elastic-constants-ketul.ppt

7/23/2019 Elastic-Constants-ketul.ppt

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ElasticElastic

ConstantConstant


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ELASTIC CONSTANTS INELASTIC CONSTANTS IN

ISOTROPIC MATERIALSISOTROPIC MATERIALS1.1.Elasticity ModulusElasticity Modulus (E)(E)

2. Poissons Ratio2. Poissons Ratio (())

3. S!a" Modulus3. S!a" Modulus (#)(#)

$. %ul& Modulus$. %ul& Modulus (')(')


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1. Modulus o Elasticity E(*oun+s Modulus)

= E

Linear-elastic

E

F

Fsimpletension

test

Units:E: [GPa]


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Slo,! o st"!ss st"ain ,lot (-ic isSlo,! o st"!ss st"ain ,lot (-ic is

,"o,o"tional to t! !lastic odulus),"o,o"tional to t! !lastic odulus)

d!,!nds on /ond st"!n+t o !tald!,!nds on /ond st"!n+t o !tal

Adapted from Fig. 6.7,

Callister 7e.

E0


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2.2. Poissons "atioPoissons "atio

Units:: dimensionless

T

L

F

Fsimpletension

test

! is t"e ratio of

trans#erse contraction strain

to longit$dinal e%tension

strain in t"e direction of

stretc"ing force.

&it"er trans#erse strain or

longit$dional strain is

negati#e, 'is positi#eT

=

LT : Transverse Strain

L : Longitudional Strain


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Virtually all common materialsVirtually all common materials

undergo a transverse contractionundergo a transverse contraction

when stretched in one direction and awhen stretched in one direction and a

transverse expansion whentransverse expansion when

compressed.compressed.

IIn an isotropic material the allowablen an isotropic material the allowable(theoretical)(theoretical) range of Poissons ratio isrange of Poissons ratio is

from !".# to $#.%& basedfrom !".# to $#.%& based onon the theorythe theory

of elasticityof elasticity..metals: ( ).**

ceramics:( ).+polmers:( ).)


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G

= G

3. S!a" Modulus #3. S!a" Modulus #

simple

torsion

test

M

M

Units:G: [GPa]


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$. %ul& Modulus '$. %ul& Modulus '

avg= KV

Vo avg

V

KVo

P

P

P

Initial Volume 'V#

Volume Change' V

Units:K: [GPa]

avgis the average

of three stressesapplied alongthree principaldirections.


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= E

= G

avg=

KV

Vo

tresses

trains

ElasticConstants

*ormal

hear

Volumetric


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Ea,l!Ea,l!4niaial Loadin+ o a P"isatic4niaial Loadin+ o a P"isatic

S,!ci!nS,!ci!n

At!"%!o"!

"#cm

"#

cm

"#cm

"#.+cm

,., cm

,., cm

5!t!"i

n! Eand

P'"###-gf


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10cm

10cm

l/2=0.2cm

d/2=0.05cm

1000 kgf

P=1000 kgf

P=1000kgf /= 10*10

100

0=10kgf/cm2

E=

=

100.0

4

= 250 kgf/cm2

long=

ll0

= =0.04

0.410

lat=dd0 = = -0.01-0.110

= --0.010.0

4

=

0.25


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For an isotroic mat!rial t"! str!ss-For an isotroic mat!rial t"! str!ss-

strain r!lations ar! as follo#s$strain r!lations ar! as follo#s$


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RELATION %67 ' 8 ERELATION %67 ' 8 E

%onsid!r a c&'! #it" a &nit (ol&m!%onsid!r a c&'! #it" a &nit (ol&m!

1

1

1

)

%

+

ca&s!s an !longation in t"!dir!ction %) and contraction in t"!dir!ctions + , %.

"! n!# dim!nsions of t"! c&'!is $

%) dir!ction is 1

% dir!ction is 1-

+ dir!ction is 1-


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00= 1= 1

Final (ol&m!Final (ol&m! fof t"! c&'! is no#$of t"! c&'! is no#$

111-1-1-1-==111-21-23322

22

= 1 - 2= 1 - 2332222-2-2223322

= 1 = 1 --22--22223322223322

is small 2, ar! small!r and can '!n!gl!ct!d.

f= 1 - 2 / = f - 0 = 1-2

6f !7&al t!nsil! str!ss!s ar! ali!d to!ac" of t"! ot"!r t#o airs of fac!s oft"! c&'! t"an t"! total c"ang! in (ol&m!#ill '! $


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8

9 =/

=

1-2

=

1-2

E

1-2

9 =E

1-2

V ' /0 ("!12) '0 ("!12) 0 ("!12) 0 ("!12)$ $

=a(g

/0


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:or!o(!r t"! r!lation:or!o(!r t"! r!lation'!t#!!n ; and E is $'!t#!!n ; and E is $

;

=

E

21

"! r!lation"! r!lation

'!t#!!n ; E and 9'!t#!!n ; E and 9

is $is $

E

1 1=

1


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For (!r soft mat!rials s&c" as ast!sFor (!r soft mat!rials s&c" as ast!s

g!ls &tti!s 9 is (!r larg!g!ls &tti!s 9 is (!r larg!

>ot! t"at as 9>ot! t"at as 9 / 0 // 0 / //). 1 & 2 *G). 1 & 2 *G

3f 4 is #er large /3f 4 is #er large /55))2 )2 ) 889o #ol$me9o #ol$me

c"angec"ange

For mat!rials lik! m!tals ?'!rs , c!rtainFor mat!rials lik! m!tals ?'!rs , c!rtain

lastics 9 m&st '! consid!r!d.lastics 9 m&st '! consid!r!d.


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:od&l&s of Elasticit $:od&l&s of Elasticit $

@ig" in co(al!nt como&nds s&c" as@ig" in co(al!nt como&nds s&c" as

diamonddiamond Ao#!r in m!tallic and ionic crstalsAo#!r in m!tallic and ionic crstals

Ao#!st in mol!c&lar amor"o&s solidsAo#!st in mol!c&lar amor"o&s solids

s&c" as lastics and r&''!r.s&c" as lastics and r&''!r.


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Elastic Constants o So!Elastic Constants o So!

Mat!"ialsMat!"ialsEsiB10EsiB10CC;Pa;Pa

;siB10;siB10CC;Pa;Pa --

Cast IronCast Iron 1C1C

110110 D.4D.4 5050 0.1D0.1D

SteelSteel 00205205

11.11. 00 0.2C0.2C

AluminuAluminu

mm 1010 D0D0 .C.C 2525 0.0.

ConcreteConcrete 1.5-5.51.5-5.5 10-10-4040

0.C2-2.00.C2-2.0 4-4-1515

0.20.2

WoodWood Aong 1.1Aong 1.1

1212

0.110.11

0.D0.D


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Tan& *ouTan& *ou

elastic-constants-ketul.ppt

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