UNIT 2. Elements of applied physical metallurgy. UNIT 2a Elements of mechanical metallurgy

Transcription

1 UNIT 2 Elements of applied physical metallugy UNIT 2a Elements of mechanical metallugy

2 The poject of Unit 2 ( x t) F, system of (genealised) foces applied to the object Initial state macoscopic object M ( x t) F, state at time t macoscopic object M

3 The defomation of the object at time t can be descibed with the displacement field ( x t) S, Ou poblem is to fomalise the elationship between the input F ( x, t) and the output S ( x, t) mediated by the object M. We can expess this elationship with the opeato αˆ Thus ou poject can be fomalised as: ( x, t) = ˆ α{ F( x t) } S,

4 The opeato αˆ can be specified with a model of the natue of M. The typical hieachy of models is epesented by the following list: 1) Continuum, homogeneous, elastic (isotopic o anisotopic) (ealm of continuum mechanics) 2) Continuum, homogeneous, plastic (isotopic o anisotopic) (ealm of continuum mechanics) 3) Continuum, heteogenous elastic (ealm of composite mechanics) 4) Continuum, heteogenous plastic (ealm of mechanical metallugy) - plasticity of pefect single cystals (heteogeneous owing to dislocations = line defects) - plasticity of single cystals with defects (heteogeneous owing to point & line defects) - plasticity of single-phase polycystals (pesence of gain boundaies & 1D and 2D defects) - plasticity of multi-phase polycystals (diffeent dimensions of gains of each phase) 5) Discontinuous (ealm of factue mechanics) Of couse: S( x t) ˆ α F( x, t) { } σ( x) = Eε( x) with : const, = 1 E = That, fo a 1D poblem, simplifies to: σ = Eε Case (2) in 1D geomety is the usual uniaxial plastic flow cuve: α ˆ2 = E ( ) ε

5 Pat 1: cystals and polycystals

6 COMMENTS ON THE KEY POINTS OF THE FIRST COURSE IN METALLURGY Chapte 1 (A vey naïve account of the) Atomistic backgound 1.1) Fom the metallic bond accoding the jellium model to cystals - cation assembly and electon jam - Aufbau of an Alkaline (s electons) metal (hinting at enegy levels) - Lennad-Jones-type foce fo the dime (see next slide) and extension to n-atom case - cohesion in metals (explanation on 2 nd next slide) with the vey cude (indeed) assumptions that: (i) the coes ae fully sceened, (ii) the electon density is low (n small and 1-2 >-R) and (iii) the electon-to-coe distance is ca. costant: -R s a/2 (with a intenuclea distance), one can wite: negative means that the Aufbau tansfomation is spontaneuos

7

8 electostatic epulsion equilibium position pue electostatic attaction

9 time 2D lattice tetame 3D lattice

10 Electical enegy: electon-cation attactive (<0) cation-cation epulsive (>0) electon-electon epulsive (>0) electon density: jam is a continuum again, jam is a continuum cations ae discete

11 n.b. 1: this is the enegy keeping the metal togethe(e el <0). n.b. 2: the shape of the cation assembly depends of E el. i.e.: the stuctue is detemined by the sepaation of the ions. n.b. 3: elesticity is elated to cohesive enegy a (poof on next slides) n.b. 4: Lennad-Jones potentials justify compessive behaviou n.b. 5: one can ty and pedict the UTS fom E el, but this yields a damatic oveestimate eal metals fail in a diffeent way (see below). n.b. 6: metal oxidation pocesses can be natually incopoated into the jellium model

12 Elesticity is elated to cohesive enegy: (toy-)poof Above, we have shown that the electonic enegy of the metal cystal is: If the cystal is stained in a lattice diection (called axial ), leaving the coodinated ones unaffected, E el enegy becomes a function of the vaiable axial lattice spacing. In the usual way, one can define an electic foce elated F el to the electic enegy E el.

