The total force acting on the central cross section, calculating using Equation 4.2, is appropriate in a nominal or average sense. However, the relationship between locally concentrated stresses and bulk nominal stresses in a nonuniform stress field may be different than characterized by the stress concentration factors considered here. Though from the perspective of obtaining simple estimates, the approach considered here and the resulting values seem reasonable. The centrifugal tensile strength data, adjusted to account for the stress concentration factor, are shown in Figure 4.4. The data of Schneebeli and Johnson (1998) were excluded on the basis of lack of specific information about the shape of the notched cross section. The authors simply describe a “sharp” notch, a description that differed from others regarding the shape of the cross section created by the pronged clip. That authors also did not describe the testing procedure in any detail compared to other studies. These concerns called into question the use of the stress concentration factors considered for the rest of the studies, and it was deemed most appropriate to neglect the data from further analysis. The adjusted strength values of Keeler and Weeks (1968) approach 1 MPa, which is near the tensile strength of pure ice (≈1.5 MPa, Schulson (2001)). According to previous compilations of tensile strength data (e.g. Mellor, 1975) the tensile strength of snow at a density of around 400 kg/m3 is still about an order of magnitude lower than that of ice. However, the data reviewed by Mellor are largely contained in this study, so the argument is somewhat circular, though no corrections for the stress concentration were made in previous studies or compilations. Given the large amount of data overlap between other studies, the data of Keeler and Weeks (1968) appear questionably high. The mean strength of all the data in Figure 4.4 at a density of about 400 kg/m3 (including the data of Keeler and Weeks (1968)) is around 20–30% of the tensile strength of pure ice, which is physically reasonable. 93 0 100 200 300 400 500 Density [kg/m3 ] 10-2 10-1 100 101 102 103 C o r r e c t e d t e n s i l e s t r e n g t h [ k P a ] Bucher (1948) deQuervain (1950) Bader (1951) Butkovitch (1956) Roch (1966) Keeler & Weeks (1968) Keeler (1969) Martinelli (1971) Sommerfeld (1974) Upadhyay (2007) Figure 4.4: Centrifugal tensile strength versus density, corrected for the stress concentration using Equation 4.4 and the stress con- centration factors in Figure 4.3. 94 Systematic error associated with reaction time Early versions of the centrifugal tensile tester had no automatic way to shut off the spinner when the sample failed nor to record the precise spin rate at failure. The operator had to read a dial indicating the spin rate when the failure of the sample was seen or heard. This procedure could have introduced a systematic bias related to the reaction time of the observer. This error can be estimated by calculating what the actual spin rate at failure may have been by using the reported spin rate minus the increase in spin rate associated with the reaction time of the observer: fcorrected = frecorded− (reaction time)× d fdt (4.5) where d f/dt is the acceleration of the spin tester. Table 4.2 shows the acceleration values for the only four studies that reported this information. If the acceleration is known and considered constant for a given data set, and a constant reaction time is assumed, then the recorded strength data can be corrected with Equation 4.5. Source d f/dt [rev/s2] Keeler and Weeks (1968) 10 Keeler (1969) 10 Martinelli (1971) 2-3 Upadhyay et al. (2007) 0.13, 0.68 Table 4.2: Sources that published the rate of acceleration d f/dt of the spin tester, allowing cal- culation of the stress rate and strain rate at failure. Equation 4.5 predicts that the systematic error decreases as the recorded value of the spin rate at failure increases (Figure 4.5), since the second term on the right hand side of the equation can be considered constant for a given data set. The slower acceleration rate of the Martinelli (1971) data result in a lower systematic error, as evident in Figure 4.5. The systematic error would be the highest in weak snow, which necessarily fails at a lower stress rate (and thus spin rate) at failure. The strength data of Keeler and Weeks (1968) are the highest in Figure 4.4 and also come from the highest reported spin tester acceleration (Table 4.2). Equation 4.5 and Figure 4.5 suggest a possible systematic bias of over 50% for the low density (and thus low strength) values of Keeler and Weeks (1968). For the strongest samples the bias falls to 5-10%. 95 Spin rate at failure (rps) Sy st em at ic bi as in te ns ile s tre ng th (p erc en t) 0 20 40 60 20 40 60 l l l l l l l l l ll l l l ll l l l l l l l l l l l l l l l lllll lllll lll lll ll l llll l llll ll llll l ll ll l l ll Keeler (1969) Keeler and Weeks (1968) Martinelli (1971) l Figure 4.5: Estimated systematic bias in tensile strength values as a function of the spin rate at failure in revolutions per second. The spin rate is related to the nominal strength via Equation 4.3. A reaction time of 0.2 seconds was assumed. No mention is made of the specific technique by which the failure was observed in most of the centrifu- gal tensile strength literature, whether an automatic shutoff was present, or whether any consideration or compensation for a systematic error associated with reaction time was made. Exceptions are Sommerfeld (1974) and Upadhyay et al. (2007) who did have optical automatic shutoff mechanisms which recorded the failure spin rate. Serious systematic errors may be present in other centrifugal data but, in the absence of any mention of the rate of acceleration of the tester, no estimate can be made of its importance. 96 Rate effects The stress rate at failure can be calculated by differentiating Equation 4.3 with respect to time t: dσN dt = 1 Ae ρpi3R2l2 f d f dt (4.6) where d f/dt is the rate of acceleration of the spin tester, which was only reported by four studies (Table 4.2). The spin rate at failure can be found from Equation 4.3 given the reported density, nominal tensile strength and dimensions of the snow sample. No information about the diameter of the samples was reported by Upadhyay et al. (2007), though the cylinder length was reported. For the following analysis I assumed that the samples of Upadhyay et al. had the same diameter as those listed above. Approximate calculations of the strain rate at failure can be made from the stress rate in Equation 4.6. These approximate calculations can be used to determine where the failure falls with respect to the creep-to- fracture transition strain rate. Below this transition, the strain rate is low enough that creep effects dominate the deformation and clean fractures cannot originate or propagate. Above this transition, elastic effects dominate over creep effects, and clean, fast fractures are observed. For dry snow at the laboratory scale, this transition occurs in tension at ∼ 10−4 s−1 (Narita, 1980, 1983). The strain rate can be calculated using the stress rate from Equation 4.6 if an assumed stress-strain behaviour is used, such as ε̇ = σ̇ E (4.7) where σ̇ = dσN/dt from Equation 4.6 and E is an effective Young’s elastic (or storage) modulus appropriate for the strain rate and sample properties. This simple linear elastic relation is not necessarily appropriate physically, since the stress rate varies parabolically in time and snow is a strongly rate-dependent material, i.e. E = E(t). This will likely lead to some decaying viscous effects in the deformation of the sample. Therefore, the relation in Equation 4.7 is not appropriate for modeling the stress rate-strain rate behaviour at any arbitrary point in time for the centrifugal tests. However, for order-of-magnitude calculations of the strain rate at failure, the simple relation in Equation 4.7 is a reasonable starting point. Critical to this calculation is the selection of a representative single value for the elastic modulus E (or effective secant modulus if considering viscous effects) for a test that starts out viscous and ends somewhat 97 less viscous, perhaps only approaching elastic right at failure. From the perspective of evaluating failure strain rates with respect to the creep-to-fracture transition rate, an upper bound estimate for E is more appropriate as it will lead to a lower bound estimate of ε̇ . This is important because the slab tensile fracture in an avalanche occurs following the propagation of shear fracture beneath the slab at a speed of around 20 m/s (McClung, 2007a). This high speed, combined with observations of clean, planar crown (tensile fracture) surfaces in slab avalanches, suggest that the tensile strain rate in the slab when it fails is above the creep-to-fracture transition. Therefore, lower bound estimates for the failure strain rate allow more conclusive decisions to be made about the applicability of test data for comparing with avalanche conditions. Cyclical loading tests conducted at a frequency of 100 Hz were reported by Sigrist (2006) for the calcu- lation of a dynamic Young’s modulus. The strain rate in these tests was on the order of 10−3 s−1. Expressed as a function of density, Young’s modulus from these data takes the form E = 1.89×10−6ρ2.94 [MPa]. (4.8) This equation was used for the calculations of the failure strain rate in Equation 4.7. The modulus values predicted by Equation 4.8 may approximate the true value right at failure but are likely too large for the early portion of the test or even for some kind of average value. They may in fact be too large by an order of magnitude. However, as discussed above, they provide lower bound strain rate estimates. Figure 4.6 shows the strain rate at failure for data from the four studies that reported the acceleration of their spin testers (Table 4.2). The strain rate at failure is clearly in the brittle range for the data from Keeler and Weeks (1968), Keeler (1969) and Martinelli (1971). In this discussion, “brittle” is taken to mean “above the creep-to-fracture transition” and should not be confused with the distinction between brittle and quasi-brittle fracture, which is primarily a distinction of relative length scales, not time or rate scales. Conversely, “ductile” is taken to mean a strain rate below the creep-to-fracture transition. Upadhyay et al. (2007) varied the acceleration rate between two different test series, as is evident from the bimodal distribution of strain rates in Figure 4.6. In both cases, however, the rate