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 7: MECHANICAL PROPERTIES

• Ceramic Materials: What special provisions/tests aremade for ceramic materials?

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

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)

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

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

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)

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)

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

• 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

• 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

11

• Elastic modulus, E

• E ~ curvature at ro

cross sectional area Ao

L

length, Lo

F

undeformed

deformed

L F Ao

= E Lo

Elastic modulus

r

larger Elastic Modulus

smaller Elastic Modulus

Energy

ro unstretched length

E is larger if Eo is larger.

PROPERTIES FROM BONDING: E

• Elastic Shear modulus, G:

12

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

130.2

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

• Simple tension:

14

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

15

• 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

16

• Stress at which noticeable plastic deformation has occurred.

when p = 0.002 tensile stress,

engineering strain,

y

p = 0.002

YIELD STRENGTH, y

17

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

18

• 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

19

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

• Plastic tensile strain at failure:

20

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

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

21

smaller toughness- unreinforced polymers

Engineering tensile strain,

Engineering tensile stress,

smaller toughness (ceramics)

larger toughness (metals, PMCs)

TOUGHNESS

• 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

23

• Room T behavior is usually elastic, with brittle failure.• 3-Point Bend Testing often used. --tensile tests are difficult for brittle materials.

FL/2 L/2

= midpoint deflection

cross section

R

b

d

rect. circ.

• Determine elastic modulus according to:

E

F

L3

4bd3

F

L3

12R4

rect. cross

section

circ. cross

section

Fx

linear-elastic behavior

F

slope =

Adapted from Fig. 12.29, Callister 6e.

MEASURING ELASTIC MODULUS

24

• 3-point bend test to measure room T strength.F

L/2 L/2cross section

R

b

d

rect. circ.

location of max tension

• Flexural strength:

rect. fs m

fail 1.5FmaxL

bd2

FmaxL

R3

xF

Fmax

max

• Typ. values:Material fs(MPa) E(GPa)

Si nitrideSi carbideAl oxideglass (soda)

700-1000550-860275-550

69

30043039069

Adapted from Fig. 12.29, Callister 6e.

Data from Table 12.5, Callister 6e.

MEASURING STRENGTH

25

• Compare to responses of other polymers: --brittle response (aligned, cross linked & networked case) --plastic response (semi-crystalline case)

Stress-strain curves adapted from Fig. 15.1, Callister 6e. Inset figures along elastomer curve (green) adapted from Fig. 15.14, Callister 6e. (Fig. 15.14 is from Z.D. Jastrzebski, The Nature and Properties of Engineering Materials, 3rd ed., John Wiley and Sons, 1987.)

TENSILE RESPONSE: ELASTOMER CASE

initial: amorphous chains are kinked, heavily cross-linked.

final: chains are straight,

still cross-linked

0

20

40

60

0 2 4 6

(MPa)

8

x

x

x

elastomer

plastic failure

brittle failure

Deformation is reversible!

26

• Decreasing T... --increases E --increases TS --decreases %EL

• Increasing strain rate... --same effects as decreasing T.

Adapted from Fig. 15.3, Callister 6e. (Fig. 15.3 is from T.S. Carswell and J.K. Nason, 'Effect of Environmental Conditions on the Mechanical Properties of Organic Plastics", Symposium on Plastics, American Society for Testing and Materials, Philadelphia, PA, 1944.)

20

40

60

80

00 0.1 0.2 0.3

4°C

20°C

40°C

60°Cto 1.3

(MPa)

Data for the semicrystalline polymer: PMMA (Plexiglas)

T AND STRAIN RATE: THERMOPLASTICS

27

• Stress relaxation test:

Er(t)

(t)o

--strain to and hold.--observe decrease in stress with time.

• Relaxation modulus:

• Data: Large drop in Er

for T > Tg.(amorphouspolystyrene)

• Sample Tg(C) values:PE (low Mw)

PE (high Mw)PVCPSPC

-110- 90+ 87+100+150

103

101

10-1

10-3

105

60 100 140 180

rigid solid (small relax)

viscous liquid (large relax)

transition region

T(°C)Tg

Er(10s) in MPa

Adapted from Fig. 15.7, Callister 6e. (Fig. 15.7 is from A.V. Tobolsky, Properties and Structures of Polymers, John Wiley and Sons, Inc., 1960.)

Selected values from Table 15.2, Callister 6e.

TIME DEPENDENT DEFORMATION

time

strain

tensile test

o

t( )

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

28

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

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

29

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

• 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.

30

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

Note: For materials selection cases related to mechanical behavior, see slides 20-4 to 20-10.

SUMMARY

Reading:

Core Problems:

Self-help Problems:

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