Dislocation Theory Dislocation Theory

Microsoft PowerPoint - Ch 5b. Dislocation theory•• Start with a brief introduction to crystal structure Start with a brief introduction to crystal structure and imperfections (defects)and imperfections (defects)
•• Good references are:Good references are: 1. Hull and Bacon1. Hull and Bacon 2. 2. WeertmanWeertman and and WeertmanWeertman 3. 3. HirthHirth and and LotheLothe (too detailed)(too detailed) 4. 4. FriedelFriedel 5. Cottrell5. Cottrell 6. 6. http://www.tf.unihttp://www.tf.uni--kiel.de/matwis/amat/def_en/index.htmlkiel.de/matwis/amat/def_en/index.html
Prof. Helmut Prof. Helmut FollFoll, University of Kiel , University of Kiel -- webweb
Plastic Deformation of Single CrystalsPlastic Deformation of Single Crystals τRSS = σ cosφ cosλ
Yielding occurs when τRSS >= τCRSS
•• Theoretical Yield Strength ~ G/2Theoretical Yield Strength ~ G/2ππ •• However, this stress value is far higher However, this stress value is far higher
than the experimental yield stress.than the experimental yield stress. •• Therefore, there must be some other Therefore, there must be some other
mechanism affecting the slip.mechanism affecting the slip. •• Thus, dislocation concept is Thus, dislocation concept is
introduced.introduced.
Copper
Nickel
* Important to note that bcc metals show wavy slip lines
b x
G is the shear modulus
b is the spacing between atoms in the direction of shear stress
a is the spacing between the two row of atoms
x is the shear translation
Theoretical shear strength
Stress to shear one plane of atoms with Stress to shear one plane of atoms with respect to adjacent parallel planerespect to adjacent parallel plane
This value is more than 2 orders of magnitude greater than the yield strength of metals
Theoretical Yield StrengthTheoretical Yield Strength (Why dislocations?)(Why dislocations?)
More rigorous analysis yields: τc = G/2π
While real materials yield at around G/1000 !!!
Deformation takes place by discrete movement of dislocations
Why Dislocations ?Why Dislocations ?
From Callister, Fundamentals of Materials Science and Engineering
Deformation by SlipDeformation by Slip •• Plastic deformation Plastic deformation
preserves the lattice preserves the lattice structure structure
•• Shearing of lattice Shearing of lattice planes against each planes against each other takes place other takes place (slip)(slip)
•• Slip occurs in Slip occurs in quantized multiples of quantized multiples of lattice vectorslattice vectors
Definition Definition -- dislocationdislocation •• A dislocation is defined as the line of A dislocation is defined as the line of
demarcation between slipped and demarcation between slipped and unslippedunslipped regions regions –– dislocation cannot end in a crystaldislocation cannot end in a crystal
•• VolterraVolterra & later & later OrowanOrowan, Polanyi and Taylor , Polanyi and Taylor (1930(1930’’s) independently put forth the s) independently put forth the dislocation conceptdislocation concept
Burgers vectorBurgers vector
•• A Burgers circuit is any A Burgers circuit is any atomatom--toto--atom path atom path taken in a crystal taken in a crystal containing dislocationscontaining dislocations
•• The circuit must pass The circuit must pass entirely through good entirely through good parts of the crystal parts of the crystal
•• The vector needed to The vector needed to complete the circuit is complete the circuit is the Burgers vectorthe Burgers vector
Vector of nonclosure of Burgers circuit
y (j)
x (i)
z (k)
Perfect crystal
W. soboyejo, Mechanical Properties of Engineered Materiasl
Edge dislocationEdge dislocation •• The Burgers vector (b) is perpendicular to The Burgers vector (b) is perpendicular to
the dislocation linethe dislocation line •• The Burgers vector (b) is parallel to the slip The Burgers vector (b) is parallel to the slip
directiondirection
Core ~ 5b
Screw dislocationScrew dislocation •• The Burgers vector (b) is parallel to the The Burgers vector (b) is parallel to the
dislocation line (t)dislocation line (t) •• The Burgers vector (b) is perpendicular to The Burgers vector (b) is perpendicular to
the glide directionthe glide direction
Screw DislocationsScrew Dislocations
As viewed from above
Top view – open (above slip plane Closed (atoms below)
Have both edge and screw componentsHave both edge and screw components
Dislocation LoopsDislocation Loops
quarter loop
half loop
full loop
EdgeEdge ScrewScrew Mixed Mixed
Motion of dislocationsMotion of dislocations
•• Glide Glide –– Pure edge dislocation glides in a Pure edge dislocation glides in a direction perpendicular to its length under direction perpendicular to its length under the application of a shear stressthe application of a shear stress
Motion of dislocationsMotion of dislocations
Glide Glide –– Pure edge Pure edge dislocation glides in dislocation glides in a direction a direction perpendicular to its perpendicular to its length under the length under the application of a application of a shear stressshear stress
Conservative motion
From Callister, Fundamentals of Materials Science and Engineering
Edge
Screw
Non-conservative motion
