elasticity.docx

8/10/2019 Elasticity.docx

1/13

Elasticity

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Discussion

basics

Elasticityis 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 byRobert HookeinThe 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 (x), "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= kx

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.
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where strainrefers to a change in some spatial dimension (length, angle, or volume)

compared to its original value and stressrefers 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 iscalled anelastic modulus(plural, moduli). Elastic moduli are properties of materials,

not objects. There are three basic types of stress and three associated moduli.

Elastic moduli

modulus (symbols)stress

(symbol)strain

(symbol)configuration

change

young's(Eor Y)

normal toopposite faces ()

length= /0

longer and thinneror shorter and fatter

shear

(Gor S)

tangential to

opposite faces ()

tangent

= x/y

rectangles become

parallelograms

bulk

(Kor B)

normal to all faces,

pressure (P)

volume

= V/V0

volume changesbut shape does not

The international standard symbols for the moduli are derived from appropriate non-English words E for lasticit (French for elasticity), Gforglissement(French for

slipping), and Kforkompression(German for compression). Some American

textbooks have decided to break with tradition and use the first letter of each

modulus in English Yfor Young's, Sfor shear, and Bfor 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 pascalin honor

ofBlaise Pascal(16231662) 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.

Strain units
http://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://physics.info/pressure/http://physics.info/pressure/http://physics.info/pressure/http://physics.info/pressure/http://en.wikipedia.org/wiki/Blaise_Pascal


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type of strain name of symbol definition unit

linear epsilon = /0 m/m = 1

shear gamma = x/y m/m = 1

volume theta = V/V0 m3/m3= 1

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

stress= modulus strain

[ Pa = Pa 1 ]

failure is an option

elastic limit, yield strength

breaking point, ultimate strength

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

scientistThomas Young(17731829). 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 lasticit (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 0and cross sectional areaAbeing

stretched by a force Fto a new length0+ .
http://en.wikipedia.org/wiki/Thomas_Young_(scientist)http://en.wikipedia.org/wiki/Thomas_Young_(scientist)http://en.wikipedia.org/wiki/Thomas_Young_(scientist)http://en.wikipedia.org/wiki/Thomas_Young_(scientist)


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[slide]

Tensile stressis the outward normal force per area (= F/A) and tensile strainis the

fractional increase in length of the rod (= /0). The proportionality constant that

relates these two quantities together is the ratio of tensile stress to tensile strain

Young's modulus.

F= E

0

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 stressand the fractional decrease in

length (= /0) is called thecompressive strain. This makes Young's modulusthe

ratio of compressive stress to compressive strain. An adjective may have changed,

but the mathematical description did not.

F= E

0

The SI units of Young's modulus is the pascal[Pa]

N

= Pam

A m

but for most materials the gigapascalis more appropriate [GPa].

1 GPa = 109Pa

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 xaxis, for example), the reverse

strain usually takes place in the perpendicular directions (the yand zaxes). The

direction of a linear stress is called the axial direction. All the directions that are

perpendicular to this are called the transverse directions.
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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 (, 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(). A

negative sign is needed to show that the changes are usually of opposite type (+

extension, vs. contraction). If we keep with the tradi tion that xis the axial direction

and yand zare the transverse directions then Poisson's ratio can be written as

= y/y0

= z/z0

x/x0 x/x0

The symbol that looks unfortunately like the Latin letter v(vee) is actually the Greek

letter (nu). It is related to the Latin letter n(en).

v n

Latin "vee" velocityGreek "nu" Poisson's

ratioLatin "en" number

Typical values for Poisson's ratio range from 0 to 0.5. Cork is an example of amaterial 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.


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Uniaxial properties of selected materials (GPa)

materialyoung'smodulus

compressivestrength

tensilestrength

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 5090 0.050

granite 52 0.145 0.0048

gold 74

iron 210

marble 0.015

marshmallow 0.000029

nickel 170


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nylon 24 0.075

oak 11 0.059 0.12

plastic, PET 2.02.7 0.055

plastic, HDPE 0.80 0.015

plastic, PVC

plastic, LDPE

plastic, PP 1.52.0 0.040

plastic, PS 3.03.5 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 0.010.10 0.0021

tin 47

titanium 120

tungsten 410

tungsten carbide 500

uranium 170


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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 tothe opposite face and changes the adjacent faces from rectangles to parallelograms.

[slide]

The coefficient that relates shear stress(= F/A) to shear strain(= x/y) is called

theshear modulusor the rigidity modulus. It is usually represented by the

symbol Gfrom the French wordglissement(slipping) although some like to

useSfrom the English word shear instead.

F= G

x

y

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 calledviscosityand is dealt with elsewhere in this book.

The inability to shear also means fluids are opaque to transverse waves likethe secondary wavesof an earthquake (also known as shear wavesor s waves).

The liquid outer core of the earth was discovered by the s wave shadow it cast on

seismometer networks. Types ofwavesare discussed elsewhere in this book.

Fluids can resist a normal stress. This means that liquids and gases are transparent

to theprimary wavesof an earthquake (also known as pressure wavesor 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 shear
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modulus 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


10/13

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 strainis measured by the fractional change in volume (= V/V0).
http://physics.info/elasticity/modulus-bulk.htmlhttp://physics.info/elasticity/modulus-bulk.htmlhttp://physics.info/elasticity/modulus-bulk.htmlhttp://physics.info/elasticity/modulus-bulk.html


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The coefficient that relates stress to strain under uniform compression is known as

the bulk modulusor compression modulus. Its traditional symbol is Kfrom the

German word kompression (compression) but some like to use Bfrom the English

word bulk which is another word for volume.

F= K

V

V0

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, thegas laws.

The reciprocal of bulk modulus is called compressibility. Its symbol is usually (beta)

but some people prefer (kappa). A material with a high compressibility experiences

a large volume change when pressure is applied.

=1

K

The SI unit of compressibility is the inverse pascal [Pa1].

Bulk properties of selected materials (GPa)

materialbulk

modulusmaterial

bulkmodulus

aluminum plastic, PET

carrot, fresh plastic, HDPE

carrot, stored 1 week plastic, PVC

concrete plastic, LDPE

concrete, high strength plastic, PP

copper plastic, PS
http://physics.info/gas-laws/http://physics.info/gas-laws/http://physics.info/gas-laws/http://physics.info/gas-laws/


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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

no gigantic animals

surface area is proportional to length2

mass and volume is proportional to length3

BMR is proportional to mass3/4

tension is proportional to length (Hooke's law)

pressure is proportional to length2(stomach, bladder stretching)

surface tension

=F

T~ 300 K unless otherwise indicated


13/13

material surface tension (N/m)

alcohol, ethyl (grain) 223.2

alcohol, isopropyl (15 ) 217.9

alcohol, methyl (wood) 225.5

water, pure 728

water, soapy

250450Surface tension for selectedliquids

Capillarity

The average diameter of the capillaries is about 20 m, although some are

only 5 m 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.

elasticity.docx

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