meng-4a

8/13/2019 MENG-4A

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Mechanical Properties

The adaptability of a material to a particular use is

determined by

its

mechanical

properties.

Properties are affected by (a) Bonding type; (b) Crystal

Structure; (c) Imperfections and (d) Processing

There are three types of deformation:

Elastic : Upon removal

of

the

applied

loads,

the

material

returns

to

its

original state.

Plastic : Upon removal of the applied loads, the material does not

return to its original state.

Fracture: Defines the separation of the material in two or more pieces

after loading.

Respond of a given material to applied loads. The

respond of the material to a given load and

loading conditions

is

known

as

deformation.

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Learning Objectives

Define engineering stress and engineering strain.

State Hooke’s law, and note the conditions under which it is

valid.

Given an

engineering

stress–strain

diagram,

determine

(a)

the modulus of elasticity, (b) the yield strength (0.002 strain

offset), and (c) the tensile strength, and (d) estimate the

percent elongation.

Name the two most common hardness‐testing techniques; note two differences between them.

Define

the

differences

between

ductile

and

brittle

materials. State the principles of impact, creep and fatigue testing.

State the principles of the ductile‐brittle transition

temperature.

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Types of Mechanical Testing

Slow application

of

stress

: Allows

dislocations

to

move

to

equilibrium positions : Tensile testing

Rapid application of stress : Ability of a material to absorb energy

as

it

fails.

Does

not

allow

dislocations

to

move

to

equilibrium

positions : Impact testing

Fracture Toughness : How does a material respond to cracks and

flaws

Fatigue : What

happens

when

loads

are

cycled?

High Temperature Loads : Creep

Mechanical Properties

StrengthTensile

Yield

Compression

Flextural

Shear Creep

Stress Rupture

ToughnessImpact Strength

Notch Sensitivity

Critical Stress Intensity Factor

StiffnessModulus of Elasticity

Shear Modulus

Flexural Modulus

Bulk Modulus

Poisson Ratio

DurabilityHardness

Wear Resistance

Fatigue Strength

FormabilityDuctility

%Reduction Area

Bend Radius

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Mechanical Tests:

Procedures that can be used to identify or measure mechanical

properties.

Example:

tensile

test,

hardness,

impact

test,

compression,

etc.

The relation between the microstructure and mechanical properties implies that in principle one can manipulate the material microstructure in order to produce new materials of desired mechanical properties (e.g. by control of the composition of alloys or of the grain size or by control of the volumetric percent and orientation of fibers

in

a

composite

material,

etc.).

We

will

learn

that

the

elastic

properties

are

almost structure insensitive, while the plastic and fracture properties are strongly structure dependent.

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Some Definitions

Tensile

stress:Where F : force, normal to

the cross‐sectional area,

A0: original cross‐sectional area

0 A

F =

Shear Stress

Fs: force, parallel to the cross-sectional

area

A0

: the cross-sectional area

unit of stress:0 A

F s=

2m

N

area

Force=

1Pa = 1 Nm

‐2

;

1MPa = 106Pa; 1GPa=109Pa

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Engineering Strain

Nominal tensile

strain

(Axial

strain)

00

0

l

l

l

l l Δ=

−=

Engineering Shear

Strain

For small strain:

θtan=

θ≅

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Poisson’s ratio

Nominal lateral strain

(transverse strain)

z

z

z l

l

0

Δ

=

x

x

x l

l

0

Δ

−=

Poisson’s ratio:

z

x

straintensile

strainlateral

ε

εν −=−=

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Dilatation (Volume strain)

Under pressure: the volume will

change

p

p p

p

V-ΔV

V

V Δ=Δ

Elastic Behavior of

Materials

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Hooke’s Law

When strains

are

small,

most

of

materials

are

linear

elastic.

