Lectures 7 and 8 Dislocations: Foundations of Plastic Deformation ME 330 Engineering Materials Please read Chapters 4 and 7 Dislocation definitions Dislocation movement Slip systems Yield and Plastic Deformation Dislocation stress fields Dislocation multiplication Twinning Diffusion Please read Chapter 5

Linear Defects - Dislocations Most insightful argument for why theoretical strength is too high Introduce extra half plane of atoms Bond breakage restricted to vicinity of extra plane (dislocation line, ) Bonds break consecutively instead of simultaneously Upper portion translated relative to lower –Same result as before with less energy Responsible for plastic deformation Interactions cause strain hardening From: Callister b

Basic Definitions For Edge and Screw Dislocations Burger’s Circuit Burger’s Vector Slip plane Dislocation Line Edge Screw Slip direction Applied Shear stress Slip direction Applied Shear stress

Dislocation Movement in Crystals Shear stresses cause dislocations to move Dislocations move on slip planes, –Generally along close packed planes Plane with highest density of atoms Dislocations move in slip directions, –Generally in close packed directions Direction on slip plane in which atoms are most closely spaced –Energy arguments will show, smallest Burger’s vector minimizes dislocation energy The combinations of planes and directions are called slip systems

Slip Systems 6 close packed planes {110} 2 close packed directions 12 slip systems BCC 1 close packed planes {0001} 3 close packed directions 3 slip systems HCP 4 close packed planes {111} 3 close packed directions 12 slip systems FCC (48 possible have been identified also possible on {112},{123} planes) BCC, FCC: ductile HCP: brittle

From Mohr’s circle, we know that pure tensile loading results in shear stresses in non-perpendicular planes Onset of plastic deformation takes place when shear stress on given slip plane (with normal oriented at  ) and slip direction (oriented at ) reaches the critical value. Resolved Shear Stress Slip Systems in Single Crystals AoAo n A slip plane  Slip Direction P resolved PP P,,,,

In metals, there are often many operative slip systems –Slip will usually occur on most favorably oriented plane and direction : –When this shear reaches a critical value, (  crss ) slip commences. –Thus, yield occurs (and plastic flow begins) when, –Slip & plastic deformation may occur simultaneously along several, equivalently oriented systems n A slip plane Yield in Single Crystals  AoAo Slip Direction P resolved PP P,,,, From: Callister, p.161

Many grains, random orientations –Each grain will have favored orientation &  crss –Slip orientations vary between grains –Grain boundaries act to impede dislocation motion Plastic deformation corresponds to grain deformation –Prior deformation, grains are equiaxed –Boundaries force continuity –Grains become deformed Polycrystalline stronger than single crystal - geometrical constraints Yield & Plastic Deformation in Polycrystalline Materials From: Hertzberg, p.78 From: Callister

Slip Plane grain2 Grain Boundary Slip Plane grain1 Grain Boundaries Dislocations have difficulty crossing grain boundaries Stresses become so high that a new dislocation can form on an adjacent slip plane in the next grain Dislocations begin to pile-up at the grain boundary From: Hertzberg, p.71

Stress Fields Tension Compression C T Extra half plane of atoms cause lattice distortions Result in tensile, compressive, and shear strains in neighboring atoms –Magnitude decreases with distance – Pure compression and tension directly above and below slip line –Over most of the effected region combination of stresses Screw dislocation –Pure shear –Simple math

Elastic Properties of Dislocations Screw Dislocations For small strains Shear strain: From Hooke’s law: Elastic strain energy: Similarly for edge dislocation For our purposes, use:  is geometrical factor

Dislocation Energetics Want to be in lowest possible energy state –When “far” apart, Burger’s vectors have no influence on each other –When together, there will be interaction between the two Attract each other if Burger’s vectors cancel Repel if Burger’s vectors are of same sign T C C T T C C T Apart Together T C T C C T C T Apart Together

Dislocation-Dislocation Interactions   perfect crystal   void Immobilized Dislocations Repelled   C C T T  C C  T T Dislocations Annihilated

