# Plastic deformation and creep in crystalline materials

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Plastic deformation and creep in crystalline materials
Chap. 11

Mechanical Properties of Materials
Stiffness Resistance to elastic deformation Young’s modulus Strength Resistance to plastic deformation Yield stress Toughness Resistance to fracture Energy to fracture ductility Ability to deform plastically Strain to fracture

Uniaxial Tensile Test (Experiment 6)
Gauge length specimen

Result of a uniaxial tensile test
STRENGTH  (Engineering stress) necking Ultimate tensile strength UTS Yield point plastic Yield strength y break elastic Area = Toughness STIFFNESS Slope = Young’s modulus (Y) DUCTILITY f (strain to fracture)  (engineering strain)

If there is a smooth transition from elastic to plastic region (no distinct yield point) then 0.2 % offset proof stress is used

Ai = instantaneous area
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 Ai = instantaneous area Eqn. 11.3 True incremental strain True strain Eqn. 11.4

Eqn. 11.5 K Strength coefficient n work hardening exponent

What happens during plastic deformation?
Externally, permanent shape change begins at sy Internally, what happens?

What happens to crystal structure after plastic deformation?

Some Possible answers Becomes random Remains the Changes to or same
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

Before Deformation After Deformation
Slip Lines Before Deformation After Deformation

Slip lines in the microstructure of plastically deformed Cu
Callister Slip lines in the microstructure of plastically deformed Cu 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
(111) z y x

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 s s: Applied tensile stress N: Slip plane normal D: Slip direction N
1 D 2 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? s

F F Resolved Shear stress  = F/ A Area=A FD = F cos 2 N
As = A cos 1 f1 D 2 Area = As F

F F No resolved shear stress on planes parallel or perpendicular to the stress axis F F 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

CRITICAL RESOVED SHEAR STRESS
N D 2 1

Schmid’s Law:  CRSS is a material constant.
If we change the direction of stress with respect to the slip plane and the slip direction cos 1 cos 2 will change. To maintain the equality which of the following changes takes place? 1.  CRSS changes. 2. y changes Schmid’s Law:  CRSS is a material constant.

Anisotropy of Yield Stress
Yield stress of a single crystal depends upon the direction of application of load cos 1 cos 2 is called the Schmid factor

Active slip system Slip system with highest Schmid factor is the active slip system

Magnitude of Critical Resolved Shear Stress
Theory (Frenkel 1926) Experiment

Potential energy Shear stress  CRSS b/2 b b d

Critical Resolved Shear Stress
Theory (GPa) 12 7 5 Experiment (MPa) 15 0.5 0.3 Ratio Theory/Exp 800 14,000 17,000 Fe (BCC) Cu (FCC) Zn (HCP)

?

Geoffrey Ingram Taylor
Solution 1934 E. Orowan Michael Polanyi Geoffrey Ingram Taylor

Solution Not a rigid body slip Part slip/ part unslipped

Boundary between slipped and unslipped parts on the slip plane
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

Remove the dislocation
Recipe for strength? Remove the dislocation

Cu Whiskers tested in tension
Stress, MPa Fig. 11.6 700 50 strain Cu Whiskers tested in tension

Effect of temperature on dislocation motion
Higher temperature makes the dislocation motion easier W Fe Si Al2O3 Ni Cu 18-8 ss Yield stress Eqn 11.15 11.16 11.17 11.18 Fig. 11.8 T/Tm 0.7

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 takes place only when the dislocation comes out of the crystal

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 Solids Pittsburgh, 1950

P b A b B b Q

b

b b Fig. 11.9 Problem 11.11

Strain Hardening or Work hardening
sy sy Stress, s=Force/Initial Area Strain, e=change in length/initial length Strength Parameters Yield Stress, sY= Stress at yield point Ultimate tensile strength, sUTS= stress at maximum Ductility Parameter % Elongation= 100 X Strain at fracture, ef Strain, e

? 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
Strain Hardening Dislocation against Dislocation A dislocation in the path of other dislocation can act as an obstacle to the motion of the latter

Sessile dislocation in an FCC crystal
] 1 10 [ 2 Energetically favourable reaction ] 1 [ 2 (001) not a favourable slip plane (CRSS is high). The dislocation immobile or sessile. Eqn ] 1 [ 2 ) 1 ( ] 1 10 [ 2 ) 11 1 ( ) 001 ( Fig

Sessile dislocation a barrier to other dislocations creating a dislocation pile-up
Sessile dislocation (barrier) ) 1 ( ) 11 1 ( Fig Piled up dislocations

Empirical relation for strain hardening or work hardening
Eq  Is the shear stress to move a dislocation in a crystal with dislocation density  o and A : empirical constants

Fig

Easy Dislocation Motion Easy Plastic Deformation
Weak Crystal Difficult Dislocation Motion Difficult Plastic Deformation Strong Crystal

Grain Boundary Grain 2 Grain1 Grain boundary

2-D Defect: Grain Boundaries
Single Crystal Polycrystal Grains of different orientations separated by grain boundaries No Grain Boundaries

Discontinuity of a slip plane across a grain boundary
Disloca-tion 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 sy: yield strength D: average grain diameter s0, k: constants

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: 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. Science 5 April 2002: Vol. 296 no pp POLYCRYSTALLINE MATERIALS Grain Boundaries and Dislocations

Solid Solutions 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

Interstitial Solid Solution
Distortion caused by a large interstitial atom Perfect Crystal

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 dislocation motion Strong crystal Alloys stronger than pure metals

Figure: Anandh Subramaniam
200 Sn (1.51) Be (1.12) Matrix = Cu (r = 1.28 Å) 150 Si (1.18) Al (1.43) (Values in parenthesis are atomic radius values in Å) 100 Ni (1.25) Zn (1.31) 50 10 20 30 40 Solute Concentration (Atom %) → Figure: Anandh Subramaniam Fig 11.13

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Alfred Wilm’s Laboratory 1906-1909
Steels harden by quenching Why not harden Al alloys also by quenching?