σ

ε

E

Tensile: σ= Ε ε

Shear: τ= G γ

Hydrostatic: – p = κ

Young’s modulus

Shear modulus

Bulk modulus

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Modulus of Elasticity ‐ Metals

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Modulus of Elasticity ‐ Ceramics

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Modulus of Elasticity ‐ Polymers

PolymersElastic Modulus (GPa)

Polyethylene (PE) 0.2

‐0.7

Polystyrene (PS) 3‐3.4

Nylon 2‐4

Polyesters 1‐

5

Rubbers 0.01‐0.1

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Physical Basis of Young’s Modulus

Review: Inter

‐atomic

forces

(attractive

and repulsive forces) dxdU F =

Define: stiffness002

2

0 x x x x dx

dF

dx

U d

S == ==

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Assume the strain is small,

)(

)(

00

0

00

r r NS

A

F

r r S F

−==

−≈

σ

Where N: number of bonds/unit

area, N=1/r 02

σ σ

Unit area

0

0

0

0

0

0

0

0

0

0

)(

)(

r

S

E

E r S

r r r

r S

r

r r

==

==−=

−=

ε

σ

ε

εQ

Young’s modulus

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Stiffness & Young’s Modulus for different bonds

Bonding type S0(Nm

‐1

) E (GPa)

Ionic(i.e: NaCl) 8‐24 32‐96

Covalent

(i.e: C‐C)

50‐180 200‐1000

Metallic 15‐75 60‐300

Hydrogen 2‐3 8‐12

Van der Waals 0.5‐1 2‐4

Material E (GPa)

Metals: 60 ~ 400

Ceramics: 10 ~ 1000

Polymers: 0.001 ~ 10

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Tensile Testing• The sample is pulled slowly

• The

sample

deforms

and

then

fails• The load and the deformation are measured

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Standard tensile

specimen

The load and deformation are easily transform into

engineering stress (s) and engineering strain (e)

A

curve

stress‐

strain

is

obtained

0 A

F =

00

0

l

l

l

l l Δ=

−=

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Parameters Obtained From Stress Strain Curve

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Parameters Obtained From Stress Strain Curve

Strength Parameters

– Modulus of

Elasticity

– Yield Strength

– Ultimate Tensile

Strength – Fracture Strength

– Fracture Energy

Ductility Parameters

– Percent Elongation

– Percent Reduction of

Area

– Strain Hardening

Parameter

Elastic means

reversible!

F

δ

Linear-

elastic

Non-Linear-

elastic

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Modulus of Elasticity

Is a measure

of

material

stiffness and relates stress

to strain in the linear

elastic range.

12

12

ε−ε

σ−σ

=δε

δσ

= E

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Yielding and Yield Strength

• Proportionality Limit

(P):

Departure

from linearity of the stress‐strain

curve

• Yielding Point

– Elastic

Limit: the

turning point which separate the

elastic and plastic regions –onset of

plastic deformation

• Yield strength: the stress at the

yielding point.

• Offset

yielding (proof

stress):

if

it

is

difficult to determine the yielding

point, then draw a parallel line

starting from the 0.2% strain, the cross

point between

the

parallel

line

and

the σ−ε curve

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Tensile Strength (TS)

The

stress

increases

after

yielding

until

a

maximum

is

reached.

It

is

also known as the Ultimate Tensile Strength (UTS), or Maximum

Uniform Strength.

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• Prior to TS, the stress in the specimen is uniformly distributed.

• After TS, necking occurs with localization of the deformation

to the

necking

area,

which

will

rapidly

go

to

failure.

Fracture Strength

σf <<σUTS Due

to

the

definition

of Engineering stress and

tensile specimen necking

o

f

f

A

P =σ

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Fracture Energy (Toughness)

Is a measure

of

the

work

required to cause the

material to fracture.

Is a function

of

strength

and

ductility.

Its magnitude is defined by

the area

under

the

stress

strain curve

Approximated by: f

UTS ysG ε

σ+σ= *

2

∫=

f

d U

ε

ε σ 0

El i R

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Elastic Recovery

•After a load is released from a

stress‐strain

test,

some

of

the

totaleformation is recovered as

elastic deformation.

–During unloading,

the

curve

traces a nearly identical straight

line path from the unloading

point•parallel to the initial elastic

portion of the curve

–The recovered

strain

is

calculated as the strain at

unloading minus the strain after

the load

is

totally

released.

