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New results for the spin of the Eulerian triad and the logarithmic spin and rate

机译：欧拉三联体自旋以及对数自旋和速率的新结果

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摘要

A new basis-free expression is derived for the logarithmic spin tensor, which assumes a unified form for all the three cases of coalescence of the three eigenvalues of the left Cauchy-Green deformation tensor B. It is shown that this new expression is endowed with particular continuity properties, i.e., each of its coefficients either accompanies the vanishing of its associated term or remains valid whenever the eigenvalues of B become repeated. These favorable properties, which are not enjoyed by the expressions derived earlier, result in the foregoing unified form of expression which could not be achieved by means of the previous expressions. At the same time, a fully explicit, basis-free expression is obtained for the spin tensor of the Eulerian triad, which is expressed straightforwardly in terms of the three basic invariants of B and hence renders computations of the eigenvalues of B unnecessary. Moreover, remarks are given towards explaining and clarifying some issues concerning the hypo-elastic equation of grade zero with the logarithmic rate, including the exact integrability and unstable stress responses associated with initial shear stresses at simple shear deformations, etc. [References: 34]
机译：对数自旋张量导出了一个新的无基表达式，对于左柯西-格林变形张量B的三个特征值的所有三种合并情况，它都采用统一形式。表明该新表达式具有特定的连续性，即每个系数要么伴随其相关项的消失，要么在B的特征值重复时保持有效。这些较早的表达式所不具有的有利特性导致上述统一的表达式形式，而这些形式是无法通过先前的表达式实现的。同时，获得了欧拉三联体自旋张量的完全明确的，无基数的表达式，该表达式直接根据B的三个基本不变式表示，因此无需计算B的特征值。此外，还提供了一些注释，以解释和澄清一些有关对数速率的零级次弹性方程的问题，包括与简单剪切变形时初始剪切应力相关的精确可积性和不稳定应力响应等。[参考文献：34]

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