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I was trying to understand this stress curve in the notch effect. $\sigma_y$ I can understand but $\sigma_x$ and $\sigma_t$ not. Could you please explain?

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    $\begingroup$ For a comprehensive book try Strength of Materials by Timoshenko - available in english or metric versions...luckily it does not change the theory... $\endgroup$ – Solar Mike Jan 26 '18 at 23:13
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    $\begingroup$ What's $\sigma_t$ (as in, the given nomenclature)? $\endgroup$ – Wasabi Jan 27 '18 at 1:10
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    $\begingroup$ Also, I edited your question removing the book-request, which would be considered off-topic in this site, and replaced it with a simple question regarding what the different stresses are. $\endgroup$ – Wasabi Jan 27 '18 at 1:11
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In these cases, it is always worth thinking about how the stress "flows" through the element. In this case, that's quite simple:

enter image description here

The stress near the center of the element is basically unaffected by the notching. The stress near the edges, however, is diverted by the notches. From the image it is already quite clear that there's quite literally a stress concentration near the notches, which explains why $\sigma_y$ peaks there.

However, if we simplify this diagram into a strut-and-tie model (ignoring the stress along the core, which simply goes straight through the middle), we end up with this:

enter image description here

Remember the convention that arrows pointing away from the node represent tension and those pointing into the node are in compression. We can therefore see that the main verticals (and the diagonals representing the stress "detours") in red are entirely in tension, as would be expected. However, the detours create transversal imbalances near the notch (in blue). At the center of the notch, both (red) diagonals point "out" of the section: their vertical components cancel out, however their horizontal components add up to a force pointing "out". This force must therefore be cancelled out by another force (blue) on the other side of the node pointing "in". Since this is an arrow pointing away from the node, it is in tension. This is $\sigma_x$.

On the outskirts of the notch (above and below it), where the stress lines start to be deflected, there is another horizontal force imbalance, however in this case the deflected tension points "in", so the balancing force must point "out". In this case, that force points into the node, so that area is under compression.

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$\sigma_x$ is tension and the reason for its curve is it is higher where there is more material to affect stretching the rod to go back to fill in the notch.

$\sigma_\tau$ is shear and it increases toward center in a hyperbolic curve because of additive stresses of vertical and horizontal tensions: $\sigma_y + \sigma_x$.

If you draw the Mohr's stress circle over a few points in that region you see it!

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