13 stess can be defined as: the meaning of F=-F el is that the stessing foce balances the cohesion foce Now, we ae inteested in Young s modulus E Y, that can be defined as: we then need dσ and dε. dσ can be deived by applying the 1 st vaiation opeato to the above expession fo σ : while, by definition: ode-of-magnitude estimate eventually:

14 Space tassellation with full coveage Bavais lattices

15

16 Take-home message of analysis of mechanical popeties in tems of pefect single cystal (without any kind of lattice defect): Quantitative desciption of mechanical popeties of cystalline mateials equies ingedients beyond pefect lattice geomety and cohesion enegy Poject: povide a continuum desciption incopoating paametically infomation about defects

17 Fist appoach to defect theoy: the enegetic scenaio at a cystal suface

18 2.1 - POINT DEFECTS: self-defects & heteo-element defects vacancy intestitial self heteo substitutional intestitial

19 A bette view of heteo-point defects (e.g. solid solutions) e.g.: Substitutional Fe Ni, Si Intestitial C, N, H, O, B Thei location within the lattice can be undestood on the basis of an E el -type of appoach

20 Themal disode bings about an equilibium concentation of point self-defects. Cu K N S ( T ) eq = [ S] [ C] N V N S Point defects (both self- and heteo-) can be viewed as: a diffeent mateial in equilibium with nomal mateial

21 As a diffeent mateial, the defect can undego mass-tanspot, typically by diffusion: E act : enegy fo a jump (go back to E el ) In an Ahenius famewok:

22 Complex defects, including both self- and heteo-point defects (aggagating in paticula combinations) Case of a 2-cation (A and B) lattice Fenkel: one cation of a specific type (say A) is dislocated yielding an A-vacancy and an A-intestitial. Schottky: coupled A-B vacancy.

23 Knowing souce (eaction) and mass-tanspot (flux) infomation in point defects, we can set up balance equations B geneated (algebaically) B in B out C t = φ φ + in out S Fo: Fickian diffusion & monomolecula eaction C 2 t = D C + S ( C)

24 2.2 - LINE DEFECTS (DISLOCATIONS): (a) GEOMETRY The bounday between slipped and unslipped egions Dislocation (ethymologically) line: atoms on half-plane α, displaced (slipped) 1 atomic distance in slip diection atoms on half-plane β, not yet slipped α β Fist of two basic types Dislocation in a slip plane Enegetics: fo a fist appoximation efe to cystal of cubes Edge dislocation poduced by slip in SC lattice. Dislocation lies along line AD, to slip diection.

25 Second of two basic types Scew dislocation poduced by slip in SC lattice: slip has occued ove aea ABCD. Dislocation lies along line AD, // to slip diection. Atomic aangement aound scew dislocation.

26 Two basic types (in fact: limiting cases) Macoscopic defomation of a cube poduced by glide of: edge (above) and scew (below) dislocations. n.b.: the same type of defomation is obtained by glide of both types of dislocations.

27 Obseving dislocations Slip lines: (Cu) 200 µm

28 20 µm Etch pits on slip bands in bass Slip lines: (Cu)

29 20 µm 2.5 µm Hexagonal netwok of dislocations in AgCl by decoation technique Netwok of dislocations in cold-woked Al (TEM)

30 Elementay dislocation enegetics (below, moe details) (a) Atom movements in the neighbouhood of a dislocation in slip (b) Movement of an edge dislocation Enegy change fom unslipped to slipped state Model: plastic defomation as tansition fom unslipped (highe-enegy owing to accumulation of elastic enegy) to slipped (lowe-enegy owing to elease of elastic enegy) state. Pocess opposed by enegy baie E, located at the tansfomation Inteface (that can be intepeted as the dislocation).

31 Assuming a sinusoidal potential pofile (esulting fom a L-J-type model, ecall τ max estimate fo pefect lattice): distance between slip planes distance between atoms in slip plane = a E b a b Stages in gowth of slipped egion Shea stess equied to move a dislocation: τ exp a b

32 Dislocations can be chaacteised/quantified by Buges vectos edge scew i) Conside and atom-to-atom cicuit aound the slip featue. ii) Join each atom along the cicuit with a vecto: iii) The vecto sum of such joining vectos is the Buges vecto: Special cases: i) No slip: ii) Edge dislocation: iii) Scew dislocation: notable beaing on disclocation movement though lattice

33 Buges vecto notation Diection indicated by diection indices and magnitude esulting fom coefficient and the diection indices as illustated below Pemise - diection indices: taslations paallel to the cystal axes yielding the desied diection [a 1 a 2 a 3 c] examples of hexagonal lattice

