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 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 6:

Mechanical Properties

Chapter 6 - 2

Mechanical Loads and Deformation

• Loads– Tension and Compression – Shear and Torsion– Stress = Force / area– What force ? Which Area ?

• Deformation - Change in the shape of a specimen.- Strain – relative change in its dimension - Which dimension ?

• Stress-Strain Behavior Property ? • Which in What ?

– Forces Statics – Stresses Strength of Materials – Property-Structure Material Science

Chapter 6 - 3

Engineering Stress-Definition• Shear stress, :

Area, A

Ft

Ft

Fs

F

F

Fs

= Fs

Ao

• Tensile stress, :

With original area (before loading) Engineering StressWith Ai (instantaneous area) True Stress T

=Ft

Ao2f

2m

Nor

in

lb=

F

Area, A

Ft

t

Chapter 6 - 4

• Simple tension: cable

Note: = M/AcR here.

Real Systems of Stress

Ao

FF

o F

A

o

FsA

M

M Ao

2R

FsAc

• Torsion (a form of shear): drive shaft

Chapter 6 - 5

• Simple compressioncompression:

Note: This is Compressive Stress ( < 0)

Other Applications-STRESS

o F

A

Ao

Chapter 6 - 6

• Bi-axial tension: • Hydrostatic compression:

Pressurized tank

< 0h

Other Applications-STRESS

Fish under water

z > 0

> 0

Chapter 6 - 7

• Tensile strain:

Engineering Strain-Definition

or L)

Lo

/2

Lo

Where L = L– Lo

* Note:With L (instantaneous): integrate with variable L True Strain = Ln(1+)

At any instant AL = AoLo initial; (Constant Volume)

ddL

L

With Lo (constant) Engineering Strain*

Integrate to get from Lo to any L

define

L

Chapter 6 - 8

Stress-Strain Testing

Typical tensile test machine

specimenextensometer

Typical tensile specimen

gauge length

Chapter 6 - 9

Stress-Strain General Behavior

• True stress

• True Strain

iT AF

oiT ln

1ln

1

T

T

Chapter 6 - 10

Stress Strain Behavior

Two behaviors: low loads versus large loads

Elastic RangeInitially, stress and strain are directly proportional to each otherWhy: atoms can be thought of as masses connected to each other

through a network of springs (Imagine) According to Hooke’s law:

the extension of a spring, x, and the applied force, F, are related by the spring constant, k:

F = - kxThus, Stress (F/A) must be linear with strain

This is Stretching of bonds

Plastic Range Non-Linear relation of stress with strain breaking of bonds and forming new bonds

Chapter 6 - 11

Linear Elastic Properties

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

• Hooke's Law:

= E F

Fsimple tension test

At low levels of stress: the shape is recoverableThe deformation is reversible

Linearity for TensionLinearity for other types of stresses ? (later)

Linear- elastic

E

Chapter 6 - 12

Elastic means reversible!For some materials: it is non-lineare.g. gray cast iron and concrete

Elastic Deformation1. Initial 2. Small load 3. Unload

F

bonds stretch

return to initial

F

Linear- elastic

Non-Linear-elastic

Anelasticity: e = f (time); the specimen continue to deform

Chapter 6 - 13

Structure-Property Relationship• Elastic modulus (E: slope of verus ) depends on bond strength of

metal• Remember curves Energy (E) versus interatomic spacing (r), • Now interatomic force F versus r ?

E ~ dFdr

roA

B

Which is more stiff ?

Chapter 6 - 14

MetalsAlloys

GraphiteCeramicsSemicond

PolymersComposites

/fibers

E(GPa)

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 6 - 15

Poisson's ratio, Upon Elongation in one direction (z) i.e. z is +ve

contraction occurs in the other two directions (x and y)

Theoretically, = ¼ , max = 0.5

Practically:

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

x

z

y

z

Define Poisson's ratio, : : how much strain occurs in the lateral directions (x& y) when strained in the (z) direction:

Note: For uniaxial stresses x = y

Chapter 6 - 16

Shear Strain-Definition

Remember: Strain is always dimensionless.

90º

y

x = x/y = tan Define

Elastic Shear modulus, G:

= G

Special relations for isotropic materials:

2(1 )EG

G

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

Approximation: For most metals G 0.4 E (show ?)

