I am taking a look at different types of strain tensor. Specifically, I am thinking about if the infinitesimal strain tensor \begin{align*} \epsilon_{ij} = \frac{1}{2} (\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}) \end{align*} is positive-definite. I have Google searched some resources, and one of them says it is positive-definite. However, I think that it is not always positive-definite, as in the one-dimensional trivial case, if $\partial u/\partial x$ is negative, then it will not be positive-definite. Other sources say that other strain tensors, like the Lagrangian strain tensor, are positive-definite. I am not sure which types of strain tensor are positive-definite and also the implications if so. (I am thinking about the strain surface being an ellipsoid or not.) Any ideas will be greatly appreciated!
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$\begingroup$ I suspect sloppy terminology is used. Often is just matters whether or not it is indefinite or not. I suspect they mean to say "positive semi-definite or negative semi-definite". $\endgroup$– Mikael ÖhmanCommented Apr 17, 2021 at 12:39
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$\begingroup$ @MikaelÖhman But is it possible for a quadratic form of the infinitesimal strain tensor to be negative? $\endgroup$– Benjamin_GalCommented Apr 17, 2021 at 15:04
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$\begingroup$ Yes I don't see why not. It can also be zero. But it will not be indefinite. I don't know what materials you have been reading, a quick googling I didn't find much that talked about this (mostly just for deformationgradients, stretch and rotation tensors, where it matters. $\endgroup$– Mikael ÖhmanCommented Apr 17, 2021 at 17:23
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$\begingroup$ @MikaelÖhman Thanks. I can’t upvote due to inadequate reputation though... $\endgroup$– Benjamin_GalCommented Apr 17, 2021 at 17:27
2 Answers
I think the material you found is just wrong. A trivial counter example is zero displacement = zero strain, which is not positive definite.
Nor do I see where such a property would be particularly useful when calculating anything. It's a different story when talking about the other tensors we deal with (such as the elasticity tensor), where it definitely is a important property.
Looking at reputable resources (wikipedia, https://www.continuummechanics.org/) i see no mention of such claims when they present different strain measures.
The strain tensor does not have to be positive definite, as mentioned before.
On the other hand, you are asking what type of strain tensor are positive definite. A positive tensor would have positive principal strains. That implies a tensor where you have elongation in 3 different directions.
