deformation of solid.ppt

7/17/2019 deformation of solid.ppt

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11

Chapter 8Chapter 8Deformation of solidsDeformation of solids

10.1 Hookes law10.1 Hookes law

10.2 Stress, Strain10.2 Stress, Strain

10.3 Elastic & plastic eha!io"r 10.3 Elastic & plastic eha!io"r



2

10.1 #eha!io"r of sprin$10.1 #eha!io"r of sprin$

Compression Sprin$

E%tension Sprin$

orsion Sprin$

'lat sprin$

(ea!e Sprin$

Spiral Sprin$



3

Hooke)s lawHooke)s law

'or s*stems that oe*'or s*stems that oe*Hooke+s law, theHooke+s law, thee%tension prod"ced ise%tension prod"ced is

directl* proportional todirectl* proportional tothe load.the load.

restorin$ force alwa*srestorin$ force alwa*s

acts in the oppositeacts in the oppositedirection of thedirection of the x x displacementdisplacement

' -k%



E%ample 1E%ample 1

/hat is the force re"ired to stretch a/hat is the force re"ired to stretch a

sprin$ whose constant !al"e is 100 msprin$ whose constant !al"e is 100 m

* an amo"nt of 0.0 m4* an amo"nt of 0.0 m4

F = kx

= 50N



E%ample 2E%ample 2

0.0153kg



5

6arallel sprin$s s*stem6arallel sprin$s s*stem

's 's

'

7ri$inal len$th 1 cm7ri$inal len$th 1 cm

k 2000 mk 2000 m-1-1

(oad, ' 100 (oad, ' 100 E%tension of each sprin$, % E%tension of each sprin$, %

otal len$th after loadin$ otal len$th after loadin$



9

Series sprin$s s*stemSeries sprin$s s*stem

/

's2

's1

's2

's1 's2 's2 /

7ri$inal len$th 1 cm7ri$inal len$th 1 cm

k 2000 m-1k 2000 m-1

(oad, ' 100 (oad, ' 100

E%tension of each sprin$, % E%tension of each sprin$, %

otal len$th after loadin$ otal len$th after loadin$

:t e"iliri"m state,



8

E%ample 1;E%ample 1;

: eam, the wei$ht of whichma* e ne$lected, iss"pported * three identicalsprin$s. /hen a wei$ht W

is h"n$ from the middle ofthe eam, the e%tension ofeach sprin$ is x .

he middle sprin$ and thewei$ht are remo!ed.

/hat is the e%tension whena wei$ht of 2W is h"n$ fromthe middle of the eam4



<


: n"mer of similar sprin$s,

each ha!in$ the same sprin$

constant, are =oined in three

arran$ements >, ? and @.

he same load is applied toeach. /hat is the order of

increasin$ e%tension for

these arran$ements4



10

10.2 Stress, Strain10.2 Stress, Strain

hehe

eha!io"r eha!io"r

of materials "nder tension orof materials "nder tension or

compressioncompression

Stren$th; the ailit* of material to withstand a

force witho"t reakin$.

Stiffness; the resistance of m. to chan$es it shape.

Elasticit*; the propert* of material that allow it toret"rn to its ori$inal shape & siAe after force

deformin$ it has een remo!ed.



11

Check *o"r "nderstandin$BCheck *o"r "nderstandin$B

Compare the de$ree of stren$th and stiffnessCompare the de$ree of stren$th and stiffness

of the material elow;of the material elow;

Stren$thStren$th StiffnessStiffness

Concrete pileConcrete pile#isc"it#isc"it

ell*ell*

lass reinforcelass reinforceplasticplastic

*lon*lon

Hi$h Hi$h

(ow Hi$h

(ow (ow

Hi$h

Hi$h (ow

edi"m



12

Stress,Stress, FF

Gs defined as force applied per "nit area ofGs defined as force applied per "nit area of

cross-section.cross-section.

nit;nit; 6ascal I6aJ or m2

ensilestress

compressi!estress

FF ':

E$; applied force, ' 30, :rea, : 0.03m2

stress 1000 m2



13


he hi$h-tensile steelhe hi$h-tensile steel

"sed to s"pport the"sed to s"pport the

roadwa* of a s"spensionroadwa* of a s"spension

rid$e reaks "nder arid$e reaks "nder astress of 1. % 10stress of 1. % 10<< 6a.6a.

'ind the minim"m cross-'ind the minim"m cross-

sectional area for a cale ifsectional area for a cale ifit is to s"pport a load ofit is to s"pport a load of

mass 2.0 % 10mass 2.0 % 1033k$.k$.

Youngjong Bridge, Korea

12.9cm2



1

Strain,Strain, KK

Gs defined as theGs defined as thee%tension per "nite%tension per "nit

len$thlen$th

nit;nit; o "nitLLL

'or e%ample;

Gnitial len$th of a sprin$ is 1cm, after appl*load, the sprin$ e%tended 3 cm.

he strain is 0.20.2

KK M( (



1

?o"n$ mod"l"s?o"n$ mod"l"s

is a meas"re of stiffness.is a meas"re of stiffness.

he ration of stress to strain is called ahe ration of stress to strain is called a

modulus of elasticity modulus of elasticity ..

