Abstract

A complex form of Hooke’s law for an anisotropic body is given, which made it possible to write down the previously known relations obtained in the simplest way. The structure of the matrix of elastic parameters and six linear invariants, which play a key role both in the connection of the stress-strain state and in the structure of the matrix of elastic parameters, have been determined. It is shown that by a certain six-dimensional unitary transformation, constructed on the basis of Hausholder matrices, the matrix of elastic moduli is reduced to the canonical form, in which the elastic moduli are invariants. Some questions of the classification of elastic materials are discussed. Formulas for the transformation of elastic moduli under three-dimensional rotation are given, as well as examples of anisotropic materials with certain properties. The possibility of constructing six invariant forms of Hooke’s law is shown.

中文翻译：

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各向异性弹性体的胡克定律的复杂形式

摘要

给出了各向异性体的胡克定律的复杂形式，这使得写下以最简单方式获得的先前已知关系成为可能。确定了弹性参数矩阵和六个线性不变量的结构，它们在应力-应变状态的连接和弹性参数矩阵的结构中都起着关键作用。结果表明，通过在Hausholder矩阵的基础上进行一定的六维unit变换，将弹性模量矩阵简化为规范形式，其中弹性模量不变。讨论了弹性材料分类的一些问题。给出了三维旋转下弹性模量的转换公式，并给出了具有一定性能的各向异性材料的例子。