AI 中文总结
本文运用泛函分析工具，揭示适用于任意空间维度等条件的希尔伯特空间四种正交分解，表征弹性体静平衡，还明确数据驱动连续介质力学中本构定律换为材料数据集时不变的内在结构。
AI 中文摘要
数学弹性力学历史悠久，文献体量庞大。令人惊讶的是，弹性力学核心呈现出丰富且清晰的结构，据我们所知，该结构很少以明确统一的形式呈现。本文运用泛函分析的标准工具，揭示了希尔伯特空间的四种正交分解，这些分解可表征小变形下弹性体的静平衡，适用于任意空间维度、任意边界条件，以及经典和数据驱动的公式化表述。关于数据驱动的连续介质力学，本文强调了当本构定律被材料数据集替代时保持不变的内在结构。
英文摘要
Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.
