Elasticity Notes
Notes
young's modulus
Imagine a piece of dough. Stretch it. It gets longer and thinner. Squash it. It gets shorter and fatter. Now imagine a piece of granite. Try the same mental experiment. The change in shape must surely occur, but to the unaided eye it's imperceptible. Some materials stretch and squash quite easily. Some do not.
The quantity that describes a material's response to stresses applied normal to opposite faces is called Young's modulus in honor of the English scientist Thomas Young (1773–1829). Young was the first person to define work as the force displacement product, the first to use the word energy in its modern sense, and the first to show that light is a wave. He was not the first to quantify the resistance of materials to tension and compression, but he became the most famous early proponent of the modulus that now bears his name. Young didn't name the modulus after himself. He called it the elastic modulus. The symbol for Young's modulus is usually E from the French word élasticité (elasticity) but some prefer Y in honor of the scientist.
Young's modulus is defined for all shapes and sizes by the same rule, but for convenience sake let's imagine a rod of length ℓ0 and cross sectional area A being stretched by a force F to a new length ℓ0 + ∆ℓ.
Tensile stress is the outward normal force per area (σ = F/A) and tensile strain is the fractional increase in length of the rod (ε = ∆ℓ/ℓ0). The proportionality constant that relates these two quantities together is the ratio of tensile stress to tensile strain —Young's modulus.
| σ = Eε |
The same relation holds for forces in the opposite direction; that is, a strain that tries to shorten an object.
Replace the adjective tensile with compressive. The normal force per area directed inward (σ = F/A) is called the compressive stress and the fractional decrease in length (ε = ∆ℓ/ℓ0) is called the compressive strain. This makes Young's modulus the ratio of compressive stress to compressive strain. The adjective may have changed, but the mathematical description did not.
| σ = Eε |
The SI units of Young's modulus is the pascal [Pa]…
| ⎡ ⎢ ⎣ | N | = Pa | m | ⎤ ⎥ ⎦ |
| A | m |
but for most materials the gigapascal is more appropriate [GPa].
1 GPa = 109 Pa