was much lower than used by previous investigators (at least those that reported the rotational acceleration) and the failures clearly fall in the ductile range. Lower modulus values in this case, which would result in higher strain rate estimates, would probably not shift the strain rates above the creep-to-fracture transition. This probably 98 explains the lower values of tensile strength reported by Upadhyay et al. (2007) (Figures 4.2 and 4.4) and suggests that the data should be classified separately from the rest of the brittle data. log(ε⋅) [ s−1 ] Ke rn e l d en sit y 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −7 −6 −5 −4 −3 −2 Keeler (1969) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Keeler and Weeks (1968) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Martinelli (1971) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Upadhyay (2007) Figure 4.6: Kernel density plot of the estimated strain rate at failure for sources that reported the rate of acceleration of the spin tester. The vertical line represents the approximate transition between creep rupture for strain rates below 10−4 s−1 and fast fracture for rates above. Temperature effects Roch (1966) reported the effect of temperature on strength, largely from data originally reported by Bucher (1948). Figure 4.7 shows the strength as a function of temperature, plotted as the ratio of the strength at the given temperature divided by the strength at the highest temperature for the given test series. Typically only one strength measurement was made at each temperature. Most of the data in Figure 4.7a indicate roughly a doubling of the strength as the temperature decreased 99 from 0◦C to about −30◦C. For reference, the tensile strength of fresh-water ice increases by just 10% as the temperature is decreased between the same limits (Schulson and Duval, 2009). This is an indication that rate-dependent viscous effects in the data of Roch (1966) are likely clouding the temperature dependence. An empirical correlation based on the data from Roch (1966) has been used by a number of investigators (e.g. Bader et al., 1951; Butkovich, 1956) to normalize experimental data to the same temperature, but this may not be appropriate in light of these viscous effects. Temperature [C]T e n si le s tre ng th / Te n si le s tre ng th o f w a rm e st s am pl e 2 4 6 −40 −30 −20 −10 0 l l ll l l ll l l l l Density 196 216 250 280 295 300 305 325 330 346 372 375 390l l l (a) Temperature [C]T e n si le S tre ng th / Te n si le s tre ng th o f w a rm e st s am pl e 1.0 1.5 2.0 2.5 3.0 3.5 −10 −8 −6 −4 −2 0 l l l l l l Density 196 216 250 280 295 300 305 325 330 346 372 375 390l l l (b) Figure 4.7: Ratio of the tensile strength at the given temperature to the strength of the warmest sample of the test series, from data reported by Roch (1966). Both graphs contain the same data, with (b) limited to temperature differences between −10◦C and 0◦C. Hardness effects Both Martinelli (1971) and de Quervain (1951) reported the rammsonde (ram) hardness for the layers in which they also measured the tensile strength. Figure 4.8a shows the nominal tensile strength versus ram hardness. The data show a generally linear trend of increasing strength with increasing hardness, though with large scatter. Figure 4.8b shows the same data plotted against the density, with plot symbols and colors binned according to the associated hardness value. The trends in Figure 4.8 suggest that, for these data, the ram hardness is as good if not better than the density for serving as a single predictor for the tensile strength. Recall from Chapter 3 that the ram hardness and the density had nearly equal and very strong correlations 100 with the nominal tensile strength in the data of Martinelli (1971). Ram hardness [N] N om in al te ns ile s tre ng th [k Pa ] 0 50 100 150 200 250 0 500 1000 1500 2000 l l l l ll l l l l l de Quervain (1950) Martinelli (1971) l (a) Density [ kg m3 ] N om in al te ns ile s tre ng th [k Pa ] 0 50 100 150 200 250 100 200 300 400 500 ll lllllll l ll ll ll ll l l l Ram hardness [N] (0,30] (30,150] (150,300] (300,2.3e+03] l (b) Figure 4.8: Nominal centrifugal tensile strength versus (a) ram hardness and (b) density for the only two sources that reported any hardness data. Size effects Nearly all of the centrifugal data are from specimens of the same size. The exception is the data of Sommerfeld (1974), who used larger specimens of a slightly different shape. The strength values of Sommer- feld’s data are on the low end of the centrifugal data. For densities greater than 200 kg/m3, Sommerfeld’s strength values are nearly an order of magnitude lower than much of the rest of the data (Figure 4.4). This difference could be in large part due to the significant size effects between Sommerfeld’s specimens and those of the rest of the studies considered here. The size effect could be explained either in statistical or deterministic terms. According to Weibull statistical theory related to brittle fracture, the mean strength σ̄N for a cross- sectional area A is related to the mean strength σ̄◦N at another size A◦ via σ̄N σ̄◦N = ( A◦ A )nd/m (4.9) where nd is the similitude dimension and m is the Weibull modulus (Bažant and Planas, 1998). Several 101 reasons why Weibull theory is probably not applicable to explain the size effect for snow slab fractures were outlined by Borstad and McClung (2009) and will be discussed further in Chapter 5. However, Equation 4.9 was applied here for simplicity and to get a rough idea if the difference between strength values between Sommerfeld (1974) and others can be attributable to some kind of size effect. Sommerfeld (1974) used large samples of effective cross-sectional area 81.6 cm2 compared to the previ- ous standard of 22.8 cm2 used by all others in Table 4.1. If Weibull theory was applicable with a modulus m = 15 (Borstad and McClung, 2009) and we considered the scaling of the cross-sectional area as a case of one-dimensional similitude (which physically corresponds to assuming that the entire cross-section fails simultaneously) then Equation 4.9 predicts a ratio of strengths of σ̄N σ̄◦N = ( 22.8 81.6 )1/15 = 0.92 (4.10) for the same snow properties and testing conditions. This is a small decrease (<10%) for the change in area and does not appear to entirely explain the low strength values reported. For low densities (<200 kg/m3) Sommerfeld’s data overlap with those of Martinelli (1971) (Figure 4.4). At higher densities, however, there appears to be nearly an order of magnitude or more difference between Sommerfeld’s data and those of other studies at similar densities. Calculated values of the Weibull modulus based on precise laboratory tests are certainly higher than 10 (Borstad and McClung, 2009). Lower values based on the coefficient of variation from imprecise in-situ tests are probably not applicable. However, some in situ field data have suggested values closer to m = 5. Using this value in Equation 4.10 leads to a prediction of about a 23% decrease in the strength. This still does not make up the observed discrepancy between Sommerfeld’s data and those of the other studies. If the failure of the cross-section is considered a two-dimensional scaling (i.e. the cross section fails as soon as a representative element within the cross section fails) then we have nd = 2. This would lead to a strength decrease of about 16% for m = 15 and 40% for m = 5 in Equation 4.10 above. In no case does it appear that a Weibull-type statistical size effect is capable of explaining the large discrepancy between the data of Sommerfeld (1974) and the rest of the centrifugal data. If a boundary layer of cracking near the notched cross section in the centrifugal tests is the origin of the tensile crack that precipitates failure, then a quasi-brittle scaling relation for failure at crack initiation might 102 be appropriate for considering the size effect between Sommerfeld’s data and the rest of the centrifugal data. A simple form for the quasibrittle size effect on the modulus of rupture (Bažant, 2005) is given by fr = fr∞ ( 1+ Db D ) (4.11) where fr is the modulus of rupture, fr∞ is the asymptotic large-size limit of strength, Db is the length scale related to the boundary layer of microcracking and D is the characteristic specimen dimension. Though this relation is for a bending test rather than a uniaxial test, it can be derived from dimensional analysis in a more general sense by considering a strain gradient in a boundary layer near the surface of a material where a tensile crack initiates. There should be a strong strain gradient in the central cross section of the centrifugal samples, therefore correspondence may be achieved which would allow the use of Equation 4.11 here. Therefore, as a complement to the size effect predictions made above using Weibull theory, Equa- tion 4.11 was applied to compare the difference in strength values predicted using D = 45.3 mm for the width of the central notched zone in the standard centrifugal specimen and D = 106.7 mm reported by Sommerfeld and Wolfe (1972). Using a boundary layer length scale Db of about 20 mm determined from ex- perimental data (Borstad and McClung, 2009) and assuming that fr∞ is a material property (Bažant, 2005), Equation 4.11 predicts about a 25% reduction in the strength of the larger samples, all else the same. In either formulation, the size effect alone does not explain the low strength values of Sommerfeld (1974). A different rate of acceleration of the spin tester is a likely partial explanation. Sommerfeld did not report the rate of acceleration of the modified (and otherwise much improved) testing machine, other than to state that the tester “accelerates rapidly.” However, insufficient information was reported to analyze the remaining discrepancy. 4.1.2 In situ tests The advent of in situ tensile tests offered a number of advantages to earlier centrifugal tests. First, the snow could be sampled in its natural state rather than extracted and transported to a lab with different environmental conditions. Second, larger specimen sizes could be tested. No laboratory facilities were necessary, which saved experimental costs, though the usual tradeoff was in the lower precision of in situ results. 