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I completed an introductory course on continuum mechanics, but we did not delve into curvilinear coordinates with CM. However, when I read some publications I see the stress-strain relationship is written usually as follows;

$\sigma^{ij} = \mathbb{C}^{ijkl}: \varepsilon_{kl}$

And I know that if the coefficients are coming from a covariant base the base itself should be contravariant. For example;

$\mathbb{\sigma} = \sigma^{ij}\ e_i\otimes e_j$

$\mathbb{C} = \mathbb{C}^{ijkl}\ e_i\otimes e_j \otimes e_k \otimes e_l $

or

$\mathbb{C} = C_{ijkl}\ e^i\otimes e^j \otimes e^k \otimes e^l $

My first question is why do these entities have to be represented with different bases? Like, the coefficients from the covariant base and the tensor base are contravariant or vice versa. BUT the two are never from the same base.

The second question how do you decide which coefficient should be from which base when formulating the constitutive law? For example in the case of;

$\sigma^{ij} = \mathbb{C}^{ijkl}: \varepsilon_{kl}$

why are the sigma and C coefficients coming from the contravariant base but the epsilon coefficient is coming from covariant base?

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    $\begingroup$ For an orthonormal basis, the covariant and contravariant forms are identical because the metric tensor is the identity. The covariant/contravariant/mixed components become important in curvilinear coordinates. See en.wikipedia.org/wiki/Tensors_in_curvilinear_coordinates for the basic math ideas. Physical interpretations are best obtained from books on gravitation. $\endgroup$ Oct 25 '21 at 20:25

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