34 e.g. example in cubic lattice in a cubic cystal means that the Buge vecto has components of 2a/3, -a/3 and a/3 along the [100], [010] and [001] diections, espectively, with a lattice paamete and magnitude: y z x

35 Specific types of movement fo edge and scew dislocations EDGE DISLOCATION: glide in slip plane, but: unde appopiate conditions it can move out of the slip plane onto a paallel plane diectly above o below it: climb. diffusion of vacancy to edge dislocation dislocation climbs up one lattice spacing diffusion-contolled pocess

36 dislocation line Buges vecto

37 SCREW DISLOCATION: glide in slip plane and coss slipping Since the line of a scew dislocation and b ae //, b not define a specific plane as with the edge dislocation (whee b is to the dislocation line): Hence, to a scew dislocation, all diections aound its axis look the same and it can glide on any plane as long as it moves // to its oiginal oientation. pat of a scew dislocation line AB coss slipping fom the pimay slip plane PQ into the plane RS

38 Reactions between dislocations (see above limiting cases ) Couples of dislocations can combine and fom a thid dislocation. The Buges vecto of the poduct dislocation will be the vecto sum of the Buge vectos of the eacting dislocations. The eaction takes places if it is enegatically favouable, i.e. if it lowes the enegy of the system: below we shall pove that: Theefoe the eaction is enegetically favouable if: The notion of disclocation eactions puts on a quantitative basis the intepetation of edge and scew dislocations as limiting cases of actual dislocations that will exhibit mixed chaacteistics.

39 Since dislocations descibe slip and slip is a popety of a cystalline object, pefeential slip planes and diections fo dislocations will be pesent within a cystal that is pefect apat fom the pesence of dislocations (the stoy will be diffeent in the pesence of hindances to slip), depending on the specific cystal geomety. Since (see above): τ exp a b density b τ high-density planes will tend to be pefeential slip planes and high-density diections, slip diections. The ensemble of the pefeential slip planes (i.e. the numbe of slip planes multiplied by the numbe of slip diections) fo a paticula cystal is denominated slip system peculia to that cystal stuctue.

40 A pactically impotant example of eaction between dislocations: patial dislocations in FCC cystals In FCC cystals slip occus on {111} planes ({111} denotes the ensemble of all (111)-type planes) in 110 diections ( 110 denotes the ensemble of all [110]-type diections). Conside the specific case of displacement: A dislocation with this Buges vecto can dissociate into two patial dislocations, accoding to the eaction: a 6 2 a 6 2 [ ] [ ] [ ] = 0.07a < 110 = 0.35a a 2 2 dissocation eaction: enegetically favouable

41 Patial dislocations in FCC cystals lead to fomation of stacking faults If a single (a/6) 112 patial dislocation passes though an FCC cystal, 112 it leaves behind a egion in which the sequence of stacking of the close-packed {111} planes does not coespond to the nomal FCC lattice: In nomal stacking, the thid plane is ove neithe the fist, no the second one. A (a/6) 112 patial dislocation changes the position of the thid plane so that it ends up diectly ove the fist one: this is in fact the HCP stacking of {111} planes and, since the natual aangement in FCC is diffeent, this fault yields a fault enegy

42 Stacking of atomically compact planes and faulted stuctues Pefect FCC packing Stacking fault in FCC Pefect HCP packing Digession: DEFORMATION BY TWINNING Twin in FCC Scheme of defomation by twinning

43 Again: Patial dislocations in FCC cystals lead to stacking faults Looking down ion (111) along [1 1-1]: Dissociation of a dislocation into two patial dislocations in an FCC cystal Pefect dislocation line having the full slip vecto b 1.

