plastic deformation and creep in crystalline materials chap. 11
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Plastic deformation and
creep in
crystalline materials
Chap. 11
Mechanical Properties of Materials
Stiffness
Strength
ductility
Toughness
Resistance to elastic deformation
Young’s modulus
Resistance to plastic deformation
Yield stress
Resistance to fracture Energy to fracture
Ability to deform plastically
Strain to fracture
Uniaxial Tensile Test (Experiment 6)
Gaugelength
specimen
Result of a uniaxial tensile test
Slope = Young’s modulus (Y)
UTS
Ultimate tensile strength
yYield strength
(Engineering stress)
(engineering strain)
f (strain to fracture)
necking
Area = Toughness
elas
tic
plastic
break
Yield point
STIFFNESS
STRENGTH
DUCTILITY
If there is a smooth transition from elastic to plastic region (no distinct yield point)
then 0.2 % offset proof stress is used
During uniaxial tensile test the length of the specimen is continually increasing and the cross-sectional area is decreasing.
True stress ≠ Engineering stress (=F/A0)
True strain ≠ Engineering strain (=L/L0)
True stress
iT A
P Ai = instantaneous area
True incremental strain L
dLd T
0
ln0
L
L
L
dLL
L
T True strain
Eqn. 11.3
Eqn. 11.4
nTT K
K Strength coefficient
n work hardening exponent
Eqn. 11.5
What happens during plastic deformation?
• Externally, permanent shape change begins at sy
• Internally, what happens?
What happens to crystal structure after plastic deformation?
?Plastic
Deformation
Some Possible answers
Remains the
same
Changes to
another
crystal
structure
Becomes random
or
amorphous
How Do We Decide?
X-ray diffraction
No change in crystal structure!
No change in internal crystal structure but change in external shape!!
How does the microstructure of polycrystal changes during plastic deformation?
EXPERIMENT 5
Comparison of undeformed Cu and deformed Cu
Slip Lines
Before Deformation After Deformation
Slip lines in the microstructure of
plastically deformed Cu
Callister
Experiment 5
Slip
Slip Planes, Slip Directions, Slip Systems
Slip Plane: Crystallographic planes
Slip Direction: Crystallographic direction
Slip System: A combination of a slip plane and a slip direction
Slip Systems in Metallic Crystals
Crystal Slip Slip Slip Plane Direction Systems
FCC {111} <110> 4x3=12(4 planes) (3 per plane)
BCC {110} <111> 6x2=12
(6 planes) (2 per plane)
HCP {001} <100> 3x1=3
(1 plane) (3 per plane)
Why slip planes are usually close packed planes?
Why slip directions are close-packed directions?
Slip Systems in FCC Crystal
x
y
z(111)
Tensile vs Shear Stress
• Plastic deformation takes place by slip
• Slip requires shear stress
• Then, how does plastic deformation take place during a tensile test?
s
ND2
1
s
s: Applied tensile stress
N: Slip plane normal
D: Slip direction
F1: angle between s and N
F2 =angle between s and D
Is there any shear stress on the slip plane in the slip direction due to the applied tensile stress?
F
ND2
f1
F
Area=A
= F/ A
FD = F cos 2
Area = As
As = A cos 1
S
DRSS A
F
1
2
cos
cos
A
F
21 coscos A
F
21 coscos RSS
Resolved Shear stress
F
F
F
F
No resolved shear stress on planes parallel or perpendicular to the stress axis
cos 2 = 0 cos 1 = 0
Plastic deformation recap
No change in crystal structure:
slip
twinning
Slip takes place on slip systems (plane + direction)
Slip planes usually close-packed planes
Slip directions usually close-packed direction
Slip requires shear stress
In uniaxial tension there is a shear component of tensile stress on the slip plane in the slip direction:
RESOLVED SHEAR STRESS
21 coscos RSSRSS
21 coscos
y
CRSS
CRITICAL RESOVED SHEAR STRESS
21 coscos yCRSS
s
ND21
s
21 coscos yCRSS
If we change the direction of stress with respect to the slip plane and the slip direction cos 1 cos 2 will change.
1. CRSS changes.
To maintain the equality which of the following changes takes place?
2. y changes
Schmid’s Law: CRSS is a material constant.
Anisotropy of Yield Stress
21coscos crss
y
Yield stress of a single crystal depends upon the direction of application of load
cos 1 cos 2 is called the Schmid factor
RSS
bb
2coscos 1
y
CRSS
Active slip system
21 coscos yCRSS
aa
2coscos 1
Slip system with highest Schmid factor is the active slip system
Magnitude of
Critical Resolved Shear Stress
Theory (Frenkel 1926)
Experiment
b
d
CRSS
Shear stress
b/2 b
Potential energy
Fe (BCC)
Cu (FCC)
Zn (HCP)
Theory
(GPa)
12
7
5
Experiment
(MPa)
15
0.5
0.3
Ratio
Theory/Exp
800
14,000
17,000
Critical Resolved Shear Stress
?
