Figure 2-2 - University of Saskatchewankasap13.usask.ca/EE271/files/06_Mechanical-08.pdf ·...

Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet

Tensile stress - strain tests are carried out by applying a tensile load in a tensile test machine.

Figure 2-2

Test of mechanical properties

http://www.youtube.com/watch?v=E5-hwTspJK0&feature=related

http://www.youtube.com/watch?v=PaMnJxsV0qM

Tensile tests

http://www.youtube.com/watch?v=E5-hwTspJK0&feature=related

http://www.youtube.com/watch?v=PaMnJxsV0qM&eurl=http://www.metlab.fi/video/index.html

Elastic deformation

F=0

F=0

L0D0

Sample shape varies in reproducible way

Elastic deformation

LD L0D0

Sample shape varies in reproducible way

Plastic deformation

LD L0D0

The variation of shape becomes irreproducible

Necking

LD L0D0

The appearance of “neck”. The changes are irreproducible.

Fracture

LD L0D0

The specimen is broken at the neck

BANG!


= engineering/nominal

tensile stress

0AF

=σ

0

0

0 LLL

LL −=

Δ=ε

= engineering strain

Stress and strain


Tensile stress-strain tests are carried out by applying a tensile load in a tensile test machine

Figure 2-2

Strain

Stress

Fracture

Plastic deformation

Elastic deformation

Necking

0AFStress =

0

0

LLLStrain −

=


= engineering/nominal

tensile stress

= true tensile stress

0AF

=σ

AF

t =σ

0

0

0 LLL

LL −=

Δ=ε

= engineering strain

= incremental true strainLL

tδδε =

= true strain∫∫ +====L

L

L

Lt L

LLL

00

)1ln()ln(0

εδδεε

True stress and true strain


Strain

Stress

True stress-strain curve

True stress-strain in the neck

Onset of necking

Necking

Fracture

M′

M

F

F′

Y′

Y

A

O O′ B

σfσy

(0.2%)

σTS

σy

εpl εf0.0020

Typical engineering stress - engineering strain characteristics from a tensile test on a ductile polycrystalline metal (e.g. aluminum alloys, brasses, bronzes, nickel etc.)

Figure 2-3

= tensile strength

= yield strength

P

stress at fracture

= plastic fracture strain


Tensile stress-strain tests are carried out by applying a tensile load in a tensile test machine

Figure 2-2

Strain

Stress

Fracture

Plastic deformation

Elastic deformation

Necking

0AFStress =

0

0

LLLStrain −

=

Hooke’s law and Young’s modulusσ = stress, ε = strain, E= Young’s modulusσ = Eε

Poisson’s ratio

Poissson’s ratio v = 1/3 for metals and v > 1/3 for rubber and polymers

strainalLongitudinstrainLateralv

z

x

__

−=−=εε

0

0

0 DDD

DD

x−

=Δ

=ε ≡

lateral strain


= sheer stressAFt=τ

Θ≈Θ=Δ

= )tan(Lxγ = sheer strain

γτ G= = sheer modulus

Shear stress, strain and modulus


VVΔ

=γ = volume strain

γKP −= = bulk modulus

Volume strain and bulk modulus


Table 1-1

Types of Elastic Deformation


Table 1-3Relationship between Elastic moduli and Poisson’s ratio for homogeneous and isotropic materials, for example, polycrystalline solids.

KGE 91

311

+=


Table 1-2Typical values of elastic moduli, Poisson’s ratio and melting temperatures for a variety of materials.

r0

r

Uni

t are

a

σσ

Elastic deformation on atomic level

r0

r

Uni

t are

a

σσ

E0 –bond energyr0 – bond length

( ) ( )0002

2

0

rrSrrdr

EdFr

−=−=⇒

( )000

)( rrdrdFrFF

r

−+=

drdEF = } ⇒

The connection between Young’s modulus and atomic bonding

0

2

2

0rdr

EdS = spring constantwhere

20

1_

__rareaUnit

bondsofNumberN ==

( ) ( ) εσ0

0

0

0

0

02

0

00

rS

NrrrN

rS

NrrrNS

ANF

=−

=−

==

0

0

rSE =Young’s modulus

Microscopic approach

Macroscopic approach


Typical stress - strain characteristics of ductile, moderately ductile and brittle materials.