103 Two primary means of measuring tensile strength in situ have been used. The first was by applying uniaxial tension by somehow gripping the snow sample and pulling on it while one end of the sample was fixed. The second was in bending, where typically a cantilevered beam was ruptured and the equations of beam mechanics were used to calculate the tensile stress at failure on the outer tensile fiber of the beam. Uniaxial tension The largest specimen sizes used to measure tensile strength were those of McClung (1979a) who used a rolling cart to slowly apply uniaxial tensile stress to naturally deposited snow. The stress was applied by gradually tilting the cart to increase the gravitational tensile body force while the upslope end of the sample was fixed relative to the downslope end. The geometry of the test specimens was similar to that in Figure 4.10 in the sense that rounded notches were cut into the samples to localize the failure. From the reported geometry of the samples, I estimated the stress concentration factor associated with these notches as about 1.5. McClung’s strength values are low, even after adjusting the reported nominal strengths using the stress concentration factor (Figure 4.9). Only a few samples exceed a (corrected) strength of 10 kPa. This is probably partly explained by the long loading times (on the order of minutes) for these tests, which likely placed the failures in the ductile (creep rupture) range of strain rates. No measurements or estimates of the strain at failure were reported. A size effect for these very large sample sizes (cross sectional area ∼ 0.12 m2) also likely led to lower strength values (Sommerfeld, 1980). Conway and Abrahamson (1984), and later Jamieson (1988) and Jamieson and Johnston (1990), devel- oped an in situ tensile test that involved inserting a slip sheet under a slab column to isolate the slab in tension. The slab was gripped on either side with frames that were pressed into the side of the snow (Figure 4.10). The frames were connected using a spreader bar, and the force to rupture the sample was applied by hand with a force gauge. Rounded notches were cut into the sample in order to localize the failure, though only the nominal strength values were reported. Conway and Abrahamson (1984) did not report the exact geometry of the notched cross section in their tests, though from their schematic drawings the stress concentration factor was estimated to be between 1.5 and 3. The error bars for their data in Figure 4.9 represent this uncertainty. Each data point represents a mean tensile strength from several tests (up to 4 tests were conducted within each layer). A total of 32 tests 104 0 50 100 150 200 250 300 350 Density [kg/m3 ] 10-2 10-1 100 101 102 Te ns ile s tr en gt h [k Pa ] Perla (1969) McClung (1979) Conway and Abrahamson (1984) Jamieson (1988) Figure 4.9: Previously reported in situ tensile and flexural strength data. All values except Perla’s were adjusted to account for the stress concentration factor associated with the notched tests. are represented by the data points plotted for Conway and Abrahamson (1984). The values measured by Jamieson (1988) were lower than those of Conway and Abrahamson (1984). This holds for both the nominal and corrected values, and is surprising given that Jamieson had more data for densities above 250 kg/m3, which was the highest density tested by Conway and Abrahamson (1984). Jamieson has more data overall, however, and the data appear more consistent than those of Conway and Abrahamson (1984). The data in Figure 4.9 from Jamieson (1988) represent a total of 457 tests in 66 different snow layers. The mean strength for each group is plotted, and the error bars represent the standard deviation in the measured strength values. Further discrimination of the strength values of Jamieson (1988) is possible using the reported snow grain types. For a given density, snow containing faceted crystals or mixed rounded and faceted crystals was 105 Figure 4.10: Schematic of in-situ uniaxial tensile test geometry, top view. The samples were isolated in tension by inserting a low-friction plate (shown here in gray) below the slab, which remained connected to the rest of the snowpack on the upslope end. The downslope end of the sample, below the notches, was gripped on either side with frames that were pressed into the slab. A spreader bar was then used to distribute the force across both sides of the sample. weaker than other snow types (Figure 4.11). Some grain forms are also indirectly correlated with density through the age of the snow layer (specifically the time since deposition and history of metamorphism). Low density snow was typically still composed of new snow forms and decomposing and fragmented forms, while older and higher density snow was mostly composed of rounded grains. Bending tests Perla (1969) developed an in-situ cantilever beam test for measuring the strength of fragile newly fallen snow. A wide cantilever was loaded by rapidly removing the supporting snow underneath. This was achieved by inserting a plate underneath the layer of interest and rapidly withdrawing it by downward pressure, leaving a freely-cantilevered beam loaded by self weight. The length of this beam was increased by inserting the plate deeper and deeper until the beam at some point failed when the plate was dropped out from under the beam. This determined the failure length of the cantilever (S). The stress at failure, defined as the “beam 106 Density [ kg m3 ] Te n si le S tre ng th [k Pa ] 0 5 10 15 100 150 200 250 300 350 l l l l l l l l ll l l l l l l l l l l l l l l l l Grain form DF FC multilayer MX PP RG l Figure 4.11: Tensile strength versus density reported by Jamieson (1988), with plot symbols cor- responding to different grain types: decomposing and fragmented (DF), faceted (FC), mixed rounded and faceted (MX), new snow or precipitation particles (PP) and rounded grains (RG). number” by Perla, is given by σ = 3S2ρg D (4.12) where S is the span, ρ is the layer density, g is the magnitude of gravitational acceleration and D is the layer thickness. Perla’s beam number data, which comprise around 280 tests, agree well with the uniaxial in situ results, though the scatter is large (Figure 4.9). Perla (1969) observed some evidence for shearing deformation from the shape of the failed cross sec- tions, which would lead to a slight overestimation of the true tensile strength of the tested layers using Equation 4.12. It should also be noted that viscous deformation was likely in many of these tests. The amount of deformation would depend on the strength of the snow, which would determine the time that the 107 beam, of progressively increasing length, was cantilevered freely. 4.1.3 Laboratory tests Similar to the in situ tests, previous laboratory tensile tests (excluding the centrifugal tests discussed above) used both uniaxial tension and bending to induce failure. The laboratory tests allowed for precise control over loading rate and environmental conditions compared to the in situ tests, though often at the expense of smaller data sets. Uniaxial tension The earliest reported tensile strength measurements were by Haefeli (1939), translated in Bader et al. (1954). Uniaxial tensile tests were conducted on homogeneous cylindrical samples of cross sectional area 26.4 cm2 and length 19 cm. The ends of the samples were frozen to metal plates with roughened surfaces. Each strength test was preceded by an elongation test of duration 1–8 days at tensile stresses ranging from 2–40 kPa. Higher stresses and longer duration of pre-stressing generally led to higher tensile strength values, though the data are not consistent nor extensive enough to draw any firm conclusions on these points. The tests were performed under load control at a rate of approximately 570 Pa/s. The time to failure for each sample was reported, with values falling in the range 26–211 seconds. These times to failure are well above the limit later specified by Bader and Kuriowa (1962) of 10 seconds or less to avoid inelastic deformation. No strain measurements were reported to allow the calculation of the strain rate. The large variability in Haefeli’s results is likely due to both the scatter in material properties, differences in pre- stressing and also likely due to the load application method, which involved the pouring of shot into a bucket (small impulse loads were reported to cause failure in some tests). Narita (1980) and Narita (1983) reported the uniaxial tensile strength of natural snow over the largest range of strain rates in the literature, spanning rates from about 5× 10−7 s−1 to about 2× 10−3 s−1. The tests were carried out in a similar manner to those of Haefeli, by freezing cylindrical samples (cross sectional area about 20 cm2) to end plates that were then pulled apart by a universal testing machine. Narita identified a creep-to-fracture transition strain rate (termed ductile-to-brittle transition at the time) from several character- istics of the data. First, the load displacement curves were distinctly different for brittle (fracture-dominated) compared to ductile (creep-dominated) failures. Second, visual observation of the failure surfaces allowed 108 characterization of the type of failure into ductile (multiple and uneven cracking) or brittle (clean and fast fracture) modes. The tensile strength, for a given density of snow, peaked at the transition strain rate (∼ 10−4 s−1). Narita reported the strength values as a function of the strain rate for different bins of density, therefore individual strength-density points cannot be reproduced. However, there is general agreement between Narita’s uniaxial values and those of Haefeli (1939) in Figure 4.12. These strength values agree well with the uniaxial centrifugal data (Figure 4.4) discussed above, much more so than the in situ data (Figure 4.9). 50 100 150 200 250 300 350 400 450 500 Density [kg/m3 ] 100 101 102 Te ns ile s tr en gt h [k Pa ] Haefeli (1939) Sigrist (2006) Narita (1980) type 'a' Narita (1983) type 'a' Figure 4.12: Previously reported tensile/flexural strength data from laboratory tests. The data shown from Narita is limited to brittle-rate fractures (type “a”), or those above the creep-to- fracture transition, with strain rates of about 10−4–10−3 s−1. 