44 Dealing with defects in a continuum famewok Alias: enegetics of dislocations scew dislocation edge dislocation

45 Fundamentals of Continuum Desciption Balance of Bulk and Suface Foces A) No stuctue (no dependence on oientation of object) hydostatic equilibium p p( x ) p = ρg ρg bulk foce (e.g. gavitational) A

46 B) With stuctue (dependence on oientation) equilibium of defomation of solids σ p( x;suface oientation) f i f esultant of bulk foces σ = f In geneal: a system of equations; stuctue system B

47 C) Just one unknown in balance equation displacement u σ ( u) f ( = u) Simplest option: linea elasticity u x i i def = ε i f i σ i ( ε ( u )) i i = E ε ( u ) ρ i + F ( u ) i 2 t u = 2 i ij i i = E ij u x i i inetial static C

48 D) Appoximations: (i) elastostatic, (ii) without bulk foces f i 2 u = ρ i + F 2 i t 0 (i) (ii) 3 equations in 6 unknowns σ x τ σ xy y τ xz τ yz σ z constitutive equations: links among stains due to mateial popeties (ν) ({ σ } ν ) σ = Oˆ ; ij kl D

49 E E) 6 6 PDE system & ways to solve it ({ σ } ν ) σ = Oˆ ; ij kl F) Ways to solve this PDE system: a convenient tansfomation of unknowns, the Beltami tenso Φ Φ Φ xx yx zx Φ Φ Φ xy yy zy Φ Φ Φ xz yz zz such that: F

50 G G) Ways to solve this PDE system: a special case of the Beltami tenso, the Maxwell stess function Φ Φ Φ zz yy xx H H) Ways to solve this PDE system: a special case of the Maxwell stess function, the Aiy stess function ( ) = Φ y x f zz, Suitable fo two-dimensional poblems.

51 2.2 - LINE DEFECTS: (b) ENERGY PREMISE: ELEMENTS OF ELASTICITY THEORY The key poblem of elasticity is: to calculate the defomation field (o, via the constitutive equations, the stess field) pevailing in an (elastic) body, esulting fom the application of a system of foces. This is, of couse, the key enegetic poblem with dislocations. compession zone dislocation tension zone

52 i) 1 st pemise: Hydostatic equilibium Conside a column of wate in hydostatic equilibium. All the foces on the wate ae in balance and the wate is motionless. On any given dop of wate, two foces ae balanced. The fist is gavity, which acts diectly on each atom and molecule inside. The gavitational foce pe unit volume isρg, whee g is the gavitational acceleation. The second foce is the sum of all the foces exeted on its suface by the suounding wate. The foce fom below is geate than the foce fom above by just the amount needed to balance gavity. The nomal foce pe unit aea is the pessue p. The aveage foce pe unit volume inside the doplet is the gadient of the pessue, so the foce balance equation is: A

53 ii) 2 Pemise: equilibium of defomable solids The momentum balance equations (in paticula thei static vesion) can be extended to moe geneal (than fluids) mateials, including solids. Fo each suface with nomal in diection i and foce in diection j, thee is a stess componentσ ij. The nine components make up the Cauchy stess tensoσ, which includes both pessue and shea. The local equilibium is expessed by: whee f is the body foce and σ epesent suface foces. B

54 iii) The geneal poblem of linea elasticity C

55 C bis

56 iv) The linea elastostatic poblem without body foces and with an isotopic homogeneous medium in stess fomulation Stess fomulation means that the suface tactions ae pescibed eveywhee on the suface bounday a) Equation of motion In the case of inteest only suface foces ae pesent, thus F x,f y,f z =0 c) Constitutive equations D n.b.: Combining the stain-displacement equations and the constitutive equations, one gets: the Beltami-Michell compatibility equations

57 The Beltami-Michell compatibility equations in engineeing notation E

58 In the case of inteest, the linea elasticity poblem thus educes to: Equilibium equations: Compatibility equations (i.e. stain-displacement equations, combined with constitutive equations): E bis

59 It can be shown hat a complete solution to the equilibium equations may be witten as: F whee Φ mn is an abitay second-ank tenso field that is continuously diffeentiable at least fou times, and is known as the Beltami stess tenso. The Maxwell stess functions ae defined by assuming that the Beltami stess tenso is esticted to be of the diagonal fom: whence: G

60 The Aiy stess function is a special case of the Maxwell stess functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stess function can theefoe be used only fo two-dimensional poblems. The stess function C is customaily epesented by φ H In pola coodinates the expessions ae:

61 NOW WE HAVE ENOUGH BACKGROUND TO GO BACK TO DISLOCATION PROBLEMS a) Enegetics of a scew dislocation. Conside the ight-handed scew dislocation with the geomety depicted below, and imagine that the cylindical egion is a potion of an infinite continuum: u z (θ=0) In cylindical coodinates, the displacements ae: u z (θ=2π)

62 Plugging the diplacement infomation into the elastostatic equations, we get: the stain field of a scew dislocation. Using the constitutive equations (with µ denoting the shea modulus) on gets the stess field of a scew dislocation: The elastic enegy of a scew dislocation can be computed as follows:

63 d b Elastic enegy pe unit length: ( ) ( ) da U R z o = ϑ σ 2 1 b ( ) ( ) ( ) ln b R b d b d b da U o R R z R z o o o = = = = π µ π µ σ σ ϑ ϑ ( ) ( ) d b A b A 2 d 2

64 Comments on the coe cutoff U = 1 2 R o σ ϑz R ( ) da( ) o d coe cutoff (avoids coe divegence) o ~atomic scale (e.g. o =b/)

65 b) Enegetics of an edge dislocation. Conside the edge dislocation with the geomety depicted below and again imagine that the cylindical egion is a potion of an infinite continuum: The displacements ae: This is a plane-stain poblem, that can be attacked with the Aiy function fomalism. The appopiate Aiy function is:

66 Plugging this Aiy function into the equilibium and compatibiilty equations: ( ) ( ) ( ) b b b θ ν π µ θ ϕ σ θ ν π µ ϕ σ θ ν π µ θ ϕ ϕ σ θ θθ cos sin 1 2 sin = = = = = + = Whence the axial stess can be deduced:

67 Hence, a hydostatic stess exists aound an edge dislocation: This can be expessed moe eicastically in catesian coodinates: That is plotted below:

68 Elastic enegy pe unit length: U σ ( ) da( ) R 1 = ϑz 2 o b

69 A R bl R ( ) δ L = b L da( ) d b R/R=b L δ=b /R

70 ( ) ( ) ( ) ( ) ( ) ( ) o R R R R R L b d R L b d R bl da U o o o ln cos = = = = = ν π µ ϑ ν π µ σ σ ϑ ϑ fo a cystallite exhibiting the shape of a ight cylinde 2R=L: ( ) 2 2 ln 1 2 b R b U o = ν π µ

71 Mixed dislocations, compound Buges vectos and the notion of dislocation loop Dislocation loop lying in a slip plane pue scew Vecto sum of all (local) bs w: pue edge >0, y: pue edge <0 x: pue ight-handed scew z: pue left-handed scew pue edge duing coss-slip only scew component has moved pue scew, can coss-slip double coss-slip Coss-slip in a FCC cystal

72 2.2 - LINE DEFECTS: (c) FORCES ON DISLOCATIONS ds b dl When an extenal foce of sufficient magnitue is applied to a cystal, the dislocations move and poduce slip. Hee we discuss the simple case of a dislocation line moving in the diection of the Buges vecto unde the influence of a unifom shea stess τ. Let an element of dislocation line ds be displaced in the diection of b, taken hee nomal to ds, by an amount dl: this displacement involves a faction of the slip plane A, amounting to: ds dl A coesponding to a factional slip of: ds dl A b

73 Since the foce F slip plane applied to the slip plane is: F slip plane = τa the wok dw expessed in the factional slip is: dw ds dl = τa b = τb A ( ds dl) Whence we can define a foce F acting on the dislocation line pe pe unit length: F def = dw ds dl = τb Extension to case of geneal oientation of Buges vecto ds dl F = f ( ξ,b,σ )

74 INTERACTIONS OF DISLOCATIONS The foce equations developed in the pevious few slides allow to estimate inteaction foces among dislocations. Hee we shall develop an explicit expession fo the simple case of paallel disocations. Moe complex cases can be handled in a simila way, though at the expense of notably lage computational effot. We condide the case of the following couple of // edge dislocations A and B: The glide foce on B, caused by the pesence of A (i.e. the foce expeienced by B, located at a distance λ fom A) in the stess field τ A of A is: F B = τ b A B = µ b 2π A b b b B A B ( 1 ν ) λ Note that:

75 Fo an abitay dislocation in a defomed cystal the detemination of the enegy E/L pe unit length is a most fomidable task. Howeve, a ough appoximation can be made. Since like-sign dislocations epel and opposite-sign ones attact, a dislocation tends to be suounded by opposite-sign ones at an aveage spacing λ. But: a pai of dislocations of opposite sign ae souces of equal and opposite stess fields that tend to cancel one anothe ove distances geate than λ. As a esult, the stain field geneated by a dislocation tends to be limited to a adius R λ. Moeove, λ can be estimated as: Whee ρ is the dislocation density (m/m 3 ).