Figure 2-4

Comparison of brittle and ductile materials

Ductility = amount of plastic deformation that is exhibited by the material at fracture

%100%100 -

%0

0 ×=×= ff

LLL

EL ε

percent elongation

%100 -

%0

0 ×=A

AAAR f

percent area reduction

Strain

Stre

ss

Brittle fracture Ductile fracture

Ductile

Moderately ductileBrittle

εfεf

<2%

Fbrittlr M

F

Ductility is measured as


Strain

Definitions of (a) resilience and (b) toughnessFigure 2-3

Stress

M

F

B

σTS

εfStrain

Stress

Y′

Y

A

O

σy

(0.2%)

σy

0

= yield strength

εy εy(0.2%)

= tensile strength

Modulus of

resilience

B’

Toughness

= the total amount of work done per unit volume to fracture

Resilience and toughness

= the extent of elastic energy stored per unit volume in the solid


EE y

yyy

22

21

21

21 σ

εσε ===

Modulus of resilience =

εσ E=

dWvol - “elementary” workF - applied forcedL - “elementary” elongationA0 - sample cross-section Strain

Stress

Y′

Y

A

O

σy

(0.2%)

σy

0 εy εy(0.2%)

Resilience, yield strength and Young’s modulus

εσdLA

FdLdWvol ==00

εσε

dWvol ∫=0

L0 - sample lengthσ - stressε- strainE – Young’s modulus


Typical stress - strain characteristics of ductile, moderately ductile and brittle materials and comparison of their toughness.

Figure 3-2

Strain

Stress

00

A

BC

D

Comparison of toughness of brittle and ductile materials

Strength

Ductility

Toughness


Typical stress - strain characteristics of ductile, moderately ductile and brittle materials and comparison of their toughness.

Figure 3-2

Strain

Stress

00

A

BC

D

Comparison of toughness of brittle and ductile materials

Strength A < B < C < D

DuctilityD < C < A=B

ToughnessD < C ≈

A < B

Typical mechanical properties obtainable from a tensile stress

Table 3.1


We want to make a spring. Which material to chose?

Table 3-3Strength

classification of metal alloys

Strength

Ductility

Toughness

Atomic mechanisms of plastic deformation


= sheer stressAFt=τ

Θ≈Θ=Δ

= )tan(Lxγ = sheer strain

γτ G= = sheer modulus

Definition of shear stress

An applied tensile stress gives rise to shear stresses !

)2sin(21

0 θστ =

Plastic deformation caused by shear stress “under microscope”

Plastic deformation in Metals and Dislocations

Edge dislocation line

(a) Dislocation is a line defect. The dislocation shown runs into the paper.

CompressionTension

(b) Around the dislocation there is a strain field as the atomic bonds have beencompressed above and stretched below the islocation line

Fig. 1.46: Dislocation in a crystal is a line defect which is accompaniedby lattice distortion and hence a lattice strain around it.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Edge dislocation

A

D

B

C

Atoms in theupper portion.

Atoms in thelower portion.

Dislocationline

(b) The screw dislocation in (a) as viewed from above.

(a) A screw dislocation in a crystal.

A

C

D

Dislocation line

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Fig. 1.47: A screw dislocation involves shearing one portion of a perfectcrystal with respect to another portion on one side of a line (AB).

Screw dislocation

Plastic deformation is due to movement of dislocations


b – Burger’s vector

Slip direction, slip plane and Burger’s vector

τ

τ τ

τ


Contribution of screw dislocations into plastic deformation

z

z

Which crystalline structure is the most ductile? Slip planes and slip directions

The glide is easiest along the plane that is most densely packed !!!

critical shear stress

12

68Filling Factor, % 74 74

Interaction of dislocations

“Pinning” of dislocations on the grain boundaries and impurities

What can stop the movement of dislocations?

