Elasticity
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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion
Elasticity is the tendency of objects to return to their original shape and size after a force deforming them has been removed.
Recall Hooke's law from mechanics. First stated formally by Robert Hooke (1635-1703) of England in The True Theory of Elasticity or Springiness (1676). Written in Latin as "ut tensio, sic vis", which translates to "as is the extension, so is the force" or in contemporary language "extension is directly proportional to force". No portrait of Hooke is known to exist.
Hooke's law can be generalized to "stress is proportional to strain"; where "strain" refers to a change in some spatial dimension (length or volume, for example) and "stress" refers to the cause of the change (force or pressure, for example).
- A force causes changes in length, thickness, or other linear dimension.
- A torque causes twist (a change in orientation of one part of an object relative to another along an axis)
- Pressure causes changes in volume, which may be isotropic (occurring equally in all directions) or anisotropic (being greater in one direction than another)
A coefficient that relates a particular type of stress to the strain that results is called a modulus (plural, moduli).
Young's Modulus (Y)
| F | = Y | ⎛ ⎝ |
Δℓ | ⎞ ⎠ |
| A | ℓ0 |
Named for Thomas Young (1773-1829) who also first stated the formulas for work and kinetic energy and also demonstrated that light is a wave.
[magnify]
Shear Modulus (S)
| F | = S | ⎛ ⎝ |
Δx | ⎞ ⎠ |
| A | y |
Gases and liquids can not have shear moduli. They have viscosity instead.
[magnify]
Bulk Modulus (B)
| F | = B | ⎛ ⎝ |
ΔV | ⎞ ⎠ |
| A | V0 |
[magnify]
elastic limits
From somewhere: Elinvar is a trademark for a kind of nickel-chromium steel used for watch springs because its elasticity is constant over a wide range of temperatures
From New Scientist: When warmed, the alloys barely expand. This rare, 'invar' behaviour is characteristic of some nickel-steel mixes that were discovered in the 1890s and are used in parts of delicate mechanisms such as wristwatches and scientific measuring instruments. This refusal to expand when warmed means that the devices are accurate across a range of temperatures.
The new compounds also show 'elinvar' behaviour - their stiffness remains constant when they are heated. This effect holds over an amazingly wide temperature range - from as low as -194 °C to over 200 °C.
| Strengths of Selected Materials (MPa) | |||
| material | compressive limit | tensile limit | young's modulus |
|---|---|---|---|
| hard steel | 552 | 827 | 207,000 |
| rubber | – | 2.1 | 1 |
| granite | 145 | 4.8 | 51,700 |
| concrete | 21 | 2.1 | 16,500 |
| oak | 59 | 117 | 11,000 |
| porcelain | 552 | 5.5 | – |
| compact bone | 170 | 120 | 17,900 |
| trabecular bone | 2.2 | – | 76 |
| Source: Physics of the Body | |||
Summary
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