Chap01 Elasticity

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CENTRE FOR SCIENCE STUDIESUNIVERSITI TUN HUSSEIN ONN MALAYSIA

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UNIT 1.0: ELASTICITY

INTRODUCTION

Elasticity is a branch of physics which studies the properties of elastic materials.A material is said to be elastic if it deformsunder stress, but then returns to itsoriginal shape when the stress is removed. Many of the necessities of everydaylife, from rubber bands to suspension bridges, depend on the elastic properties ofmaterials.

In this chapter we will introduce a simple principle called Hookes aw that helpsus to predict the deformations of elastic material and the concepts of stress,strain and elastic modulus. !ome e"amples and e"ercises are given along thetopic to improve your understanding on the topics discussed.

LEARNING OBJECTIVES

#he ob$ectives of this unit are to impart student%

i. &ith the basic knowledge in elasticity.ii. &ith the concept of elasticity in engineering course.

LEARNING OUTCOMES

After completing this unit, students should be able to%

i. Analy'e situations in which a body is deformed by tension, compression,pressure or shear.

ii. Manipulate what happens when a body is stretched so much that itdeforms or breaks.

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http://en.wikipedia.org/wiki/Deformationhttp://en.wikipedia.org/wiki/Stress_(physics)http://en.wikipedia.org/wiki/Stress_(physics)http://en.wikipedia.org/wiki/Deformation


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Stretching force, F

x

x

Compressive force, F

No forces acts


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1.1 HOOKES LAW

#he Hooke(s law of elasticitystates the relationship of a deformed elastic materialwhen acts by e"ternal forces. It is more convenient to write Hookes law, after)obert Hooke *+-/+01-2 as a proportionality of stress and strain.

*Hookes aw2 *+.+2

Hookes law states that the strain produced is directly proportional to thestress applied, provided the stress is below the elastic limit.

#he elongation of an ideal spring is proportional to the stretching force. !uppose

an ideal spring in e3uilibrium with no e"ternal forces acted on it, F4 1. &hen acompressive or tensile force acted on the spring, it will change the length of thespring as shown in 5igure +.+.

Figu! 1.1 An ideal spring acted by e"ternal force

According to Hookes law, the deformation is proportional to the deforming forcesas long as they are not too large,

F4 kx *+.62

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http://en.wikipedia.org/wiki/Theory_of_elasticityhttp://en.wikipedia.org/wiki/Theory_of_elasticity


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&here Fis the restoring force e"erted by the spring, kis a spring constant and xis the distance the spring elongated or compressed. #he unit of kis 7ewton per

metre, 7m

/+

.

Example 1.1

An ideal spring is placed on a hori'ontal wooden tabletop. A tensile force of .1 7is pulling one end of the spring causing it to elongate +1 cm. &hat is,

*a2 #he spring constant.*b2 #he displacement of the spring if a +1 7 of compressive force pushing it.

Solution

*a2 !ince the spring is elongate +1 cm, from Hookes law the spring constantis,

*b2 #he compressive force of +1 7 causing the spring to displace *compress2,

Example 1.2

&hat is the ma"imum load that could be suspended from a hanging spring tostretch it .1 cm from e3uilibrium8 #he spring constant, kis +11 7m/+.

Solution

5rom E3. +.6, the ma"imum load is e3ual to the restoring force of the spring.

1." STRESS# AND STRAIN#

A bar, rod or wire will stretch when its ends are pulled9 hence show its elasticbehavior. 5igure +.6 shows a metal rod that initially has uniform cross/sectional A

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lo

l

F

Area, A

lo

A


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and length, l. A force Fis applied perpendicularly at the end of the rod, causing itin tension.

*a2

*b2

Figu! 1." *a2 Metal rod with initial length, lo. *b2 #ensile stress acts

perpendicularly to the metal rod causing elongation, l.

&e define tensile stress at the cross section as the ratio of the force to, F thecross sectional area, A9

*+.-2

#he !I unit for stress is the :ascal or 7m /6.

Stress is characterized as the strenth o! the !orces causin thede!ormation o! an elastic material, on a "!orce per unit area basis.

;esides that, the deformation of an ob$ect under tensile stress will cause tensile

strain. As in 5igure +.6, the rod is stretches to a length l 4 lo < lwhen under

tension.

#he tensile strain o! the rod $or elastic ob%ect& is e'ual to the !ractional

chane in lenth, which is the ratio o! the elonation l to the oriinal

lenth lo(

*+.=2

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#ensile strain is a ratio of two lengths, always measured in the same units, hencehas no units.

Example 1.)