•• Climb Climb –– Edge dislocations can move from one Edge dislocations can move from one plane to another plane lying above by climb. plane to another plane lying above by climb. It is a diffusion based process and occurs at high It is a diffusion based process and occurs at high temperatures (creep)temperatures (creep)
•• Generally observed when the dislocation Generally observed when the dislocation experiences an obstacle to its motion in its slip experiences an obstacle to its motion in its slip planeplane
Dislocation ClimbDislocation Climb
Change of glide plane
⊥ Characteristics Edge Screw
Relation between t and b perpendicular parallel Slip Direction parallel to b parallel to b
Change of glide plane climb cross-slip Direction of ⊥ motion relative to b parallel perpendicular • Direction of ⊥ motion relative to t perpendicular perpendicular
Narrow Narrow vsvs Wide DislocationsWide Dislocations
τP-N = 2G 1-ν exp(-
2πw b ) where w=
Peierls-Nabarro Stress
Mixed dislocationsMixed dislocations
•• Have both edge and screw componentsHave both edge and screw components •• Most of the dislocations encountered in real Most of the dislocations encountered in real
crystals are of this typecrystals are of this type
Glide Loop
Prismatic Loop
•• Surface method (Etch Pits)Surface method (Etch Pits)
•• Decoration methodDecoration method
Surface MethodSurface Method
Reference – Gilman, Johnston and Sears, J. Appl. Physics, 29, 747-754,1958
Dislocations intersecting the surface etch at a different rate than the surrounding matrix.
If the etching is fast they appear as pits, else they are observed as small hillocks
Decoration of dislocationsDecoration of dislocations By suitable heat treatment precipitation of foreign atoms can be induced along the dislocations
e.g. Dislocations in KCl are revealed by adding AgCl to the melt prior to crystal growth.
Adjacent optical micrograph shows silver particles decorating dislocations in KCl
Reference – Amelinckx, Acta Metall., 6, 34, 1958
Transmission Electron MicroscopyTransmission Electron Microscopy •• Most widely used techniqueMost widely used technique •• Uses the principle of diffractionUses the principle of diffraction •• For perfectly flat TEM specimen free of defects, For perfectly flat TEM specimen free of defects,
the image is homogeneous with no variations in the image is homogeneous with no variations in intensity intensity
•• Strain field around the dislocation contributes to Strain field around the dislocation contributes to its detection in TEMits detection in TEM
Velocity of dislocationsVelocity of dislocations •• The velocity of a dislocation is dependent on The velocity of a dislocation is dependent on
a) applied shear stressa) applied shear stress b) purity of crystalb) purity of crystal c) temperaturec) temperature d) type of dislocationd) type of dislocation
Dislocation DensityDislocation Density ρ is the total length of dislocations per unit volume (m-2)
p.123
Plastic strain (Plastic strain (εε)) due to dislocation motiondue to dislocation motion ((OrowanOrowan Equation)Equation)
xb Lh
i
N
i
ργ
γ
x Lh b Lh xb
h N
Plastic strain (Plastic strain (εε)) and dislocation and dislocation motion (motion (OrowanOrowan Equation)Equation)
•• Dislocation motion causes plastic strainDislocation motion causes plastic strain
vbρε =
xbρε =
is the plastic strain
mρ is the mobile dislocation density
vx , are the average distance traveled by the dislocation and the average velocity, respectively
b is the Burgers vector of the dislocation
ε
Dislocations in Dislocations in REALREAL CrystalsCrystals
fcc
bcc
hcp
εij or eij
Elastic Stresses around dislocationsElastic Stresses around dislocations (screw dislocation)(screw dislocation)
•• Strain fieldStrain field
•• Stress field of a screw dislocationStress field of a screw dislocation
•• No tensile or compressive componentsNo tensile or compressive components
•• Field exhibits complete radial symmetryField exhibits complete radial symmetry
z
==
Stresses around Edge DislocationStresses around Edge Dislocation
•• The stress field has both shear and The stress field has both shear and dilatational componentsdilatational components
)( 22
Screw dislocations 1. No normal stresses
2. Only shear stresses in the glide plane
Edge dislocations 1. In the glide plane (y=0): only shear stresses exist
2. For y>o, σx<0 – compression & for y<0, σx>0 – tension (see figure)
x
y
Elastic Strain Energy Density Elastic Strain Energy Density of a dislocationof a dislocation
F
lengthunit per 2GbE α=⊥
Elastic strain energy of a dislocation is about 3 eV for each atom plane threaded by the dislocation
Or The energy required to create 1 cm long dislocation line is more than 3 MeV
This value is >>> kT and thus dislocations cannot be created by thermal energy (dislocations do not occur in thermal equilibrium)
* contrast with vacancies *
•• bb11 bb22 + b+ b33 & dislocation dissociation is & dislocation dissociation is energetically favorable if benergetically favorable if b11
22 > b> b22 22 + b+ b33
22
When Burgers vector addition or dissociation is When Burgers vector addition or dissociation is made, each of the components need to be added made, each of the components need to be added or subtracted separately or subtracted separately –– conservation of b.conservation of b.
