Snapping Shrimp, Shooting Rubber Bands, and Mass Dampers
How understanding how energy is converted helps us understand how structures will deform.
Strain Energy
Energy exists in a variety of forms — kinetic energy, potential energy, chemical energy, electrical energy, thermal energy — and much of what we observe in the world around us is the result of one form of energy converting to another. A falling object is converting its gravitational potential energy into kinetic energy. Since energy is conserved during this exchange from one form to another, this presents a way to store energy in one form and release it in another. Structures tend to store potential energy by deforming. External forces do work by causing a displacement, which we measure as strain. If the deformations are elastic, the structure will return to its original shape once the forces are removed, and therefore these deformations are a form of a potential energy known as strain energy. Storing energy in elastic deformations is responsible for slingshots, car suspensions, archery bows, bounding kangaroos, and the lethal attack of snapping shrimp:
In class, we learned about several types of strain energy based on the external forces or moments applied. To understand stretching energy, consider a slingshot. Pulling back on the rubber band causes it to stretch, converting the force you apply to elastic strain in the material. Upon release, the rubber contracts back to its original length, and if there’s an object in your slingshot then that potential energy in the rubber will be converted to kinetic energy that propels the object. As BU’s Prof. Bird has shown, even the simple act of shooting a rubber band has a remarkably complex way to release its strain energy — it forms a beautiful, self-similar shape that grows upon release of the rubber band:
Consider what we learned in class yesterday about the various ways to store strain energy, and see if you can rationalize why the modern compound bow is superior at storing and releasing strain energy than the longbow that was used in the late Middle Ages. Here’s how these bows look strung and unstrung
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As engineers, one of the most important things that strain energy is used for is understanding when structures break, a subfield of solid mechanics known as fracture mechanics. We’ll return to this topic at a later date!
Stretching vs. Bending
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For structures, often the difficulty is determining how the work done by external forces will be stored in the structure. To develop some intuition into this problem, it is helpful the evaluate the energy stored in simple deformations. In general, strain energy is half the product of stress and strain integrated over the volume of the object.
If a simple structure, like a rod, is subjected to uniaxial tension or compression, then the stress can be found using Hooke’s law. This results in a stretching energy that depends on material properties, geometry, and the amount the rod has stretched, and this energy scales as
where E is the elastic modulus, h is thickness, b is width, L is length, and ε is the strain. Similarly, we can see how that rod stores energy when bent by a moment. The bending energy also depends on materials and geometry, but in a different way:
where κ is the curvature strain. To determine how a structure will deform, it is helpful to compare the relative cost of bending vs. the cost of stretching. Nature is frugal, and prefers paths and deformations that minimize the total potential energy. When comparing bending to stretching, we find that
As the object’s thickness gets smaller, this ratio quickly becomes very small. That means:
Thin objects like to bend, and don’t like to stretch.
We see this all around us everyday. It’s easy to bend a sheet of paper, but nearly impossible to stretch it. When we drape a towel over a rack, it may bend and fold over itself, but it certainly won’t stretch and elongate until it meets the floor. This is also why straws bend when you compress them, and soda cans buckle when stood upon — while you’re trying to compress these structures by applying uniaxial compression, the structures much prefer to bend instead. A helpful framework to think of structural instabilities is that they often are an example of bending when you least expect it. This balance of bending vs. stretching is responsible for the curly shapes of leaves, the wrinkles on our skin, and the shape of a blooming lily:
This line of thinking also helps simplify our analysis. Statements like “we will assume the rod to be inextensible” is fancy way of saying, “stretching is so costly, it’ll never happen, so let’s just ignore it completely.”
And, as we will see, many of the recent advances in mechanical metamaterials center around the idea of helping thin sheets stretch. For example, folds in origami and cuts in kirigami enable thin sheets to stretch by sacrificing parts of the sheet and reducing its resistance to deformation.
Callback to Class: Mass Dampers
As we discussed pressure waves and shear waves emitted from earthquakes, a few questions came up about how engineers design tall buildings to withstand forces from high winds and (mild) earthquakes. A common solution is to use a Tuned Mass Damper at the top of the building, which acts to dampen the harmonic oscillations of the building. Buildings tend to sway at their resonant frequency, and so the oscillation frequency of the tuned mass is tuned to be similar to the building’s resonant frequency. Watch and learn how these work:
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