Numerical modeling of tensile fracture initiation and propagation in snow slabs using nonlocal damage mechanics
Highlights
► Tensile cracking related to slab avalanches is a quasi-brittle phenomenon. ► A nonlocal damage model simulated experimental fracture data. ► Simulations produced crack initiation and crack propagation. ► Simulations verified fracture parameters calculated from experiments.
Introduction
When fractured in tension, heterogeneous materials such as snow, ice, concrete, rocks, and ceramics typically develop a relatively large and diffuse zone of microcracking prior to the coalescence and propagation of a traction-free macrocrack and ultimate failure. This diffuse cracking behavior has been observed directly using spatially located acoustic emission data for materials such as concrete (Otsuka and Date, 2000), for kilometer-scale ice shelf rifts using seismic signals (Bassis et al., 2007), and is supported by observed acoustic emission rates observed in snow fracture experiments (St. Lawrence and Bradley, 1975, St. Lawrence et al., 1973).
The diffuse nature of microcracking in heterogeneous materials serves to blunt the primary crack tip and leads to a number of important and observable features in a fracture test, including inelastic stress–strain response prior to peak load and strain softening following peak load. Strain softening has been observed in snow fracture experiments in both shear shear (McClung, 1977, Schweizer, 1998) and tension (Borstad, 2011, Sigrist, 2006).
Diffuse cracking and strain softening also necessitate the introduction of an intrinsic material length scale related to the material microstructure, in addition to a fracture energy or toughness term, to fully describe the fracture (Cotterell and Mai, 1996). Snow is considered to have a fracture process zone—an intrinsic length scale that characterizes the region of softening damage ahead of a crack—on the order of 10–100 times the grain size (Bažant et al., 2003, Borstad and McClung, 2009, McClung and Schweizer, 2006, Sigrist, 2006), which explains strength size effects which do not conform to the predictions of Linear Elastic Fracture Mechanics (LEFM) (Borstad, 2011, Sigrist et al., 2005b).
Any fracturing material that has a characteristic length or exhibits strain softening is considered quasi-brittle (Bažant and Planas, 1998), in contrast to fully brittle materials which obey LEFM and have no intrinsic length scale. To date, no experimental evidence has been presented which demonstrates that snow fractures in either shear or tension according to the predictions of LEFM. Furthermore, the equivalent elastic or effective crack (quasi-brittle) fracture theories of Bažant and Planas (1998) and Bažant (2005) all contain LEFM as a large-size asymptotic limit. This makes a quasi-brittle approach to analyzing snow fracture a more general and appropriate foundation in the absence of direct experimental evidence of the nature of stresses and strains in the vicinity of cracks in a highly porous material such as snow.
Numerical modeling of quasi-brittle fracture is considerably more difficult than fully brittle or LEFM models, for which closed-form analytical solutions exist for many types of cracking problems. Early quasi-brittle numerical models suffered from a number of problems, notably spurious localization of damage and vanishing energy dissipation with mesh refinement (e.g. Pijaudier-Cabot and Bažant, 1987) and loss of ellipticity of the governing differential equations due to strain softening and strain localization, which make the boundary value problem ill-posed (Jirásek and Patzák, 2002). These problems have been overcome with more recent continuum damage models which are formulated nonlocally, that is which spatially smear the microstructural heterogeneity and the associated diffusion of microcracking over an intrinsic material length scale. This is done by calculating traditional continuum properties such as stress and strain using spatial fields around the point of interest rather than as idealized point properties. Nonlocal damage formulations have proven successful in replicating observed experimental features in quasi-brittle fracture tests (such as strain softening and size effects), without mesh sensitivity problems, for many quasi-brittle materials (Bažant, 2005, Bažant and Jirásek, 2002). However, this approach has never been applied to the fracture of snow.
The nonlocal isotropic damage model was applied to replicate the tensile failure of dry snow slab specimens. Seventeen beam bending experiments were carried out using snow samples from the same homogeneous layer of natural snow. All experimental conditions were held constant except for the presence or absence of a notch at the bottom of the beam. Material properties such as Young's modulus, tensile strength and tensile failure strain were derived from the measured load–displacement data and used to determine parameters for the numerical model. Finite element meshes were created for both the notched and unnotched experimental geometries, and the simulations were run using a displacement-controlled boundary condition as in the experiments. The model results agreed well with many key global features of the experimental load–displacement curves that cannot be explained using a fully brittle (LEFM) failure model, especially the initiation of a crack from a smooth boundary.
The physical motivation for the quasi-brittle modeling approach is first introduced, followed by the experimental procedures and model formulation. The experimental results are discussed in detail, and the model results are then superimposed on the measurements. Sensitivity analyses of several uncertain model parameters are presented. The spatial distribution of damage at peak load in both the notched and unnotched simulations is shown. A concluding discussion is presented around model limitations, applications, and suggestions for future work to constrain model parameters for different types of snow.
Section snippets
Experimental evidence of quasi-brittle behavior in snow
Evidence for quasi-brittle behavior in snow is growing increasingly abundant. Given that fully brittle (LEFM) assumptions are still often used as a basis for models of snow fractures related to avalanches (Heierli and Zaiser, 2008, Heierli et al., 2008), and since LEFM and quasi-brittle theories lead to very different large-size predictions of strength and energy release (Bažant and Planas, 1998), a review of quasi-brittle experimental observations and conditions under which LEFM is appropriate
Experimental methods
The experiments considered here were carried out on March 26, 2009 at Rogers Pass in the Columbia Mountains of British Columbia. Seventeen beam-shaped snow samples were extracted from the natural snow cover and then transported to a nearby cold laboratory. The samples were 50 cm long by 10 cm deep by 10 cm wide and were cut from the snowpack using rectangular boxes. The beam width was limited by the testing machine, but the beam span and depth were chosen on the basis of prior experience. The
Experimental results
Experimental load-crosshead and load-midspan displacement curves are shown in Fig. 3. The notched experiments failed at peak loads about one-third the level of the unnotched experiments. The measured crosshead displacement (Fig. 3a) includes the effects of any settling or crushing of the snow at the supports or below the central loading plate. The midspan deflection below the beam was not affected by any crushing at the point of load application, though crushing at the supports still influenced
Discussion
A desire for relative numerical simplicity combined with physical reasoning related to the microstructure of snow motivated the adoption of the nonlocal isotropic damage model here. Comparison of this model with other quasi-brittle tensile failure models (local and nonlocal) should be conducted to determine the best model for general applicability to snow. For example, a model like the rotating crack model (Jirásek and Zimmerman, 1998) might better address the stress locking observed in the
Conclusions
The nonlocal isotropic damage model was applied for the first time to explain tensile fracture of snow. Experimental fracture data for a single type of snow were used to calibrate model parameters. The model was capable of simulating the propagation of a crack from a stress concentration as well as the initiation of a crack from a smooth boundary using the same model parameters. The simulated load–displacement curves agreed well with the experimental curves and displayed quasi-brittle features,
Acknowledgements
We are grateful for the financial support of the Natural Sciences and Engineering Research Council of Canada, Canadian Mountain Holidays, and the University of British Columbia and for the in-kind support of the Avalanche Control Section of Parks Canada at Rogers Pass.
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