figure 2-2 - university of saskatchewankasap13.usask.ca/ee271/files/06_mechanical-08.pdf ·...
TRANSCRIPT
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
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!
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
= engineering/nominal
tensile stress
0AF
=σ
0
0
0 LLL
LL −=
Δ=ε
= engineering strain
Stress and strain
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
Strain
Stress
Fracture
Plastic deformation
Elastic deformation
Necking
0AFStress =
0
0
LLLStrain −
=
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
= 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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
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
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
= sheer stressAFt=τ
Θ≈Θ=Δ
= )tan(Lxγ = sheer strain
γτ G= = sheer modulus
Shear stress, strain and modulus
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
VVΔ
=γ = volume strain
γKP −= = bulk modulus
Volume strain and bulk modulus
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Table 1-1
Types of Elastic Deformation
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Table 1-3Relationship between Elastic moduli and Poisson’s ratio for homogeneous and isotropic materials, for example, polycrystalline solids.
KGE 91
311
+=
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
= 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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
b – Burger’s vector
Slip direction, slip plane and Burger’s vector
τ
τ τ
τ
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Temperature stimulates the movement of dislocations
Cold work
Cold work : yield strength tensile strength
resilience ductility
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Practical realization of cold work
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
%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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Copper Brass
Iron Steel
Plastic deformation of polycrystalline material
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
)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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
( )⎥⎦⎤
⎢⎣⎡ −−
==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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Essential Mechanical Properties (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Typical stress - strain characteristics of ductile, moderately ductile and brittle materials.
Figure 2-4
Strain
Stress
Brittle fracture
Ductile fracture
Ductile
Moderately ductileBrittle
εfεf
<2%0
Fbrittle M
F
0
Two Mechanisms of Fracture : Brittle and Ductile Fracture
Typical stress - strain characteristics of ductile, moderately ductile and brittle materials.
Figure 2-4Strain
Stre
ss
Brittle fracture Ductile fracture
Ductile
Moderately ductileBrittle
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
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Ductile fracture
τ >τcr
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Fatigue = failure under cyclic stress
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Endurance limit
Endurance limit = maximum amplitude of stress that can be cycledinfinitely without fracturing the specimen
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Fatigue in Metals
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Temperature dependence of creep rate
TB > TA
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Stress dependence of creep rate
σB > σA
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
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
Mechanical Properties II (© S.O. Kasap, 1990-2001: v.1.5) An e-Booklet
Impact Energy vs. Temperature