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For shell elements, is the elasticity tensor for linear-elastic & isotropic materials the same in a local curve-linear (convected) system vs. a local Cartesian system?

I wonder because intuitively, for a shell element & linear-elastic isotropic material, the only direction that matters (material-property wise) is the z direction. And in this case, the local z axis is aligned with the curve-linear coordinate system. But how do we mathematically show it?

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A linear elastic stress-strain law can be expressed in coordinate free form as $$ \boldsymbol{\sigma} = \mathsf{C} : \boldsymbol{\varepsilon} $$ where $\boldsymbol{\sigma}$ is the Cauchy stress (a second-order tensor), $\mathsf{C}$ is the stiffness tensor (a fourth-order tensor), and $\boldsymbol{\varepsilon}$ is the small strain tensor (a second-order tensor).

Expressed in terms of the covariant components, the second-order tensors can be written in the form $$ \boldsymbol{\sigma} = \sigma_{ij}\, \mathbf{g}^i \otimes \mathbf{g}^j $$ where $\mathbf{g}^i$ are the reciprocal basis vectors in curvilinear coordinates.

Similarly, the fourth-order tensor can be expressed as $$ \mathsf{C} = C_{ijkl}\, \mathbf{g}^i \otimes \mathbf{g}^j \otimes \mathbf{g}^k \otimes \mathbf{g}^l $$

In general curvilinear coordinates, the expressions for the stiffness matrix become extremely complicated. However, these can be simplified by using a locally orthonormal basis. In that case, the reciprocal basis simplifies to a locally Euclidean basis, i.e., $$ \mathsf{C} = C_{ijkl}\, \mathbf{e}^i \otimes \mathbf{e}^j \otimes \mathbf{e}^k \otimes \mathbf{e}^l $$

Most shell elements use stiffness matrices that are expressed in a local orthonormal basis at each point.

Since the stresses are often (but not always) assumed to be zero through the thickness (the $z$-direction in the question?), it is sufficient to use linear elastic properties in the in-plane directions with corrections for the zero-stress assumption. You will, of course, have to make sure that the correct coordinate system is used at each point.

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