Chapter 6 - 17

From Elastic to Plastic Behavior

What happens if we continue to apply tensile loading beyond theelastic limit? (i.e., stretching atomic bonds to the point of breaking)

Plastic deformation:• stress and strain are not proportional• the deformation is not reversible• deformation occurs by breaking and re-arrangement of atomic bonds (in

Proportional Limit or elastic limit, is the point where The stress and strain values at this point are known as the proportional-limit stress and strain, respectively.This is the point beyond which Hooke's law can no longer be used – no spring

Chapter 6 - 18

Plastic means permanent!Not recoverable - irreversible

Plastic Deformation (Metals)

F

linear elastic

linear elastic

plastic

1. Initial 2. Small load 3. Unload

planes still sheared

F

elastic + plastic

bonds stretch & planes shear

plastic

Chapter 6 - 19

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

Plastic (Permanent) Deformation

• Simple tension test:

engineering stress,

engineering strain,

Elastic+Plastic at larger stress

permanent (plastic) after load is removed

p

plastic strain

Elastic initially

Chapter 6 - 20

• Stress at which noticeable plastic deformation has occurred.

when p = 0.002

Yield Strength, y

y = yield strengthtensile stress,

engineering strain,

y

p = 0.002

Proportionality Limit (P)Initial deviation from linearityHard to determine use yield strength

Why do we need y ?For design (to prevent plastic deformation)

Chapter 6 - 21

Yield Strength – Clear Case

For a low-carbon steel• The stress vs. strain curve includes both an upper and lower yield point.• The yield strength is defined in this case as

the average stress at the lower yield point

Chapter 6 - 22

Room T values

Yield Strength : ComparisonGraphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Yie

ld s

tren

gth,

y

(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 6 - 23

Tensile Strength, TS

For Metals: TS occurs when noticeable necking starts.

y

Typical response of a metal

F = fracture or ultimate strength

Neck – acts as stress concentrator

Later: types of fracture eng

inee

ring

TS

str

ess

engineering strain

• Maximum stress on engineering stress-strain curve.

Chapter 6 - 24

Tensile Strength : Comparison

Si crystal<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Ten

sile

str

engt

h, T

S

(MP

a)

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

Room Temp. valuesBased on data in Table B4,Callister 7e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workeqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.

Chapter 6 - 25

Types of Failure (from Ch.8)

Ductile fracture is usually desirable!Details in Chapter 8

Very Ductile

ModeratelyDuctile BrittleFracture

behavior:

Large ModerateElongation: SmallDuctile:

warning before fractureBrittle:

No warning

Chapter 6 - 26

• Evolution to failure:

Resulting fracture surfaces (steel)

50 mm

particlesserve as voidnucleationsites.

50 mm

100 mm

Moderately Ductile Failure

necking

void nucleation

void growth and linkage

shearing at surface fracture

Chapter 6 - 27

Ductile vs. Brittle Failure

cup-and-cone fracture brittle fracture

Chapter 6 - 28

Ductility

2- Percentage Area Reduction 100xA

AARA%

o

fo -=

Percentage tensile strain at failure: x 100L

LLEL%

o

of

smaller %EL

larger %ELLf

Ao AfLo

Distinguish Behavior ? ductile versus brittle ?Define a parameter - ductility: measures the amount of plastic deformation that a material goes through by the time it breaks.1- Elongation:

Ductility increases with temperature

Chapter 6 - 29

Energy absorbed by material up to fracture per unit volume(Energy to break a unit volume of material at low strain rate)

• Approximation: the area under the stress-strain curve.

Toughness

Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy

very small toughness (unreinforced polymers)

small toughness (ceramics)

large toughness (metals)

Chapter 6 - 30

Effect of Testing Temperature on Mechanical Behavior

The yield and tensile strengths ………… with increasing temperature.

Ductility ……………. with temperature.

Stiffness …….. with temperature

Toughness …….. with temperature

Chapter 6 - 31

Resilience, Ur

Ability of a material to store energy and (release it upon unloading) – Energy stored best in elastic region

If we assume a linear stress-strain curve:

yyr2

1U

y dUr 0

Units: J/m3

Show that

For springs: better to absorb large energy i.e. Ur should be large This needs large y and low E

Chapter 6 - 32

Example

a) Modulus of elasticity

b) Yield strength

c) Tensile Strength

d) Fractural Strength

e) Ductility (% Elongation)

f) Resilience

g) Toughness

From the tensile σ - ε behavior for a specimen of brass shown in the figure, determine the following:

Chapter 6 - 33

Hardness

Material Resistance to localized plastic deformationResistance to surface indentation (surface property)

Historically; the ability of material to scratch anotherLarge hardness means: - resistance to plastic deformation or cracking in compression. - better wear properties.

