CH6 CALLISTER

8/13/2019 CH6 CALLISTER

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Chapter 6 - 1

ISSUES TO ADDRESS... Stress and strain : What are they and why are

they used instead of load and deformation?

Elastic behavior: When loads are small, how muchdeformation occurs? What materials deform least?

Plastic behavior: At what point does permanentdeformation occur? What materials are most

resistant to permanent deformation? Toughness and ductility : What are they and how

do we measure them?

Chapter 6:Mechanical Properties


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Chapter 6 - 2

Elastic means reversible !

Elastic Deformation1. Initial 2. Small load 3. Unload

F

d

bondsstretch

return toinitial

F

d

Linear-elastic

Non-Linear- elastic


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Chapter 6 - 3

Plastic means permanent !

Plastic Deformation (Metals)

F

d linearelastic

linearelastic

d plastic

1. Initial 2. Small load 3. Unload

p lanesstillsheared

F

d elastic + plastic

bondsstretch& planesshear

d plastic


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Chapter 6 - 4

Stress has units:

N/m2 or lb f /in2

Engineering Stress Shear stress, t :

Area, A

F t

F t

F s

F

F

F s

t = F s A o

Tensile stress, s :

original areabefore loading

Area, A

F t

F t

s = F t A o

2f

2mNor

inlb=


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Chapter 6 - 5

Simple tension: cable

Note: t = M / Ac R here.

Common States of Stress

A o = cross sectionalarea (when unloaded)

F F

o s =

F A

o t =

F s A

s s

M

M A o

2R

F s A c

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


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Chapter 6 - 6

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

o s = F A

Simple compression:

Note: compressivestructure member(s < 0 here).(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (1)

A o

Balanced Rock, ArchesNational Park


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Chapter 6 - 7

Bi-axial tension: Hydrostatic compression:

Pressurized tank

s < 0 h

(photo courtesyP.M. Anderson)

(photo courtesy

P.M. Anderson)

OTHER COMMON STRESS STATES (2)

Fish under water

s z > 0

s q > 0


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Chapter 6 - 8

Tensile strain: Lateral strain:

Shear strain:

Strain is alwaysdimensionless.

Engineering Strain

q

90

90 - q y

x q g = x /y = tan

e = d

Lo

- d e L = L w o

Adapted from Fig. 6.1 (a) and (c), Callister 7e.

d /2

d L /2

Lowo


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Chapter 6 - 9

Stress-Strain Testing Typical tensile test

machine

Adapted from Fig. 6.3, Callister 7e. (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.)

specimen extensometer

Typical tensile

specimen

Adapted fromFig. 6.2,

Callister 7e.

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 = E e s

Linear-elastic

E

e

F

F simpletensiontest


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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

e n = - L e

metals : n ~ 0.33ceramics : n ~ 0.25polymers : 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 7e.


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Chapter 6 - 13

Elastic Shearmodulus, G :

t G

g t = G g

Other Elastic Properties

simpletorsiontest

M

M

Special relations for isotropic materials:

2(1 n) E

G = 3(1 - 2n)

E K =

Elastic Bulkmodulus, K:

pressure

test: Init.vol = V o.Vol chg.= V

P

P P P = - K

VVo

P V

K V o


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Chapter 6 - 14

Metals

Alloys

GraphiteCeramicsSemicond

Polymers Composites

/fibers

E (GPa)

Based on data in Table B2,Callister 7e .Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.

Youngs Moduli: Comparison

10 9 Pa

0.2

8

0.6

1

Magnesium, Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, Ni Molybdenum

G raphite

Si crystal

Glass - soda

Concrete

Si nitride Al oxide

PC

Wood( grain)

AFRE( fibers) *

CFRE *

GFRE*

Glass fibers only

Carbon fibers only

A ramid fibers only

Epoxy only

0.4

0.8

2

46

10

2 0

4 06 08 010 0

200

600800

10 001200

400

Tin

Cu alloys

Tungsten

Si carbide

Diamond

PTF E

HDP E

LDPE

PP

Polyester

PS PET

C FRE( fibers) *

G FRE( fibers)*

G FRE(|| fibers)*

A FRE(|| fibers)*

C FRE(|| fibers)*


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Chapter 6 - 15

Simple tension:

d = FL o E A o

d L = - n Fw o E A o

Material, geometric, and loading parameters all contribute to deflection.

Larger elastic moduli minimize elastic deflection.

Useful Linear Elastic Relationships

F

A o d /2

d L /2

Lo wo

Simple torsion:

a = 2 ML o p r o

4 G

M = momenta = angle of twist

2r o

Lo


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Chapter 6 - 16

(at lower temperatures, i.e. T < T melt /3)Plastic (Permanent) Deformation

Simple tension test:

engineering stress, s

engineering strain, e

Elastic+Plasticat larger stress

permanent (plastic)after load is removed

e p

plastic strain

Elasticinitially

Adapted from Fig. 6.10 (a),Callister 7e.


