Dislocations. Mostly from Introduction to Dislocations, 3 rd Edition by D. Hull and DJ Bacon

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1 Dislocations Mostly from Introduction to Dislocations, 3 rd Edition by D. Hull and DJ Bacon

2 Testing of Pure Mg single crystal with (0001) plane at 45º to the stress axis

3 Theoretical Shear Strength

4 Theoretical Shear Strength Frenkel 1926 Sinusoidal periodicity for energy max max x / sin a sin 2x b max 2x b x / a 2 More realistic, t = m/30 m Mg = 2,500,000 psi => t ~10 5 psi

5 Theoretical Tensile Yield Strength Gage length = 2r 0 Displacement = 0.25r 0 Strain, e = 1/8 Theoretical Strength, s 0 ~ E e = E/8 More realistic estimates give, s 0 ~ E/15

7 In 1934, Taylor, Orowan, and Polamyi independently proposed a lattice defect which is the cause for the low yield strengths

9 Shear stress Shear stress Shear stress Slip plane Unit step Edge dislocation Edge Dislocation Symbol: or T

10 Screw Dislocation Symbol:

11 Mixed Dislocations

12 The Bugers Vector: b Dislocation is a lattice line defect that defines the boundary between slipped & unslipped part of a crystal. b characterizes the magnitude & direction of a given dislocation. Determination: 1. Choose, arbitrarily, a positive direction. 2. Draw a counterclockwise (RHS) circuit of atom-toatom steps. 3. Vector b connecting the end point to the starting point is Burgers vector of the dislocation.

13 Assume that the direction out of paper plane is +ve. Draw a circuit in the same manner as a rotating righthand screw screw advancing in the positive direction.

14 Assume that the right to left direction is +ve. Draw a circuit in the same manner as a rotating righthand screw screw advancing in the positive direction.

15 FS/RH Burgers Circuit Dislocation line going into the board

17 The Burgers vector is given by the line integral, taken in the RHS relative to (the dislocation line direction), of the elastic displacement u around the dislocation b u dl l C Dislocation is characterized by two parameters, b and

18 Differences Edge dislocation 1. Dislocation line lies perpendicular to b b. =0 2. Moves in the b direction. Screw Dislocation 1. Dislocation line lies parallel to b b. =b (+ for RH screw and for LH screw) 2. Moves perpendicular to the b direction.

19 Mixed Dislocations b is invariant along the dislocation line b s ( b ξ)ξ b e ξ (bξ)

20 A dislocation whose Burgers vector is a lattice translation vector is known as a Perfect or unit dislocation. The dislocation density, r, is defined as the total length of dislocation line per unit volume of crystal. (Units = 1/mm 2 ) Or, no. of dislocations intersecting a unit area. Well-annealed crystals, r ~ 10 4 to 10 5 /cm 2 Heavily cold-worked metal, r ~10 11 /cm 2

21 Important Points Slip plane is the plane that contains both and b For an edge dislocation, since and b are perpendicular to each other, the slip plane is uniquely defined. Can t cross-slip! Need nonconservative (climb) motion For a screw dislocation, the slip plane can be any plane containing the dislocation. Can glide in any direction that is parallel to the original orientation.

22 Cross-Slip

23 Dislocation Loops Dislocations can terminate at a free surface or a grain boundary, but never within a grain. Dislocations must form network with branches terminating at the surface or must form loops.