Interaction of dislocations

“Pinning” of dislocations on

the grain boundaries and

impurities

Strain (work) hardeningdue to plastic deformation of polycrystalline material


Temperature stimulates the movement of dislocations

Cold work

Cold work : yield strength tensile strength

resilience ductility


Practical realization of cold work


%100%0

0 ×−

=A

AACW Cold work is the percentage change in the cross- sectional area as a result of plastic deformation

Volume = const no change in material density

Numerical definition of cold work


Copper Brass

Iron Steel

Plastic deformation of polycrystalline material


)exp(0 RTH

RR recryst−=

where R- gas constant andT - absolute temperature

Recrystallization Rate

Trecryst ≅

0.4 T melting

Ductility

Yield strength

Recrystallization

Recrystallization temperature =recrystallization is complete in

1 hour


Trecryst ≅

0.4 T melting

<= above 4500C – hot working

Typical recrystallization temperatures

Impurity or Solid Solution Hardening

Dispersion or Precipitation Hardening

dK

yy +=0

σσ

where d- average grain size (diameter)σy0

,K -parameters of material

(Hall-Petch equation)

Grain Size Hardening

Strain Hardening (Cold Work)

Mechanical strengthening / hardening mechanisms


2

2

854.1

)21sin(2

_ dF

d

FAreaSurface

LoadVHN =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Θ

== where F is the load (kgf)d=(d1 +d2 )/2

is average diagonal (mm)Θ -apex angle

Vicker Hardness Number

Hardness = resistance to indentation or penetration

Indenter

Impression


( )⎥⎦⎤

⎢⎣⎡ −−

==22

2_ dDDD

FAreaSurface

FBHNπ

where F is the load (kgf)D is the diameter of the ball (mm)d is indentation number

Brinell Hardness Number

1000 kgf / 30 sec


Hardness and Strength

VHN ≅

3 σyVicker hardness

σTS ≅ 3.45 BHN Brinell hardness

[σTS ] = MPa [BHN] = kgf/mm2

t measured in function of load

Rockwell Hardness Test

Mechanical Failure


Strain

Stress

True stress-strain curve

True stress-strain in the neck

Onset of necking

Necking

Fracture

M′

M

F

F′

Y′

Y

A

O O′ B

σfσy

(0.2%)

σTS

σy

εpl εf0.0020

Typical engineering stress - engineering strain characteristics from a tensile test on a ductile polycrystalline metal (e.g. aluminum alloys, brasses, bronzes, nickel etc.)

Figure 2-3

= tensile strength

= yield strength

P

stress at fracture

= plastic fracture strain



Figure 2-4

Strain

Stress

Brittle fracture

Ductile fracture

Ductile


εfεf

<2%0

Fbrittle M

F

0

Two Mechanisms of Fracture : Brittle and Ductile Fracture


Figure 2-4Strain

Stre

ss

Brittle fracture Ductile fracture

Ductile


εfεf

<2%

Fbrittle M

F

Brittle Fracture. Stress amplification on the crack

0σσ tm K=Kt = strength concentration factor

σ0 = stress inside the volume

σm = maximum stress factor

Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet

rc

m 02σσ ≈

for sharp crack withc>>r

Brittle Fracture. Stress amplification on the crack

c = 0.2 μm or 200 nmr = 0.1 nm ⇒

Kt ≈

90Example:

cE

cb πγσ 2

=

where σcb = critical applied stress E = Young’s modulus

c = length of the crack

γ = surface tension (surface energy per unit surface area)

Critical applied stress

Griffith’s theory

where σcb = critical applied stress E = Young’s modulus

c = length of the crack

γ = surface tension (surface energy per unit surface area)

Crack propagation

cE

cb πγσ 2

=

Griffith’s theory

rc

m 02σσ ≈

for sharp crack withc>>r


Ductile fracture

τ >τcr


cE

cb πγσ 2

=

γ– surface tension(surface energy per unit

surface area)

Critical applied stressIn brittle material

cEGc

cd πσ 2

=

Gc– toughness(plastic work done per unit

surface area)

In ductile material

Mechanical experiments

tensile experiment = reaction under varying stressfatigue = reaction to cyclic stresscreep = reaction to constant applied stressimpact = reaction to impact


Fatigue = failure under cyclic stress


Endurance limit

Endurance limit = maximum amplitude of stress that can be cycledinfinitely without fracturing the specimen


Fatigue in Metals


Creep

Creep = slow permanent deformation with time under permanent load

Creep regions

Strain hardening

Creep regions

Strain hardening

Creep regions

Disentanglement and un-pinning of dislocations

Creep regions

Propagation of Cracks


Temperature dependence of creep rate

TB > TA


Stress dependence of creep rate

σB > σA


Impact Energy and Toughness

Impact Energy = the energy absorbed by a metal specimen to fracture in a standard impact test

Toughness = the total amount of plastic work done per unit volume to fracture


Impact Energy vs. Temperature

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