Effective, anisotropic elasticity tensor of snow, firn, and bubbly ice

The study aims to determine the effective elastic properties of snow, firn, and bubbly ice based on microstructural quantities. Anisotropy, one of these quantities (the other being ice volume fraction) in snow and ice, has two types: geometrical and crystallographic, resulting in snow's macroscopic anisotropic elastic behavior. The research focuses on the impact of geometrical anisotropy on potential ice volume fractions in snow and ice. 391 micro-CT images from various locations, including laboratories, the Alps, the Arctic, and Antarctica, were analyzed to achieve this. The analysis involved microstructure-based finite element simulations, which inherently consider microstructure and calculate the elasticity tensor. Hashin-Shtrikman bounds were utilized to predict the elastic properties of the microstructure samples. These bounds effectively captured the nonlinear interplay between geometrical anisotropy, captured by the Eshelby tensor and density. HS bounds have the advantage of the correct limiting behavior for low to high-ice volume fractions. We derived parameterization for five transversely isotropic elasticity tensor components, requiring only two free parameters. This parameterization was valid for ice volume fractions ranging from 0.06 to 0.93. The analysis employing the Thomsen parameter highlighted the dominance of geometrical anisotropy until an ice volume fraction of 0.7. However, to fully comprehend the elasticity of bubbly ice, a comprehensive approach is necessary to integrate coupled elastic theories that account for both geometrical and crystallographic anisotropy. This dataset includes a Jupyter notebook with all the necessary functions required to predict the elasticity tensor of snow for the given ice volume fraction and anisotropy. Also, the code contains the least squares optimization function to compute the elasticity tensor for the six components of stress and strain. For example, we consider our dataset to calculate the samples' elasticity tensor and reproduce Fig. 7 from the paper. We take the stress and strain values obtained from load states as input for this example. Also, a .csv file contains all the microstructural information: ice volume fraction, anisotropy, correlation functions, voxels size, and no. of voxels of the samples and the elasticity tensor obtained from finite element simulations and from present work parameterization.本研究旨在基于微观结构参数，测定雪、粒雪与气泡冰的有效弹性性质。雪与冰的各向异性（另一参数为冰体积分数）包含几何各向异性与晶体各向异性两类，二者共同造就了雪的宏观各向异性弹性行为。本研究聚焦于几何各向异性对雪与冰中潜在冰体积分数的影响。研究共分析了来自实验室、阿尔卑斯山脉、北极以及南极等不同区域的391幅显微CT（micro-CT）图像。 分析过程采用了基于微观结构的有限元模拟方法，该方法可自然纳入微观结构信息并计算弹性张量；同时利用Hashin-Shtrikman界（Hashin-Shtrikman bounds）预测微观结构样本的弹性性质。该界能够有效捕捉由Eshelby张量（Eshelby tensor）与密度表征的几何各向异性之间的非线性相互作用，且在低至高冰体积分数范围内均具备正确的极限行为特性。 本研究推导得到了仅需两个自由参数的横向各向同性弹性张量五分量参数化模型，该模型适用于冰体积分数范围为0.06至0.93的样本。通过Thomsen参数（Thomsen parameter）开展的分析表明，在冰体积分数达到0.7之前，几何各向异性占据主导地位。但要全面理解气泡冰的弹性行为，需采用综合方法，整合兼顾几何与晶体各向异性的耦合弹性理论。 本数据集包含一个Jupyter笔记本（Jupyter notebook），其中涵盖了针对给定冰体积分数与各向异性参数预测雪的弹性张量所需的全部功能函数；该代码还包含最小二乘优化函数，可用于计算应力与应变六个分量对应的弹性张量。例如，本研究利用本数据集计算样本弹性张量，并复现了论文中的图7，本次示例以载荷状态下得到的应力与应变值作为输入。 此外，数据集还包含一个.csv文件，其中存储了所有样本的微观结构信息：冰体积分数、各向异性参数、相关函数、体素尺寸、体素总数，以及通过有限元模拟得到的弹性张量与本研究提出的参数化方法得到的弹性张量。