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

Elastic means reversible!

Elastic Deformation

2. Small load

F

d

bonds

stretch

1. Initial 3. Unload

return to

initial

F

d

Linear-elastic

Non-Linear-elastic

Chapter 6 - 3

Plastic means permanent!

Plastic Deformation (Metals)

F

d

linear elastic

linear elastic

dplastic

1. Initial 2. Small load 3. Unload

planesstill sheared

F

delastic + plastic

bonds stretch & planes shear

dplastic

Chapter 6 - 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

Ao2

f

2m

Nor

in

lb=

Area, Ao

Ft

Ft

Chapter 6 - 5

• Simple tension: cable

Note: t = M/AcR here.

Common States of Stress

o

s =F

A

o

t =Fs

A

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 6 - 6

Tensile Compressive

ShearTorsion

Chapter 6 - 7

(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM

o

s =F

A

• Simple compression:

Note: compressive

structure member

(s < 0 here).(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (i)

Ao

Balanced Rock, Arches National Park

Chapter 6 - 8

• Bi-axial tension: • Hydrostatic compression:

Pressurized tank

s < 0h

(photo courtesy

P.M. Anderson)

(photo courtesy

P.M. Anderson)

OTHER COMMON STRESS STATES (ii)

Fish under water

sz > 0

sq > 0

Chapter 6 - 9

• Tensile strain: • Lateral strain:

Strain is always

dimensionless.

Engineering Strain

• Shear strain:

q

90º

90º - qy

x qg = x/y = tan

e =d

Lo

Adapted from Fig. 6.1(a) and (c), Callister & Rethwisch 8e.

d/2

Lowo

-deL =

L

wo

dL/2

Chapter 6 -

Poisson’s Ratio

10

Chapter 6 - 11

Stress-Strain Testing

• Typical tensile test

machine

Adapted from Fig. 6.3, Callister & Rethwisch 8e. (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.)

specimenextensometer

• Typical tensile

specimen

Adapted from

Fig. 6.2,

Callister &

Rethwisch 8e.

gauge length

Chapter 6 - 12

Metals

Alloys

Graphite

Ceramics

Semicond

PolymersComposites

/fibers

E(GPa)

Based on data in Table B.2,

Callister & Rethwisch 8e.

Composite data based on

reinforced epoxy with 60 vol%

of aligned

carbon (CFRE),

aramid (AFRE), or

glass (GFRE)

fibers.

Young’s Moduli: Comparison

109 Pa

0.2

8

0.6

1

Magnesium,

Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, Ni

Molybdenum

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 - 13

(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. 6.10(a),

Callister & Rethwisch 8e.

permanent (plastic) after load is removed

Chapter 6 - 14

• Stress at which noticeable plastic deformation has

occurred.when ep = 0.002

Yield Strength, sy

sy = yield strength

Note: for 2 inch sample

e = 0.002 = z/z

z = 0.004 in

Adapted from Fig. 6.10(a),

Callister & Rethwisch 8e.

tensile stress, s

engineering strain, e

sy

ep = 0.002

Chapter 6 - 15

Room temperature

values

Based on data in Table B.4,

Callister & Rethwisch 8e.

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

Yield Strength : ComparisonGraphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Yie

ld s

tre

ng

th,s

y(M

Pa)

PVC

Ha

rd to

me

asu

re,

sin

ce

in t

en

sio

n, fr

actu

re u

su

ally

occu

rs b

efo

re y

ield

.

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

200

300

400

500600700

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 to

me

asu

re,

in c

era

mic

ma

trix

an

d e

po

xy m

atr

ix c

om

po

sites, sin

ce

in te

nsio

n, fr

actu

re u

su

ally

occu

rs b

efo

re y

ield

.

HDPEPP

humid

dry

PC

PET

¨

Chapter 6 - 16

Tensile Strength, TS

• Metals: occurs when noticeable necking starts.

• Polymers: occurs when polymer backbone chains are

aligned and about to break.

Adapted from Fig. 6.11,

Callister & Rethwisch 8e.

sy

strain

Typical response of a metal

F = fracture or

ultimate

strength

Neck – acts

as stress

concentrator

en

gin

eering

TSstr

ess

engineering strain

• Maximum stress on engineering stress-strain curve.

Chapter 6 - 17

Tensile Strength: Comparison

Si crystal<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Ten

sile

str

en

gth

, T

S(M

Pa)

PVC

Nylon 6,6

10

100

200

300

1000

Al (6061) a

Al (6061) ag

Cu (71500) hr

Ta (pure)Ti (pure) a

Steel (1020)

Steel (4140) a

Steel (4140) qt

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

Cu (71500) cw

LDPE

PP

PC PET

20

3040

2000

3000

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 B.4,

Callister & Rethwisch 8e.

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

AFRE, GFRE, & CFRE =

aramid, glass, & carbon

fiber-reinforced epoxy

composites, with 60 vol%

fibers.

Room temperature

values

Chapter 6 - 18

• Plastic tensile strain at failure:

Ductility

• Another ductility measure: 100xA

AARA%

o

fo-

=

x 100L

LLEL%

o

of-

=

Lf

AoAf

Lo

Adapted from Fig. 6.13,

Callister & Rethwisch 8e.

Engineering tensile strain, e

Engineering

tensile

stress, s

smaller %EL

larger %EL

Chapter 6 - 19

• Energy to break a unit volume of material

• Approximate by the area under the stress-strain curve.

Toughness

Brittle fracture: elastic energy

Ductile fracture: elastic + plastic energy

Adapted from Fig. 6.13,

Callister & Rethwisch 8e.

very small toughness (unreinforced polymers)

Engineering tensile strain, e

Engineering

tensile

stress, s

small toughness (ceramics)

large toughness (metals)

Chapter 6 - 20

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. 6.15,

Callister & Rethwisch 8e.

yyr2

1U es@

e

es=y

dUr 0

Chapter 6 - 21

Elastic Strain Recovery

Adapted from Fig. 6.17,

Callister & Rethwisch 8e.

Str

ess

Strain

3. Reapplyload

2. Unload

D

Elastic strain

recovery

1. Load

syo

syi

Chapter 6 - 22

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 6 - 23

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 6 - 24

Hardness: MeasurementTable 6.5

Chapter 6 - 25

True Stress & Strain

Note: S.A. changes when sample stretched

• True stress

• True strain

iT AF=s

oiT ln=e

e=e

es=s

1ln

1

T

T

Adapted from Fig. 6.16,

Callister & Rethwisch 8e.

If no volume change

Chapter 6 - 26

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 hardeningsy0

sy1

Chapter 6 - 27

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

s =n

xi - x 2

n-1

1

2

n

xx n

n

=

where n is the number of data points

Chapter 6 - 28

• Design uncertainties mean we do not push the limit.

• Factor of safety, N

N

y

working

s=s

Often N is

between

1.2 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

N

y

working

s=s 1045 plain

carbon steel: sy = 310 MPa

TS = 565 MPa

F = 220,000N

d

Lo

d = 0.067 m = 6.7 cm

Chapter 6 - 29

• 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.

Chapter 6 - 30

Core Problems: 6.4, 6.8,6.17, 6.22, 6.26,

6.37, 6.45, 6.53, 6.55

Self-help Problems:6.13, 6.50, 6.62

ANNOUNCEMENTS

Reading:6.1-6.12