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covariant contravariant stress tensor

Covariant and contravariant stress tensors: difference between the Cauchy stress and the 2nd Piola-Kirrchoff stress

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I am currently reading a book about "Material Inhomogeneities in Elasticity" written by Gérard Maugin.

He brings up the covariancy and contravariancy of some well-known stress tensors and in particular that of the Cauchy stress tensor. I was not very familiar with the concept of covariance, so i read up on this very specific tensorial notion. 

I have been able to demonstrate easily that the deformation gradient F is a mixed tensor (covariant and contravariant).

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