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Resilience

Resilience is

the

capacity

of

a

material

to

absorb

energy

when

it

is

deformed elastically and then, upon unloading, to have this energy

recovered.

∫= y d U r ε

ε σ 0

Modulus of resilience Ur

If it

is

in

a linear

elastic

region,

E E

U y y

y y yr

22

1

2

12

σ σ σ ε σ =⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ==

D ili

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Ductility

Ductility

is

a

measure

of

the

degree

of

plastic

deformation

at

fracture

– expressed as percent elongation

– also expressed as percent area reduction

– lO and AO are the original gauge length and original cross‐

section area

respectively

– lf and Af are length and area at fracture

100*)(%0

0

l l l f −=EL

100*)(%0

0

A A A f −=AR

Percentage elongation

and

percentage

area

reduction

are

UNITLESS

A smaller gauge length will produce a

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larger overall %elongation due to the

contribution from necking. Therefore

%elongation should

be

reported

with

original gauge length.

%Reduction is not affected by sample

size, thus

it

is

a better

measure

of

ductility

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Typical mechanical properties for some metals

and alloys

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• Nonlinear Elastic

Behavior

• Gray cast iron,

concrete, many

polymers

• Not possible to

determine a

modulus of

elasticity

– Either tangent

or secant

modulus is

normally used.

True Stress

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True stress is the stress determined by the instantaneous load acting

on the instantaneous cross‐sectional area.

Engineering stress‐strain curve beyond maximum point (M) seems to

indicate that the material is becoming weaker.

Not

true,

rather

it

becomes

stronger. Since cross‐sectional area is decreasing at the neck then it reduces

load bearing capacity of the material

True stress

is

related

to

engineering

stress:

Assuming material volume remains constant

A

A

A

P

A

A

A

P

A

P o

oo

o

T ** ===

σ ll

A A oo

=

)1(1 ε+=+δ=+δ==oo

o

o

o

A A

ll

l

ll )1()1( ε+σ=ε+=σ

o

T A P

True Strain

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The rate of instantaneous increase in the instantaneous gauge length.

The equations are valid only to the onset of necking; beyond this point

true stress and strain should be computed from actual load, area and

gauge length.

)1ln(

lnln

ln

ε ε

ε

ε

+=

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ Δ+⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ+=

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

== ∫

T

oo

o

o

o

T

o

i

T

d

l

l

l

l

l

ll

l

l

l

l

)1()1( ε+σ=ε+=σo

T A

P

Corrected takes into Corrected takes into

account complex stress account complex stress

state with in neck region.state with in neck region.

Strain Hardening Parameter

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Strain Hardening Parameter

(n)

Strain hardening

parameter

0<n<1

T

T

T

T

d

d n

ε

σ

ε

σ =

n

T T K ε σ =

True Stress Strain Curve

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True Stress-Strain Curve

σ = F/Ao ε = (li-lo/lo)

σT = F/Ai εT = ln(li/lo)

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Instability in Tension

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Instability in TensionNecking or localized deformation begins at maximum load, where the

increase in

stress

due

to

decrease

in

the

cross

‐sectional

area

of

the

specimen becomes greater than the increase in the load‐carrying

ability of the metal due to strain hardening.

This conditions of instability leading to localized deformation is defined

by the condition δ P = 0.

A P T σ = 0=+= T T A A P δσ δ σ δ

T

A

A

L

Lδε

δ δ =−= AL L AV oo ==

From the constancy‐of ‐volume relationship, T

T

A

A

σ

δσ δ =−

so that

at

the

point

of

tensile

instability

T

T

T σ δε

δσ =

T

n

T

n

T

T

T n

T T n K Kn K ε ε ε δε

δσ ε σ =⇒=== −1

Instability occurs when ε = nBut

The necking criterion can be expressed more explicitly if engineering

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g p p y g g

strain is used.