A steel rod 6.1 m long has a cross sectional area of 1.-1 cm6. #he rod is nowhung by one end from a support structure, and a 1 kg weight is hung from therods lower end, causing it to elongate +.> mm. &hat is the stress and strain ofthe rod8

Solution

1.$ ELASTIC MODULUS

&hen the stress and strain is small enough, we often find that the two are directlyproportional. &e call the proportionality constant an elastic modulus, as in E3.+.+. #able +.+ lists appro"imate elastic modulus for some materials.

T%&'! 1.1 Appro"imate elastic modulus for some materials.

M%(!i%' Y)u*g+ M),u'u+#*- " +1+1 %/

Bu' M),u'u+#+- " +1+1 %/

S!% M),u'u+#S- " +1+1 %/

ead +. =.+ 1.

?rown glass .1 .1 6.

Aluminum 0.1 0. 6.

;rass @.1 .1 -.

?opper ++.1 +=.1 =.=

!teel 61.1 +.1 0.

Iron 6+.1 +.1 0.07ickel 6+.1 +0.1 0.>

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2. m

F F


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1.2 YOUNGS MODULUS# *

5or a sufficiently small tensile stress, stress and strain are proportional. #hus thecorresponding elastic modulus is called oungs modulus, Y9

*+.2

E3. +. can be simplified as9

#ensile stress, 4 Y*#ensile strain, 2

#he !I unit of oungs modulus is :a or 7m /6. A material with a large value of Yisrelatively unstretchable9 a large stress is needed for a given strain. 5or e"ample,the value of Yfor iron is much larger than aluminum, as shown in #able +.+.

*ouns modulus can be thouht o! as the inherent sti!!ness o! a material(it measures the resistance o! the material to elonation or compression.

Example 1.

A circular steel wire with length 6.1 m must stretch no more than 1.6 cm when atensile force of =11 7 is applied to one end of the wire. &hat minimum diameteris re3uired for the wire8

Solution

o4 6.1 m, l 4 1.6 cm 4 1.6 " +1/6m, 5 4 =11 7, steel4 61 " +1

+1:a

5rom Hookes law,

!


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F " #A

F " #A

F " #AF " #A

F " #A

F " #A

F " #A

F " #A

$o

$

$


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!tress 4 Y*strain2

Example 1.-

)efer to E"ample +.-, what is the oungs modulus of the steel rod8

Solution

;y using E3. +., the Yis9

1.3 BULK MODULUS# +

&hen a diver plunges into the ocean, the water e"erts nearly uniform pressureeverywhere on the diver, and s3uee'es him to a slightly smaller volume. #his is adifferent situation from the tensile and compressive stress and strain asdiscussed in the earlier section. #he stress is now uniform pressure on all sides,and the resulting deformation is a volume change. &e use the term bulk stress*or volume stress2 and bulk strain *or volume strain2 to describe these

3uantities.

If an ob$ect is immersed in a fluid, the fluid e"erts a force on any part of theob$ects surface9 this force is perpendicular to the surface, resulting pressure onthe ob$ect as shown in 5igure +.-. :ressure plays important role in a volumedeformation.

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Figu! 1.$ Bolume change due to volume stress.

*+.2

#he volume strain is the ratio of the volume change Vto the original volume Vo9

*+.02

&hen Hookes law is obeyed, an increase in pressure *bulk stress2 produces aproportional bulk strain.

#he correspondin elastic modulus !or bulk stress and bulk strain is called+ulk modulus, +.

*+.>2

E3. +.> can be simplified as9

;ulk stress 4 / B*;ulk strain2

#he minus sign in the e3uation because an increase of pressure always causesdecrease in volume.

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Example 1.

A hydraulic press contains 1.-1 m

-

of oil. 5ind the decrease in the volume of theoil when it is sub$ected to a pressure increase P 4 6.1 " +10:a. #he bulk

modulus of the oil is B4 .1 " +1@:a.

Solution

1.4 SHEAR MODULUS# S

#he third kind of stress/strain is called shear. ?onsider a book placed on atabletop. If we push hori'ontally on the top cover of the book while pushing inopposite direction on the bottom cover to hold it in place, the book is deformed asshown in 5igure .=.

Figu! 1.2 A book under shear stress.

Shear de!ormation is the result o! a pair o! e'ual and opposite !orces thatact parallel to two opposite sur!aces.