A dislocation with a unit strength has a minimum A dislocation with a unit strength has a minimum energy when Burgers vector is parallel to the closeenergy when Burgers vector is parallel to the close-- packed directions packed directions ---------- in FCC b is along <110>in FCC b is along <110>
InIn--Class WorkClass Work (dislocation reactions)(dislocation reactions)
Show that the reaction is valid. (Vectorially correct and energetically favorable)
InIn--Class WorkClass Work
b. What are the line vectors of these dislocations if they are edge type?
Y
2121 )()( llll fb =τ
Work done by stress: (τ l1 l2)b Work by resistive force: (f l1)l2 Force per unit length,
Applied shear stress, τ exerts a force on the dislocation Work done by the shear stress is equated to the work done by the frictional/resistance force (principle of virtual work) f is force per unit length
Forces on dislocationsForces on dislocations (due to applied stress)(due to applied stress)
General Equation for Force on a Dislocation per unit length is given by Peach-Koehler Formula
f = τbi
Forces on dislocations due to Forces on dislocations due to external stressesexternal stresses
(using Peach(using Peach--Koehler formula)Koehler formula)
•• Edge dislocation (bi, k) due to Edge dislocation (bi, k) due to σσxyxy (same as done earlier)(same as done earlier)
•• Edge dislocation (bi, k) due to Edge dislocation (bi, k) due to σσxxxx
•• Screw dislocation (Screw dislocation (bkbk, k) due to , k) due to σσxyxy
•• Screw dislocation (Screw dislocation (bkbk, k) due to , k) due to σσxzxz
•• Screw dislocation (Screw dislocation (bkbk, k) due to , k) due to σσxxxx
Comment on how may the dislocation move
SummarySummary Effects of External Stresses on DislocationsEffects of External Stresses on Dislocations
•• Normal stresses (Normal stresses (σσijijδδijij) have no influence on Screw ) have no influence on Screw dislocations; i.e. only shear stresses have forces on them dislocations; i.e. only shear stresses have forces on them ------ onlyonly ττijij’’ss on the glide plane of the dislocation have nonon the glide plane of the dislocation have non--zero zero forces : if b=forces : if b=bkbk and t=k, thenand t=k, then ττxyxy has no effecthas no effect
•• Normal stresses on edge dislocations induce forces Normal stresses on edge dislocations induce forces perpendicular to the glide/slip planeperpendicular to the glide/slip plane ------ dislocation has to climbdislocation has to climb ((nonconservativenonconservative process)process)
•• A A glide loopglide loop expands or contracts under the appropriate expands or contracts under the appropriate applied shear stress applied shear stress
Forces between dislocations
y
x
r
Forces between dislocationsForces between dislocations •• Screw DislocationsScrew Dislocations
Like dislocations repel each other Like dislocations repel each other whereas unlike dislocations, i.e., of the opposite sign whereas unlike dislocations, i.e., of the opposite sign attract each otherattract each other
•• Edge DislocationsEdge Dislocations (t along z(t along z--axis)axis) YY--force force –– repulsion (like dislocations) and attraction repulsion (like dislocations) and attraction (unlike dislocations)(unlike dislocations) XX--force force –– attraction for x<y and repulsion for x>yattraction for x<y and repulsion for x>y
X-force
θ b D
LATB Ti3Al2.5V
Force on a curved dislocationForce on a curved dislocation
•• A dislocation wants to be straight to minimize line tensionA dislocation wants to be straight to minimize line tension
•• A force is required to keep a dislocation curved (R):A force is required to keep a dislocation curved (R):
F
Origin and Multiplication of dislocationsOrigin and Multiplication of dislocations
•• Dislocations arise during growth of crystal from melt or Dislocations arise during growth of crystal from melt or vapor phase due to gradients of temperature and vapor phase due to gradients of temperature and composition.composition.
