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Chapter 6 - 2

Elastic means reversible!

Elastic Deformation

2. Small load

F

d

bonds

stretch

1. Initial 3. Unload

return to

initial

F

d

Linear-

elastic

Non-Linear-elastic

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Chapter 6 - 3

Plastic means permanent!

Plastic Deformation (Metals)

F

d

linearelastic

linearelastic

dplastic

1. Initial 2. Small load 3. Unload

planesstillsheared

F

delastic + plastic

bondsstretch& planesshear

dplastic

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Chapter 6 - 4

Stress has units:

N/m2 or lbf/in2

Engineering Stress

Shearstress, t:

Area,Ao

Ft

Ft

Fs

F

F

Fs

t =Fs

Ao

Tensile stress, s:

original area

before loading

s = FtAo

2f

2mNor

inlb=

Area,Ao

Ft

Ft

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Chapter 6 - 5

Simple tension: cable

Note: t = M/AcRhere.

Common States of Stress

os=

F

A

o

t =Fs

A

ss

M

M Ao

2R

FsAc

Torsion (a form of shear): drive shaftSki lift (photo courtesyP.M. Anderson)

Ao = cross sectional

area (when unloaded)

FF

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Chapter 6 - 6

(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM

os= F

A

Simple compression:

Note: compressivestructure member

(s < 0 here).(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (i)

Ao

Balanced Rock, ArchesNational Park

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Chapter 6 - 8

Tensile strain: Lateral strain:

Strain is always

dimensionless.

Engineering Strain

Shearstrain:

q

90

90 -qy

x qg = x/y= tan

e = d

Lo

Adapted from Fig. 6.1(a) and (c), Callister & Rethwisch 8e.

d/2

Lowo

-deL=

L

wo

dL/2

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Chapter 6 - 9

Stress-Strain Testing

Typical tensile test

machine

Adapted from Fig. 6.3, Callister & Rethwisch 8e. (Fig. 6.3 is taken from H.W.

Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials,

Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons, New York, 1965.)

specimenextensometer

Typical tensile

specimen

Adapted from

Fig. 6.2,

Callister &Rethwisch 8e.

gauge

length

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Chapter 6 - 10

Linear Elastic Properties

Modulus of Elasticity, E:(also known as Young's modulus)

Hooke's Law:

s = Ee s

Linear-elastic

E

e

F

Fsimpletensiontest

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Chapter 6 - 11

Poisson's ratio, n

Poisson's ratio, n:

Units:

E: [GPa] or [psi]

n: dimensionless

n > 0.50 density increases

n < 0.50 density decreases(voids form)

eL

e

-n

en= - L

e

metals: n ~ 0.33

ceramics: n ~ 0.25

polymers: n ~ 0.40

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Chapter 6 - 12

Mechanical Properties

Slope of stress strain plot (which is

proportional to the elastic modulus) dependson bond strength of metal

Adapted from Fig. 6.7,

Callister & Rethwisch 8e.

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Chapter 6 - 13

Elastic Shear

modulus, G:

t

Gg

t = Gg

Other Elastic Properties

simple

torsion

test

M

M

Special relations for isotropic materials:

2(1n)

EG=

3(12n)

EK =

Elastic Bulk

modulus, K:

pressure

test: Init.vol =Vo.

Vol chg.

= V

P

P PP= -K

VVo

P

V

KVo

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Chapter 6 - 14

Metals

Alloys

Graphite

Ceramics

Semicond

PolymersComposites

/fibers

E(GPa)

Based on data in Table B.2,


Composite data based on

reinforced epoxy with 60 vol%

of aligned

carbon (CFRE),

aramid (AFRE), or

glass (GFRE)

fibers.