'or tensile stress and strain, the ratio is'or tensile stress and strain, the ratio iscalledcalled ?o"n$ mod"l"s?o"n$ mod"l"s, E., E.

nit;nit; 6ascal I6aJ or m2

L A

FL

E ∆

==ε

σ



15


: wire of diameter 1.0mm and len$th 2.3 m is : wire of diameter 1.0mm and len$th 2.3 m is

made of copper. 0 load is applied on themade of copper. 0 load is applied on the

wire. 'ind ;wire. 'ind ;

he radi"s of the cross-sectionhe radi"s of the cross-section

he strain if the wire is stretched * 0.8mmhe strain if the wire is stretched * 0.8mm

he stresshe stress

he force necessar* to ca"se this stresshe force necessar* to ca"se this stress



19


: wire is stretched * 8 mm when a load of

50 is applied.

/hat will e the e%tension of a wire of the

same material ha!in$ fo"r times the cross-

sectional area and twice the ori$inal len$th,

when the same load is applied4

A 2 mm B mm C 8 mm D 15 mm



18


*lon reaks when the stress within itreaches 1% 10< 6a.

/hich ran$e incl"des the hea!iest load

that co"ld e lifted * a n*lon thread ofdiameter 1 mm4

A 2 to 20

B 20 to 200 C 200 to 2000

D 2000 to 20 000



1<

E%periment to determine EE%periment to determine E

he lon$er the wire, the more it e%tends forhe lon$er the wire, the more it e%tends fora $i!en tension.a $i!en tension.

/hen wei$ht is applied, the wire stretches,/hen wei$ht is applied, the wire stretches,

and the marker mo!es.and the marker mo!es. marker

he e%tension e"al tohe e%tension e"al to

the chan$e of markerthe chan$e of marker

readin$.readin$.

/ire diameter is/ire diameter is

meas"red * m.m.s.$meas"red * m.m.s.$

'"ll article in pa$e 109, ook of im #reitha"pt



20

Some interestin$ !al"es ofSome interestin$ !al"es of E E

D: N 10D: N 1088 6a6a

spa$hetti Idr*J N 10spa$hetti Idr*J N 10<< 6a6a

cotton thread N 10cotton thread N 101010

6a6aplant cell walls N 10plant cell walls N 101111 6a6a

So, can *o" determine ?o"n$ mod"l"s !al"e of *o"r hair4So, can *o" determine ?o"n$ mod"l"s !al"e of *o"r hair4



21

Stress a$ainst strain $raphStress a$ainst strain $raph

radient of the $raphradient of the $raph ?o"n$ mod"l"s

Stress m-2

Strain



22


here are fo"r wireshere are fo"r wires

of different material. Gfof different material. Gf

the wires arethe wires are

stretched * the samestretched * the sameforce, which wire willforce, which wire will

ha!e the $reatestha!e the $reatest

e%tension4e%tension4

Stress

Strain

:

#

C

D

: : ## CC DD

(en$thm(en$thm 1.01.0 2.02.0 2.02.0 1.01.0

Cross sectionalCross sectional

areammareamm22

0.00.0 1.001.00 0.00.0 1.001.00



23

Strain IelasticJ ener$*Strain IelasticJ ener$*

/hen an o=ect is/hen an o=ect isstretched, work isstretched, work isdone on the o=ect,done on the o=ect,and strain ener$* isand strain ener$* isstored within which isstored within which is

a form of potentiala form of potentialener$*.ener$*.

he strain ener$* inhe strain ener$* inthe form of elasticthe form of elasticdeformation is mostl*deformation is mostl*

reco!erale in the formreco!erale in the formof mechanical work.of mechanical work.

'orce

E%tension m

FxW

FdxW e

2

1

0

=

=

∫ :rea that oe*

Hooke)s law

enerall*



2


he $raph shows theeha!io"r of a sample of ametal when it is stretched "ntilit starts to "nder$o plastic

deformation. /hat is the totalwork done in stretchin$ thesample from Aero e%tension to12.0 mm4

Simplif* the calc"lation *

treatin$ the re$ion >? as astrai$ht line.

A 3.30 B 3. C 3.50 D 5.50

'orce



2

E%ample 2;E%ample 2;he tension in a sprin$ of nat"ral len$th l 0 is first

increased from Aero to T 1, ca"sin$ the len$th toincrease to l 1. he tension is then red"ced to T 2,

ca"sin$ the len$th to decrease to l 2 .

A (6 B O6 C SP D 6(

/hich area of the$raph represents the

work done * the

sprin$ d"rin$ this

red"ction in len$th4



25


he $raph shows how thehe $raph shows how the

len$th of a partic"larlen$th of a partic"lar

r"er cord !aries asr"er cord !aries as

force is applied.force is applied.

