Thursday, April 30, 2020

On the flexural rigidity of a slice of pizza

People who eat pizza (not the deep dish casserole style from Chicago, but normal pizza), unbeknownst to most of them, have developed an intuition for a key concept in elasticity and solid mechanics. 

I hope that all right-thinking people agree that pizza slice droop (left hand image) is a problem to be avoided.  Cheese, sauce, and toppings are all in serious danger of sliding off the slice and into the diner's lap if the tip of the slice flops down.  Why does the slice tend to droop?   If you hold the edge of the crust and try to "cantilever" the slice out into space, the weight of the sauce/toppings exerts downward force, and therefore a torque that tries to droop the crust.  

A simple way to avoid this problem is shown in the right-hand image (shamelessly stolen from here).  By bending the pizza slice, with a radius of curvature around an axis that runs from the crust to the slice tip, the same pizza slice becomes much stiffer against bending.   Why does this work?  Despite what the Perimeter Institute says here, I really don't think that differential geometry has much to do with this problem, except in the sense that there are constraints on what the crust can do if its volume is approximately conserved.  

The reason the curved pizza slice is stiffer turns out to be the same reason that an I-beam is stiffer than a square rod of the same cross-sectional area.  Imagine an I-beam with a heavy weight (its own, for example) that would tend to make it droop.  In drooping a tiny bit, the top of the I-beam would get stretched out, elongated along the \(z\) direction - it would be in tension.  The bottom of the I-beam would get squeezed, contracted along the \(z\) direction - it would be in compression.  Somewhere in the middle, the "neutral axis", the material would be neither stretched nor squeezed.  We can pick coordinates such that the line \(y=0\) is the neutral axis, and in the linear limit, the amount of stretching (strain) at a distance \(y\) away from the neutral axis would just be proportional to \(y\).  In the world of linear elasticity, the amount of restoring force per unit area ("normal stress") exhibited by the material is directly proportional to the amount of strain, so the normal stress \(\sigma_{zz} \propto y\).  If we add up all the little contributions of patches of area \(\mathrm{d}A\) to the restoring torque around the neutral axis, we get something proportional to \(\int y^2 \mathrm{d}A\).  The bottom line:  All other things being equal, "beams" with cross-sectional area far away from the neutral axis resist bending torques more than beams with area close to the neutral axis.

Now think of the pizza slice as a beam.  (We will approximate the pizza crust as a homogeneous elastic solid - not crazy, though really it's some kind of mechanical metamaterial carbohydrate foam.)  When the pizza slice is flat, the farthest that some constituent bit of crust can be from the neutral axis is half the thickness of the crust.  When the pizza slice is curved, however, much more of its area is farther from the neutral axis - the curved slice will then resist bending much better, even made from the same thickness of starchy goodness as the flat slice.

(Behold the benefits of my engineering education.) 


Pizza Perusing Physicist said...

Did you have me in mind when you wrote this? :-)

Anonymous said...

Funny I had an argument with a mathematician/theoretical physicist about exactly this! They appeal to a theorem by Gauss, but that misses the point I feel. I've had my fair share of soggy homemade pizza where folding doesn't work very well :), but diff geom. tells me my pizza is topologically equivalent to a well-made pie where folding actually works.

Douglas Natelson said...

PPP, it had crossed my mind :-)
Anon, I agree. Actual continuum mechanics has to be important. Trying the fold technique with a cloth does not make the cloth able to hold up arbitrary amounts of weight. I would argue that elasticity sets the "energy barrier" that has to be overcome to flip the sign of the gaussian curvature of the crust.

Unknown said...

In the first example you induce a negative bending moment (top in tension) around the z-axis that of course allows the pizza to flow downwards. In the second example not only you changed the axis for bending (now y axis) but the top is in compression so nothing flows. The rest about the I beam etc are not applicable. Sorry.

Douglas Natelson said...

Unknown, I don’t think you’re interpreting my description properly. My coordinates have the z axis running from the edge of the crust held by the fingertips toward the tip of the slice, the x axis running transverse (the width of the slice), and the y axis vertically (along the direction of gravity). The weight of the toppings etc induces a bending moment pointing along the positive x direction. That causes the top of the pizza crust (or ibeam) to be in tension. In the ibeam case, maybe my drawing isn’t clear, but the coordinates are the same. That is supposed to be what is happening at an imagined cross section of the beam. The beam extends along z (the normal to the exposed cross section is +z), the transverse direction is x, and the vertical-along-gravity direction is y. The weight of the part of the beam that you can’t see exerts a bending moment (not drawn explicitly) pointing along +x, just as in the pizza case. Sorry if the geometry was unclear.