Why Close Packed Systems? Energy Argument for Preferred Slip Systems Example: Look at slip in an FCC crystal To move towards new atomic position – of {111}, move distance a/2, in x- & y-direction, 0 in z- direction – of {001}, move distance a in y-direction, 0 in x- & z-direction Energy of resultant dislocation – a x y z

Partial Dislocations Energy Argument for Existence A single dislocation can dissociate into two “partial” dislocations if: Alternately, two can combine to one only if: Example: Look at slip in an FCC crystal a {111} slip plane

–Slip can occur by the Burgers vector b 1 = a/2(1,1,0) –Or by two partials: b 2 =a/6(2,1,1) and b 3 =a/6(1,2,-1) – where b 1 =b 2 +b 3 Motion of edge dislocation in FCC

Strengthening Mechanisms Introduce barriers to dislocation movement –Increase dislocation density –Pin dislocations with solutes –Second phases –Grain boundaries Strengthening comes from stress field interactions between dislocations –Higher density leads to more repulsions –Larger stress needed to move dislocations through lattice

Dislocation Multiplication  Pinning points - dislocation, solute particles Frank-Reed source  Dislocation loops

Twinning Annealing twins form in FCC crystals Mechanical twinning occurs only in BCC and HCP crystals High rates of loading –Mechanical shock –Impact –Slip is restricted Bulk plastic deformation small compared to slip Reorients crystal planes –May put new slip systems in favorable orientations AAA BBB CCC AAA BBB AAA BBB CCC BBB AAA AAA BBB AAA BBB AAA AAA BBB AAA BBB AAA AAA BBB BBB AAA AAA From: Hull & Bacon, p.15 From: Callister

E F F Updating the Model  (MPa)  (%) E

New Concepts & Terms Yield and Plastic Deformation –Resolved shear stress –Single crystal vs. polycrystalline –Grain boundary effects – Dislocation stress fields –Annhilation –Repulsion –Immobilization Dislocation multiplication –Frank-Reed source –Dislocation loops Twinning Dislocation definitions –Burger’s circuit –Burger’s vector –Slip –Dislocation line –Slip plane –Dislocation density Dislocation movement –Positive & negative edge dislocation –Right & left-hand screw dislocation –Glide –Climb Slip systems –Slip plane –Slip direction

Dislocation website Simulations of Farid Abraham of IBM Almaden Research, in collaboration with LLNL personel Mark Duchaineau and Tomas Diaz De La Rubia.

“The phenomenon of material transport by atomic motion” Step-wise migration of atoms between lattice sites Impurity diffusion: atoms of different species interdiffuse Self-diffusion: diffusion in pure metals Example of diffusion: Diffusion couple Diffusion CuNi Pure Cu Pure Ni Cu-Ni Alloy Heat below T m for “long” time From: Callister p. 94

Vacancy Diffusion Mechanisms Conditions for migration –Open site available –Energy Break current bonds Cause lattice distortions Vacancy Diffusion –Move to vacant lattice site –Function of number of available vacancies –Impurities may substitute for hosts Interstitial Diffusion –Move to vacant interstitial position –Usually for small impurities –Often more rapid than vacancy mode Interstitials are smaller More open site Interstitial atom Vacancy Interstitial atom

Diffusive Time Dependence Diffusion flux (J): rate of mass transfer M: mass or number of atoms diffusing A: cross-sectional area Steady-state: constant with time –Concentration profile is linear –Concentration gradient is constant From: Callister p. 96

Time Varying Diffusion (Nonsteady-state) Most common in engineering sense More difficult to solve Fick’s second law: Solution will vary with time and position Must specify boundary conditions and remember differential equations Note: At long times –Concentration profile becomes ~linear –Near steady-state

Diffusive Temperature Dependence Diffusion coefficients are strongly temperature dependent Q d : activation energy for diffusion R: gas constant T: absolute temperature Second point to remember Arrhenius rate equation –Creep –Oxidation –Corrosion Multiplication of Bacteria Spoilage of Milk Diffusion

Next... Strengthening Mechanisms –Grain boundary Refinement –Solid-solution Hardening –Precipitation Hardening –Cold Work Annealing Please read chapters