76 LINE TENSION The line tension has units of enegy pe unit length (i.e. N) and it is the 1D analogue of 2D suface tension. The line tension Γ poduces a foce tending to staighten the dislocation line, that will emain cuved only if thee is a shea stess τ which poduces a foce F=τb on the dislocation esisting the line tension. dislocation line Deduction of the shea foce tem in balance F shea We have poved that the foce pe unit length F acting on a dislocation b owing to the application of shea τ is: F = dw ds dl = τb Theefoe the outwad foce acting on the dislocation element ds is: F ds = τb ds n.b.: [ F ds = F ] = N shea

77 Deduction of the line tension tem in balance The angle subtended by the dislocation element ds is dθ: d ϑ = ds R dislocation line The inwad foce on the dislocation line due to line tension is: F line tension = 2Γsin dϑ 2 Γ dϑ F line tension Balance of foces establishes the equilibium: F shea whence: = F τb ds = Γ dϑ = Γ line tension Γ τ = gµ b 0. 7µ b br ds R taking Γ elastic enegy of dislocation pe unit length

78 2.2 - LINE DEFECTS: (d) MOTION IN IMPERFECT CRYSTALS Dislocation motion, in pactice, becomes hindeed by a ange of obstacles and beak-away fom obstacles becomes ate-contolling. The simplest way to conside the effect of obstacles, is that they pin dislocations and cause thei bow-out. On the one hand, we have poved above that a foce F on disl on a dislocation can be witten as: In the case of small bow-out, the foce on a dislocation segment of length λ can be expessed as (see Figue):

79 On the othe hand, in the line tension appoximation, the foce F line acting at the pinning point is: φ F pin F line F pin φ φ φ = Fline cos = τ lineb cos = ( gµ b) b cos = gµ b φ cos 2 But the foce on the dislocation F on disl is tansmitted to the pinning points via line tension foces F pin acting on the pinning points, thus:

80 If thee exists a citical stess σ c fo beakaway of the dislocation fom the pinning point, fo: the bow-out is stable and the dislocation will be held pinned. A SELECTION OF STRENGTHENING MECHANISMS i) Solid solution stengthening (by alloying: single atoms of alloying element)

81 ii) Dispesion stengthening (by alloying: cystalline domains of alloying element) iii) Wok hadening (by cold woking: inceases dislocation density via souces, see below) iv) Gain efining: multiplicity of gains, elated to changes in oientation of cystal planes, highe enegy equied to activate slip systems. Hall-Petch elationship: R s = a + b D Slip plane Gain bounday Gain A Gain B

82 INTERSECTION OF DISLOCATIONS Since even well-annealed cystals contain many dislocations, it is quite common that a dislocation, moving along its slip plane, will intesect othe dislocations cossing that paticula slip plane. Dislocation intesections play an impotant ole in the stain-hadening pocess. The intesection of two dislocations gives ise to a shap beak a few atom spacings in length in the dislocation line, that can be of two types: a jog: a shap beak in the dislocation, moving it out of the slip plane; a kink: a shap beak in the dislocation line, which emains in the slip plane

83 Example 1: intesection of two edge dislocations with Buges vectos Dislocation XY moving on plane P XY cuts though dislocation AD moving on plane P AD. Thei intesection poduces the jog PP in dislocation AD. Jog // to b 1 has length b 1 and has Buges vecto b 2, since it to the dislocation line APP D.

