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?
Need to understand the terms you are using. A local curvilinear system is in a sense a local cartesian system, the axes are orthogonal and the origin moves. You use the term convected which leaves me wondering if you are distinguishing between the two that only the curvilinear system origin moves? If so then the stress tensor would not be the same in both cases and I would not describe the cartesian system as "local". Help me to better understand your question.
The elasticity tensor for linear-elastic & isotropic material by definition is invariant in relation to rotation about any axis.
The stress tensor consists of 6 components. There are 3 invariants of the stress tensor, the first consisting of the sum of the normal stresses and the other two consisting of the sum of orderly combinations of normal stress and shear stress. But at no time can we say that any of the individual components of the stress tensor are invariant to the axes of rotation.
It seems that the original question was about the elasticity tensor.
So please define your interpretation of elasticity tensor and how that may differ from my response.
I have been responding regarding a 4th order tensor. Per chance, are you referring to a second order tensor?
I agree that for an isotropic material the elastic constants are invariant to the axes of rotation by definition but that does not make the components of the stress tensor which is subject to the stress boundary conditions invariant.
Hooke's law for continuous media is
S=C.E, where E={eij} - deformation tensor, S={sij} - stress tensor, C={cijkl} - elasticity tensor. Of course, sij and eij depend on the coordinate system, but in the case of linear-elastic & isotropic material cijkl are invariant.
I have never heard of the of the constants being referred to as a stress tensor. But if you choose to label as such, that I suppose is your choice to do so but I doubt that this is the intent of the original question.
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