Discussion

basics

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. 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 extension, so 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 coefficient that relates a particular type of stress to the strain that results is called a modulus (plural, moduli).

tension

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.

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• 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 ℃ to over 200 ℃.

shear

Shear Modulus (S)

F	= S	⎛ ⎝	Δx	⎞ ⎠
A			y

Gases and liquids can not have shear moduli. They have viscosity instead.

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compression

Bulk Modulus (B)

F	= B	⎛ ⎝	ΔV	⎞ ⎠
A			V0

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Solid Properties of Selected Materials (MPa)
material	young's modulus	compressive strength	tensile strength	shear modulus	shear strength	bulk modulus
concrete	16,500	21	2.1
bone, compact	17,900	170	120
bone, trabecular	76	2.2
granite	51,700	145	4.8
marshmallow	0.029
oak	11,000	59	117
porcelain		552	5.5
steel, hard	207,000	552	827
rubber	1		2.1

failure

• elastic limits
• compression
• tension
• flexing
• shear

scaling

• no gigantic animals
• surface area is proportional to length²
• mass and volume is proportional to length³
• BMR is proportional to mass^3/4
• tension is proportional to length (Hooke's law)
• pressure is proportional to length² (stomach, bladder stretching)

surface tension

γ =	F
	ℓ

Surface Tension for Selected Materials (T ~ 300 K unless otherwise indicated)
material	surface tension (μN/m)
alcohol, ethyl (grain)	223.2
alcohol, isopropyl (15 ℃)	217.9
alcohol, methyl (wood)	225.5
water, pure	728
water, soapy	250–450

Capillarity

• The average diameter of the capillaries is about 20 μm, although some are only 5 μm in diameter. there are about 190 km of capillaries in 1 kg of muscle, the surface area of the capillaries in 1 kg of muscle is about 12 m².