24 At Nodes, b 1 +b 2 +b 3 = 0 n i1 b i 0 If dislocations are of the same sense, b 1 = b 2 +b 3

25 Elastic Properties of Dislocations Screw Dislocation Shear strain, g Z = b/2r Shear stress, t Z = mb/2r Only applicable for r > 5-10Å

26 Edge Dislocation

27 xx yy by 2(1 ) by 2(1 ) Edge Dislocation (3x 2 y 2 ) (x 2 y 2 ) 2 (x 2 y 2 ) (x 2 y 2 ) 2 bx (x 2 y 2 ) xy yx 2(1 ) (x 2 y 2 ) 2 zz ( xx yy ) xz zx zy yz 0 p xx 1 yy zz b 2 (1 ) x 2 y y 2

28 Energy of Dislocations E screw 1 2 r r 1 0 z bdr E 1 screw 2 r 1 b2 dr b 2 ln r 1 r 0 2 r 4 ln r 0 E edge b ln r 1 r 0 Mixed dislocations, E b 2 a=0.5-1

29 E el (edge) > E el (screw) by 1/(1-n) E el ~4 nj/m or 1 aj (6 ev) for each atom plane threaded by the dislocation E core ~ aj Varies as the dislocation moves Crystals contain many dislocations Crystals contain many dislocations Dislocations tend to form configurations in which the superimposed long range stress fields cancel out R~half of the average spacing for dislocations arranged in random

30 Consequences of Eb 2 Minimize b, hence slip occurs usually in close-packed directions! Dislocation dissociation when b 1 2 b 2 2 b 3 2 Known as Frank s Rule

31 Line Tension, T (energy per unit length) is increase in energy per unit increase in the dislocation line length T b 2 Tries to straighten the dislocation, so as to minimize the energy Stress required to keep a dislocation curved, t T/bR mb/r R-radius of curvature Glide force acting on a dislocation, f, due to applied shear stress, t f (/unit length) = tb

32 Stress on a Curved Dislocation Net force acts in OA direction Disln. will remain curved if there is a stress t o opposing it d =dl/r Outward force = t o b dl Inward force = 2T sin (d/2) =T d for small d Equilibrium, t o b dl =T d T= agb 2 t o = a G b/r

33 Glide force acting on a dislocation Derivation: Use PVW Work done = (t x area) x b = t l 1 l 2 b (1) Works against force, f, opposing against its motion = (f l 1 )l 2 Note that f defined as force per unit length Equating f = t b

34 Forces between Dislocations

35 Forces between Dislocations Screw dislocations of opposite sign attract and those of the same sign repel. The magnitude of the attractive/repulsive force is inversely proportional to the distance between them. Edge dislocations: Dislocations on the same slip plane, attract each other if they are of opposite signs and repel if of the same sign. Complicated if they are on different planes

36 Dislocations of same sign, but on parallel slip planes, attract each other if x<y Can form arrays with dislocations stacked one the other to form Low-angle tilt boundaries.

37 Stable Configurations

38 Small Angle Grain Boundary

39 Image Forces A dislocation near the surface is attracted towards the free surface. (Repelled by a rigid surface layer.) Analysis for a infinite, straight dislocation line that is parallel to the surface at x = d.

40 For a free surface, tractions s xx, s yx, and s xz should be zero at x = 0. Met if a opposite sign (imaginary) dislocation exists at x = -d. For screw dislocation, at x> ) ( 1 ) ( 1 2 y d x y d x Gby zx ) ( ) ( ) ( ) ( 2 y d x d x y d x d x Gb zy

41 Force per unit length in x direction, F x, induced by the surface is obtained by F x For an edge dislocation, zy b xd, y0 2 Gb 4d Gb F x 4 (1 ) d Image forces decrease with increasing d. May remove dislocations from surface. Important factor in TEM of dislocations and in thin films! 2

42 Partial Dislocations As noted earlier, a dislocation can dissociate to minimize energy. Has to satisfy, b 12 > b 22 +b 3 2 Dislocations b 2 and b 3 are known as Shockley partial dislocations. In FCC, <110> type dislocation dissociates into two <112> type dislocation a 2 a a 6

44 In HCP, Partials repel each other! The equilibrium separation distance depends on the stacking fault energy (SFE)! Aside FCC crystal structure is created by stacking (111) planes in ABCABCAB..manner HCP by stacking in ABABABAB.. manner 3 1 3

