Why Do Ductile Materials Neck in the Middle?

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For materials that have a propensity for necking, assuming they're heterogeneously structured/have equal strength throughout and are without defects- why do these materials have a propensity for necking in the center when forces are applied to both ends?

I'm trying to use physics to explain why, when you pull a piece of pizza dough apart, it will thin out/neck in the center, and then, if you keep pulling, it will eventually break at that center point.
 
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That sounds like an interesting problem but does it really happen? I just tried stretching thin strips of paper but in most cases the strip broke close to one of the ends where I was gripping the paper. Of course paper is not pizza dough. Please let us know the results of any experiment that you try out.
 
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I just tried stretching thin strips of paper but in most cases the strip broke close to one of the ends where I was gripping the paper. Of course paper is not pizza dough.
The fibers in paper might stretch a tiny bit when pulled apart, but, for the most part, it isn't ductile. Here's a good photo of taffy being stretched:

https://littlehousebliss.files.wordpress.com/2014/07/taffy-after-pulling.jpg

This is typical for taffy and dough- and chewing gum, and molten glass- off the top of my head.
 
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Isn't it true for ductile materials in general, including metal? Here are a couple examples:



That being said, I have no formal education in material properties, and I might just be misunderstanding the discussion.
 
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From your videos necking does appear to start close to the middle of the specimen but in the video I linked necking is a fair distance from the centre (go to 5.17). It would be nice to get access to other results.

One thing that should be considered is that the tested samples had a wider cross section at each end. On the plus side I think this prevents necking and breakage at the places where the sample is gripped. On the minus side I think that these two discontinuities in cross sectional area can cause some sort of discontinuity in the crystal structure along the length of the bar which is possibly, assuming everything else is perfect, symmetrical about the centre. It would be interesting to test this possible effect further by using either longer bars and or bars where the two ends have different cross sectional areas.

I think the biggest problem is that ductile metals and indeed pizza dough are not heterogeneously structured and of equal strength throughout. The only materials I'm familiar with which come close to having such characteristics are intrinsic semiconductors and glass of the type used in fibre optic cables.

I think there are experts in this forum who could give better advice than us.
 
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I don't know how the error occurred above but this shows the result i referred to.
 
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I think the biggest problem is that ductile metals and indeed pizza dough are not heterogeneously structured and of equal strength throughout.
After being properly kneaded and proofed, pizza dough, if formed into a column and pulled apart, will fail in the center point at least 99 out of 100 times. It may not be homogenously* structured enough to happen 100% of the time, but it's enough to happen almost all the time- and it's the 99 times- when a material is of equal strength throughout that it fails in the middle- that's what I'm trying to understand.

*I had been using 'heterogeneous' when I meant 'homogeneous.'
 
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When you tested the dough was it stretched sideways so that the dough was kept approximately horizontal and straight? I mention this because I imagine the dough bending under its own weight and forming a curve with the central part of the dough being at the lowest point. If this happens the tension will not be even at all points at any particular cross section of the dough but can be a maximum at the bottom part of the central section. It's analogous to bending a stick of wood, the outside of the bend will have the greatest stretch and the greatest tension.

(Whoops I used heterogeneous as well)
 
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When you tested the dough was it stretched sideways so that the dough was kept approximately horizontal and straight?
That's a good question. Yes, gravity would play a role if pulling horizontally, but I'm talking about pulling vertically, like the steel in the video.
 

Stephen Tashi

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Can we find a simple model for a homogeneous material that exhibits necking?

Suppose we model a rod as chain of "identical" coiled springs connected end-to-end along the x-axis. Pulling on the ends of the rod (slowly) would cause the coils of the springs to form narrower coils, but there is no reason one spring would coil narrower than another.

Suppose we model a rod as a two parallel chains of such springs and connect each junction of two coils on chain A with a vertical spring from that junction to the corresponding junction of chain B. I see no reason why stretching the rod would pull the two x-axis chains of springs closer together near the middle (assuming there is no initial tension in the springs that go in the y-direction.)

