The Cake Practical

The theme of this month's Accretionary Wedge blog carnival is "Time to Think Out of the Box", in terms of approaches to teaching. I thought I'd contribute an activity I've been running in my classes which comes directly out of a box - a cake mix box.

We spend a lot of time in my Honours tectonics class (that's equivalent to a senior seminar in Tectonics in the states) talking about the lithosphere.

We discuss and in some cases, calculate:

• the flexural thickness (relatively thin, <10km>
• thermal definitions (thickness varies, but temperature is really a proxy for rheology),
  
• definitions based on a sheared layer (high seismic anisotropy) separating the lithosphere from the asthenosphere,
  
• more abstract definitions based on calculated rheologic transitions (using more complex proxies than just temperature)
  
• discussion expanded this year to include looking at sub-continental lithospheric mantle which has been a stable geochemical reservoir for a very long time.. therefore not likely communicating with the convecting mantle (shout out to my colleague Steve), a distinction which is obviously important and in some places observable but was previously not on my radar.

However, for the students to really grok* this they have to have a good grasp on the concept that the same material can behave both as either an elastic or ductile solid under different conditions of pressure and temperature. There begins the search for an analog material.

The Cake Practical

As it turns out, cake deforms elastically at low stress and non-recoverably (let's call it viscous) at higher stress. This is a cheap tolerable proxy for Maxwell behavior. I bake some thin sheet cakes - from an ordinary cake mix - each pair of students gets about 100cm² piece of 3cm thick cake. The technical staff of the department was kind enough to provide me with a few "core samplers" = pieces of 1" pvc, about 2" long, with a nice bevel cut around one end to sharpen it for cutting into the cake. They also need an ordinary metric ruler and a watch or cell phone with a stopwatch.

The students cut as many sample cores from the cake as they can. This leads to a bit of waste, invariably eaten, thereby increasing the general level of blood sugar excitement in the room.

Fernando shows his glee at performing rheological experiments
on rock analog materials. Photo: William Cheng.

I give a bit of a talk about elastic at low stress, viscous at higher stress rheologic models and remind them that elastic deformation is linearly related to stress by the Young's modulus:

σ = E * e
where σ is stress in Pascals, E is the Young's Modulus in Pascals, and e is the linear strain (change in length / original length).

They are also reminded that viscosity is the relationship between stress and strain rate, not linear strain, by the relation:

σ = η * ε

Where η is the viscosity in Pa*s and ε is the strain rate in s^-1.

We assume a Maxwell rheology wherein:

σ = η ε + E e

and also assume that if we do experiments at very low stress, viscous strain negligible (all strain is elastic) and therefore set η ε = 0 at very low stress. This enables the students to isolate the Young's modulus (E).

Load is applied by placing other food items (of labeled mass, e.g. small cans and jars) on top of the cake cores. The students measure the surface area of contact and calculate applied stress.**

The students are asked to design and execute experiments to determine:

• the Young's Modulus

Cake sample after elastic rebound (there's a bit of a delay there - hysteresis loop?)
Photo: William Cheng

This turns out to be amazingly reproducible. For my cake this year, everybody seemed to get a result between 5-8 kPa. Or, since the concept of significant figures doesn't seem to have taken hold, 6449.33352 Pa.... that's another issue.

• the elastic limit

100g jar of capers produces about 1.2 kPa load on the cake core.
Amazingly, at 50% shortening the cake core still rebounds elastically.
Must be all those eggs I put in.
Photo: William Cheng

This parameter is bracketed by increasing the load on the sample until the sample no longer rebounds elastically to its original height after the load is removed. For the cans and jars I brought, all groups bounded the elastic limit at between 100g capers and 400g organic kidney beans. With our cake core samples this is between about 1-6 kPa. Next time I would get more intermediate weights.

• the viscosity (for deformation above the elastic limit)
  

Under the 400g can o' beans, the cake has gone viscous and very little rebound is
observed. No conch shell or painted stick reported either...
(Photo: William Cheng)

The viscosity is the most unreliable part of this experiment, mostly because the students have to estimate the timescale of deformation and that timescale is very short. Perhaps larger pieces of cake would address this....

Anyway this prac gets good reviews. The discussions of the lithosphere seem to go well afterwards, although I will let you know for sure after the exams in October. I have used the same concept for a more open-ended inquiry and had students investigating the shear modulus by sheathing the cores in plastic wrap and shearing them, investigating the temperature effects by freezing them and microwaving them [don't tell the boss I broke into his lab for a little N₂ (liq)]. Obviously no amount of temperature increase is going to allow dislocation glide in chocolate so the metaphor rapidly breaks down. However - I never have to clean anything up with this lab... except wiping a few crumbs from the table tops.

Hoping someone out there will improve on this or suggest improvements... leave me a comment if you have any ideas!

*I was introduced to the Grok concept by my algebra II teacher in high school who made us chant (daily) the quadratic formula "so that if any of you two reproduce, your children will be born knowing it". Well, Mr. Steiger, x equals the opposite of B plus or minus the square root of B squared minus four-A-C over two-A.

** Yes I'm aware that the labeled weight on the can only refers to the food inside and does not include the weight of the container. Trust me this is not the largest source of error here and anyway, saves time and hassle when the students have only a short time to complete their experiments.