109 Bending tests Sigrist (2006) calculated the modulus of rupture from unnotched three point bending tests (see Figure 4.13a for a schematic of the test geometry). The crosshead speed was 0.33 cm/s which resulted in strain rates in the outer fiber of 10−2 to 10−1 s−1. The beam span to depth ratio S/D was 2 and the temperature was -9.5 ± 0.2 ◦C for all tests. The sample length was 50 cm (load span 40 cm), the beam depth D was 20 cm and the width was 10 cm. Sigrist reported the modulus of rupture as the tensile strength, but this only holds for very slender beams in pure bending. For short beams with concentrated central loads, such as in the common laboratory three point bending test, the elastic stress distribution in the central cross section is altered from that predicted in pure bending. Timoshenko and Goodier (1951) related the elastic tensile stress to the nominal stress in three point bending as a function of the span to depth ratio, leading to the following expression for the tensile strength ft in terms of the measured modulus of rupture fr: ft = fr ( 1−0.1773D S ) . (4.13) This expression was used to correct Sigrist’s data as presented in Figure 4.12. The bending data overlap somewhat with the uniaxial lab data for higher densities. In general, however, the beam tests resulted in lower strength values. There may be a small size effect for comparing the results of Sigrist (2006) to the uniaxial lab data. Furthermore, uniaxial tensile strength usually differs from flexural strength for the same material, with flexural strength being around 50% greater than tensile strength for concrete (e.g. Banthia and Sheng, 1996). Summary The results of around 2000 previous tensile strength tests have been reviewed in the first part of this chapter, summarized by test type, loading geometry and sample properties. The published values, many of them corrected to account for neglected stress concentrations, span four orders of magnitude from 0.1 kPa to 1000 kPa over a density range of 30–500 kg/m3. The next section (Section 4.2) contains new strength data collected from laboratory bending tests in the present study. In a similar manner to the foregoing section, the influence of sample properties and testing conditions on the results are shown. 110 4.2 New Tensile Strength Data A total of 245 unnotched beam bending tests were conducted for the calculation of tensile strength (techni- cally, the flexural strength) in the present study. The tests were performed over the course of 20 days in the cold laboratory in the winters of 2007–2008 and 2008–2009. The tests were split among three point bending (n = 149) with the testing machine oriented vertically and four point bending (n = 96) with the machine oriented horizontally. All tests were weight compensated. Descriptions of the characteristics of the snow and the testing conditions for each individual test series are indicated in Table 4.3. The majority of the results in the present section were obtained with a crosshead speed (V) of 1.25 cm/s and beam depth (D) of 10 cm. However, some of the unnotched test results here were drawn from series of tests on a particular day that were conducted to explore secondary variables such as loading rate, specimen size, or notched versus unnotched tests. For the analysis here only the unnotched data from such test series were used, and this explains the small number unnotched tests on some dates (Table 4.3). This section begins with the derivation and definition of tensile strength for the data using beam theory. The strength results as a function of hardness and density are then discussed and placed in context with the published laboratory strength data discussed earlier in the chapter. In some test series, the influence of secondary variables such as grain size, loading rate and specimen size are explored. 111 Code Date n ρ̄ [kg/m3] R B̄ [N] T̄ [◦C] F, E [mm]1 V [cm/s] D [cm] S/D Type A 080115 7 185 ± 2 3.3 N/A -11.1 ± 0.3 RG, 0.5 / DF, 1 1.25 10 3 4PB B 080117 24 318 ± 2 4 7.1 ± 0.82 -7.1 ± 0.5 FCxr, 0.5-1 1.25 5,10 3 4PB C 080118 20 327 ± 2 4.3 8.5 ± 0.92 -8.5 ± 1.9 FCxr, 0.5-1 1.25 5,10,20 3 4PB D 080119 20 294 ± 4 4.3 5.1 ± 0.62 -7.8 ± 0.5 RG, 0.5 1.25 5,10,20 3 4PB E 080128 7 269 ± 153 3.7 4.0 ± 1.13 -11.2 ± 0.7 FC, 1 1.25 10 3 4PB F 080130 25 343 ± 173 4.3 11.9 ± 2.93 -9.0 ± 0.4 FCxr, 1 0.01-1.25 10 3 4PB G 080302 7 243 ± 133 4 3.8 ± 1.23 -9.7 ± 1.2 RG, 0.5 0.05-1.25 5,10,20 3 3PB H 090118 29 185 ± 1 3.3 3.2 ± 0.5 -9.0 ± 0.8 RG, 0.5-1 / DF, 1-2 1.25 10 2 3PB I 090119 8 303 ± 3 4.3 11.7 ± 1.8 -6.7 ± 0.4 FCxr, 0.5-1 1.25 10 2.5 3PB J 090121 9 317 ± 3 4.3 17 ± 52 -7.5 ± 0.5 RG, 0.3-0.5 1.25 5,10,15 2 3PB K* 090125 20 331 ± 3 4.3 14.7 ± 1.23 -8.5 ± 0.5 RGxf, 0.5-1 1.25 10 2 3PB L 090129 5 326 ± 5 4.3 12.2 ± 0.9 -6.7 ± 0.3 FCxr, 0.5 1.25 10 2.5 3PB M 090202 4 227 ± 4 3 2.0 ± 0.2 -6.0 ± 0.8 FCxr, 0.5-1 1.25 10 2.5 3PB N 090205 1 238 ± 20 3.3 2.3 -5.7 FCxr, 0.5-1 1.25 10 2.5 3PB O 090215 12 296 ± 3 4 5.4 ± 0.74 -6.4 ± 0.4 FCxr, 0.5-1 1.25 5,10,15,20 2 3PB P 090301 9 152 ± 2 2 05 -6.5 ± 0.6 DF, 0.5-1 1.25 10 2.5 3PB Q 090321 10 334 ± 3 4 9.2 ± 1.2 -4.8 ± 0.7 RG, 1 0.125-1.25 10 2.5 3PB R 090323 16 337 ± 3 4 9.9 ± 0.8 -4.7 ± 0.9 RG, 1 0.0125-1.25 10 2.5 3PB S 090326 8 154 ± 2 3 2.1 ± 0.3 -5.1 ± 0.4 RG, 0.5 / DF, 1 1.25 10 2.5 3PB T 090405 4 241 ± 4 3.7 5.7 ± 0.9 -3.9 ± 1.1 RG, 0.5 1.25 10 2.5 3PB Table 4.3: Series of unnotched beam bending experiments used for the calculation of tensile strength. Date is in yymmdd format. Other column variables include the number of tests (n), mean snow density (ρ̄), mean blade hardness index (B̄), mean snow temperature (T̄ ), grain forms and grain size (F and E, respectively), testing machine crosshead speed (V), beam depth (D), and beam span to depth ratio (S/D). *the sample width varied in this test series, taking values of 5,10,15 and 20 cm (the standard sample thickness for all other tests was 10 cm). 1Following the International Classification for Seasonal Snow on the Ground (Fierz et al., 2009). Key: RG = rounded grains, DF = decomposing and fragmented crystals, FCxr = mixed rounded and faceted crystals; RGxf = rounded grains becoming faceted. 2Mean blade hardness index for entire test series (individual values of B not paired with individual strength tests). 3Mean of repeated in situ measurements in layer from which samples were extracted (not measured in lab). 4Blade hardness index not measured for smallest samples (D = 5 cm). 5Blade hardness gauge recorded 0 N, actual resistance in the range 0 < B < 1.7 N. 112 4.2.1 Strength calculation from beam theory For unnotched three or four point bending, the “modulus of rupture” (flexural strength) can be defined as fr = 6M bD2 (4.14) where M is the bending moment in the central cross section of the beam, b is the beam width and D is the beam depth (Bažant and Planas, 1998). The bending moment in the central cross section for a three point bending test (Figure 4.13a) is M = PS 4 (4.15) where P is the applied central load and S is the load span. In four point bending, the bending moment in the central portion of the beam is constant between the two loading points, and is expressed by M = Pa 2 (4.16) where a is the distance between a load point and the adjacent support point (Figure 4.13b). (a) (b) Figure 4.13: Schematic of unnotched three point bending (a) and four point bending (b) test for determining the tensile strength. Using Equation 4.15 in Equation 4.14, the modulus of rupture for a three point bending test is fr = 3PS 2bD2 (4.17) 113 where P is here understood as the peak load measured in the test. Due to the effect of the concentrated central load in a three point bending test, the elastic tensile stress distribution in the central cross section is slightly different from that predicted by Equation 4.17. This difference diminishes as the slenderness (S/D) of the beam increases, i.e. as the beam asymptotically approaches a pure bending solution. The correction to simple beam theory that accounts for the concentrated central load (Timoshenko and Goodier, 1951) takes the form σx = 3PS 2bD2 −0.266 P bD (4.18) where σx is the tensile stress in the outer fiber of the beam. If the maximum elastic tensile stress at failure is equated with the tensile strength (σx = fr at peak load), then combination of Equations 4.17 and 4.18 leads to ft = fr ( 1−0.1773D S ) [3PB]. (4.19) This equation coincides with Equation 4.13 used previously to correct the modulus of rupture data of Sigrist (2006). Note that the flat plates used in the bending tests (Figure 4.13, not necessarily to scale), necessary to prevent excessive crushing of snow at the contact points, are different from the rounded geometry of the concentrated load considered by Timoshenko and Goodier (1951) for arriving at Equation 4.19. This may lead to an actual stress distribution in the snow samples that lies somewhere between the predictions of simple beam theory (Equation 4.17) or the correction represented by Equation 4.19. The solutions above allow for consistent definitions of tensile strength for the comparison of data from different sources that used different span to depth ratios. Given the difficulty in extracting and handling snow specimens, it is not possible to test very slender beams of snow. This makes a consistent definition of tensile or flexural strength important, as most beam data for snow, including those in the present study, were collected for deep beams (small span to depth ratios). Sigrist (2006) used beams of S/D = 2 primarily, and the new data presented here came from beams that varied in the range S/D = 2–3. These ratios lead to tensile strength values that are 6% (S/D = 3) to 9% (S/D = 2) lower than predicted using the simple modulus of rupture (Equation 4.17). The maximum tensile stress in the outer fiber of a beam loaded in three point bending is only about 2% greater than the tensile stress a distance of 1/4D in from the outer fiber (Timoshenko and Goodier, 1951). 114 Therefore the bottom quarter of the central cross section, where the tensile crack coalesces, experiences roughly constant tensile stress in three point bending. The central axis of the beam experiences compressive stress, therefore the neutral axis is shifted toward the outer tensile face of the beam. Due to the short and deep beams tested and reviewed here, some shearing effects are also present in the beams. Though the shear is zero in the outer tensile fiber of the beam, the tensile crack is assumed to initiate in a boundary layer of finite thickness at the bottom of the beam where some shear stresses are present. However, these are expected to be minimal in the center of the beam where the crack coalesces. In four point bending (Figure 4.13b), the tensile strength can be calculated from simple beam theory since the portion of the beam between the central loading points is under pure bending (no shear). The tensile strength is equal to the modulus of rupture in this case. Equating the tensile strength with the maximum elastic tensile stress in the outer fiber of the central cross section of the beam, we have from Equations 4.14 and 4.16 ft = fr = 3Pa bD2 . (4.20) For all four point bending tests, the loading was done at the third points of the beam. Therefore a= S/3 and Equation 4.20 can be written as ft = PS bD2 [4PB]. (4.21) The beam theory presented here also contains the assumption that the elastic modulus of the material is the same in compression and tension. This may not be an appropriate assumption for a highly porous material such as snow, for which experiments have shown that strain and Poisson’s ratio under constant load are much different for compression and tension (e.g. Haefeli, 1939). This speaks to potential theoretical difficulties in interpreting the results of bending tests, despite their experimental advantages over uniaxial tests. However, for simplicity and consistency with other studies, the framework of simple beam theory is retained here. 4.2.2 Pertinent variables and range of values The range of values recorded for the pertinent variables related to the tensile strength tests is shown in Figure 4.14. Recall from the Chapter 3 that the variable that correlated the best with tensile strength was the blade hardness index, followed by the density. Weaker correlations (that were still statistically significant) 115 Ke rn e l d en sit y 0. 