Eureka ! Hardness has Increased !!
T Wilm’s Plan for hardening Al-4%Cu alloy Hold 550ºC Heat Quench Check hardness time Sorry! No increase in 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 !!

Hardness initially increases: age hardening
Peak hardness Hardness Overaging As- quenched hardness time Hardness initially increases: age hardening Attains a peak value Decreases subsequently: Overaging

: solid solution of Cu in FCC Al +
Tsolvus : solid solution of Cu in FCC Al + : intermetallic compound CuAl2 4 supersaturated saturated  Precipitation of  in  FCC FCC Tetragonal 4 wt%Cu 0.5 wt%Cu 54 wt%Cu

TTT diagram of precipitation of  in 
Stable  Tsolvus  start unstable   finsh + As-quenched  Aging 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

Aging temperature hardness 100ºC 20ºC 180ºC Fig. 9.15 Aging time 0.1 1 10 100 (days) Peak hardness is less at higher aging temperature Peak hardness is obtained in shorter time at higher aging temperature

T U + I 1  start  finsh 180 ºC 100 ºC Aging hardness Stable 
Tsolvus  start unstable   finsh + 180 ºC 100 ºC As-quenched  Aging I 1 hardness 180ºC 100ºC 20ºC

Hardness increases as as a function of time
As-quenched hardness No change in microstructure - Wilm!

Numerous fine precipitates form with time
Not visible in optical micrograph X-Ray Diffraction (XRD) Transmission Electron Microscopy (TEM) Guinier-Preston Zones, 1938

“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 Either cut through the precipitate particles (small precipitate) Or they can bypass the precipitates

Precipitate cutting before after Fig a, c

Dislocation bypassing the precipitate
Fig b and d

Obstacles to the movement of dislocations cause strengthening
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)

Q1: How do glaciers move?

2. Electric Bulb “Genius is one percent inspiration and ninety-nine percent perspiration” -T.A. Edison Q2: How do bulbs fuse?

Rolls-Royce Plc Q3: What does the Rolls-Royce plc make?

Ans: CREEP Q: What is common to all the three?
Glaciers move due to creep of snow. Bulbs fuse due to creep of W filament. Life of jet engine depends of creep of the turbine blades.

Creep Creep is time dependent plastic deformation at constant load or stress Difference between normal plastic deformation and creep ? It is a “high temperature” deformation Tm is the m.p. in K.

CREEP

Fig

Creep Mechanisms of crystalline materials
Cross-slip Dislocation climb Creep Vacancy diffusion Grain boundary sliding

b 1 2 3 Slip plane 2 Slip plane 1 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 plane 2

Dislocation climb Edge dislocations piled up against an obstacle can climb to another slip plane and cause plastic deformation [as f(t), in response to stress] Rate controlling step is the diffusion of vacancies

Nabarro-Herring creep → high T → lattice diffusion
Diffusional creep Coble creep → low T → Due to GB diffusion 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

Grain boundary sliding
At low temperatures the grain boundaries are ‘stronger’ than the crystal interior and impede the motion of dislocations Being a higher energy region, the grain boundaries melt before the crystal interior Above the equicohesive temperature grain boundaries are weaker than grain and slide past one another to cause plastic deformation

Single crystal blade: best creep resistance Pigtail: a helical channel which gradually eliminates most columnar grains Starter: initiates columnar grains as in Directional Solidification (DS)

Coarser grains. -> Less grain boundaries
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!

Improvements due to blade manufacturing technique:

Improvements due to engineering design: Blade cooling
Engineering Materials 1: Ashby and Jones

Thermal Barrier Coating (TBC)
NiCrAlY or NiCoCrAlY Ceramic top coat: Yittria stabilized Zirconia (YSZ) Low thermal conductivity High thermal expansion High M.P Reduction in surface temp oC Operating temp > M.P. (~1300 oC)

Creep Resistant Materials
Higher operating temperatures gives better efficiency for a heat engine High melting point → E.g. Ceramics Dispersion hardening → ThO2 dispersed Ni (~0.9 Tm) Creep resistance Solid solution strengthening Single crystal / aligned (oriented) grains

Cost, fabrication ease, density etc. are other factors which determine
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 larger particles grow  interparticle separation ↑) Ni-base superalloys have Ni3(Ti,Al) precipitates which form a low energy 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
Substitu-tional solute Stacking fault G-P zone Disloca-tions Interstitial solute Vacancy (Diffusion) Grain Boundary

Strength depends upon defects
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
Physics and chemistry 1A0 1nm Metallurgy and Materials Science 1mm 1mm Engineering: Civil, Mechanical, etc. 1m

Properties depend upon microstructure
Real Moral of the Story Properties depend upon microstructure Structure Sensitive vs Structure Insensitive Properties

Duc Franccois de la Rochefoucald (1613-1680)
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 ( )