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.
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Morin, L., Gilormini, P. & Derrien, K. Generalized Euclidean Distances for Elasticity Tensors. J Elast 138, 221–232 (2020). https://doi.org/10.1007/s10659-019-09741-z
Mathematics Subject Classification
- Elasticity tensor
- Log-Euclidean distance
- Closest isotropic tensor