Chapter 6-

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 much deformation occurs? What materials deform least?

• Plastic behavior: At what point do dislocations cause permanent deformation? What materials are most resistant to permanent deformation?

1

• Toughness and ductility: What are they and how do we measure them?

CHAPTER 6: MECHANICAL PROPERTIES

Chapter 6-

F

bonds stretch

return to initial

2

1. Initial 2. Small load 3. Unload

Elastic means reversible!

F

Linear- elastic

Non-Linear-elastic

ELASTIC DEFORMATION

Chapter 6- 3

1. Initial 2. Small load 3. Unload

Plastic means permanent!

F

linear elastic

linear elastic

plastic

planes still sheared

F

elastic + plastic

bonds stretch & planes shear

plastic

PLASTIC DEFORMATION (METALS)

Chapter 6- 4

• Tensile stress, : • Shear stress, :

Area, A

Ft

Ft

FtAo

original area before loading

Area, A

Ft

Ft

Fs

F

F

Fs

FsAo

Stress has units:N/m2 or lb/in2

ENGINEERING STRESS

Chapter 6- 5

• Simple tension: cable

o

FA

• Simple shear: drive shaft

Ao = cross sectional Area (when unloaded)

FF

o

FsA

Note: = M/AcR here.

Ski lift (photo courtesy P.M. Anderson)

COMMON STATES OF STRESS

M

M Ao

2R

FsAc

Chapter 6-

Canyon Bridge, Los Alamos, NM

6

• Simple compression:

Ao

Balanced Rock, Arches National Park o

FA

Note: compressivestructure member( < 0 here).

(photo courtesy P.M. Anderson)

(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (1)

Chapter 6- 7

• Bi-axial tension: • Hydrostatic compression:

Fish under waterPressurized tank

z > 0

> 0

< 0h

(photo courtesyP.M. Anderson)

(photo courtesyP.M. Anderson)

OTHER COMMON STRESS STATES (2)

Chapter 6- 8

• Tensile strain: • Lateral strain:

• Shear strain:/2

/2

/2 -

/2

/2

/2

L/2L/2

Lowo

Lo

L L

wo

= tan Strain is alwaysdimensionless.

ENGINEERING STRAIN

Chapter 6-

• Typical tensile specimen

9

• Other types of tests: --compression: brittle materials (e.g., concrete) --torsion: cylindrical tubes, shafts.

gauge length

(portion of sample with reduced cross section)=

• Typical tensile test machine

load cell

extensometerspecimen

moving cross head

Adapted from Fig. 6.2, Callister 6e.

Adapted from Fig. 6.3, Callister 6e. (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.)

STRESS-STRAIN TESTING

Chapter 6-

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

10

• Hooke's Law:

= E • Poisson's ratio, :

metals: ~ 0.33 ceramics: ~0.25 polymers: ~0.40

L

L

1-

F

Fsimple tension test

Linear- elastic

1E

Units:E: [GPa] or [psi]: dimensionless

LINEAR ELASTIC PROPERTIES

Chapter 6-

• Elastic Shear modulus, G:

11

1G

= G

• Elastic Bulk modulus, K:

P= -KVVo

P

V

1-K Vo

• Special relations for isotropic materials:

P

P P

M

M

G

E2(1 )

K E

3(1 2)

simpletorsiontest

pressuretest: Init.vol =Vo. Vol chg. = V

OTHER ELASTIC PROPERTIES

Chapter 6- 120.2

8

0.6

1

Magnesium,Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, NiMolybdenum

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

46

10

20

406080

100

200

600800

10001200

400

Tin

Cu alloys

Tungsten

<100>

<111>

Si carbide

Diamond

PTF E

HDPE

LDPE

PP

Polyester

PSPET

CFRE( fibers)*

GFRE( fibers)*

GFRE(|| fibers)*

AFRE(|| fibers)*

CFRE(|| fibers)*

MetalsAlloys

GraphiteCeramicsSemicond

PolymersComposites

/fibers

E(GPa)

Eceramics > Emetals >> Epolymers

109 Pa

Based on data in Table B2,Callister 6e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.

YOUNG’S MODULI: COMPARISON

Chapter 6-

• Simple tension:

13

FLo

EAo

L

Fwo

EAo

/2

/2L/2L/2

Lowo

F

Ao

• Simple torsion:

M=moment =angle of twist

2ro

Lo

2MLoro

4G

• Material, geometric, and loading parameters all contribute to deflection.• Larger elastic moduli minimize elastic deflection.

