Covariant and contravariant stress tensors: difference between the Cauchy stress and the 2nd Piola-Kirrchoff stress
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).
But i still don't understand why the Cauchy stress tensor σ is said to be covariant. I know that covariancy and contravariancy has to do with how the components of the tensor are trasformed along with a change of coordinates. But it is not clear to me why the Cauchy stress tensor σ is covariant while the 2nd Piola Kirchoff stress tensor S is contravariant.
Since σ is defined as force per unit surface in the deformed configuration while S corresponds to a pull-back onto the undeformed configuration, i believe that the covariancy or the contravariancy has to do with the configuration accounted for by the diffent stress tensors introduced above.
I am looking for a demonstration or an explanation showing and proving why the Cauchy stress tensor is covariant whereas the 2nd Piola-Kirrchoff stress tensor is contravariant.
Let me know what you know or think about that.