46 HCP FCC

47 Extended Dislocations Stacking sequence, ABCACABCA

48 Stacking Fault A 2-D defect Has surface energy, g. Small compared to GB energy. SFE and the repulsive force between the partials determine the equilibrium separation distance, d, between two partial dislocations. ( b 2 b3) d 2 High SFE, low width and vice-versa

49 SFE depends on material, composition, etc.. SFE determines the ability of dislocations to cross-slip and in-turn work hardening characteristics. High SFE, lower separation between partials, easier cross-slip, lower work hardening exponent, n. Materials exhibit wavy slip. Low SFE, higher separation between partials, difficult for cross-slip, and higher n. Material exhibits planar slip.

51 SFE and Slip Character Metal SFE (mj/m 2 ) n Slip Character Stainless <10 ~0.45 Planar Steel Cu ~90 ~0.3 Planar/ Wavy Al ~ Wavy

52 Lomer-Cottrell Lock A sessile dislocation that occurs in FCC metals. Also known as Stair-Rod Dislocations

53 Cottrell Lock in BCC Metals Two perfect dislocations interact to form a sessile [100] type dislocation. a 0 a a The [001] dislocation lies on a {001} plane and large stresses are required to move it. Helps in Cleavage fracture through crack nucleation. (Acts as a microcrack.)

54 Superlattice Dislocations Found in Intermetallic compounds Superlattic = long range order

55 (111)

Need another dislocation to close the antiphase domain.

56 Movement of one dislocation generates unfavorable lattice distortion. Creates a antiphase domain. Need another dislocation to close the antiphase domain. Dislocations have to move in pairs (superdislocation) The region (2D) between them in known as Antiphase boundary (APB)

57 APB is a 2-D defect. APB width depends on the balance between the APB energy and the repulsive force between dislocations. APB energy depends on the degree of order in the lattice. APBs lead to planar slip character. In some precipitation hardened alloys such as Al-Li alloys, the ordered precipitates force planar slip, and hence lowered ductility.

58 Kinks Dislocation step in the slip plane Kink in an edge dislocation has screw character Kink in a screw dislocation has edge character Easy to remove

59 Jogs Steps in dislocation normal to the slip plane Jogs always have an edge character

60 Dislocation-Dislocation Intersection Summary When two edge dislns. interact jogs are created. These jogs do not impede the dislocation motion.

61 A screw disln. interacts with another screw or edge disln. to produce edge jogs that impede the motion.

62 Nonconservative motion of jogs is required to move jagged screw dislocations.

63 0 x b 2 b 2 x 0 x 0

64 Disregistry of atoms Defined as displacement difference Du between two atoms on adjacent sites above and below the slip plane Du = u(b) u(a) Width, w, of a dislocation is defined as the distance over which Du is one half of its max value, i.e. b/4 Du b/4 A measure of the size of the dislocation core wherein the elasticity theory fails

65 f(x) = d(du)/dx Distribution of Burgers Vector Core width, b-5b, depends on interatomic potential and crystal structure

66 The Peierls Barrier The core disregistry imparts resistance to dislocation motion. Peierls (1940) and Nabarro (1947) assumed a sisusoidal force relations to obtain analytical solutions for Du. P-N calculated the dislocation energy per unit length as a function of the dislocation Position. 2 Max. fluctuation, Gb 2w E P N exp( (1 ) Max. slope of E p is critical force per unit length required to move the dislocation. b )

67 The Peierls-Nabarro Stress, t P-N The t P-N characterizes the lattice s resistance to dislocation motion from one low energy position to another. Magnitude depends on 1) The width, W, of the dislocation. W represents the distance over which the lattice is distorted. 2) Distance between similar planes, a.

68 t P-N m exp(-2w/b) W = a/(1-) t P-N decreases with increasing a. Close packed planes preferred.

