# What causes stress concentration (aka stress risers/raisers) at corners?

I've read a few explanations about why stress concentration occurs at sharp corners but I don't find the explanations intuitive.

Can anyone explain it perhaps using an analogy such as atoms "holding hands" with neighbor atoms or a similar easy to understand analogy?

Most explanations I've shown just show the result of a finite-element-analysis based stress analysis. This shows the emergent property but doesn't explain the underlying mechanism that causes the stress to concentrate at corners.

Better understanding the cause of the crashes of the De Havilland Comet jet aircraft due to metal fatigue at the corners of their rectangular windows is the context for this question. See:

http://www.cracked.com/article_19623_6-small-math-errors-that-caused-huge-disasters.html

The answer I'm looking for should not contain any math.

• Is it analogous to the electrostatic effect in which charge concentrates where a conductor has a corner? – user4552 Aug 10 '13 at 0:15

I made this drawing of my intuitive view of stress concentrations. The bottom object is exactly the same as the top, except the dark gray material has been added, resulting in a nasty stress concentration. In other words, adding this material has made the part weaker. How could that be possible?

My rough intuitive view is: I think of the blue portion of the material as forming sort of a "lever" which the dark gray material is pulling on. The "lever" portion of the material transmits all this force to a single point (the tip of the crack), attempting to pull the material apart there. The dark gray material is serving to pull on the tip of the crack, in a way that wouldn't be possible without it.

Or, if I want to be a little more accurate, I think of little finite elements in the red region that are all in tension. In the blue region, there's no tension pulling the finite elements upward, so they must be held up by shear forces coming from the right. Those shear forces add to the stress at the tip of the crack.

In contrast, if look at the "normal design" case at the top of the image, the finite elements are all mostly in tension for the whole length of the part. There's still some shear force. But there's no one place where a large area of tension gets converted to shear forces acting on a single point.

And that's why rounded holes are good.

Keep in mind that stress is a response to strain. Stress is not applied, rather strain is applied and stress is the material's response to that applied strain. Stress accounts for the forces generated by particles of a material in order to resist an external force pulling them apart. Now with this context consider as mentioned below,

Imagine that a group of atoms are forming a grid like pattern like this,

ooooooooooooooooooooooooooooooooo >

ooooooooooooooooooooooooooooooooo >

ooooooooo..........................ooooooooooo >

ooooooooooooooooooooooooooooooooo >

ooooooooooooooooooooooooooooooooo >

Each atom attracts all neighboring atoms. In the center of the grid there is a hole i.e. no material therefore no atoms. Now if a uniform force pulls this grid in the direction indicated by ">" then atoms will resist the force by holding each other firmly. But where there is a hole, atoms are not present to resist the force pulling them apart therefore atoms of the adjacent rows near the hole will have to pull each other more firmly in order to compensate for the atoms shortage in the hole. This is stress concentration. It is concentration of resisting forces around a deformity in an attempt to prevent pulling apart of whole structure.

Stress concentration is the aggravated resistance to pulling that a group of atoms have got to offer in order to compensate for a defect in material.

This is the intuitive explanation as far as my understanding is concerned.

The phenomena of stress concentration arises from the analysis of the stress field surrounding a hole, corner, or crack in a continuum. The underlying physical mechanism is really just Newtonian mechanics, subject to the assumption that the mechanical behavior of the material can be represented by a constitutive law, such as an elastic solid. Then when you solve the equations subject to the boundary conditions, the near field stress solution close to the corner is more intense than the far-field stress solution far from the corner.

If you doubt the effect is real, and want to see a non-mathematical demonstration, I suggest you search the web for images and videos of "stress concentration photoelasticity" The optical properties of test pieces made of photoelastic materials change under stress and are routinely used to make the local stress concentration around corners 'visible.'

Effects such as stress corrosion cracking, fatigue, and yielding may manifest close to corners because the stress is driving a microscopic stress-activated process, such as dislocation motion or cleavage. The details are dependent upon the system you are interested, often difficult to pin down, and the subject of much materials science research.

• Mark, thanks for the answer. I don't doubt the effect is real. I'm looking for an intuition about why the equations yield the answer they do. – Dan Sandberg Aug 10 '13 at 12:47