通过结构张量构建各向异性超弹性不可约完整性基
Construction of minimal integrity basis for anisotropic hyperelasticity via structural tensors
AI总结：
本文提出通过结构张量构建各向异性超弹性不可约完整性基的方法，用于建模各向异性材料。
摘要：
我们提出了一种直接的分析-数值方法，用于确定各向异性超弹性的多项式完备且不可约标量值不变量集。通过应用所提出的技术，我们利用结构张量概念，为超弹性中所有常见的各向异性获得不可约完整性基，即不变量是由变形度量（对称二阶张量）和描述材料对称性的结构张量集合形成。我们的工作涵盖了由经典7个晶系产生的11种各向异性结果，以及从圆柱形、球形和二十面体对称系统导出的4种额外非晶向异性结果。通过Molien级数以及文献中已建立的标量值不变量集结果，证明了所提出完整性基的多项式完备性和不可约性。此外，我们推导了指定对称群的结构张量集合与仅使用单个结构张量描述之间的关系。两者均可通过应用所提出的分析-数值方法来构建不可约完整性基。所提供的不变量集对于通过结构张量概念使用经典模型以及基于机器学习的现代方法建模各向异性材料具有重要意义。除了所展示的结果外，本文还旨在为建模各向异性材料的复杂领域提供入门概述。
英文摘要：
We present minimal integrity bases for all common anisotropies in hyperelasticity via the structural tensor concept, which can be used to formulate any algebraic invariant function in the elements of the respective bases. Hence, the provided minimal integrity bases are of great interest for formulating a concise but general anisotropic material model. Our work covers results for the 11 types of anisotropy that arise from the classical 7 crystal systems, as well as findings for 4 additional non-crystal anisotropies derived from the cylindrical, spherical, and icosahedral symmetry systems. By using well-known results from literature about structural tensors, isotropic invariants, and isotropic extension, functional bases are directly determined. A simple analytical-numerical approach is employed to identify polynomial relations between the invariants of these functional bases, thereby enabling the construction of functional bases of reduced cardinality. After that, we show that the determined reduced functional bases are also minimal integrity bases by identifying polynomial relations to known integrity bases from literature. Furthermore, fundamental concepts from invariant theory, including the Hironaka decomposition of invariant rings and the closely related Hilbert series, are employed to further validate the results. Alongside the presented findings, this article also aims to provide an introductory overview of the complex field of modeling anisotropic materials, especially for researchers with an engineering background.
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