( ) T

O

o

T T

T

L

L

L L

L Lσ ε

δε

δσ

δε

δσ

δ

δ

δε

δσ

δε

δε

δε

δσ

δε

δσ =+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ === 1

/

/

ε δε

δ

+=

1

T σ σ

σT

ε

1 ε

Example: The stress‐strain curve below was obtained from a

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commercial steel with a 8mm diameter cross section. Calculate K and

n in the equationn

T T K ε σ =

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Ductile material Significant Fracture Behavior

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Ductile material – Significant

plastic deformation and

energy absorption

(toughness) before fracture.

Characteristic feature of

ductile material ‐ necking

Brittle material – Little

plastic

deformation

or

energy absorption before

fracture.

Characteristic feature

of

brittle materials – fracture

surface perpendicular to the

stress.

Fracture Behavior

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Steel

Before and

after

fracture

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Ductile Fracture (Dislocation Mediated): Extensive plastic

deformation Necking formation of small cavities enlargement of

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deformation. Necking, formation of small cavities, enlargement of

cavities, formation of cup‐and‐cone. Typical fibrous structure with

“dimples”.

(a)Necking,

(b)Cavity Formation,

(c)Cavity coalescence to

form a crack,

(d)Crack propagation,

(e) Fracture

Crack grows

90o

to

applied stress

45O‐ maximum

shear stress

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Scanning Electron Microscopy:

Fractographic studies at

high

resolution. Spherical “dimples”

correspond to micro‐cavities that

initiate crack

formation.

Brittle Fracture (Limited Dislocation Mobility): very little

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Brittle Fracture (Limited Dislocation Mobility): very little

deformation, rapid crack propagation. Direction of crack

propagation perpendicular

to

applied

load.

Crack

often

propagates by cleavage ‐ breaking of atomic bonds along

specific crystallographic planes (cleavage planes).

Brittle fracture in a mild steel

Intergranular fracture: Crack

ti i l i b d i

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Transgranular fracture: Cracks

pass through grains. Fracture

surface has

faceted

texture

because of different

orientation of cleavage planes

in grains.

propagation is along grain boundaries

(grain boundaries are weakened or

embrittled by impurities segregation

etc.)

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3‐point Bending tests

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3

22

3

R L F

bd

L F

f fs

f fs

π σ

σ

=

=

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Ductile Brittle

Torsion Test

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• Ductile material twist

• Brittle material

fractures

G φ

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G I

TL

P =φ L

Gr MAX

φ

τ =

Impact Test

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p

(testing fracture characteristics under high strain rates)

Notched-bar impact tests are used to measure the impact

energy (energy required to fracture a test piece under

impact load), also called notch toughness. It determines thetendency of the material to behave in a brittle manner.

Due to the non-equilibrium impact conditions this test will

detect differences between materials which are notobservable in tensile test.

We can compare the absorption energy capacity before

fracture of different materials.

Two classes of specimens have been standardized for

notched-impact testing, Charpy (mainly in the US) and Izod

(mainly in the UK)

Charpy v‐notch Test

A 10mm square section material

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A 10mm square section material

is tested,

having

a 45

o

notched, 2mm deep.

CharpyIzod

h’

h

Energy ~ h ‐

h’

The impact toughness is

determined from

finding

the

difference in potential energy

before and after the hammer

has fractured

the

material.

Units are J (Joules) when

testing Metals, J/cm2 when

testing polymers

(Polymers

will stretch, metals will snap).

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Impact Test Examples

Material Charpy Impact

Strength, in

Joules

Steel 20

Titanium 20Aluminum 14

Magnesium 6

Low‐Grade Plastic 4

Ductile‐to‐brittle transition

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As

temperature

decreases

a

ductile

material

can

become

brittle ‐

ductile‐to‐brittle transition.

FCC metals show high impact energy values that do not change

appreciably with changes in temperature.

BCC metals, polymers and ceramic materials show a transition

temperature below which the material behaves in a brittle

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temperature, below which the material behaves in a brittle

manner. The

transition

temperature

varies

over

a wide

range

of

temperatures. For metals and polymers is between ‐130 to 93oC.

For ceramics is over 530oC.

In low alloy and plain carbon steels, the transition temperature is

set to an impact energ of 20J or to the temperat re

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set to an impact energy of 20J or to the temperature

corresponding to

50%

brittle

fracture.

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