00 .1 0. 20 .3 5 10 15 20 l l ll Beam depth D [cm] 0 1 2 3 0.4 0.6 0.8 1.0 1.2 l l Grain Size E [mm] 0. 00 0. 02 0. 04 0 20 40 60 80 l ll llllll llllllllll l lllll ll l l l llllllllll l l Tensile strength ft [kPa] 0. 00 0. 04 0. 08 −5 0 5 10 15 20 l ll l lll llll ll llll l llll lll l ll l lll Blade hardness index B [N] 0. 00 00 .0 06 100 200 300 400 llllll lllllll lll l llllllll ll Density ρ [ kg m3 ] 0. 0 0. 5 1. 0 2.0 2.5 3.0 ll l Span to depth ratio S/D 0. 00 0. 10 0. 20 −10 −5 0 llll llll llll l lllllllll ll lll l ll lllll l l Temperature T [ °C ] 0. 0 1. 0 2. 0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 lll l l Crosshead speed V [10x cm/s] Figure 4.14: Kernel density plots of the variables associated with flexural strength tests. were observed between strength and secondary variables including the grain size, beam depth and crosshead speed. The blade hardness index B was highly correlated with the density, and the strength of the correlation was nearly as high as that between the tensile strength and B. The temperature in the cold lab was adjusted to approximately match the temperature of the snow layer from which the samples were extracted, therefore the temperature data in Figure 4.14 approximately represents the natural temperatures of the snow layers at the time of sampling. The heavy weighting of the data at densities higher than 300 kg/m3 is evidence of the limited selection 116 of homogeneous snow layers that were available to choose from for sampling, rather than a voluntary bias toward stronger and denser layers. Few layers at densities around 200 kg/m3 were available that were greater than 10 cm in thickness, approximately homogeneous in density and hardness as a function of depth, and strong enough to permit sample extraction and handling. This is partly related to the elevation of the primary Rogers Pass study plot from which most samples were obtained (1320 m.a.s.l.). 4.2.3 Tensile strength versus blade hardness index The relation between tensile strength and the blade hardness index is nearly linear (Figure 4.15). The data have a similar pattern as that between the centrifugal tensile strength and ram hardness (Figure 4.8a). Only data for which the blade hardness index was directly paired with a strength test is shown. For small samples with D = 5 cm, no blade hardness measurements were taken because of the risk of penetrating all the way through the sample and damaging the force gauge. In other instances, following the bending test, the snow sample fell off of the support plates and was damaged. This prevented a representative hardness measurement from being taken. In a few cases the force gauge battery died or the measurement was simply forgotten. For identifying individual data sets with particular variables or testing conditions, Figure 4.16 identifies groups of data against the date codes in Table 4.3. Blade hardness index [N] Te n si le s tre ng th [k Pa ] 10 20 30 40 50 0 5 10 l ll ll ll l llllll lll lll ll l ll l l l l l ll ll l l ll lll ll l l ll l l ll l l l ll l l ll l ll l l lll ll l ll l Figure 4.15: Tensile strength versus blade hardness index, including only strength tests that were paired directly with a blade hardness measurement (n = 101, Table 4.3). 117 Blade hardness index [N] Te n si le s tre ng th [k Pa ] 10 20 30 40 50 0 5 10 ll l ll l l l l l l ll l l l l l ll l l Date code H I L M N O P Q R S Tl l Figure 4.16: Tensile strength versus blade hardness index, with different plot symbols indicating individual test series. Only tests which directly paired strength and blade hardness index are plotted (n = 101). 4.2.4 Tensile strength versus density The strength scales with the density in a similar manner as previous results from laboratory tests (Figure 4.17). Though the scatter is large when expressing strength as a function of density, the flexural tests of the present study agree well with the uniaxial tests of Narita (1980, 1983) at similar densities. Most of Haefeli’s uniaxial data are higher than those of the present study, though Haefeli may have stored the samples for longer periods and the pre-stressing of samples prior to fracture testing likely influence the ultimate strength. The data from the present study are higher than those from similar tests conducted by Sigrist (2006). This difference may be attributable to a systematic tendency for the snow sampled in the present study to be of higher hardness than that sampled by Sigrist. This may have resulted from the higher elevations from 118 50 100 150 200 250 300 350 400 450 500 Density [kg/m3 ] 100 101 102 Te ns ile s tr en gt h [k Pa ] Present study Haefeli (1939) Sigrist (2006) Narita (1980) type 'a' Narita (1983) type 'a' Figure 4.17: Data from the present study in the context of previously measured tensile strength from laboratory tests. which snow was sampled in Sigrist’s study (1562 and 2668 m.a.s.l) compared to the primary study plot in the present study at 1320 m.a.s.l. Most of the tests from the present study were conducted at a crosshead speed around four times greater than the standard speed used by Sigrist, though this systematic rate effect should lead to lower strength values for the present study, all else the same (lower strength for higher strain rate, as will be seen below). The grouping of the test data by date of testing is shown in Figure 4.18, analogous to Figure 4.16. In many test series, there appears to be less scatter in density values (Figure 4.18) than blade hardness index values (Figure 4.16). This may be an indication that the blade hardness has greater sensitivity to the spatial variability of snow structure than density. However, several series are characterized by high scatter in density values as well. The tensile strength varies by a factor of about three for a density of around 300 kg/m3, a 119 greater degree of scatter than in the strength at any given blade hardness index (Figure 4.16). Density [ kg m3 ] Te n si le s tre ng th [k Pa ] 20 40 60 150 200 250 300 350 ll l l l l l l l l l l l l l l l l l l l l l l l l l l l ll ll l ll Date code A B C D H I J K L M N O P Q R S T l l l Figure 4.18: Tensile strength versus density grouped by date of testing, excluding three series for which density was not measured in the lab (n = 206). The date codes reference Table 4.3. 4.2.5 Influence of grain size Cohesive snow with smaller grains is typically stronger than coarse-grained snow, all else the same. The grain size does not systematically explain the variability in the strength as a function of the blade hardness index (Figure 4.19a). This indicates that the blade hardness index may be implicitly accounting for the dependence of grain size on a structural property such as strength. The grain size is not a particularly helpful variable for explaining the strength-density data either (Figure 4.19b). The larger grain sizes at low densities (less than 200 kg/m3, series H and S) were due to the presence of decomposed and fragmented crystals in the young snow layers. However, the large grain sizes at higher 120 densities (Series Q and R represent the coarse grains (1 mm) at densities around 340 kg/m3 in Figure 4.19b) were all correlated with faceted or mixed rounded and faceted crystal forms. Faceted crystal forms typically grow in size at the expense of strength (McClung and Schaerer, 2006), and the in-situ uniaxial strength data of Jamieson (1988) confirm this. Blade hardness index [N] Te n si le s tre ng th [k Pa ] 20 40 60 0 5 10 ll lll ll l ll l l l l l ll lll l l ll ll l ll l l l l Grain size [mm] 0.5 1l Density [ kg m3 ] Te n si le s tre ng th [k Pa ] 20 40 60 150 200 250 300 350 ll ll ll ll l l l ll ll lll ll l ll l l l l l l l ll l ll lll ll l l ll l l ll l lll ll l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l ll l lll l l ll ll l l llll l l ll Grain size [mm] 0.5 1l Figure 4.19: Tensile strength versus blade hardness index (n = 101) and density (n = 206), show- ing the influence of the snow grain size. 4.2.6 Loading rate effects Most strength tests were conducted at the fastest possible crosshead speed (1.25 cm/s) to minimize viscous effects as much as possible. However, two of the test series (F and R) were designed to explore the rate dependence of tensile strength. Figure 4.20 shows the influence of the loading rate, expressed as the nominal strain rate in the outer fiber of the beam calculated using the following expressions from beam theory (e.g. Timoshenko, 1940) ε̇N = 6DV S2 [3PB] (4.22) and ε̇N = 4DV S2 [4PB] (4.23) 121 where D is the beam depth, V is the crosshead speed, and S is the loading span. Note that these expressions are only approximate given the low span-to-depth ratios of the present study. Shearing deformation in the deep beams would lead to deformations and strains in the outer fiber of the beam that deviate from simple beam theory. However, since the beam geometry did not change in each of the two test series considered here, the simple expressions listed above give a self-consistent (though approximate) estimate of the rate dependence of the flexural strength measurements. Series F Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 30 35 40 45 50 0.001 0.01 l l l l l l l l l l l l l ll l l l l l l l l (a) Series R Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 30 35 40 45 50 0.001 0.01 0.1 l l l l l l l l l l l l l l l l (b) Figure 4.20: Tensile strength versus nominal strain rate for samples all taken from the same layer and all other experimental conditions the same. The slopes of the linear regressions