84 The jog esulting fom the intesection of two edge dislocations has an edge oientation and theefoe it can glide with the est of the jogged dislocation A(PP )D. Dislocation XY slides the half cystal containing dislocation AD

85 Example 2: intesection of two edge dislocations with // Buges vectos Both dislocations ae jogged: PP, QQ. Befoe intesection Afte intesection

86 b 1 b 2

87 b 1 b 2

88 Effect of blue on ed: half cystal slid by blue dislocation b 1 b 2 b 2 half cystal unslid

89 b 1 b 2

90 Effect of ed on blue: b 1 half cystal slid by ed dislocation half cystal unslid b 1 b 2

91 Joint effects of ed on blue & blue on ed : b 1 b 1 b 2 b 2

92 Both jogs have a scew oientation (// to b) and ae called kinks. b 2 b 1

93 Example 3: intesection of a scew dislocation with an edge dislocation Poduces jogs with edge oientations in both edge and scew dislocations. Example 4: intesection of two scew dislocations Poduces jogs with edge oientations in both scew dislocations.

94 SYNOPSIS OF NATURE OF STEPS FROM INTERSECTIONS OF ELEMENTARY DISLOCATIONS edge scew //b: scew edge edge b: edge scew edge edge

95 Impact of dislocation intesection on plastic defomation i) Jogs poduced by intesection of two edge dislocations of eithe oientation ae able to glide eadily, because they lie in the slip planes of the oiginal dislocations. The only diffeence is that instead of gliding along a single plane, they do so ove a stepped suface.

96 ii) Jogs poduced by intesection involving at least one scew dislocation all have and edge oientation. Since an edge dislocation can glide feely only in a plane containing its line and its Buges vecto, the only way the jog can move by slip is along the axis of the scew dislocation. Movement of jog on Scew dislocation: constained to plain AA BB The only way the scew dislocation can slip to a new position such as MNN O taking its jog with it, is by climb. Motion of scew dislocations impeded by jogs.

97 Climb of jogged scew dislocations Section of edge jog in a scew dislocation gliding on the plane of the slide in diection BD. Edge climb (alias: non-consevative movement) must emove o add matte. geneate vacancies geneate intestitials vacancy intestitial atom climb(a B) + slip(b C) foms ow of vacancies climb(a D) + slip(d E) foms ow of intestitials

98 Vacancy jogs and intestitial jogs tend to annichilate mutually duing dilocation motion, a net concentation of jogs of the the same type will tend to accumulate. Because of thei mutual epulsion, they will tend to spead out along the dislocation line at appoximately evenly spaced intevals. Unde applied shea stess acting in the slip diection, the jogs will act as pinning points. The dislocation will thus tend to bow out between jogs. At some citical shea, non-conseevative climb will occu and the dislocations will move, leaving behind a tail of vacancies o intestitials.

99 On the one hand, an estimate of the enegy to fom a vacancy o intestitial fom a jog is: 2 ( enegy to fom defect) ( enegy pe unit length) ( length of defect) Gb b On the othe hand, the enegy to move the jog fowad by the length of a defect is: enegy foce dw whence: = τ b ds dl W Gb 3 ( length of defect) ( spacing between jogs) τ τb b l b 2 l τ Gb The shea stess equied to geneate a defect and move the dislocation by meely mechanical means (i.e. without themal activation of climb): l

100 2.2(e) DISLOCATION SOURCES: the Fank-Read case Suppose that the pinned dislocation is ca. flat, i.e. φ/2 π/2. Since: gµ b φ gµ b σ = cos σ max = (i.e.: φ = λ 2 λ 0) σ > σ max If the stess is inceased to σ max, the semi-cicula equilibium position is eached. If the stess is inceased futhe, beyond the equilibium value: the equilibium conditions no longe holds and the dislocation keeps gowing unde the foce: F = τb

101 but, if the pinning conditions still hold, i.e. in the case: F =τb l D D the dislocation will stat spialling aound the pinning points and when the two lobes D and D touch one anothe, being equal and opposite dislocations, will annihiliate geneating a complete, unpinned loop, while the potion contained by the pinning points can again outbow and epeat the pocess.

102

103 Dislocation pile-ups and continuous dislocations Dislocations fequently pile up on slip planes at obstacles such as gain boundaies and second phases. slip plane shea: τ F obstacte F othe dislocations

104 back stess yielding on othe side of obstacle cacking of obstacle distibution of dislocations along slip plane n L τ Gb giant dislocation: nb continuous dislocation: b dn