USEFUL LINEAR ELASTIC RELATIONS

Chapter 6- 14

• Simple tension test:

(at lower temperatures, T < Tmelt/3)

tensile stress,

engineering strain,

Elastic initially

Elastic+Plastic at larger stress

permanent (plastic) after load is removed

pplastic strain

PLASTIC (PERMANENT) DEFORMATION

Chapter 6- 15

• Stress at which noticeable plastic deformation has occurred.

when p = 0.002 tensile stress,

engineering strain,

y

p = 0.002

YIELD STRENGTH, y

Chapter 6- 16

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

Yield

str

en

gth

, y (M

Pa)

PVC

Ha

rd t

o m

ea

sure,

si

nce in

ten

sion

, fr

actu

re u

sually

occu

rs b

efo

re y

ield

.

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

200

300

400500600700

1000

2000

Tin (pure)

Al (6061)a

Al (6061)ag

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

Steel (1020)cdSteel (4140)a

Steel (4140)qt

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

Mo (pure)Cu (71500)cw

Ha

rd t

o m

ea

sure

, in

cera

mic

matr

ix a

nd

ep

oxy m

atr

ix c

om

posi

tes,

sin

ce

in

ten

sion

, fr

actu

re u

sually

occu

rs b

efo

re y

ield

.HDPEPP

humid

dryPC

PET

¨

Room T values

y(ceramics) >>y(metals) >> y(polymers)

Based on data in Table B4,Callister 6e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

YIELD STRENGTH: COMPARISON

Chapter 6- 17

• Maximum possible engineering stress in tension.

• Metals: occurs when noticeable necking starts.• Ceramics: occurs when crack propagation starts.• Polymers: occurs when polymer backbones are aligned and about to break.

Adapted from Fig. 6.11, Callister 6e.

TENSILE STRENGTH, TS

strain

en

gin

eeri

ng

s

tress

TS

Typical response of a metal

Chapter 6- 18

Room T valuesSi crystal

<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

Ten

sile

str

en

gth

, TS

(MPa

)

PVC

Nylon 6,6

10

100

200300

1000

Al (6061)a

Al (6061)agCu (71500)hr

Ta (pure)Ti (pure)aSteel (1020)

Steel (4140)a

Steel (4140)qt

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

Cu (71500)cw

LDPE

PP

PC PET

20

3040

20003000

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 fib

C fibersAramid fib TS(ceram)

~TS(met)

~ TS(comp) >> TS(poly)

Based on data in Table B4,Callister 6e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.

TENSILE STRENGTH: COMPARISON

Chapter 6-

• Plastic tensile strain at failure:

19

Engineering tensile strain,

Engineering tensile stress,

smaller %EL (brittle if %EL<5%)

larger %EL (ductile if %EL>5%)

• Another ductility measure: %AR

Ao A fAo

x100

• Note: %AR and %EL are often comparable. --Reason: crystal slip does not change material volume. --%AR > %EL possible if internal voids form in neck.

Lo LfAo Af

%EL

L f LoLo

x100

Adapted from Fig. 6.13, Callister 6e.

DUCTILITY, %EL

Chapter 6-

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

20

smaller toughness- unreinforced polymers

Engineering tensile strain,

Engineering tensile stress,

smaller toughness (ceramics)

larger toughness (metals, PMCs)

TOUGHNESS

Chapter 6-

• Resistance to permanently indenting the surface.• Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties.

21

e.g., 10mm sphere

apply known force (1 to 1000g)

measure size of indent after removing load

dDSmaller indents mean larger hardness.

increasing hardness

most plastics

brasses Al alloys

easy to machine steels file hard

cutting tools

nitrided steels diamond

Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Propertiesand Applications of Plastics, p. 202, John Wiley and Sons, 1957.)

HARDNESS

Chapter 6-

• An increase in y due to plastic deformation.

22

• Curve fit to the stress-strain response:

large hardening

small hardeningunlo

ad

relo

ad

y 0

y 1

T C T n“true” stress (F/A) “true” strain: ln(L/Lo)

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

HARDENING

Chapter 6-

• Design uncertainties mean we do not push the limit.• Factor of safety, N

23

working

y

N

Often N isbetween1.2 and 4

• Ex: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below. Use a factor of safety of 5.

1045 plain carbon steel: y=310MPa TS=565MPa

F = 220,000N

d

Lo working

y

N

220,000N

d2 /4

5

DESIGN OR SAFETY FACTORS

Chapter 6-

• Stress and strain: These are size-independent measures 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).• Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive)

uniaxial stress reaches y.

24

• Toughness: The energy needed to break a unit volume of material.• Ductility: The plastic strain at failure.

SUMMARY

Chapter 6-

Reading: Chapter 6

Homework:

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