had p- values of 0.012 (a) and 0.015 (b). The strength decreases with increasing strain rate, with the rate varying over about two orders of mag- nitude for both series in Figure 4.20. The linear regressions through the data have statistically significant slopes at the α = 0.05 level for both series. The trends in Figure 4.20 are consistent with the data of Mellor and Smith (1966); Narita (1980, 1983) which show that strength decreases with increasing strain rate above the creep-to-fracture transition (around 10−4 s−1 for snow in tension). The data in Figure 4.20 are all above this transition strain rate. There are little published data from quasi-static tests at strain rates higher than about 10−2 s−1 to com- pare with the highest strain rates in the present study (on the order of 10−1 s−1). Some of the data pre- 122 sented by Narita (1983) suggest that the initial decrease in strength values above the creep-to-fracture transition may level off at higher strain rates, in agreement with observations made by Mellor and Smith (1966). This leveling-off is not apparent in the data from the present study, though such behaviour cannot be ruled out. For higher strain rates it may be necessary to consider data from cyclical loading tests (e.g. Camponovo and Schweizer, 2001). In very high frequency dynamic or cyclical tests, Young’s modulus in- creases with frequency (Mellor, 1975) and therefore the strength may be expected to eventually increase again with increasing strain rate or loading rate. The data from the present study suggest that this transition to dynamics effects must be higher than the highest strain rates achieved in the experiments, greater than 10−1 s−1. 4.2.7 Specimen size effects Four different test series were carried out on unnotched beam samples with only the specimen size (beam depth) varied. The strength significantly decreased with increasing beam depth D for all but Series O (Figure 4.21). The greatest decrease in strength, expressed as a percentage decrease from the smallest to largest size, was for series J. In this series, the decrease in strength was slightly less than a factor of two as the size increased by a factor of three. As noted in Chapter 2, only about one in four of the largest samples (D = 20 cm) which was extracted was successfully transported to the lab and tested. Most failed during extraction and removal from the cutter box or transportation to the lab. This fact raises the question of whether the samples that were successfully tested may have been damaged to an extent not sufficient to cause failure nor to be noticed. If this were the case, then the largest samples may have failed at lower nominal strength values for reasons unrelated to a purely deterministic or statistical size effect. However, for the three size effect test series in Figure 4.21 which included samples of beam depth D = 20 cm, only Series C (Figure 4.21a) would have the statistical significance of its size effect slope changed by excluding the largest samples. The values of the slopes do change, however, which has implications for physical theories which explain the observed size effects. The loading rate was held constant for each of these test series, but the different beam depths led to different nominal strain rates in the outer fiber of the beam. Equations 4.22 and 4.23 indicate that the nominal strain rate from simple beam theory is proportional to the beam depth. Figure 4.22 indicates the change in nominal strain rate with changing beam depth. The strength data show a statistically significant correlation 123 Series C Beam depth D [cm] Te n si le s tre ng th [k Pa ] 30 32 34 36 38 40 5 10 15 20 l l l l l l l l l l l l l l l l l l (a) Series D Beam depth D [cm] Te n si le s tre ng th [k Pa ] 15 20 25 30 35 5 10 15 20 l l l l l l l l ll l l ll l l l l l (b) Series J Beam depth D [cm] Te n si le s tre ng th [k Pa ] 35 40 45 50 55 60 65 6 8 10 12 14 l l l ll l l l l (c) Series O Beam depth D [cm] Te n si le s tre ng th [k Pa ] 16 17 18 19 20 5 10 15 20 l l l l l l l l l l l l (d) Figure 4.21: Tensile strength versus beam depth for four different size-effect test series. The slopes of the linear regressions had p-values of 0.001 (a), <0.001 (b), 0.002 (c) and 0.49 (d). 124 with the nominal strain rate for all but Series O, which was also the series that did not have a significant size effect. These results suggest that a rate effect on strength may be complicating the interpretation of the size effect (or vice-versa). Given the slower nominal strain rate of the largest samples, these samples may actually have failed at higher nominal strength values according to the rate-dependence of tensile strength observed by others (Narita, 1980). Thus the rate effects between the samples of different sizes may have actually weakened the size effect as observed in Figure 4.21, and is a reason that Bažant and Gettu (1992) suggest that a condition of constant time-to-failure tp be used in size effect testing instead of constant loading rate. This trend (weakening of the size effect due to rate effects) would be the opposite as that which may have been caused by the damage and weakening of the largest samples during transportation and handling, which may have strengthened the apparent size effect. Statistical and deterministic explanations for the size effects considered here, and their connection with material and fracture parameters, will be explored further in the next chapter. Due to the concerns given above about the size effect being complicated by the possible damage of large samples and the rate dif- ferences for different samples at constant crosshead speed, preference is given to methods of calculating fracture parameters that use medium-sized samples only. 125 Series C Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 30 32 34 36 38 40 0.04 0.06 0.08 0.10 l l l l l l l Beam depth D [cm] 5 10 20l (a) Series D Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 15 20 25 30 35 0.04 0.06 0.08 0.10 l l l l ll l Beam depth D [cm] 5 10 20l (b) Series J Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 35 40 45 50 55 60 65 0.15 0.20 0.25 0.30 0.35 l l l Beam depth D [cm] 5 10 15l (c) Series O Nominal strain rate ε⋅N [ s−1 ] Te n si le s tre ng th [k Pa ] 16 17 18 19 20 0.10 0.15 0.20 0.25 0.30 0.35 l l l l l Beam depth D [cm] 5 10 15 20l l (d) Figure 4.22: Tensile strength versus nominal strain rate, grouped by beam depth, for the same size effect data as in Figure 4.21. All four test series had a constant crosshead speed of 1.25 cm/s. The slopes of the linear regressions had p-values of 0.004 (a), <0.001 (b), 0.016 (c), and 0.62 (d). These data indicate that part of the size effect on tensile strength (Figure 4.21) could be attributable to a nominal rate effect between samples of different sizes. 126 Summary The tensile strength data from the present study agree well with previous quasi-static laboratory data. Of all the variables related to the measurement of strength, the hardness and the density are the two most important. The blade hardness index is the best single variable for graphical representation and statistical correlation with tensile strength in the data from the present study. The snow type (structure) must be the same, together with other experimental controls, to ascertain the influence of secondary variables such as strain rate, temperature, and specimen size. Rate effects cannot be separated from size effects when constant-speed displacement controlled tests at different specimen sizes are conducted. The next section contains several models developed to explain the mean tensile strength data considered thus far. The snow density and hardness are used, separately, as the predictor variables in these models, and comparison is made between the results using each variable. 4.3 Models of Tensile Strength Regression models of strength allow for the systematic and reproducible use of strength data in analyti- cal or numerical models. Least squares regression models using a single predictor variable, typically the snow density, are the most common for snow strength models. Least squares regression contains assump- tions about the form of a model and the distribution of model errors, assumptions which should be checked and commented on when reporting regression results. Many techniques exist, such as transformation of variables, weighted regression, and variance modeling, for addressing violations of these assumptions. Vi- olations of model assumptions can also indicate when the choice of a predictor variable, such as density, is inappropriate for explaining the mean structure of a dependent variable. Appendix B contains more detail about the implicit assumptions in least squares regression, common goodness of fit graphical and statistical techniques, and remedies for model violations applied in this section. Only univariate models are considered here, primarily for comparison with previous models of the ten- sile strength of snow, nearly all of which are univariate functions of density. The most common model formulation for strength as a function of density is a power law. Models of this form are explored for explaining the data from the present study and those of several other studies. The ram hardness data of (Martinelli, 1971) is shown to be at least as good as density as a predictor variable for tensile strength. The 127 blade hardness index, according to several goodness of fit measures, is also a better predictor than density for the strength data from the present study. 