Chapter 7: Mechanical Properties

Chapter 7 - 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 much deformation occurs? What materials deform least?

• Plastic behavior: At what point does permanent deformation occur? What materials are most resistant to permanent deformation?

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

Chapter 7: Mechanical Properties

Chapter 7 - 2

Elastic means reversible!

Elastic Deformation2. Small load

F

d

bonds stretch

1. Initial 3. Unload

return to initial

F

d

Linear- elastic

Non-Linear-elastic

Chapter 7 - 3

Plastic means permanent!

Plastic Deformation (Metals)

F

dlinear elastic

linear elastic

dplastic

1. Initial 2. Small load 3. Unload

planes still sheared

F

delastic + plastic

bonds stretch & planes shear

dplastic

Chapter 7 - 4

Stress has units: N/m2 or lbf /in2

Engineering Stress• Shear stress, t:

Area, Ao

Ft

Ft

Fs

F

F

Fs

t = Fs

Ao

• Tensile stress, s:

original area before loading

s =Ft

Ao2f

2m

Nor

in

lb=

Area, Ao

Ft

Ft

Chapter 7 - 5

• Simple tension: cable

Note: t = M/AcR here.

Common States of Stress

os= F

A

ot =

FsA

ss

M

M Ao

2R

FsAc

• Torsion (a form of shear): drive shaftSki lift (photo courtesy P.M. Anderson)

Ao = cross sectional

area (when unloaded)

FF

Chapter 7 - 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, Arches National Park

Chapter 7 - 7

• Bi-axial tension: • Hydrostatic compression:

Pressurized tank

s < 0h

(photo courtesyP.M. Anderson)

(photo courtesyP.M. Anderson)

OTHER COMMON STRESS STATES (ii)

Fish under water

sz > 0

sq > 0

Chapter 7 - 8

• Tensile strain: • Lateral strain:

Strain is alwaysdimensionless.

Engineering Strain

• Shear strain:

q

90º

90º - qy

x qg = Dx/y = tan

e = d

Lo

Adapted from Fig. 7.1 (a) and (c), Callister & Rethwisch 3e.

d/2

Lowo

-deL= L

wo

dL/2

Chapter 7 - 9

Stress-Strain Testing• Typical tensile test machine

Adapted from Fig. 7.3, Callister & Rethwisch 3e. (Fig. 7.3 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. 7.2,Callister & Rethwisch 3e.

gauge length

Chapter 7 - 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

Fsimple tension test

Chapter 7 - 11

Poisson's ratio, n

• Poisson's ratio, n:

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

> 0.50 density increases

< 0.50 density decreases (voids form)

eL

e

-n

en= - L

e

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

Chapter 7 - 12

Mechanical Properties• Slope of stress strain plot (which is

proportional to the elastic modulus) depends on bond strength of metal

Adapted from Fig. 7.7, Callister & Rethwisch 3e.

Chapter 7 - 13

• Elastic Shear modulus, G:

tG

gt = G g

Other Elastic Properties

simpletorsiontest

M

M

• Special relations for isotropic materials:

2(1 + n)EG=

3(1 - 2n)

EK=

• Elastic Bulk modulus, K:

pressuretest: Init.

vol =Vo. Vol chg. = DV

P

P PP = -K

DVVo

P

DV

K Vo

Chapter 7 - 14

MetalsAlloys

GraphiteCeramicsSemicond

PolymersComposites

/fibers

E(GPa)

Based on data in Table B.2,Callister & Rethwisch 3e. Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.

Young’s Moduli: Comparison

109 Pa

0.2

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

4

6

10

20

40

6080

100

200

600800

10001200

400

Tin

Cu alloys

Tungsten

<100>

<111>

Si carbide

Diamond

PTFE

HDPE

LDPE

PP

Polyester

PSPET

CFRE( fibers) *

GFRE( fibers)*

GFRE(|| fibers)*

AFRE(|| fibers)*

CFRE(|| fibers)*

Chapter 7 - 15

• Simple tension:

d= FLo

EAo

dL= -nFw o

EAo

• 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=2MLo

ro4G

M = moment a = angle of twist

2ro

Lo

Chapter 7 - 16

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

• Simple tension test:

engineering stress, s

engineering strain, e

Elastic+Plastic at larger stress

ep

plastic strain

Elastic initially

Adapted from Fig. 7.10 (a),Callister & Rethwisch 3e.

permanent (plastic) after load is removed

Chapter 7 - 17

• Stress at which noticeable plastic deformation has occurred.

when ep = 0.002

Yield Strength, sy

y = yield strength

Note: for 2 inch sample

= 0.002 = z/z

z = 0.004 in

Adapted from Fig. 7.10 (a),Callister & Rethwisch 3e.

tensile stress, s

engineering strain, e

sy

ep = 0.002

Chapter 7 - 18

Room temperature values

Based on data in Table B.4,Callister & Rethwisch 3e. a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

Yield Strength : ComparisonGraphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Yie

ld s

tren

gth,

sy

(MP

a)

PVC

Har

d to

mea

sure

,

sin

ce in

te

nsi

on

, fr

act

ure

usu

ally

occ

urs

be

fore

yie

ld.

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

Har

d to

mea

sure

, in

ce

ram

ic m

atr

ix a

nd

ep

oxy

ma

trix

co

mp

osi

tes,

sin

cein

te

nsi

on

, fr

act

ure

usu

ally

occ

urs

be

fore

yie

ld.

HDPEPP

humid

dry

PC

PET

¨

Chapter 7 - 19

Tensile Strength, TS

• Metals: occurs when noticeable necking starts.• Polymers: occurs when polymer backbone chains are aligned and about to break.