e.g., 10 mm sphere

(1) apply known force (2) measure size of indent after removing load

D

Smaller indents mean larger hardness

d

How: indenter with a load and a specimenThe indenter must be harder than the specimen, otherwise flattening

Hardness is related to the size (depth) of the indentation

Specimen: smooth surface

Chapter 6 - 34

Hardness

increasing hardness

most plastics

brasses Al alloys

easy to machine steels file hard

cutting tools

nitrided steels diamond

Importance of Hardness Test• Easy • Nondestructive• Can be used to get other data (e.g. T.S.) (both are resistance to plastic deformation)

There are different scales for hardness- they vary in:Shape of indenter: (1) ball, (2) conical diamond and (3) square based diamondLoad : 100 kg, 150 kg …etc/

Rockwell: uses indenter (1) and (2) - for rapid and routine testsBrinell: uses indenter (1) for materials with moderate hardnessVickers: uses indenter (3) for all ranges – more accurate

HB = Brinell HardnessTS (psia) = 500 x HBTS (MPa) = 3.45 x HB

Chapter 6 - 35

Hardness: MeasurementTable 6.5

Chapter 6 - 36

Hardening

• Curve fit to the stress-strain response:

T K T 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 y due to plastic deformation.

large hardening

small hardeningy 0

y 1

Chapter 6 - 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.

Data for Design - Safety Factors

4

0002202 /d

N,

5

Ny

working

1045 plain

carbon steel: y = 310 MPa

TS = 565 MPa

F = 220,000N

d

Lo

d = 0.067 m = 6.7 cm

Chapter 6 - 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 y.

Chapter 6 - 39

• Impact Test – Rapid loading • Fatigue Test – Cyclic loading • Creep Test – Effect of time

Other Tests from Chapter 8

Chapter 6 - 40

Impact Testing

final height initial height

(Charpy)•Measures Toughness•“energy absorbed by a specimen up to fracture” •Upon Rapid loading

Energy absorbed = change in potential energy of the hammer

Chapter 6 - 41

Note: Loading Rate

• Increased loading rate... -- increases y and TS -- decreases %EL

• Why? An increased rate gives less time for dislocations to move past obstacles.

y

y

TS

TS

larger

smaller

Chapter 6 - 42

• Increasing temperature... --increases %EL and Kc

• Ductile-to-Brittle Transition Temperature (DBTT)...

Impact Results and Effect of Temperature

BCC metals (e.g., iron at T < 914°C)

Imp

act

Ene

rgy

Temperature

High strength materials (y > E/150)

polymers

More Ductile Brittle

Ductile-to-brittle transition temperature

FCC metals (e.g., Cu, Ni)

Adapted from Fig. 8.15, Callister 7e.

Chapter 6 - 43

• Pre-WWII: The Titanic • WWII: Liberty ships

• Problem: Used a type of steel with a DBTT ~ Room temp.

Design Strategy:Stay Above The DBTT!

Ship-cyclic loadingfrom waves.

Large Impact

Chapter 6 - 44

Fatigue• Fatigue = failure under cyclic stress.causes ~ 90% of mechanical engineering failures

• Stress varies with time. key parameters are S, m, and frequency

max

min

time

mS

tension on bottom

compression on top

countermotor

flex coupling

specimen

bearing bearing

Count number of cycles to failure

Stress amplitude S or a

a= max+(-min) 2

Chapter 6 - 45

Fatigue limit, Sfat: no fatigue if S < Sfat

Fatigue limit OREndurance limit T.S./2

Fatigue Results and Design Parameters

Sfat

case for steel (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

Sometimes, the fatigue limit is zero!

case for Al (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

Ferrous alloysFerrous alloys

Nonferrous alloysNonferrous alloys

Chapter 6 - 46

CreepTime-dependent deformation

Sample deformation at a constant stress () vs. time

Important at elevated temperature

i.e. at T > 40% of Tmelting

0 t

How ?• Subject the specimen to constant load• Measure Length as f(time)• Then get = (L-Lo)/Lo = f(time)• Plot the curve

Chapter 6 - 47

Results of Creep

Primary Creep: slope (creep rate) decreases with time.

Secondary Creep: steady-state i.e., constant slope.

Tertiary Creep: slope (creep rate) increases with time, i.e. acceleration of rate.

Chapter 6 - 48

• Occurs at elevated temperature, T > 0.4 Tm

Effect of Temperature on Creep

elastic

primarysecondary

tertiary