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Chapter 6 - 17

Stress at which n o t i c e a b l e plastic deformation hasoccurred.

when e p = 0.002

Yield Strength, s y

s y = yield strength

Note: for 2 inch sample

e = 0.002 = z /z

z = 0.004 in

Adapted from Fig. 6.10 (a),Callister 7e.

tensile stress, s

engineering strain, e

s y

e p = 0.002


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Chapter 6 - 18

Room T values

Based on data in Table B4,Callister 7e .a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

Yield Strength : ComparisonGraphite/Ceramics/Semicond

Metals/ Alloys

Composites/fibers Polymers

Y i e l d s t r e n g

t h ,

s y ( M P a

)

PVC

H

a r d

t o m e a s u r e

,

s i n c e

i n t e n s

i o n , f r a

c t u r e u s u a

l l y o c c u r s

b e f o r e y i e

l d .

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

2 00

3 00

4 005 006 007 00

10 00

2 0 00

Tin (pure)

Al (6061) a

Al (6061) ag

Cu (71500) hr Ta (pure) Ti (pure) a Steel (1020) hr

Steel (1020) cd

Steel (4140) a

Steel (4140) qt

Ti (5Al-2.5Sn) a W (pure)

Mo (pure) Cu (71500) cw

H a r d

t o m e a s u r e ,

i n c e r a m

i c m a

t r i x a n

d e p o x y m a

t r i x c o m p o s

i t e s , s

i n c e

i n t e n s i o n ,

f r a c

t u r e u s u a

l l y o c c u r s

b e f o r e y i e

l d .

H DPE PP

humid

dry

PC PET


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Chapter 6 - 19

Tensile Strength, TS

Metals : occurs when noticeable necking starts. Polymers : occurs when polymer backbone chains are

aligned and about to break.


s y

strain

Typical response of a metal

F = fracture orultimatestrength

Neck actsas stressconcentrator

e n g

i n e e r i n g

TS

s t r e s s

engineering strain

Maximum stress on engineering stress -strain curve.


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Chapter 6 - 20

Tensile Strength : Comparison

Si crystal

Graphite/Ceramics/Semicond

Metals/ Alloys

Composites/fibers Polymers

T e n s

i l e s t r e n g

t h ,

T S

( M P a

)

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) a W (pure)

Cu (71500) cw

L DPE

PP

PC PET

20

3040

20003000

5000

Graphite

Al oxide

Concrete

Diamond

Glass-soda

Si nitride

H DPE

wood ( fiber)

wood(|| fiber)

1

GFRE (|| fiber)

GFRE ( fiber)

C FRE (|| fiber)

C FRE ( fiber)

A FRE (|| fiber)

A FRE( fiber)

E-glass fib C fibers Aramid fib

Room Temp. valuesBased on data in Table B4,Callister 7e .a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

AFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%

fibers.


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Chapter 6 - 21

Plastic tensile strain at failure:


Ductility

Another ductility measure: 100x A

A ARA%

o

fo -=

x 100 L

L L EL %

o

o f - =

Engineering tensile strain, e

E ngineeringtensile

stress, s

smaller % EL

larger % EL Lf Ao A f Lo


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Chapter 6 - 22

Energy to break a unit volume of material Approximate by the area under the stress -strain

curve.

Toughness

Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy

very small toughness(unreinforced polymers)

Engineering tensile strain, e

E ngineeringtensile

stress, s

small toughness (ceramics)

large toughness (metals)



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Chapter 6 - 23

Resilience, U r

Ability of a material to store energy Energy stored best in elastic region

If we assume a linearstress-strain curve thissimplifies to

Adapted from Fig. 6.15,

Callister 7e.

y y r 2 1

U e s @

ees= y d U r 0


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Chapter 6 - 24

Elastic Strain Recovery



25/33

Chapter 6 - 25

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

d D Smaller indentsmean largerhardness.

increasing hardness

mostplastics

brasses Al alloys

easy to machinesteels file hard

cuttingtools

nitridedsteels diamond


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Chapter 6 - 26

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 - 27

Hardness: MeasurementTable 6.5


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Chapter 6 - 28

True Stress & StrainNote: S.A. changes when sample stretched

True stress

True Strain

i T AF =s

oi T ln=e e+=e

e+s=s1ln

1

T

T



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Chapter 6 - 29

Hardening

Curve fit to the stress -strain response:

s T = K e T ntrue stress ( F / A) true strain: ln( L/Lo)

hardening exponent:n = 0.15 (some steels)to n = 0.5 (some coppers)

An increase in s y due to plastic deformation.

s

e

large hardening

small hardening s y 0

s y 1


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Chapter 6 - 30

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 2

1

2

1 --=

n x x s i

n

n x x n

n

=

where n is the number of data points


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Chapter 6 - 31

Design uncertainties mean we do not push the limit.

Factor of safety, N

N y

working

s=s

Often N isbetween1.2 and 4

Example: Calculate a diameter, d , to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use afactor of safety of 5.

Design or Safety Factors

40002202 / d

N ,p

5N

y working

s=s 1045 plain

carbon steel:s y = 310 MPaTS = 565 MPa

F = 220,000N

d

Lo

d = 0.067 m = 6.7 cm


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Chapter 6 - 32

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 ( E or G).

Toughness : The energy needed to break a unitvolume of material.

Ductility : The plastic strain at failure.

Summary

Plastic behavior: This permanent deformationbehavior occurs when the tensile (or compressive)uniaxial stress reaches s y .


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Chapter 6 - 33

Core Problems:

Self-help Problems:

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