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#he shear stress is the magnitude of the shear force divided by the area of thesurface on which the force acts9

*+.@2

!hear strain is the ratio of the relative displacement " to the separation of the

two surfaces9

*+.+12

#he shear stress is proportional to the shear strain as long as the stress is nottoo large. #he constant of proportionality is the +!% 5),u'u+# S9

*+.++2

E3. +.++ can be simplified as9

!hear stress 4 S*!hear strain2

Example 1./

An outdoor sculpture made of brass base plate9 e"periences shear forces as aresult of an earth3uake. #he frame is 1.>1 m s3uare and 1.1 cm thick. Howlarge a force must be e"erted on each of its edges if the displacement is 1.+mm8

Solution

5rom #able +.+ the shear modulus of brass, Sbrass4 -. " +1+1:a.

7ote that Lrepresents the 1.>1 m length of each side of the s3uare plate, andthe area Ais the product of the 1.> m length and 1.1 cm thickness.

Csing E3. +.++ the shearing force is9

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.& m

.& m

.5 cm

F

F

Area, A


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1.6 OISSON RATIO# v

0oisson ratio, v is de!ined as the ratio o! transversal strain overlonitudinal strain.

*+.+62

5igure +. shows a bo" e"periencing tensile stress causing it to elongate in the Fdirection, from Loto L, and reduce in the width, from Woto W. #he relationship ofthis transversal strain to longitudinal strain is called :oisson ratio.

Figu! 1.3 A bo" e"periencing transversal strain and longitudinal strain.

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*+.+-2

SUMMARY

+. Hookes law states that the strain produced is directly proportional to the

stress applied, provided the stress is below the elastic limit.

6. !tress is characteri'ed as the strength of the forces causing thedeformation of an elastic material, on a Dforce per unit area basis.

-. #he tensile strain of the rod *or elastic ob$ect2 is e3ual to the fractional

change in length, which is the ratio of the elongation l to the original

length lo9

=. oungs modulus can be thought of as the inherent stiffness of a material9it measures the resistance of the material to elongation r compression.

. #he corresponding elastic modulus for bulk stress and bulk strain is called;ulk modulus, ;.

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. !hear deformation is the result of a pair of e3ual and opposite forces thatact parallel to two opposite surfaces.

0. :oisson ratio, v is defined as the ratio of transversal strain overlongitudinal strain.

E7ERCISES

+. oungs modulus for diamond is about 61 times as large as that of marble.oes that tell you which is stronger8 If not, what does it tell you8

6. A 6. m long steel beam is placed vertically in the basement of a building.#his is to ensure the floors as well the building above from sagging. #heload on the beam is .> " +1=7 and the cross/sectional area of the beam

is 0. " +1/-

m6

. &hat is the vertical compression of the beam8*Ans% 1.1@0 mm2

-. A 1.1 m string with cross/sectional area +.1 " +1/ m6 has oungsmodulus of 6.1 " +1@7m/6. ;y how much must you stretch the string toobtain a tension of 61 78*Ans% .1 mm2

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=. &hat is the ma"imum load that could be suspended from a copper wire oflength +.1 m and diameter 6.1 mm without permanently deforming the

wire8 ?opper has an elastic limit of 6.1 " +1

>

7m

/6

and a tensile strength of=.1 " +1>7m/6.*Ans% +-11 72

. A steel rod is compressed by the application of forces with magnitude 5 ateach end. &hat magnitude of forces re3uired to compress by the sameamount9*a2 a steel bar of the same cross/sectional area but one half the

length8 *b2 a steel bar of the same length but one half the radius8. An anchor, made of cast iron with volume 1.6-1 m-has bulk modulus of

1 " +1@:a is lowered over the side of the ship to the bottom of theharbor. #he pressure below the harbor is greater than sea level pressureby +.0 " +1:a. &hat is the change in the volume of the anchor8*Ans% /.0+ cm-2

0. A copper sphere of bulk modulus +-1 F:a is sub$ected to +11 M:a ofpressure. ;y what fraction does the volume of the sphere change8*Ans% 0.0 " +1/=2

>. A cube of gelatin with dimensions of .1 cm on each side is displaced1.= cm by a tangential force. If the shear modulus of gelatin is @=1 M:a,what is the magnitude of the force8*Ans% 1.-1 72

REFERENCES

+. :re/C #e"t !#:M% :hysics Bolume +, ?heong 5oon ?hoong, :earson/ongman, 611.

6. ?ollege :hysics. International Edition, Alan Fiambattista, ;etty Mc?arthy)ichardson and )obert ?. )ichardson, McFraw/Hill, 611=.

-. Cniversity :hysics% &ith Modern :hysics. +6thEd., oung H. and ). A.5reedman, :earson Addison/&esley, 611>.

=. :hysics for !cientists G Engineers with Modern :hysics. - rdEd., ouglas?. Fiancoli, :rentice Hall, 6111.

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Chap01 Elasticity