Youngs Moduli: Comparison

109 Pa

0.2

8

0.6

1

Magnesium,

Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, Ni

Molybdenum

Graphite

Si crystal

Glass -soda

Concrete

Si nitrideAl oxide

PC

Wood( grain)

AFRE( fibers) *

CFRE*

GFRE*

Glass fibers only

Carbon fibers only

Aramid fibers only

Epoxy only

0.4

0.8

2

4

6

10

20

40

6080

100

200

600800

10001200

400

Tin

Cu alloys

Tungsten

Si carbide

Diamond

PTFE

HDPE

LDPE

PP

Polyester

PSPET

CFRE( fibers) *

GFRE( fibers)*

GFRE(|| fibers)*

AFRE(|| fibers)*

CFRE(|| fibers)*

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Chapter 6 - 15

Simple tension:

d = FLoEAo

dL

= - nFwoEAo

Material, geometric, and loading parameters all

contribute to deflection.

Larger elastic moduli minimize elastic deflection.

Useful Linear Elastic Relationships

F

Aod/2

dL/2

Lowo

Simple torsion:

a = 2MLoro

4G

M = momenta = angle of twist

2ro

Lo

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Chapter 6 - 16

(at lower temperatures, i.e. T< Tmelt/3)

Plastic (Permanent) Deformation

Simple tension test:

engineering stress, s

engineering strain, e

Elastic+Plasticat larger stress

ep

plastic strain

Elasticinitially

Adapted from Fig. 6.10(a),


permanent (plastic)after load is removed

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Chapter 6 - 17

Stress at which not iceable plastic deformation has

occurred.when ep = 0.002

Yield Strength,sy

sy= yield strength

Note: for 2 inch sample

e = 0.002 = z/z

z= 0.004 in

Adapted from Fig. 6.10(a),


tensile stress, s

engineering strain, e

sy

ep = 0.002

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Chapter 6 - 18

Room temperaturevalues



a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

Yield Strength : ComparisonGraphite/Ceramics/Semicond

Metals/Alloys

Composites/fibers

Polymers

Yieldstreng

th,

sy

(MPa

)

PVC

H

ard

tomeasure

,

since

intens

ion,

fra

ctureusua

llyoccurs

beforey

ield

.

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

200

300

400

500600700

1000

2000

Tin (pure)

Al (6061) a

Al (6061) ag

Cu (71500) hrTa (pure)Ti (pure) aSteel (1020) hr

Steel (1020) cd

Steel (4140) a

Steel (4140) qt

Ti (5Al-2.5Sn) aW (pure)

Mo (pure)Cu (71500) cw

Hard

tomeasure,

inceram

icma

trixan

depoxyma

trixcompo

sites,

since

intens

ion,

frac

tureusua

llyoccurs

befo

rey

ield

.

HDPEPP

humid

dry

PC

PET

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Chapter 6 -

VMSE: Virtual Tensile Testing

19

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Chapter 6 - 20

Tensile Strength, TS

Metals: occurs when noticeable necking starts.

Polymers: occurs when polymer backbone chains are

aligned and about to break.

Adapted from Fig. 6.11,Callister & Rethwisch 8e.

sy

strain

Typical response of a metal

F= fracture or

ultimate

strength

Neck acts

as stress

concentratorenginee

ring

TS

stres

s

engineering strain

Maximum stress on engineering stress-strain curve.

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Chapter 6 - 21

Tensile Strength: Comparison

Si crystal

Graphite/Ceramics/Semicond

Metals/Alloys

Composites/fibers

Polymers

Tens

ilest

reng

th,

TS

(MPa

)

PVC

Nylon 6,6

10

100

200300

1000

Al (6061) a

Al (6061) ag

Cu (71500) hr

Ta (pure)Ti (pure) a

Steel (1020)

Steel (4140) a

Steel (4140) qt

Ti (5Al-2.5Sn) aW (pure)

Cu (71500) cw

LDPE

PP

PC PET

20

3040

2000

3000

5000

Graphite

Al oxide

Concrete

Diamond

Glass-soda

Si nitride

HDPE

wood ( fiber)

wood(|| fiber)

1

GFRE(|| fiber)

GFRE( fiber)

CFRE(|| fiber)

CFRE( fiber)

AFRE(|| fiber)

AFRE( fiber)

E-glass fibC fibersAramid fib



a = annealed

hr = hot rolled

ag = aged

cd = cold drawncw = cold worked

qt = quenched & tempered

AFRE, GFRE, & CFRE =

aramid, glass, & carbon

fiber-reinforced epoxy

composites, with 60 vol%

fibers.

