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In this table of elastic constants, indices m = 1 and n = 4 result in negative values. What could the reason be?

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This looks like something unquestionably from a minerology textbook, the content of which I was never great at learning (or testing in). Regardless, here's my take on your info and question.

First, make sure you understand 2 terms enough before proceeding: 1] indices (for a mineral) and 2] elastic constants (@f.thorpe's comment motivates this).

Now that you have said terms defined in this context and know (approximately) what they mean, let's proceed with a simple, short take for an answer...

When rocks (or more specifically minerals) are "deformed (i.e. "strained") in labs, in earth, or seismic energy, they "compress" and/or "extended" in different ways (and directions). Now, here's where I carefully throw out an important and hard-to-pronounce-at-first term: mechanical anisotropy. As a short answer, this is likely what's going on. However, for more info, read on...

Anisotropy is basically everywhere. Optics, seismics, ...mechanics - you name it! For this example, though, it's the non-linear relationship between "stress" (i.e. "load") and "strain" (again, a ratio of deformation - keep in mind that it's unitless). Think of it this way: when I push the keys on my keyboard to try and provide an answer here, they are (at a very small scale) mechanically shortened vertically and extended laterally. However, if I did the same action with typing and instead the key compressed laterally (whoa, don't believe me - here's a common material that does this), that could be quantified as negative elastic constant (note though that "elastic constant(s) is a general term includes a lot of more specific terms provided here).

If you examine that last link (even just the intro), I think that'll help guide more understanding here.

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  • $\begingroup$ so what do you mean ? What could be the possible hypothesis that it has negative value in particular direction only? $\endgroup$
    – mark22
    Oct 25, 2023 at 3:07
  • $\begingroup$ I'll add more and respond to the first question once we have a reference to the image above. So, please supply that. As for your second question, can you please rephrase it? I don't understand it with its current wording. $\endgroup$
    – nate
    Oct 26, 2023 at 15:19

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