69 W depends on atomic structure and atomic bonding (spherical vs. directional). Close packed structures (spherical), W is high, and hence t P-N is low. When bonding forces are highly directional (BCC, ionic and covalent bonded materials), W is low, and hence t P-N is large. The t P-N is sensitive to thermal energy in the lattice and, hence, to the test temperature. Low T, High t P-N. Increase in T, t P-N decreases.

70 W and Temperature Sensitivity Material Crystal Type W Peierls Stress y temp. Sensitivity Metal FCC Wide Very Small Metal BCC Narrow Moderate Strong Ceramic Ionic Narrow Large Strong P (xg) Negligible < Ceramic Covalent Very Narrow Very Large Strong 10-2

71 Deformation due to Dislocation Motion (Orowan s equation) Deformation is related to no. of dislns that move and the distance traveled by them. Cube (dx 1, dx 2, dx 3 ) sheared by N dislns. Plastic strain, g 13 =Nb/dx 3 Density of dislocations, r=ndx 2 / dx 1 dx 2 dx 3 g 13 =rbdx 1

72 In real materials, deformation generated by each dislocation is related to the average distance traveled, l, by it. bl 13 Correction parameter, k, to account for multiple slip, misalignment, etc. kbl p Known as Taylor-Orowan or Orowan s eqn. Note: r mobile < r total As r increases, more and more dislocations are locked. Taking the time derivative, d dt p kb dl dt or p kb

73 Origin of Dislocations Thermodynamically stable density of dislocations is zero in a stress-free crystal Dislocations in freshly grown crystals are due to 1) dislns. in seed crystals 2) Nucleation during the growth process a) heterogeneous nucleation due to internal stresses thermal gradients, change in composition, or change in lattice structure b) impingement of growing interfaces c) formation and movement of disln. loops formed by the vacancy platelets

74 Dislocation Multiplication Dislocation density increases by 6 to 8 orders of magnitude when a fully annealed metal is coldworked heavily. Explained by dislocation multiplication mechanisms. Most common is the Frank-Read source-- A dislocation segment pinned on either end. Shear stress will cause the dislocation to bow out. Radius, R Gb/t When R = l/2, the loop becomes unstable

75 Frank-Read Source

77 Frank-Read Source in a Si Crystal

78 Cross-slip is another Mechanism for Dislocation multiplication

79 Slip and Twinning

81 HCP If c/a < 1.633, then True in Zr(1.593) and Ti(1.587). Can t explain in Co (1.623), Mg (1.623), & Be (1.568). If c/a > 1.633, then basal slip plane {0001}. Examples: Zn (1.856) and Cd (1.886).

82 Von Mises Criterion Ductility of a material depends on its ability to withstand a general homogeneous strain involving an arbitrary shape change Von Mises postulated that the above is possible when there are 5 independent slip systems. Six independent components of strain tensor, e xx, e yy, e zz, e xy, e xz, and e zy But, e xx +e yy +e zz = 0 (constant vol. process) Hence, 5 independent slip systems.

83 Independent Slip System Defined as one producing crystal shape change that can t be reproduced by any other slip system. Taylor (1938): Of the 12 possible slip systems in FCC and BCC, 5 are independent. Hence, BCC and FCC metals show good ductility. For HCP, only two out of three (basal or prism) slip systems are independent. Hence, HCP metals lack ductility and need twinning.

84 Geometry of Slip Single crystal subjected to stress. A slip-plane =A 0 /cos P resolved = Pcosl Resolved shear stress, t rss = (P/A 0 ) coscosl Plastic deformation starts when slip gets activated, i.e., t rss = t crss Critical RSS The product coscosl is known as Schmid factor

85 Yield Behavior of Anthracene Crystals

86 Slip Plane Rotation L L i 0 sin 0 sin i L i, L o -gage lengths before and after plastic flow

87 Stereographic Projection

88 Standard Projection for Cubic Crystals

89 The relative orientation of a crystal can be given with respect to any triangle within the stereographic projection.

90 Primary slip system (slip system with highest Schmid factor) : Rotation occurs along great circle, towards 101 pole. As the crystal rotates, l decreases and increases If, l 0 >45º> 0, rotation will increase the Schmid factor. Yielding at lower loads. Geometric Softening. If, l 0 <45º< 0, Geometric Hardening. (Different from work hardening, which is a result of dislocation-dislocation interactions.)