Room temperaturevalues

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Chapter 6 - 22

Plastic tensile strain at failure:

Ductility

Another ductility measure: 100xA

AARA%

o

fo-

=

x 100

L

LLEL%

o

of -

=

LfAo AfLo



Engineering tensile strain, e

Engineering

tensile

stress, s

smaller %EL

larger %EL

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Chapter 6 - 23

Energy to break a unit volume of material

Approximate by the area under the stress-strain curve.

Toughness

Brittle fracture: elastic energy

Ductile fracture: elastic + plastic energy



very small toughness(unreinforced polymers)

Engineering tensile strain, e

Engineeringtensile

stress, s

small toughness (ceramics)

large toughness (metals)

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Chapter 6 - 24

Resilience, Ur

Ability of a material to store energy

Energy stored best in elastic region

If we assume a linear

stress-strain curve this

simplifies to



yyr2

1U es@

e

es= y

dUr0

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Chapter 6 - 25

Elastic Strain Recovery



Stre

ss

Strain

3. Reapplyload

2. Unload

D

Elastic strain

recovery

1. Load

syo

syi

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Chapter 6 - 26

Hardness

Resistance to permanently indenting the surface.

Large hardness means:-- resistance to plastic deformation or cracking in

compression.

-- better wear properties.

e.g.,10 mm sphere

apply known force measure sizeof indent afterremoving load

dDSmaller indentsmean largerhardness.

increasing hardness

mostplastics

brassesAl alloys

easy to machinesteels file hard

cuttingtools

nitridedsteels diamond

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Chapter 6 - 27

Hardness: Measurement

Rockwell No major sample damage

Each scale runs to 130 but only useful in range

20-100.

Minor load 10 kg Major load 60 (A), 100 (B) & 150 (C) kg

A = diamond, B = 1/16 in. ball, C = diamond

HB = Brinell Hardness TS (psia) = 500 x HB

TS (MPa) = 3.45 x HB

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Chapter 6 - 28

Hardness: MeasurementTable 6.5

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Chapter 6 - 29

True Stress & Strain

Note: S.A. changes when sample stretched

True stress

True strain

iT AF=s

oiT ln=e

e+=e

e+s=s

1ln

1

T

T



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Chapter 6 - 31

Variability in Material Properties

Elastic modulus is material property

Critical properties depend largely on sample flaws

(defects, etc.). Large sample to sample variability.

Statistics

Mean

Standard Deviation

s =

n

xi-x

2

n- 1

1

2

n

xx

n

n

=

where n is the number of data points

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Chapter 6 - 32

Design uncertainties mean we do not push the limit.

Factor of safety, N

N

y

working

s=s

Often Nisbetween

1.2 and 4

Example: Calculate a diameter, d, to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use a

factor of safety of 5.

Design or Safety Factors

220 ,000 N

d2/ 4

5N

y

working

s=s 1045 plain

carbon steel:

sy= 310 MPa

TS = 565 MPa

F= 220,000N

d

Lo

d= 0.067 m = 6.7 cm

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Chapter 6 - 33

Stress and strain: These are size-independentmeasures of load and displacement, respectively.

Elastic behavior: This reversible behavior often

shows a linear relation between stress and strain.

To minimize deformation, select a material with a

large elastic modulus (Eor G).

Toughness: The energy needed to break a unit

volume of material.

Ductility: The plastic strain at failure.

Summary

Plastic behavior: This permanent deformation

behavior occurs when the tensile (or compressive)

uniaxial stress reaches sy.

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