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CHAPTER 7: MECHANICAL PROPERTIES

CHAPTER 7: MECHANICAL PROPERTIES

ISSUES TO ADDRESS...

• Stress and strain

• Elastic behavior

• Plastic behavior

• Toughness and ductility

• Ceramic Materials

StressStrainElasticityStrengthTensileElongationDuctileFractureTensionFlexuralPlasticity

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

7.2 STRESS & STRAIN

Tensile loadCompressive load

Shear strain = tan

Torsional deformationangle of twist,

o

FA

Stress () for tension and compression

Lo

Strain () for tension and compression

Shear stress

o

FsA

5

• Simple tension: cable

o

FA

• Simple shear: drive shaft

Ao = cross sectional Area (when unloaded)

FF

o

FsA

Note: = M/Ac

Ski lift (photo courtesy P.M. Anderson)

7.2 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

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

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.)

7.2 STRESS-STRAIN TESTING

Normal and shear stresses on an arbitrary plane

Stress is a function of the orientation

On plane p-p’ the stress is not pure tensile

There are two componentsTensile or normal stress ’ (normal to the pp’ plane)Shear stress ’ (parallel to the pp’ plane)

• 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

ELASTIC DEFORMATIONS7.3 Stress-strain behavior

11

• Elastic modulus, E

Energy ~ 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

c07f08

c07tf01

7.4 ANESLATICITY

Assumed:Time-independent elastic deformationApplied stress produces instantaneous elastic strainRemains constant while elasticity stress is appliedAt release of load, strain is recovered

In real life:Time-dependent elastic strain component: AnelasticityTime-dependent microscopic and atomistic processesFor metals is smallSignificant for polymeric materials: Viscoelastic behavior

7.5 ELASTIC PROPERTIES OF MATERIALS

Poisson’s ratio = -x/z = -y/z

G

E2(1 )

For isotropic materials

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

II. MECHANICAL BEHAVIOR—METALS

F

bonds stretch

return to initial

2

1. Initial 2. Small load 3. Unload

Elastic means reversible!

F

Linear- elastic

Non-Linear-elastic

II. ELASTIC DEFORMATION

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

II. PLASTIC (PERMANENT) 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

II. PLASTIC DEFORMATION (METALS)

16

• YIELD STRENGTH, y

Stress at which noticeable plastic deformation has occurred.

when p = 0.002 tensile stress,

engineering strain,

y

p = 0.002

7.6 Tensile properties

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

7.6 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.


7.6 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.

7.6 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%)

ductility as percent reduction in area

%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


7.6 DUCTILITY, %ELDegree of plastic deformation at fracture

Brittle, when very little plastic deformation

c07tf02

c07f14

Stress-strain of iron at several temperatures

RESILIENCE

Capacity to absorb energy when deformed elastically and then upon unloadign, to have this energy recovered

Modulus of Resilience

For a linear elastic region:

y

dU r

0

yyrU 2

1

• Ability to absorb energy up to fracture

21

smaller toughness- unreinforced polymers

Engineering tensile strain,

Engineering tensile stress,

smaller toughness (ceramics)

larger toughness (metals, PMCs)

7.6 TOUGHNESS

Usually ductile materials are tougher than brittle onesAreas below the curves

7.7 True stress & strain

Decline in stress necessary to continue deformation past MLooks like metal become weakerActually, it is increasing in strengthCross sectional area decreases rapidly within the neck regionReduction in the load-bearing capacity of the specimenStress should consider deformation

HARDENING: 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

7.7 True stress & strain

nTT K n = hardening exponent

n = 0.15 (some steels)n = 0.5 (some copper)

c07tf04

7.8 Elastic Recovery After Plastic Deformation

7.9 Compressive, Shear, and Torsional Deformation

Similar to tensile counterpart

No maximum for compression

Necking does not occur

Mode of fracture different from that of tension

III. MECHANICAL BEHAVIOR—CERAMICSLimited applicability, catastrophic fracture in a brittle manner, little energy absorption

7.10 FLEXURAL STRENGTH

Tensile tests are difficultdifficult to prepare geometryeasy to fractureceramics fail at 0.1% strainbending stressrod specimen is usedthree of four point loading techniqueflexure test

7.10 MEASURING STRENGTH• Flexural strength= modulus of rupture= fracture strength = bend strength

• Type values:Material fs(MPa) E(GPa)

Si nitrideSi carbideAl oxideglass (soda)

700-1000550-860275-550

69

30043039069

Data from Table 12.5, Callister 6e.

22

3

bd

LFffs

3R

LFffs

7.11 Elastic Behavior (for ceramics)

Similar to tensile test for metals

Linear stress-strain

Moduli of elasticity for ceramics are slightly higher than for metals

No plastic deformation priorto fracture

7.12 INFLUENCE OF POROSITY ON THE MECHANICAL PROPERTIES OF CERAMICS

Powder as precursorCompaction to desire shapePores or voids elimination incompleteResidual porosity remainsDeleterious influence on elasticity and strengthVolume fraction porosity P

Eo = modulus of elasticity of the non porous material-Pores reduce the area -Pores are stress concentrators-tensile stress doubles in an isolated spherical pore

Aluminum oxideE = Eo(1 – 1.9P + 0.9P2)

Aluminum oxide

fs = oe-nP

• 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.)

IV MECHANICAL BEHAVIOR—POLYMERS

7.13 STRESS—STRAIN BEHAVIOR

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

7.13 T & STRAIN RATE: THERMOPLASTICS

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

c07f25

7.14 Macroscopic Deformation

Semicrystaline polymer

7.15 Viscoelasticity Deformation

Amorphous polymer:Glass at low TViscous liquid at higher TSmall deformation at low T may be elastic Hooke’s lawRubbery solid at intermediate TA combination of glass and viscous/liquidViscoelasticityElastic deformation is instantaneousUpon release, deformation is totally recovered

c07f26

7.15 Viscoelasticity Deformation

Totally elastic

Viscoelastic

Viscous

Load

Relaxation Modulus for viscoelastic polymers:

or

ttE

)(

)(

Amorphous polystyreneA viscoelastic polymer

Polystyrene configurations

amorphous

Lightly crosslinked atactic

Almost totally crystalline isotactic

Viscoelastic creepCreep modulus Ec(t)

)()(

ttE o

c

V. Hardness & Other Mechanical Property Considerations

7.16 Hardness

Measure of material resistance to localized plastic deformationEarly tests: Mohs scale 1 for talc and 10 for diamond

Depth or size of an indentation

Tests:Mohs HardnessRockwell HardnessBrinell HardnessKnoop & Vickers Microindentation Hardness

c07tf05

c07tf06a

c07tf06b

Hardness Conversion

Correlation between Hardness and Tensile Strength

Tensile strength and Hardness measure metal resistance to plastic deformation

For example:

TS(Mpa) = 3.45 × HB

or

TS(psi) = 500 × HB

c07tf07

7.17 Hardness of Ceramic Materials

7.18 Tear Strength & Hardness of Polymers

Thin films in packagingTear Strength: Energy required to tear apart a cut specimen of a standard geometry

VI. Property Variability and Design/Safety Factors

7.19 Variability of Material Properties: Average and standard deviation

• 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

7.20 DESIGN/SAFETY FACTORS

d = 47.5 mm

• 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.

SUMMARY

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