Talk:Hooke's law
|
|
I think the comment about hashing would be better placed on the anagram page.
Charles Matthews 12:13, 20 Mar 2004 (UTC)
Would it be appropriate to mention here that Hooke's Law is only a linear approximation (albeit one that is very good up to the elastic limit)? CyborgTosser 01:04, 23 Aug 2004 (UTC)
Tensor notation
Hi,
Does anyone know whether in the generalized form, some indices are covariant and some other are contravariant? So should we write something like
- <math>\sigma_{ij} = C^{ij}_{kl} \cdot \varepsilon^{kl}<math>
(random proposition)?
I never saw something like this in my books, nobody cares in continuum mechanics, but if we want to be strict and stick to the math conventions...
Cdang 15:34, 26 Nov 2004 (UTC)
Probably more like
- <math>\sigma_{ij} = C^{kl}_{ij} \cdot \varepsilon_{kl}<math>
actually.
(PS There is something suspicious about the strain tensor, though, as I read in another encyclopedia.)
Charles Matthews 16:37, 26 Nov 2004 (UTC)
- I got another answer from fr:, someone proposes σji because it should be covariant in j (number of the face) and contravariant in i (axes of the force). Any other opinion ?
- other opinion: the force is of same type than the conjugate momentum and thus of opposite type than the coordinate; the faces are also "dual" to the coordinates (via the completely antisymmetric tensor of rank 3), thus both indices should be of the same type. (But maybe I made an error in my reasoning...). But without considering opinions, one could just check its transformation properties under rotations...— MFH: Talk 18:12, 20 Jun 2005 (UTC)
- Concerning your suspicion about the strain tensor, pleasewrite them down at the talk page.
- Cdang 13:44, 6 Dec 2004 (UTC)
As it is written currently on strain tensor, the latter should have one upper and one lower index (but maybe rather xj is meant to be there instead of xj, and idem for u).
Next, upper and lower indices are different only if the metric is different from δij, else this tensor and its inverse allow to raise and lower indices. However, in 3 dimensions people usually consider xj and xj the same. Without fixing a convention about the index position of coordinated and derivatives, it is quite useless to ask this question for more complicated derived tensors (imho). — MFH: Talk 17:38, 20 Jun 2005 (UTC)