•• Emission from grain boundaries is an important source of Emission from grain boundaries is an important source of dislocations in the early stages of plastic deformationdislocations in the early stages of plastic deformation
•• Heterogeneous nucleation of dislocations is possible from Heterogeneous nucleation of dislocations is possible from second phase particlessecond phase particles
FrankFrank--Read MechanismRead Mechanism
FrankFrank--Read MechanismRead Mechanism •• Frank Read mechanism Frank Read mechanism –– The scheme by The scheme by
which dislocations could be generated from which dislocations could be generated from existing dislocationsexisting dislocations
•• The minimum shear stress required for the The minimum shear stress required for the operation of a Frank Read source is given byoperation of a Frank Read source is given by
l Gb
Dislocation PileDislocation Pile--upsups •• Dislocations generated from a FrankDislocations generated from a Frank--Read source pile up Read source pile up
at barriers such as grain boundaries, second phases and at barriers such as grain boundaries, second phases and sessile dislocations sessile dislocations
•• This leads to a high stress concentration on the leading This leads to a high stress concentration on the leading dislocation in the piledislocation in the pile--upup
•• The stress concentration is relieved by plastic The stress concentration is relieved by plastic deformation deformation
Partial DislocationsPartial Dislocations Perfect vs Imperfect Dislocations
FCC
Partial (Imperfect) dislocations
Shockley Partials in FCCShockley Partials in FCC •• The {111} slip plane in F.C.C has a ABCABCAThe {111} slip plane in F.C.C has a ABCABCA…… type of type of
stackingstacking •• During slip an A, B or a C layer will move to occupy During slip an A, B or a C layer will move to occupy
another of a similar position in the direction <110>another of a similar position in the direction <110> •• Sometimes this movement may not be energetically Sometimes this movement may not be energetically
favorablefavorable •• Decrease in energy due to dissociation should be greater Decrease in energy due to dissociation should be greater
than the interfacial energythan the interfacial energy ABCABCACABCABC
Stacking Fault
Frank partials in F.C.C.Frank partials in F.C.C. •• The Burgers vector is 1/3[111]The Burgers vector is 1/3[111]
(perpendicular to the closed packed plane)(perpendicular to the closed packed plane)
•• A disk of vacancies condenses on one A disk of vacancies condenses on one of the planes to form a closed of the planes to form a closed dislocation loop of Frank partialdislocation loop of Frank partial
•• It is a sessile dislocation because It is a sessile dislocation because Burgers vector is perpendicular to the Burgers vector is perpendicular to the plane of the loopplane of the loop
Fig. 5-11
Shockley and Frank Partial DislocationsShockley and Frank Partial Dislocations
Stacking Fault WidthStacking Fault Width (Stacking Fault Energy)(Stacking Fault Energy)
The 2 partials repel each other while the stacking fault energy, γ, provides a force γ per unit length pulling the dislocations together so that the stacking fault width , w:
πγ4
2Gbw = ABCABCACABCABC
Stacking Fault
Higher the γ, smaller the stacking fault width Al (high γ) – no stacking faults
Cu, SS (low γ) – many stacking faults ---------------------
BCC – high γ – no stacking faults
Note: stacking faults make materials stronger -difficult for the extended dislocation to slip –
Ex. Cu+Al
]110[ 2 1]111[
→+
Examples: Irradiated SS and after heat treatment Quenched Al-Mg and after annealing
After Quenching Al3.5Mg After Annealing
Similar to radiation exposure in stainless steel followed by annealing
See #6 in Class-work
Burgers vector of the intersecting Burgers vector of the intersecting dislocationdislocation
•• A A jogjog forms when the step is out of the forms when the step is out of the slip planeslip plane
•• A A kinkkink is a sharp break in the dislocation is a sharp break in the dislocation line which line which remains in the slip planeremains in the slip plane
Jog
Intersection of 2 Edge dislocationsIntersection of 2 Edge dislocations •• 2 Edge dislocations with 2 Edge dislocations with
perpendicular Burgers vectors perpendicular Burgers vectors (Fig. 5(Fig. 5--19) 19) –– jog formationjog formation
•• 2 Edge dislocations with parallel 2 Edge dislocations with parallel Burgers vectors Burgers vectors (Fig. 5(Fig. 5--20) 20) –– kink formationkink formation
Intersection of an Edge Dislocation with a Intersection of an Edge Dislocation with a Screw Dislocations Screw Dislocations (Fig. 5(Fig. 5--21 & 521 & 5--22)22)
Jogs (PP’ & QQ’)
Jogged Screw Dislocation


+


→


−−−
• HCP
• BCC
]011[]101[]101[ 222 aaa =+
plane glide (100) with ]110[ along liesn dislocatioproduct The #2 – class-work Lomer-Cottrell Lock
LomerLomer--Cottrell Lock Cottrell Lock -- FCCFCC ((StairStair--rod dislocation)rod dislocation)
partials attached with 1]1[0 along [011] b n withdislocatio a form react tolock Cottrell-Lomer in the partials Leading
6 a=
Slip Systems: {110}<110>
like atoms)