Adapted from Fig. 7.11, Callister & Rethwisch 3e.

y

strain

Typical response of a metal

F = fracture or

ultimate

strength

Neck – acts as stress concentrator e

ngin

eerin

g TS

str

ess

engineering strain

• Maximum stress on engineering stress-strain curve.

Chapter 7 - 20

Tensile Strength: Comparison

Si crystal<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Tens

ile s

tren

gth,

TS

(M

Pa)

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

Based on data in Table B4,Callister & Rethwisch 3e. a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.

Room temperature values

Chapter 7 - 21

• Plastic tensile strain at failure:

Ductility

• Another ductility measure: 100xA

AARA%

o

fo -=

x 100L

LLEL%

o

of -=

Lf

Ao AfLo

Adapted from Fig. 7.13, Callister & Rethwisch 3e.

Engineering tensile strain, e

Engineering tensile stress, s

smaller %EL

larger %EL

Chapter 7 - 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

Adapted from Fig. 7.13, Callister & Rethwisch 3e.

very small toughness (unreinforced polymers)

Engineering tensile strain, e

Engineering tensile stress, s

small toughness (ceramics)

large toughness (metals)

Chapter 7 - 23

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

Adapted from Fig. 7.15, Callister & Rethwisch 3e.

yyr2

1U es@

y dUr 0

Chapter 7 - 24

Elastic Strain Recovery

Adapted from Fig. 7.17, Callister & Rethwisch 3e.

Str

ess

Strain

3. Reapplyload

2. Unload

D

Elastic strainrecovery

1. Load

syo

syi

Chapter 7 - 25

Mechanical Properties

Ceramic materials are more brittle than metals. Why is this so?

• Consider mechanism of deformation– In crystalline, by dislocation motion– In highly ionic solids, dislocation motion is difficult

• few slip systems• resistance to motion of ions of like charge (e.g., anions)

past one another

Chapter 7 - 26

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

Adapted from Fig. 7.18, Callister & Rethwisch 3e.

Flexural Tests – Measurement of Elastic Modulus

FL/2 L/2

d = midpoint deflection

cross section

R

b

d

rect. circ.

• Determine elastic modulus according to:

Fx

linear-elastic behaviord

F

dslope =

3

3

4bd

LFE

(rect. cross section)

4

3

12 R

LFE

(circ. cross section)

Chapter 7 - 27

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

Adapted from Fig. 7.18, Callister & Rethwisch 3e.

Flexural Tests – Measurement of Flexural Strength

FL/2 L/2

d = midpoint deflection

cross section

R

b

d

rect. circ.

location of max tension

• Flexural strength: • Typical values:

Data from Table 7.2, Callister & Rethwisch 3e.

Si nitrideSi carbideAl oxideglass (soda-lime)

250-1000100-820275-700

69

30434539369

Material sfs (MPa) E(GPa)

22

3

bd

LFffs (rect. cross section)

(circ. cross section)3R

LFffs

Chapter 7 - 2828

Mechanical Properties of Polymers – Stress-Strain Behavior

• Fracture strengths of polymers ~ 10% of those for metals

• Deformation strains for polymers > 1000%

– for most metals, deformation strains < 10%

brittle polymer

plasticelastomer

elastic moduli – less than for metals Adapted from Fig. 7.22,

Callister & Rethwisch 3e.

Chapter 7 - 2929

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

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

Adapted from Fig. 7.24, Callister & Rethwisch 3e. (Fig. 7.24 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.)

Influence of T and Strain Rate on Thermoplastics

20

40

60

80

00 0.1 0.2 0.3

4°C

20°C

40°C

60°Cto 1.3

s(MPa)

e

Plots forsemicrystalline PMMA (Plexiglas)

Chapter 7 -

• Representative Tg values (C):PE (low density)PE (high density)PVCPSPC

- 110- 90+ 87+100+150

Selected values from Table 11.3, Callister & Rethwisch 3e.

3030

• Stress relaxation test:

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

or

ttE

)(

)(

• Relaxation modulus:

Time-Dependent Deformation

time

strain

tensile test

eo

s(t)

• There is a large decrease in Er

for T > Tg. (amorphouspolystyrene)

Adapted from Fig. 7.28, Callister & Rethwisch 3e. (Fig. 7.28 is from A.V. Tobolsky, Properties and Structures of Polymers, John Wiley and Sons, Inc., 1960.)

103

101

10-1

10-3

105

60 100 140 180

rigid solid (small relax)

transition region

T(°C)Tg

Er (10 s) in MPa

viscous liquid (large relax)

Chapter 7 - 31

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 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

Chapter 7 - 32

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

Chapter 7 - 33

Hardness: MeasurementTable 7.5

Chapter 7 - 34

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

• True stress

• True strainiT AF

oiT ln

1ln

1

T

T

Adapted from Fig. 7.16, Callister & Rethwisch 3e.

Chapter 7 - 35

Hardening

• Curve fit to the stress-strain response:

sT =K eT( ) n

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

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

• An increase in sy due to plastic deformation.s

e

large hardening

small hardeningsy 0

sy 1

Chapter 7 - 36

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

xxs i

n

n

xx n

n

where n is the number of data points

Chapter 7 - 37

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

Ny

working

Often N isbetween1.2 and 4

• Example: 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.

Design or Safety Factors

220,000N

d2 / 4 5

Ny

working

1045 plain

carbon steel: sy = 310 MPa

TS = 565 MPa

F = 220,000N

d

Lo

d = 0.067 m = 6.7 cm

Chapter 7 - 38

• 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).

• 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.