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I know that if we are given a stress tensor that is diagonal, the sign on the diagonal entries tell us whether we have traction or compression.

Now, imagine that we are given a non diagonal stress tensor, and we wanna know the only traction and only compression directions. Those are of course the principal directions, that is, the directions given by the eigenvectors, the eigenvalues being the tensions on those directions.

But i see a problem here, the eigenvectors don't define a direction because being $\vec{v}$ an eigenvector, $-\vec{v}$ is also an eigenvector. So the sign of the eigenvalues doesn't specify the directions because I don't know whether $\vec{v}$ or $-\vec{v}$ is the positive direction I am meant to take.

So, in this case, how to figure out if it is pure traction or pure compression?

Yossarian
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2 Answers2

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It is also matter of convention.

Consider a portion $V$ (of a continuous body $B$) bounded by a closed surface $\partial V$ and a point $p$ on $\partial V$. If $n$ is the outward unit normal to $\partial V$ at $p$: $$f(p, n)_i = \sum_{j=1}^3\sigma(p)_{ij}n^j$$
is the surface density of force acting on $p$ and due to the part of $B$ outside of $V$. (One could adopt the other convention, where $n$ is the inward unit vector.)

With this definition if $f$ is parallel to $-n$, $f$ is compressive.

Consequently, always referring to that definition, if $\sigma$ is diagonal, a positive eigenvalue means traction along the corresponding eigenvector $n$ interpreted as the outward unit vector to on closed surface, and a negative eigenvalue means compression along the corresponding eigenvector.

The direction of $n$ as an eigevector is irrelevant, it just indicates where the closed surface is placed.

Indeed the stress tensor of a fluid in equilibrium is $-p \delta_{ij}$ with $p>0$. It implies that every portion of the fluid is compressed.

Valter Moretti
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The tensor, written in the basis of the eigen vectors will be diagonal. So you can use the positivi/negativity of its diagonal elements to infer your traction/compression.

In other words, I'd use the eigenvalue to know if in a given direction (dictated by the eigen vector) you have traction or compression.

Antonio Ragagnin
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