The aim of this short paper is to provide, for elasticity tensors, generalized Euclidean distances that preserve the property of invariance by inversion. First, the elasticity law is expressed under a non-dimensional form by means of a gauge, which leads to an expression of elasticity (stiffness or compliance) tensors without units. Based on the difference between functions of the dimensionless tensors, generalized Euclidean distances are then introduced. A subclass of functions is proposed, which permits the retrieval of the classical log-Euclidean distance and the derivation of new distances, namely the arctan-Euclidean and power-Euclidean distances. Finally, these distances are applied to the determination of the closest isotropic tensor to a given elasticity tensor.

中文翻译：

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弹性张量的广义欧几里德距离

这篇短文的目的是为弹性张量提供广义欧几里得距离，通过反演保持不变性。首先，弹性定律是通过规范在无量纲形式下表达的，这导致没有单位的弹性（刚度或柔量）张量的表达。然后基于无量纲张量函数之间的差异，引入广义欧几里德距离。提出了一个函数子类，它允许检索经典对数欧几里得距离和推导新距离，即反正切-欧几里得距离和幂-欧几里得距离。最后，这些距离用于确定与给定弹性张量最近的各向同性张量。