1934
E. Orowan Michael Polanyi
Geoffrey Ingram Taylor
Solution
Solution
• Not a rigid body slip
• Part slip/ part unslipped
Slip Not-yet-slipped
Boundary between slipped and unslipped parts
on the slip plane
Dislocation Line (One-Dimensional Defect)
Movement of an Edge Dislocation
From
W.D. Callister
Materials Science
and Engineering
Plastic Deformation Summary
• Plastic deformation slip
• Slip dislocations
• Plastic deformation requires movement of dislocations on the slip plane
Recipe for strength?
Remove the dislocation
700
50
Stress, MPa
strain
Cu Whiskers tested in tension
Fig. 11.6
Effect of temperature on dislocation motion
Higher temperature makes the dislocation motion easier
WFe S
i
Al2O3
Ni
Cu
18-8 ss
Yie
ld s
tres
s
T/Tm0 0.7
Fig. 11.8
Eqn. 11.14
11.15
11.16
11.17
11.18
Recipe for strength
Remove the dislocation: Possible but Impractical
Alternative:
Make the dislocation motion DIFFICULT
Strengthening Mechanisms
• Strain hardening
• Grain refinement
• Solid solution hardening
• Precipitation hardening
Movement of an Edge Dislocation
A unit slip takesplace only whenthe dislocation
comes out of thecrystal
During plastic deformation dislocation density
of a crystal should go down
Experimental Result
Dislocation Density of a crystal actually goes up
Well-annealed crystal: 1010 m-2
Lightly cold-worked: 1012 m-2
Heavily cold-worked: 1016 m-2?
Dislocation Sources
F.C. Frank and W.T. Read
Symposium on
Plastic Deformation of Crystalline SolidsPittsburgh, 1950
A
B
P
Q
b
b
b
http://zig.onera.fr/~douin/index.html
b
http://zig.onera.fr/~douin/index.html
bb
Fig. 11.9
Problem 11.11
Strain Hardening or Work hardening
Strain, e
sy
sy
During plastic deformation dislocation density increases.
Dislocations are the cause of weakness of real crystals
Thus as a result of plastic deformation the crystal should weaken.
However, plastic deformation increases the yield strength of the crystal: strain hardening or work hardening
?
Dislocation against Dislocation
A dislocation in the path of other dislocation can act as an obstacle to the motion
of the latter
Strain Hardening
]110[2
1
)111(
]110[2
1
)111(
)001(
]011[2
1
]110[2
1
]011[2
1
Sessile dislocation in an FCC crystal
Eqn. 11.20
222
222 aaa
]110[
]110[2
1
(001) not a favourable slip plane (CRSS is high).
The dislocation immobile or sessile.
Energetically favourable reaction
Fig. 11.10
)111(
)111(
Sessile dislocation a barrier to other dislocations creating a dislocation pile-up
Piled up dislocations
Sessile dislocation (barrier)
Fig. 11.10
Empirical relation for strain hardening or work hardening
A 0
Is the shear stress to move a dislocation in a
crystal with dislocation density
o and A : empirical constants
Eq. 11.21
Fig. 11.11
Dislocation Motion Plastic Deformation
Difficult
Dislocation Motion
Difficult
Plastic Deformation
Strong Crystal
Easy Dislocation Motion Easy Plastic Deformation
Weak Crystal
Grain Boundary
Grain1
Grain 2
Grain boundary
2-D Defect: Grain Boundaries
Single Crystal Polycrystal
No Grain BoundariesGrains of different orientations separated by grain boundaries
Discontinuity of a slip plane across a grain boundary
Disloca-tion
Slip plane
Grain Boundary
Grain Boundary Strengthening
• Slip plane discontinuity at grain boundary
• A dislocation cannot glide across a grain boundary
• Higher stresses required for deformation• Finer the grains, greater the strength
Coarse Grains Fine Grains
Grain Size Strengthening
Hall-Petch Relation
ykD
0
sy: yield strengthD: average grain diameters0, k: constants
Science 5 April 2002: Vol. 296 no. 5565 pp. 66-67 POLYCRYSTALLINE MATERIALSGrain Boundaries and Dislocations
The hardness of coarse-grained materials is inversely proportional to the square root of the grain size. But as Van Swygenhoven explains in her Perspective, at nanometer scale grain sizes this relation no longer holds. Atomistic simulations are providing key insights into the structural and mechanical properties of nanocrystalline metals, shedding light on the distinct mechanism by which these materials deform.
