Elasticity

Discussion

basics

Elasticity is the property of solid materials to return to their original shape and size after the forces deforming them have 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).

ut tensio, sic vis   as extension, so force

In contemporary language, we'd say this as something like

extension is directly proportional to force

only we'd replace the words "extension" with the symbol (Δx), "force" with the symbol (F), and "is directly proportional to" with an equals sign (=) and a constant of proportionality (k). To show that the springy object was trying to return to its original state, we'd also tack a negative sign () in front. In other words, we'd write the equation.

F = − k Δx

This is Hooke's law for a spring — a simple object that's essentially one-dimensional. Hooke's law can be generalized to …

stress is proportional to strain

where strain refers to a change in some spatial dimension (length, angle, or volume) compared to its original value and stress refers to the cause of the change (a force applied to a surface).

The coefficient that relates a particular type of stress to the strain that results is called a modulus (plural, moduli). Elastic moduli are properties of materials, not objects. There are three basic types of stress and three associated moduli.

modulus (symbols) type of stress type of strain configuration change
young's
(Y, E)
normal to opposite faces linear
(length)
longer and thinner or shorter and fatter
shear
(S, G)
tangential to opposite faces shear
(tangent of angle)
rectangles become parallelograms
bulk
(B, K)
normal to all faces
(uniform compression)
volume volume changes but shape does not
Elastic moduli

I have chosen to use symbols that are popular in American textbooks. Each modulus is represented by the first letter of its common English name — Y for young's, S for shear, and B for bulk. The international standard symbols are derived from appropriate non-English words — E for élasticité (French for elasticity), G for glissement (French for slipping), and K for kompression (German for compression).

Stresses on solids are always described as a force divided by an area. The direction of the forces may change, but the units do not. The SI unit of stress is the newton per square meter, which is given the special name pascal in honor of Blaise Pascal (1623–1662) the French mathematician (Pascal's triangle), physicist (Pascal's principle), inventor (Pascal's calculator), and philosopher (Pascal's wager).


Pa =  N
m2

Strains are always unitless.

type of strain definition unit
linear ε = Δℓ/ℓ0 m/m = 1
shear γ = Δx/y m/m = 1
volume θ = ΔV/V0 m3/m3 = 1
Strain units

Which means that pascal is also the SI unit for all three moduli.

  stress  =  modulus  ×  strain  
[ Pa  =  Pa  ×  1 ]

Young's modulus

Imagine a piece of dough. Stretch it. (Try to pull it apart.) It gets longer and thinner. Squash it. (Push it from opposite sides.) 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 (Y) in honor of the British 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. He called it the modulus of elasticity (E), but it is not often called this today.

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 0 and cross sectional area A being stretched by a force F to a new length 0 + Δℓ.


[slide]

The tensile stress is the outward normal force per area (F/A) and the tensile strain is the fractional increase in length of the rod (Δℓ/ℓ0). The proportionality constant that relates these two quantities together is the ratio of tensile stress to tensile strain —young's modulus (a.k.a. the elastic modulus).

F = Y Δℓ
A0

The same relation holds for forces in the opposite direction; that is, a strain that tries to shorten an object.


[slide]

Replace the adjective tensile with compressive. The normal force per area directed inward (F/A) is called the compressive stress and the fractional decrease in length (Δℓ/ℓ0) is called the compressive strain. This makes young's modulus the ratio of compressive stress to compressive strain. An adjective may have changed, but the mathematical description did not.

F = Y Δℓ
A0

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

Extension and contraction are opposite types of linear strain. Extension means to get longer. Contraction means to get shorter. Whenever a material is extended or contracted by a linear stress in one direction (the x axis, for example), the reverse strain usually takes place in the perpendicular directions (the y and z axes). The direction of a linear stress is called the axial direction. All the directions that are perpendicular to this are called transverse directions.

An axial extension is usually accompanied by a transverse contraction. Stretching a piece of dough makes it get thinner as well as longer. This is the way Chinese hand-pulled noodles (拉面, la mian) are made. Likewise, an axial contraction is usually accompanied by a transverse extension. Flattening a piece of dough makes it get wider and longer as well as thinner. This is the way Italian fresh pasta is made.

The ratio of transverse strain to axial strain is known as poisson's ratio (ν). A negative sign is needed to show that the changes are usually of opposite type (+ extension, vs. − contraction). If we keep with the tradition that x is the axial direction and y and z are the transverse directions then poisson's ratio can be written as …

ν = −  Δy/y0  = −  Δz/z0
Δx/x0 Δx/x0

The symbol that looks unfortunately like the Latin letter v (vee) is actually the Greek letter ν (nu). It is related to the Latin letter n (en).

v ν n
Latin "vee" — velocity Greek "nu" — poisson's ratio Latin "en" — number

Typical values for poisson's ratio range from 0 to 0.5. Cork is an example of a material with a very low poisson's ratio (nearly zero). When a cork is pushed into a wine bottle, it gets shorter but not thicker. (There is some axial strain, but barely any transverse strain.) Rubber on the other hand, has a very high poisson's ratio (nearly 0.5). When a rubber stopper is pushed into a chemical flask, the stopper gets shorter by some amount and wider by nearly half that amount. (The axial strain is accompanied by a large transverse strain.) Corks can be pounded into bottles with a mallet. Pounding a rubber stopper into a flask with a mallet is likely to end in disaster.

Surprisingly, negative poisson's ratios are also possible. Such materials are said to be auxetic. They grow larger in the transverse direction when stretched and smaller when compressed. Most auxetic materials are polymers with a crumpled, foamy structure. Pulling the foam causes the crumples to unfold and the whole network expands in the transverse direction.

shear modulus

Shear Modulus (S) also known as the rigidity modulus

F = S Δx
Ay

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


[slide]

bulk modulus

Bulk Modulus (B)

F = B ΔV
AV0


[slide]

Solid Properties of Selected Materials (GPa)
material young's
modulus
compressive
strength
tensile
strength
shear
modulus
shear
strength
bulk
modulus
aluminum 70 0.040        
carrot, fresh 0.00136   0.000504      
carrot, stored 1 week 0.00103   0.000507      
concrete 17 0.021 0.0021      
concrete, high strength 30 0.040        
copper 130 0.22        
bone, compact 18 0.17 0.12      
bone, spongy 76 0.0022        
brass 110 0.25        
diamond 1100          
glass 50–90 0.050        
granite 52 0.145 0.0048      
gold 74          
iron 210          
marble   0.015        
marshmallow 0.000029          
nickel 170          
nylon 2–4 0.075        
oak 11 0.059 0.12      
plastic, ♳ PET 2.0–2.7 0.055        
plastic, ♴ HDPE 0.80 0.015        
plastic, ♵ PVC            
plastic, ♶ LDPE            
plastic, ♷ PP 1.5–2.0 0.040        
plastic, ♸ PS 3.0–3.5 0.040        
plutonium 97          
porcelain   0.55 0.0055      
silicon 110          
silicon carbide 450          
steel, stainless   0.86        
steel, structural 200 0.40 0.83      
steel, high strength   0.76        
rubber 0.01–0.10   0.0021      
tin 47          
titanium 120          
tungsten 410          
tungsten carbide 500          
uranium 170          

failure

scaling

surface tension

γ = F
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
Surface Tension for Selected Materials
(T ~ 300 K unless otherwise indicated)

Capillarity