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 in The True Theory of Elasticity or Springiness (1676) …

ut tensio, sic vis

which can be translated literally into …

As extension, so force.

or translated formally into …

Extension is directly proportional to force.

Most likely we'd replace the word "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), then, to show that the springy object was trying to return to its original state, we'd add a negative sign (). 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 an elastic modulus (plural, moduli). Elastic moduli are properties of materials, not objects. There are three basic types of stress and three associated moduli.

modulus (symbols) stress
(symbol)
strain
(symbol)
configuration
change
young's
(E or Y)
normal to
opposite faces (σ)
length
ε = Δℓ/ℓ0
longer and thinner
or shorter and fatter
shear
(G or S)
tangential to
opposite faces (τ)
tangent
γ = Δx/y
rectangles become
parallelograms
bulk
(K or B)
normal to all faces,
pressure (P)
volume
θ = ΔV/V0
volume changes
but shape does not
Elastic moduli

The international standard symbols for the moduli 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). Some American textbooks have decided to break with tradition and use the first letter of each modulus in English — Y for Young's, S for shear, and B for bulk.

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 name of symbol definition unit
linear epsilon ε = Δℓ/ℓ0 m/m = 1
shear gamma γ = Δx/y m/m = 1
volume theta 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 ]

failure is an option

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 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. Young didn't name the modulus after himself. He called it the elastic modulus, but this term should be used moduli in general as was mentioned above. The symbol for Young's modulus is usually E from the French word élasticité (elasticity) but some prefer Y in honor of the man himself.

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]

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 = Δℓ/ℓ0). The proportionality constant that relates these two quantities together is the ratio of tensile stress to tensile strain —Young's modulus.

F = E Δℓ
A0
σ = Eε

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 (σ = F/A) is called the compressive stress and the fractional decrease in length (ε = 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 = E Δℓ
A0
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

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 the 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 glass 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.

Uniaxial properties of selected materials (GPa)
material young's
modulus
compressive
strength
tensile
strength
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    

shear modulus

A force applied tangentially (or transversely or laterally) to the face of an object is called a shear stress. The deformation that results is called shear strain. Applying a shear stress to one face of a rectangular box slides that face in a direction parallel to the opposite face and changes the adjacent faces from rectangles to parallelograms.

[slide]

The coefficient that relates shear stress (τ = F/A) to shear strain (γ = ∆x/y) is called the shear modulus or the rigidity modulus. It is usually represented by the symbol G from the French word glissement (slipping) although some like to use S from the English word shear instead.

F = G Δx
Ay
τ = Gγ

Fluids (liquids, gases, and plasmas) cannot resist a shear stress. They flow rather than deform. The quantity that describes how fluids flow in response to shear stresses is called viscosity and is dealt with elsewhere in this book.

The inability to shear also means fluids are opaque to transverse waves like the secondary waves of an earthquake (also known as shear waves or s waves). The liquid outer core of the earth was discovered by the s wave shadow it cast on seismometer networks. Types of waves are discussed elsewhere in this book.

Fluids can resist a normal stress. This means that liquids and gases are transparent to the primary waves of an earthquake (also known as pressure waves or p waves). The solid inner core of the earth was detected in p wave signals that made it all the way from one side of the earth through the liquid outer core to the other side. P waves are also audible. You can hear them when they transmit into the air.

The resistance of a material to a normal stress is described by the bulk modulus, which is the next topic in this section.

Shear properties of selected materials (GPa)
material shear
modulus
shear
strength
aluminum    
concrete    
concrete, high strength    
copper    
bone, compact    
bone, spongy    
brass    
diamond    
glass    
granite    
gold    
iron    
marble    
marshmallow    
nickel    
nylon    
oak    
plastic, ♳ PET    
plastic, ♴ HDPE    
plastic, ♵ PVC    
plastic, ♶ LDPE    
plastic, ♷ PP    
plastic, ♸ PS    
plutonium    
porcelain    
silicon    
silicon carbide    
steel, stainless    
steel, structural    
steel, high strength    
rubber    
tin    
titanium    
tungsten    
tungsten carbide    
uranium    

bulk modulus

A force applied uniformly over the surface of an object will compress it uniformly. This changes the volume of the object without changing its shape.

[slide]

The stress in this case is simply described as a pressure (P = F/A). The resulting volume strain is measured by the fractional change in volume (θ = ∆V/V0). The coefficient that relates stress to strain under uniform compression is known as the bulk modulus or compression modulus. Its traditional symbol is K from the German word kompression (compression) but some like to use B from the English word bulk — which is another word for volume.

F = K ΔV
AV0
P = Κθ

The bulk modulus is a property of materials in any phase but it is more common to discuss the bulk modulus for solids than other materials. Gases have a bulk modulus that varies with initial pressure, which makes it more of a subject for thermodynamics, in particular, the gas laws.

The reciprocal of bulk modulus is called compressibility. Its symbol is usually β (beta) but some people prefer κ (kappa). A material with a high compressibility experiences a large volume change when pressure is applied.

β =  1
K

The SI unit of compressibility is the inverse pascal [Pa−1].

Bulk properties of selected materials (GPa)
material bulk
modulus
material bulk
modulus
aluminum   plastic, ♳ PET  
carrot, fresh   plastic, ♴ HDPE  
carrot, stored 1 week   plastic, ♵ PVC  
concrete   plastic, ♶ LDPE  
concrete, high strength   plastic, ♷ PP  
copper   plastic, ♸ PS  
bone, compact   plutonium  
bone, spongy   porcelain  
brass   silicon  
diamond   silicon carbide  
glass   steel, stainless  
granite   steel, structural  
gold   steel, high strength  
iron   rubber  
marble   tin  
marshmallow   titanium  
nickel   tungsten  
nylon   tungsten carbide  
oak   uranium  

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 liquids T ~ 300 K unless otherwise indicated

Capillarity