I did not mention this in the class but in the interest of recent developments of nanotechnology I feel you should at least be aware of this:
• Mixture of two or more metals• Solute atoms: a zero dimensional defect or
a point defect• Two types:
– 1. Interstitial solid solution– 2. Substitutional solid solution
Solid Solutions
Interstitial Solid Solution
Perfect CrystalDistortion caused by a
large interstitial atom
Substitutional Solid Solution
Small solute atom
Large solute atom
Solute atom: a zero-dimensional point defect
Solid Solution Strengthening
Strains in the surrounding
crystal
Solute atoms
Obstacle to dislocationmotion
Strongcrystal
Alloys stronger than pure metals
Fig 11.13
Solute Concentration (Atom %) →
50
100
150
10 20 30 40
200
0
Matrix = Cu (r = 1.28 Å)Be (1.12)
Si (1.18)
Sn (1.51)
Ni (1.25)
Zn (1.31)
Al (1.43)
(Values in parenthesis are atomic radius values in Å)
Figure: Anandh Subramaniam
Airbus A380 to be launched on October 2007
A shop inside Airbus A380
Alfred Wilm’s Laboratory 1906-1909
Steels harden by quenching
Why not harden Al alloys also by quenching?
time
Wilm’s Plan for hardening Al-4%Cu alloy
Sorry! No increase in hardness.
550ºC
T
Hea
t
Quench
Hold
Check hardness
Eureka ! Hardness
has Increased !!
One of the greatest technological achievements of 20th century
Hardness increases as a function of time: AGE HARDENING
Property = f (microstructure)
Wilm checked the microstructure of his age-hardened alloys.
Result: NO CHANGE in the microstructure !!
As- quenched hardness
Hardness
time
Peak hardness
Overaging
Hardness initially increases: age hardening
Attains a peak value
Decreases subsequently: Overaging
+: solid solution of Cu in FCC Al
: intermetallic compound CuAl2 4
Tsolvus
supersaturated saturated +
FCC FCC Tetragonal
4 wt%Cu 0.5 wt%Cu 54 wt%Cu
Precipitation of in
Stable
unstable Tsolvus
As-quenched
start finsh
+
Aging
TTT diagram of precipitation of in
A fine distribution of precipitates in matrix causes hardening
Completion of precipitation corresponds to peak hardness
-grains
As quenched
-grains +
Aged Peak aged
Dense distribution of fine
overaged
Sparse distribution of coarse
Driving force for coarsening
/ interfacial energy
0.1 1 10 100
hardness
Aging time
(days)
180ºC
100ºC 20ºC
Aging temperature
Peak hardness is less at higher aging temperature
Peak hardness is obtained in shorter time at higher aging temperature
Fig. 9.15
U
I
T Stable
unstable
As-quenched
start finsh
+
Aging
Tsolvus
1
hardness180ºC
100ºC 20ºC
100 ºC
180 ºC
Hardness increases as as a function of time
No change in microstructure - Wilm!
hardness
time
As-quenched hardness
Guinier-Preston Zones, 1938
Numerous fine precipitates form with time
Not visible in optical micrograph
X-Ray Diffraction (XRD) Transmission Electron Microscopy (TEM)
“It seems justifiable at the moment to conclude that the process of age
hardening in this alloys is associated with the segregation of copper atoms on
the (100) planes of the crystal as suggested by C.H. Desch in The
Chemistry of Solids, 1934”
Preston, 1938, “The Diffraction of X-rays by Age-Hardening Aluminium Copper Alloys
Precipitation Hardening
Precipitates are obstacles to the motion of dislocation
Solute atoms Pebbles
Precipitates boulders
Cake with nuts
Age-hardening = Precipitation hardening
Dislocation-precipitate interaction
Dislocation can
1. Either cut through the precipitate particles (small precipitate)
2. Or they can bypass the precipitates
before after
Precipitate cutting
Fig 11.14 a, c
Dislocation bypassing the precipitate
Fig. 11.14 b and dL
b
Movement of one-dimensional defects called dislocations causes plastic
deformation
Obstacles to the movement of dislocations cause
strengthening
Strengthening Mechanisms
Name Obstacle Type
Solid solution hardening Solute atoms (0-D)
Strain hardening Dislocations (1-D)
Grain refinement Grain boundaries (2-D)
Precipitation hardening Precipitates (3-D)
95
Q1: How do glaciers move?
98
“Genius is one percent
inspiration and ninety-nine
percent perspiration”
-T.A. Edison
Q2: How do bulbs fuse?
2. Electric Bulb
99
Rolls-Royce Plc
Q3: What does the Rolls-Royce plc make?
101
Q: What is common to all the three?