Now suppose we join the two chains of springs that go in the x-direction using other springs that are not parallel to the y-axis. For example, we could join chains of springs A and B with springs that go from junction A[n] to B[n+1] and springs that go from junction A[n+1] to B[n]. If this configuration is stretched then do the two chains of springs in the x-direction remain parallel?

In tensile testing or pizza dough pulling, the very ends of the specimen are constrained so the specimen can't change its cross sectional area at its ends. To model that with chains of springs, we impose a similar constraint on the y-distance between the two x-axis chains of springs at their ends. Intuitively (to me) the force needed to keep the two springs apart that that constant y-distance increases as the rod is stretched. So the x-axis chains of springs won't remain parallel because if they did then, at their ends, they would not exert any force in the y-direction.
 

Nidum

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Flat tensile specimen v3.png



FEA showing VonMises stress in a flat test specimen under axial load .
 
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Stephen Tashi

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Necking1.jpg
Necking2.jpg
A rubber band and paperclip demo:
 
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It likely has to do with the specimens (pizza dough or metal test coupons) being gripped from the outside. The force being applied from the outside is transmitted as tension down the outer surface, and as shear towards the center.
There is some small deflection from the outer surface towards the center which could show as a dimple on a metal sample if the ends of a test coupon were polished to a mirror finish.
This longitudinal shear of the outer surface diminishes toward the mid point of any specimen. At the center the entire cross section will be in tension. At this cross section the first thinning is most likely to occur. Once thinning begins it quickly becomes the the path of least resistance. Effects like work hardening make the thinned section more broad.
 
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The observed effect has to do with load balancing and boundary conditions. When a classic material is stretched axially, the microstructure creates off-axis forces to shrink the cross-section and preserve the mass density per the Poisson effect. If these microforces are not balanced, density waves will form and rapidly travel the length of the structure until the microforces are balanced. Without boundary conditions (i.e. electromagnetic loading), this would lead to a uniform contraction of the cross-section. Clamp-type boundary conditions prevent the constrained parts of the structure from stretching, thus also constraining the cross-section at the boundaries. The result is that for symmetric loading of a symmetric sample, the cross-section as a function of length must be continuous throughout and equal at the ends, resulting in and exponential function with the minimum being located at the center. The smaller the cross-section becomes, the more stress is concentrated and the more defects are likely to form, resulting in necking. Hence, this is why classic materials have a propensity for necking in the center when forces are applied to both ends.

Realistically, preexisting defects in the microstructure and/or asymmetric loading will cause the necking and fracture to occur elsewhere.

And having said all that, uncooked dough may have fluid behavior, so trying to analyze it as if it were an elastic solid may not be correct as that would ignore the viscous flow of dough from the top to the bottom of the sample. How long was a sample allowed to remain in the vertical state before being pulled?
 
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... At this cross section the first thinning is most likely to occur. Once thinning begins it quickly becomes the the path of least resistance...

.
I've been mulling this over and have awakened a couple mornings thinking about this. I think analyzing a round bar in tension as a number of concentric thin wall tubes with a very small center rod all in tension provides an advantage.
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For a thin wall tube beginning as a right circular cylinder and placed in axial tension, an increase in axial length with the least increase in surface area should require the least total strain. This suggests the cylinders will all deform towards a catenoid shape, as a minimal surface.
Any shape other than a catenoid would require more total strain. As such, with a catenoid having the least cross sectional area in the middle, once the catenoid shape has been approximated, additional thinning will most easily occur in the center.
.

Work hardening should result in a flattening of the center forming a somewhat oblate catenoid.

I haven't been able to locate any papers or texts to support this conjecture. I would appreciate meaningful input on this especially if you have knowledge of any round ductile tensile specimens approaching catenoid shapes prior to failure.
 

Gandhar NImkar

For materials that have a propensity for necking, assuming they're heterogeneously structured/have equal strength throughout and are without defects- why do these materials have a propensity for necking in the center when forces are applied to both ends?

I'm trying to use physics to explain why, when you pull a piece of pizza dough apart, it will thin out/neck in the center, and then, if you keep pulling, it will eventually break at that center point.
Here is a example of why necking happens. See if this help you out.
 

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