/hat is the ma%im"m/hat is the ma%im"m

strain ener$* in thisstrain ener$* in this

deformed r"er cord4deformed r"er cord4

: : 2. 2. ## .0 .0 CC 9. 9. DD 10 10



29

raphraph M( - '

he $raph ofhe $raph of

e%periment "sin$e%periment "sin$

copper and steelcopper and steel

are shown plottedare shown plottedon the same a%es.on the same a%es.

/hich material is/hich material is

stiffer4stiffer4

Steel ; lower $radient of $raph M( - '



28

10.3 Elastic & plastic eha!io"r 10.3 Elastic & plastic eha!io"r

E%tensionE%tension Q elasticQ elasticlimitlimit

Pet"rn to ori$inalPet"rn to ori$inallen$thlen$th

7e* Hooke)s law7e* Hooke)s lawe%cept some pol*mer e%cept some pol*mer

Does not chan$eDoes not chan$e

internal str"ct"reinternal str"ct"re :toms are p"lled apart :toms are p"lled apartwith distance Q atomicwith distance Q atomicradi"sradi"s

E%tensionE%tension R elasticR elasticlimitlimit

Does not ret"rn toDoes not ret"rn toori$inal len$thori$inal len$th

Does not oe*Does not oe*Hooke)s lawHooke)s law

Gnternal str"ct"reGnternal str"ct"re

chan$edchan$ed :toms slip o!er one :toms slip o!er oneanother res"ltin$another res"ltin$pile-"ppile-"p



2<

E ; limit of proportionalit*E ; limit of proportionalit*

? ; *ield point Iplastic deformation e$inJ? ; *ield point Iplastic deformation e$inJ

; "ltimate tensile stress Ima%J ; "ltimate tensile stress Ima%J

' ; reakin$ point' ; reakin$ point

U

FE

E

Y

raph stress - strainraph stress - strain



30

Stress-strain $raph for BStress-strain $raph for B

lass I#rittleJlass I#rittleJ *pical metal ID"ctileJ*pical metal ID"ctileJ

*pical r"er*pical r"er



31

*pical r"er *pical r"er

: r"er and : r"er and

s"spended !erticall* iss"spended !erticall* isloaded with increasin$loaded with increasin$

load to a ma%im"m. heload to a ma%im"m. he

load is then red"cedload is then red"ced

$rad"all* to Aero. he$rad"all* to Aero. he

$raph elow shows the$raph elow shows the

res"lts otained.res"lts otained.

/ithin the elastic limit,

r"er does ret"rn toAero e%tension when

"nloaded completel*.

So, can *o" sketch the

$raph4

H*steresis loop



32

6reliminar* anal*sis tools compare6reliminar* anal*sis tools compare

different conceptsdifferent concepts

http;www.dp$riffiths.comima$escosmos-stress.=p$http;www.dp$riffiths.comima$escosmos-stress.=p$

http://www.dpgriffiths.com/images/cosmos-stress.jpg






33


State whether copper is aState whether copper is ad"ctile, rittle or pol*mericd"ctile, rittle or pol*mericmaterial.material.

ark with the letter ( onark with the letter ( onthe dia$ram, the point onthe dia$ram, the point onthe line e*ond whichthe line e*ond whichHooke)s law does notHooke)s law does notappl*.appl*.

State how the sprin$ constant for the wireState how the sprin$ constant for the wirema* e otained from the ao!e fi$"re.ma* e otained from the ao!e fi$"re.

E%tension of a copper wire



3


/hat is the "ltimate tensile stress of amaterial4

A the stress at which the material ecomes

d"ctileB the stress at which the material reaks

C the stress at which the material deforms

plasticall*D the stress at which the material reaches

its elastic limit



3


he fi$"re elow is thehe fi$"re elow is the

stress-strain $raph forstress-strain $raph for

three specimensthree specimens X X ,, Y Y , and, and

Z Z which ha!e the samewhich ha!e the samedimensions. /hich of thedimensions. /hich of the

followin$ co"ld e thefollowin$ co"ld e the

materials are made of4materials are made of4

P"er, copper, $lass4P"er, copper, $lass4

Stress

Strain

X

Y

Z



35

E%ample ;E%ample ;

Gn descriin$ the eha!io"r of a sprin$, thesprin$ constant is "sed. Different loads are "sedto e%tend the sprin$ * different amo"nts.

o find the sprin$ constant, which "antities arere"ired4

: the elastic limit and the loads

# the elastic limit, e%tensions and the len$th of the

sprin$C the loads and the e%tensions of the sprin$

D the loads and the len$th of the sprin$



39

E%ample E%ample

: s"spended copper wire is $rad"all* loaded"ntil it is stretched ="st e*ond the elastic limit,

and it is then $rad"all* "nloaded.

/hich $raph Iwith arrows indicatin$ the

se"enceJ est ill"strates the !ariation of the

tensile stress with lon$it"dinal strain4

deformation of solid.ppt

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