4.3.1 Density power law models Snow properties such as strength are commonly expressed as functions of the bulk snow density, or the fraction of the density of pure ice. Power law models are perhaps the most common, though exponential models using the porosity or void ratio, which can be expressed as functions of density, have also been used (e.g. Ballard and Feldt, 1966; Ballard and McGaw, 1966; Mellor and Smith, 1966; Keeler and Weeks, 1968; Keeler, 1969). The most common power law function for the tensile strength has the form ft = a ( ρ ρi )b (4.24) where ρi is the density of freshwater ice and a and b are the model parameters to be determined by fitting the relation through the strength-density data. Theoretically the value of a could be fixed at the tensile strength of pure ice, which is around 1.5–2 MPa (Schulson and Duval, 2009), leading to a single-parameter relation. However, a single monotonic expression such as Equation 4.24 is not expected to hold over the entire range of snow densities. Over the range of densities relevant for most avalanche applications (∼100–350 kg/m3) the strength is determined by the structure of the ice matrix, which as discussed previously is only loosely characterized by the bulk density. At higher densities than typical alpine snow, such as in multi-year polar or glacier firn snow, the snow will behave more as a porous solid which is likely to be governed more by the density. Therefore the parameters a and b should not be considered universal or material constants, but rather are limited to the specific range of snow densities that they arise from. However, the value of a should probably still fall within an order of magnitude of the tensile strength of ice, providing a rough check on the results. Figure 4.23 shows regression fits of the form of Equation 4.24 through 17 of the data sources from this chapter. The strength values cover nearly three orders of magnitude, from less than 0.1 kPa for very low density snow to over 100 kPa for the highest densities typically found in seasonal alpine snow. For the lowest density snow, there is considerable agreement between the different data sources. The strength of very low density snow does not exceed 1 kPa until the density exceeds 100 kg/m3. At higher densities, the lowest 128 strength values come from Sommerfeld (1974), who had the largest sized specimens (though the loading rate is unknown). The in situ values of Perla (1969), McClung (1979a) and Jamieson (1988) agree very well and are the lowest excluding Sommerfeld’s. The data of Keeler and Weeks (1968) are questionably far from the rest of the data, and raise questions about the presence of an unknown systematic error. The data of Martinelli (1971) cover the widest range of densities and are perhaps the best representative single data set of tensile strength, though these data are higher than all of the in-situ data for densities of 100 kg/m3 and above. The slope of the data of Butkovich (1956) looks far too steep compared to the rest of the data, and could be due to the smaller number of data points and the small density range tested. Conversely, the slope of Haefeli’s data (in Bader et al., 1954) appears very flat, similar to the creep-fracture data of Upadhyay et al. (2007), which calls into question whether Haefeli’s tests were also heavily influenced by creep. The model values from the present study fall right in the middle of the locus of values from other studies. 129 Density [ kg m3 ] Te n si le s tre ng th [k Pa ] 0.1 1 10 100 1000 100 200 300 400 500 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Bader (1951) Bucher (1948) Butkovitch (1956) Conway and Abrahamson (1984) de Quervain (1950) Haefeli (1939) Jamieson (1988) Keeler (1969) Keeler and Weeks (1968) Martinelli (1971) McClung (1979) Perla (1969) Present study Roch (1966) Sigrist (2006) Sommerfeld (1974) Upadhyay (2007) l l l l Figure 4.23: Nonlinear regression of the form of Equation 4.24 through individual data sets. Data from other sources have been corrected to represent the ultimate tensile strength (rather than the nominal strength) where appropriate. The black dashed line is drawn to represent a lower- bound estimate of the uniaxial tensile strength of pure ice (1.5 MPa). 130 New data from the present study A subset of the tensile strength tests from Table 4.3, with the same specimen size and width (D = b = 10 cm) and loading rate (V = 1.25 cm/s), were considered for regression modeling as a function of density. The censored data set contained 123 tests. Fitting Equation 4.24 through this data led to a fit of the form ft = (200±30) ( ρ ρi )1.6±0.1 [kPa]. (4.25) Both regression parameters a and b were statistically significant at the α = 0.05 level (p-values < 0.001 for both). The model residuals were not constant, displaying an increase with increasing density (Figure 4.24a). The residuals were neither normally distributed (Figure 4.24b) nor independent (Figure 4.24c). The fit had an R2 = 0.65. See Appendix B for the definition of R2 for nonlinear regressions, as the standard coefficient of determination in linear least-squares regression (referred to here using lower case r2) differs. A Box-Cox profile likelihood was calculated (e.g. Ritz and Streibig, 2008) as a hypothesis test for whether a transformation of the data (such as a log-transform) would improve the fit of the model with respect to the residual structure. The 95% confidence interval for the parameter λ , which indexes the appropriate family of transformations, indicated that a transformation of the data would not significantly improve the fit. 131 Fitted values St an da rd ize d re sid ua ls −1 0 1 2 10 15 20 25 30 35 40 l ll l ll l l ll l lll ll l l ll l l l l ll l l l l l l l l l l ll l l l l ll l l l l l ll l l l ll l l l l l l l l ll lll l l l l l lll l l l l l lll l l ll ll l l ll (a) l ll l ll l l ll l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l ll l l l l l l lll l l l l l l ll l ll l ll l l ll l l l l l −2 −1 0 1 2 − 10 0 10 20 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.016 (b) ll l lll l l l l ll l l l l l ll l l ll l l l l l l l l l l l l l l l l l l lll ll l ll l l l l l lll l l l ll l l l l l l l l l l l l l l l lll l l l l l l llll l lll l l l l ll l l l l l l lll l l l −10 0 10 20 − 10 0 10 20 Residuals La gg ed re sid ua ls Runs test p−value < 0.001 (c) Figure 4.24: Residual plots for assessing the goodness of fit of Equation 4.25. Standardized resid- uals versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro-Wilk test is that the residuals are normally distributed. The null hypothesis of the runs test is that the residuals are independent. In the absence of variable transformation, an alternative approach was taken in an attempt to improve the model fit before discarding density as an inappropriate predictor for the strength data. The positively 132 correlated errors in Equation 4.25 (Figure 4.24c) were interpreted as a result of the indirect grouping of the data as a result of the experimental design. The test data in this study were grouped by date, and on a particular date all snow samples were taken from the same snow layer. This explains the clustering of the data by date of testing (Figure 4.18), with the date serving as a proxy for the snow layer of interest on that particular date. This sort of grouping typically produces positively correlated errors in regression modeling (Rawlings et al., 1998). It is not appropriate to assume independent and constant variance in a regression model if the data are grouped, have different numbers of observations in each group, and the dependent variables are correlated within each group (Rawlings et al., 1998), all of which are the case for the data considered here. The strong autocorrelation among model residuals was addressed by computing the means of density and strength for each date of testing. These means were used in a subsequent weighted regression of the same form as Equation 4.24. The number of tests on a particular date divided by the variance of the strength values on that date was used a weighting factor for each of the subsequent 20 group means. The resulting model fit took the form ft = (360±160) ( ρ ρice )2.3±0.4 [kPa]. (4.26) In this case the parameter a is not statistically significant at the α = 0.05 level (p-value = 0.06), though b is (p-value < 0.001). This fit had an R2 = 0.50, lower than the previous model fit. The residuals of Equation 4.26 showed no obvious pattern as a function of density (Figure 4.25a) but the assumptions of normally distributed residuals (Figure 4.25b) and independent residuals (Figure 4.25c) were not satisfied at the α = 0.05 level. These plots indicate that the model represented by Equation 4.26 is a slightly improved fit in most regards over Equation 4.25. 133 Fitted values St an da rd ize d re sid ua ls −2 −1 0 1 2 10 20 30 l l l l l l l l l l ll l l ll l l l l (a) l l l l ll l l l l l l l l l l l l l l −2 −1 0 1 2 − 5 0 5 10 15 20 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.047 (b) l l l ll l l l l l l l l l l l l l l −5 0 5 10 15 20 − 5 0 5 10 15 20 Residuals La gg ed re sid ua ls Runs test p−value = 0.022 (c) Figure 4.25: Residual plots for assessing the goodness of fit of the weighted regression through the strength-density group means (Equation 4.26). Standardized residuals versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro-Wilk test is that the residuals are normally distributed. The null hypothesis of the runs test is that the residuals are independent. A Box-Cox profile likelihood indicated that a square-root transition might improve the residual structure of the group-means model of Equation 4.26. This transformation was performed and the data was re-fit to 134 the power law model, leading to the following expression: ft = (350±140) ( ρ ρi )2.4±0.4 [kPa]. (4.27) Both model parameters were statistically significant at the α = 0.05 level, and the fit had an improved R2 = 0.62. The parameters did not change much compared to the original model of Equation 4.26. The trans- formation resulted in normally-distributed residuals (Figure 4.26b) but did not improve the autocorrelation of the residuals (Figure 4.26c). The distribution of residuals indicated a slightly worse fit, as the residuals decreased with increasing density (Figure 4.26a) moreso than in the un-transformed model (Figure 4.25a). 135 Fitted values St an da rd ize d re sid ua ls −2 −1 0 1 2 3 4 5 6 l l l l l l l l l l ll l l l l l l l l (a) l l l l l l l l l l l l l l l l l l l l −2 −1 0 1 2 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 1. 5 2. 0 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.1 (b) l l l l l l l l l l l l l l l l l l l −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 1. 5 2. 