92 When the crystal axis crosses over into the triangle II, slip system (conjugate slip system) becomes activated. Now, the crystal starts rotating towards [011] Back and forth movement of crystal orientation between primary and conjugate slip systems, moves it along the [001] tie line until load axis is parallel to Cross-slip Slip system: Critical Slip System:

93 Resolved shear stress vs. shear strain response of a FCC crystal oriented for single slip I: Easy glide (primary slip) II: Linear hardening (secondary slip) III: Parabolic hardening (dynamic recovery & Cross-slip) cos 1 cos 0 l sin l 2 1 / l / l 0 2 sin 2 0 cos 0 Suresh: p. 31

94 Twinning

96 Differences between Slip & Twinning Slip Simple translation of one rigid portion over the other. No shape change or orientation change. Multiples of unit displacement, b. 1-D defect Twinning A shape change is involved with parts oriented differently Atom movements on all planes in fractional amounts, with d distance from the twin boundary. A 2-D defect

97 Mechanical (or deformation) twins: A result of superimposed plastic deformation. ex: Zn Annealing Twins: A result of pre-existing plastic deformation. Example: Brass -Form during recrystallization and grain growth of heavily deformed materials. -Form due to interaction of new grains with stacking faults. -Generally observed in low SFE metals (high stacking fault probability). -Increase with increased cold work.

100 Twin Geometry X, Y, Z to X, Y, Z X= X, Z=Z, Y= Y+SZ where S is the shear strain. Equation for the distorted sphere: X 2 +Y 2 +Z 2 =1 X 2 +Y 2 +2SZY+Z 2 (1+S 2 ) =1 Ellipsoid with major axis inclined to 1 by. AO is shortened and BO is elongated. Compositional plane (K 1 ) and plane OC (K 2 ) are undistorted.

101 S = 2cot2f

102 Elongation Potential of Twinning Total deformation to be expected from a completely twinned crystal is l l 4 1 S S S where S is the shear strain. Quite small in HCP crystals. However, the rotation of the crystal within the twins may make them favorable for slip deformation.

103 Twin Shape b increases with decreasing shear strain (Cahn 1954) Twin initiation stress is much larger than that required to propagate it.

104 Twinning in HCP Crystals 2+2 =180 cot2 =S/2 S=(tan 2-1)/tan tan =c/3a S [( c / a) 2 3)] 3a 3c c/a = 3, twinning does not occur in {1012} mode Twin deformation is opposite in sense for c/a >< 3

105 Be (c/a=1.568) Compression of basal plane and tension of the prism plane. If basal plane is oriented parallel to the loading direction, twinning will only occur under compression

106 Zn (c/a = 1.856) Zn will twin when applied stress causes tension in basal plane or compression in prism.

107 c/a=1.856 c/a=1.624

108 c/a <3 = 3 >3

109 Commonality between Martensite and Twinning Both form as a result of atoms in a finite parent phase crystal realigning as new crystal lattices. In twins, the realignment reproduces the original crystal structure with a new orientation. In martensite, new orientation as well as new crystal structure (e.g. FCC to BCT in Steels.) In both cases, the change in shape distorts the surrounding matrix. Both look alike, either small lenses or plates. Driving force: Applied shear stress for twins Free energy Change for Martensite

110 Deformation twins in Polycrystalline Zr. Martensite plates in a Fe-1.5C-5.1Ni steel

Chapter Outline Dislocations and Strengthening Mechanisms. Introduction

Chapter Outline Dislocations and Strengthening Mechanisms What is happening in material during plastic deformation? Dislocations and Plastic Deformation Motion of dislocations in response to stress Slip