CHAPTER 6: MECHANICAL PROPERTIES

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 dislocationscause permanent deformation? What materials aremost resistant to permanent deformation?

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

ELASTIC DEFORMATION

F

δ

bonds stretch

return to initial

1. Initial 2. Small load 3. Unload

Elastic means reversible!

F

δ

Linear- elastic

Non-Linear-elastic

PLASTIC DEFORMATION (METALS)

1. Initial 2. Small load 3. Unload

Plastic means permanent!

F

δlinear elastic

linear elastic

δplastic

planes still sheared

F

δelastic + plastic

bonds stretch & planes shear

δplastic

ENGINEERING STRESS

• Tensile stress, σ: • Shear stress, τ:

Area, A

Ft

Ft

σ =FtAo

original area before loading

Area, A

Ft

Ft

Fs

F

F

Fs

τ = FsAo

Stress has units:N/m2 or lb/in2

COMMON STATES OF STRESS

• Simple tension: cable

oσ =

FA

• Simple shear: drive shaft

Ao = cross sectional

Area (when unloaded)

FF

σσ

oτ =

FsA

τ

Note: τ = M/AcR here.

Ski lift (photo courtesy P.M. Anderson)

M

M Ao

2R

FsAc

OTHER COMMON STRESS STATES (1)

Canyon Bridge, Los Alamos, NM

• Simple compression:

Ao

Balanced Rock, Arches National Park o

σ =FA

Note: compressivestructure member(σ < 0 here).

(photo courtesy P.M. Anderson)

(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (2)

• Bi-axial tension: • Hydrostatic compression:

Fish under waterPressurized tank

σ < 0h

(photo courtesyP.M. Anderson)

σz > 0

σθ > 0(photo courtesyP.M. Anderson)

ENGINEERING STRAIN

• Tensile strain: • Lateral strain:

• Shear strain:θ/2

π/2

π/2 - θ

θ/2

δ/2

δ/2

δL/2δL/2

Lowo

ε = δLo

εL =−δL

wo

γ = tan θ Strain is alwaysdimensionless.

STRESS-STRAIN TESTING

• Typical tensile specimen

• Other types of tests:--compression: brittle

materials (e.g., concrete)--torsion: cylindrical tubes,

shafts.

gauge length

(portion of sample with reduced cross section)=

• Typical tensiletest 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.)

LINEAR ELASTIC PROPERTIES

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

• Hooke's Law:

σ = E ε• Poisson's ratio, ν:

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

εν = − Lε

εL

ε

1-ν

F

Fsimple tension test

σ

Linear- elastic

1E

ε

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

OTHER ELASTIC PROPERTIES

• Elastic Shearmodulus, G:

τ

1G

γτ = G γ

• Elastic Bulkmodulus, K:

P= -K∆VVo

P

∆V

1-K Vo

• Special relations for isotropic materials:

P

P P

M

M

G =

E2(1+ ν)

K =E

3(1− 2ν)

simpletorsiontest

pressuretest: Init.vol =Vo. Vol chg.= ∆V

YOUNG’S MODULI: COMPARISON

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

46

10

20

406080

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)*

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 Modulus: Typical Temperature Dependence

Melting Point of Al

More on Shear StressA0

A’

θθσθθ

θ

θτ

θ

θ

σ

cossincossin

cos

sin'

sin

cos'

00

0

0

==⎟⎠⎞

⎜⎝⎛

==

=

=

=

AF

AF

AF

FF

AA

AF

S

S

Schmidt Factor

For uniaxial deformation: maximum shear stress at 45°

Plastic Deformation

σy = Yield Strength(typically definedat ε = 0.002)

Discontinuous YieldingorYield Point PhenomenaDefine yield strengthat lower yield point

Typical metals: plastic deformation onset for strains > 0.005Typical σy: 35 MPa (soft Al); 1400 MPa (high strength steel)

Elastic Strain Recovery

σy0 = Yield Strength(typically definedat ε = 0.002)Initial loading

σyi = Yield StrengthAfter unloadingand reloading

Note: for many metals YS↑ after plastic deformation

HARDENING

• An increase in σy due to plastic deformation.

• Curve fit to the stress-strain response:

σ

ε

large hardening

small hardeningu

nlo

ad

relo

ad

σy 0

σy 1

σT = C εT( )n“true” stress (F/A) “true” strain: ln(L/Lo)

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

Tensile Strength

Tensile Strength TSMaximum stress

“Necking”

Typical TS for metals50 MPa (soft Al)3000 MPa = 450,000 psi (high strength steel)

Typical Stress-Strain Curves

TENSILE STRENGTH: COMPARISON

Room T valuesSi crystal

<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

Te

nsi

le s

tre

ng

th, T

S (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 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.

Brittle and Ductile Materials

Ductility

( )

( )100%

100%

0

0

0

0

×−

=

×−

=

AAA

RA

lll

EL

f

f

( )

( )100%

100%

0

0

0

0

×−

=

×−

=

AAA

RA

lll

EL

f

f

lf = length at fractureAf = section area at fracture

Typical Ductility for soft metals %EL25 – 75%

TOUGHNESS

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

curve.

smaller toughness- unreinforced polymers

Engineering tensile strain, ε

Engineering tensile stress, σ

smaller toughness (ceramics)

larger toughness (metals, PMCs)

True Stress and True Strain

( )( )εε

εσσ

ε

σ

+=+=

=

=

1ln1

ln0

T

T

iT

iT

ll

AF

True Stress: Load F divided by instantaneous section area Ai

True Strain: from integration: ∫=∫fl

l ldld

00

εε

HARDNESS

• Resistance to permanently indenting the surface.• Large hardness means:

--resistance to plastic deformation or cracking incompression.

--better wear properties.

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

Hardness

DESIGN OR SAFETY FACTORS

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

σworking =

σyN

Often N isbetween1.2 and 4

• Ex: Calculate a diameter, d, to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use a factor of safety of 5.

1045 plain carbon steel: σy=310MPa

TS=565MPa

F = 220,000N

d

Lo σworking =

σyN

220,000N

π d2 / 4⎛ ⎝ ⎜ ⎞

⎠ ⎟ 5