0 Residuals La gg ed re sid ua ls Runs test p−value = 0.022 (c) Figure 4.26: Residual plots for assessing the goodness of fit of the weighted regression through square-root transformed strength-density group means (Equation 4.27). Standardized residu- als versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro-Wilk test is that the residuals are normally distributed. The null hypothesis of the runs test is that the residuals are independent. The model fits of Equations 4.25, 4.26, and 4.27 are shown together in Figure 4.27. The weighted regressions through group means of Equations 4.26 and 4.27 led to greater curvature in the mean function, 136 expressed by the higher power law exponent. Owing to the smaller variance of strength values in Series A and P compared to Series H, the weighted regression models (Equation 4.26 and 4.27) shifted the mean function downward relative to Equation 4.25. This is in spite of the fact that Series H contained the largest number of tests of any series. The single model residual for Equation 4.26 that lies outside of 2 standard deviations from the mean (Figure 4.25a) corresponds to Series H in Figure 4.27. This is a drawback of the otherwise improved fit represented by Equation 4.26. Density [ kg m3 ] Te n si le s tre ng th [k Pa ] 10 20 30 40 50 60 150 200 250 300 ll l l l l l l l l l l l lll l Date code A B C D E F G H I J K L M N O P Q R S T l l l l Figure 4.27: Tensile strength versus density, grouped by date of testing, with regression model fits. The top dashed curve is the original model of Equation 4.25 through individual data points, the solid curve is the regression fit of Equation 4.26 through group means and the bottom dotted curve that of Equation 4.27 through the square-root transformed group means. The date codes are further explained in Table 4.3. None of the regression models capture the data in a wholly satisfactory manner from a visual perspec- tive. The square-root transformation made a marginal difference to the visual appearance of the model of 137 Regression through full data set Weighted regression through group means Equation 4.25 Equation 4.26 Variable rs p rs p grain size E 0.09 0.3 -0.32 0.17 beam slenderness S/D -0.47 <0.001 -0.27 0.25 ∗blade hardness index B 0.33 0.002 0.34 0.16 temperature T -0.32 0.001 -0.17 0.48 Table 4.4: Spearman’s correlation coefficients rs and p-values for the residuals of the tensile strength versus density models tested against other variables. ∗Correlations excluded Series A for which the blade hardness index was not measured. Equation 4.27. The dotted line of Equation 4.27 is hardly discernible from the original model, the solid line of Equation 4.26. From the numerous goodness of fit statistics and the relative simplicity of the model, Equation 4.26 is considered the best model among the three. Correlations between model residuals and other variables not represented in the regression models (in other words, variables other than density) offer a final check on the goodness of fit of the regression mod- els. The original regression model through individual data points (Equation 4.25) and the first weighted regression model through group means (Equation 4.26) are compared in Table 4.4. For the model through individual data points, the residuals were significantly correlated with the span to depth ratio of the beams, the blade hardness index, and the temperature. The direction of these correlations are as expected: more slen- der beams and colder snow were correlated with lower strength, and higher hardness snow was correlated with higher strength (for Equation 4.25). The model through group means (Equation 4.26) did not have any statistically significant residual correlations. This is an indication that this model is a better representation of the mean structure of the tensile strength data, at least when expressed as a function of density. Therefore the best regression model here is that of Equation 4.26, re-written here for convenience: ft = 360(ρ/ρi)2.3, with ft in kPa. Jamieson’s data Jamieson (1988) reported a nonlinear regression of the form of Equation 4.24 through his in situ uniaxial tensile strength data. His data were not corrected for the difference between the nominal and maximum stress at failure, and took the form 80(ρ/ρi)2.4. I refit this data, composed of the mean uniaxial tensile strength of 43 layers, excluding layers composed of faceted or mixed rounded and faceted crystals. The number 138 of tests in each snow layer divided by the variance of strength values were used to weight the individual means in the regression, as performed above in Equation 4.26. Residual plots of the initial fit showed a variance structure that was largely homogeneous and normally distributed. A Box-Cox profile likelihood of the model indicated that a square root transformation of both sides would improve the fit (Jamieson’s fit was through log-transformed data). The resulting model took the form ft = (150±25) ( ρ ρi )2.4±0.1 [kPa]. (4.28) Both regression parameters were statistically significant at the α = 0.05 level, and the fit had an R2 = 0.94. The residual plots for Equation 4.28 indicate that the variance structure is mostly homogeneous. The residuals slightly decrease with increasing density (Figure 4.28a) but are normally distributed (Figure 4.28b) and independent (Figure 4.28c). These findings indicate that the density alone can adequately represent the strength data of Jamieson (1988). The model of Equation 4.28 is slightly less than a factor of 2 greater than that reported by Jamieson (1988). This can be attributed to the stress concentration factor that was calculated and accounted for here. For the sample dimensions reported by Jamieson (1988) (test geometry in Figure 4.10), the stress concentration factor took values in the range 1.7–2.1. Individual values were calculated for each layer in the data set, since the geometric dimensions for each layer were published. 139 Fitted values St an da rd ize d re sid ua ls −2 −1 0 1 2 3 1.0 1.5 2.0 2.5 3.0 3.5 l l l l l l l l l l l l l l l l ll l l l l l l l ll l l l l l l ll l l l l l l l l (a) l l l l l l l l l l l l l l l l ll l l l l ll l ll l l l l l l l l l l l l l l l l −2 −1 0 1 2 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.49 (b) l l l l l l l l l l l l l l l ll l l l l l l l ll l l l l l l l l l l l l l l l l −0.4 −0.2 0.0 0.2 0.4 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 Residuals La gg ed re sid ua ls Runs test p−value = 0.65 (c) Figure 4.28: Residual plots for assessing the goodness of fit of the log-transformed strength- density data of Jamieson (1988) (Equation 4.28). Standardized residuals versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro-Wilk test is that the residuals are normally distributed. The null hypothesis of the runs test is that the residuals are independent. 140 Sigrist’s data Sigrist (2006) kept all variables other than the sample density the same for his tensile strength test series. I fit his data, corrected using the relationship between the tensile strength and modulus of rupture in Equation 4.19, to the general power law expression of Equation 4.24. The initial fit showed strongly heterogeneous residuals, with the variance increasing sharply with the mean. The residuals were non-normally distributed, but did pass the runs test for independence. This is likely the result of the smaller data set that was designed just to measure the tensile strength, so all other experimental variables were held exactly the same across the data set. I next transformed Sigrist’s data using a log transformation of both sides, and the resulting model fit was improved compared to the initial fit. The fit through the transformed data took the form (230±30) ( ρ ρi )2.4±0.1 [kPa]. (4.29) The fit had an R2 = 0.91, and both regression parameters were statistically significant at the α = 0.05 level. The residual structure was still heterogeneous, with increasing variance with increasing density (Figure 4.29a). The residuals remained normally distributed (Figure 4.29b) and independent (Figure 4.29c). Sigrist (2006) reported similar parameter estimates ( ft = 240(ρ/ρi)2.44). The primary discrepancy is the correction applied here for the difference between the modulus of rupture and the tensile strength in three point bending, a correction which reduces the elastic tensile stress in the outer fiber of the beam. 141 Fitted values St an da rd ize d re sid ua ls −2 −1 0 1 2 0 1 2 3 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l (a) l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l ll l l l l l l l l l l l l l l l l l −2 −1 0 1 2 − 0. 6 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 0. 6 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.16 (b) ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 − 0. 6 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 0. 6 Residuals La gg ed re sid ua ls Runs test p−value = 0.78 (c) Figure 4.29: Residual plots for assessing the goodness of fit of Equation 4.29 for the data of Sigrist (2006). Standardized residuals versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro- Wilk test is that the residuals are normally distributed. The null hypothesis of the runs test is that the residuals are independent. 142 Martinelli’s data I fit the centrifugal data of Martinelli (1971), corrected for the stress concentration, as a function of density using a model of the form of Equation 4.24. The initial fit showed strong heteroscedasticity and autocor- relation of the residuals. A power law transformation, determined using the Box-Cox profile likelihood approach, led to a best fit of the form ft = (3400±600) ( ρ ρi )3.4±0.2 (4.30) with approximate 95% confidence intervals on the parameters and ft in kPa. The fit had an R2 = 0.88. The parameter a is about an order of magnitude higher than those of the previous models, but the power law exponent is also greater by about one, which accounts for this order-of-magnitude discrepancy. Figure 4.30 shows the goodness of fit plots for this model. It shows normality in the model residuals and no apparent pattern in the variances, but the residuals were autocorrelated. The model fit through the data is shown in Figure 4.31 for eventual comparison with the same strength data modeled using the ram hardness. 143 Fitted values St an da rd ize d re sid ua ls −2 −1 0 1 2 0.5 1.0 1.5 2.0 2.5 3.0 l l l l ll l l l l l l l l ll l l l l l ll l l l ll l l l l l l l l ll l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l ll l l l l l ll ll l (a) l l l l l l l l l l l l ll l l l l l ll l l lll l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l −2 −1 0 1 2 − 0. 6 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 Theoretical Quantiles Sa m pl e Qu an tile s Shapiro−Wilk test p−value = 0.1 (b) l l l l l l l l l l l l l l l l l l l l l l l l lll l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l l l ll l l l −0.6 −0.4 −0.2 0.0 0.2 0.4 − 0. 6 − 0. 4 − 0. 2 0. 0 0. 2 0. 4 Residuals La gg ed re sid ua ls Runs test p−value < 0.001 (c) Figure 4.30: Residual plots for assessing the goodness of fit of Equation 4.30 for the data of Martinelli (1971). Standardized residuals versus fitted values (a), normal quantile plot of the residuals (b) and autocorrelation plot of the residuals (c). The null hypothesis of the Shapiro- Wilk test is that
UBC Theses and Dissertations
Tensile strength and fracture mechanics of cohesive dry snow related to slab avalanches Borstad, Christopher P. 2011
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