Ans: CREEP
1. Glaciers move due to creep of snow.
2. Bulbs fuse due to creep of W filament.
3. Life of jet engine depends of creep of the turbine blades.
Creep
Creep is time dependent plastic deformation at constant load or stress
It is a “high temperature” deformation
mTT 4.0 Tm is the m.p. in K.
Difference between normal plastic deformation and creep ?
CREEP
Fig. 11.15
CreepDislocation climb
Vacancy diffusion
Cross-slip
Grain boundary sliding
Creep Mechanisms of crystalline materials
Cross-slip
In the low temperature of creep → screw dislocations can cross-slip (by thermal activation) and can give rise to plastic strain [as f(t)]
1 2
3
b Slip plane 1
Slip
plan
e 2
Dislocation climb
Edge dislocations piled up against an obstacle can climb to another slipplane and cause plastic deformation [as f(t), in response to stress]
Rate controlling step is the diffusion of vacancies
Diffusional creep
In response to the applied stress vacancies preferentially move from surfaces/interfaces (GB) of specimen transverse to the stress axis to
surfaces/interfaces parallel to the stress axis→ causing elongation This process like dislocation creep is controlled by the diffusion of
vacancies → but diffusional does not require dislocations to operate
Flow of vacancies
Coble creep → low T → Due to GB diffusion
Nabarro-Herring creep → high T → lattice diffusion
Grain boundary sliding
At low temperatures the grain boundaries are ‘stronger’ than the crystalinterior and impede the motion of dislocations
Being a higher energy region, the grain boundaries melt before the crystalinterior
Above the equicohesive temperature grain boundaries are weaker than grain and slide past one another to cause plastic deformation
110
Starter: initiates columnar grains as in Directional Solidification (DS)
Pigtail: a helical channel which gradually eliminates most columnar grains
Single crystal turbine blade
Single crystal blade: best creep resistance
111
Coarser grains -> Less grain boundaries-> Better for creep application
Single Crystal-> No grain boundaries-> Best for creep application
Nanocrystalline materials -> not good for creep applications!
112
Improvements due to blade manufacturing technique:
Show turbine blades
113
Improvements due to engineering design: Blade cooling
Engineering Materials 1: Ashby and Jones
114
NiCrAlY or NiCoCrAlY
Thermal Barrier Coating (TBC)
Ceramic top coat:
Yittria stabilized Zirconia (YSZ)
1. Low thermal conductivity
2. High thermal expansion
3. High M.Pwww.matsceng.ohio-state.edu
Reduction in surface temp 100-300 oC
Operating temp > M.P. (~1300 oC)
Creep Resistant Materials
Higher operating temperatures gives better efficiency for a heat engine
Creep resistance
Dispersion hardening → ThO2 dispersed Ni (~0.9 Tm)
Solid solution strengthening
High melting point → E.g. Ceramics
Single crystal / aligned (oriented) grains
Cost, fabrication ease, density etc. are other factors which determine the final choice of a material
Commonly used materials → Fe, Ni, Co base alloys
Precipitation hardening (instead of dispersion hardening) is not a good method as particles coarsen (smaller particles dissolve and largerparticles grow interparticle separation ↑)
Ni-base superalloys have Ni3(Ti,Al) precipitates which form a lowenergy interface with the matrix low driving force for coarsening
Cold work cannot be used for increasing creep resistance as recrystallization can occur which will produced strain free crystals
Fine grain size is not desirable for creep resistance → grain boundary sliding can cause creep elongation / cavitation► Single crystals (single crystal Ti turbine blades in gas turbine
engine have been used)► Aligned / oriented polycrystals
No Dislocations
Ultra Strong Crystals
Whiskers
Composite Materials
Various Crystal Defects
Disloca-tions
Grain Boundary
G-P zone
Substitu-tional solute
Interstitial solute
Stacking fault
Vacancy (Diffusion)
Moral of the Story
Strength depends upon defects
Microstructure
• Structural features observed under a microscope– Phases and their distribution– Grains and grain boundaries– Twin boundaries– Stacking faults– Dislocations
Hierarchy of Structures
en g in eerin g s tru c tu re
m ac ros tru c tu re
m icros truc ture
c rys ta l s tru c tu re
a tom ic s tru c tu re
n u c lear s tru c tu re
Physics and chemistry
Metallurgy and Materials Science
Engineering: Civil, Mechanical, etc. 1m
1mm
1mm
1nm
1A0
Real Moral of the Story
Properties depend upon microstructure
Structure Sensitive vs
Structure Insensitive Properties
For true understanding comprehension of detail is imperative. Since such
detail is well nigh infinite our knowledge is always
superficial and imperfect.
Duc Franccois de la Rochefoucald(1613-1680)
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