Fabric tensors and effective properties of granular materials with application to snow
Shertzer, Richard Hayden
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Granular materials e.g., gravel, sand, snow, and metallic powders are important to many engineering analysis and design problems. Such materials are not always randomly arranged, even in a natural environment. For example, applied strain can transform a randomly distributed assembly into a more regular arrangement. Deviations from random arrangements are described via material symmetry. A random collection exhibits textural isotropy whereas regular patterns are anisotropic. Among natural materials, snow is perhaps unique because thermal factors commonly induce microstructural changes, including material symmetry. This process temperature gradient metamorphism produces snow layers that can exhibit anisotropy. To adequately describe the behavior of such layers, mathematical models must account for potential anisotropy. This feature is absent from models specifically developed for snow, and, in most granular models in general. Material symmetry is quantified with fabric tensors in the constitutive models proposed here. Fabric tensors statistically characterize directional features in the microstructure. For example, the collective orientation of intergranular bonds impacts processes like conduction and loading. Anisotropic, microstructural models are analytically developed here for the conductivity, diffusivity, permeability, and stiffness of granular materials. The methodology utilizes homogenization an algorithm linking microscopic and macroscopic scales. Idealized geometries and constitutive assumptions are also applied at the microscopic scale. Fabric tensors tying the granular arrangement to affected material properties are a natural analysis outcome. The proposed conductivity model is compared to measured data. Dry dense snow underwent temperature gradient metamorphism in a lab. Both the measured heat transfer coefficient and a developing ice structure favored the direction of the applied gradient. Periodic tomography was used to calculate microstructural variables required by the conductivity model. Through the fabric tensor, model evolution coincides with measured changes in the heat transfer coefficient. The model also predicts a different conductivity in directions orthogonal to the gradient due to developing anisotropy. Models that do not consider directional microstructural features cannot predict such behavior because they are strictly valid for isotropic materials. The conclusions are that anisotropy in snow can be significant, fabric tensors can characterize such symmetry, and constitutive models incorporating fabric tensors offer a more complete description of material behavior.