Lectures on: Introduction to and fundamentals of discrete dislocations and dislocation dynamics. Theoretical concepts and computational methods

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1 Lectures on: Introduction to and fundamentals of discrete dislocations and dislocation dynamics. Theoretical concepts and computational methods Hussein M. Zi School of Mechanical and Materials Engineering Washington State University Pullman, W zi@wsu.edu Summer school Generalized Continua and Dislocation Theory Theoretical Concepts, Computational Methods nd Experimental Verification July 9-3, 2007 International Centre for Mechanical Science Udine, Italy

2 Contents Lecture : The Theory of Straight Dislocations Zi Lecture 2: The Theory of Curved Dislocations Zi Lecture 3: Dislocation-Dislocation & Dislocation-Defect Interactions -Zi Lecture 4: Dislocations in Crystal Structures - Zi Lecture 5: Dislocation Dynamics - I: Equation of Motion, effective mass - Zi Lecture 6: Dislocation Dynamics - II: Computational Methods - Zi Lecture 7 : Dislocation Dynamics - Classes of Prolems Zi

3 Lecture 3: Dislocation-Dislocation & Dislocation-Defect Interactions Forces on a dislocation- Peach-Khoeler Force Interaction with externally applied stress Dislocation- Dislocation Interaction Surface Effect Dislocation- defect interaction

4 Dislocations interact with: *point defects (vacancies, interstitials, particles, *Surfaces: free surface, grain oundaries

5 Elastic ody with internal stress E W dv T. uds Total V Elastic energy S Potential Energy T n T; traction: i ij j Force; f k E x Total k

6 Consider a lock of material with a screw dislocation pply shear stress s yz as shown. The dislocation moves from its initial configuration x to final configuration x 2 Work done y the system: E yz L x x ). ( 2 Recall; force is equal to the spatial gradient of work, therefore, Force per unit length: F i ( yz Lx) x F / L yzi yz Li Peach-Koehler Force

7 General expression of Peach-Koehler Force Suppose a dislocation line L has moved a distance δx due to a force f, then the change in interaction energy is E int V f k x k ds lso the interaction energy etween the applied stress and the dislocation cut can e written as E int s ij ( ds Equating the aove equations yields f k i j klj ); ( ds) t l ji xds ( xt) ds i t δx ds ds Vector form F t (. ) Peach-Koehler Force Peach & Koehler, Physical Review, 80, 436,950

8 Dislocation Dislocation Interaction Consider the interaction etween two screws dislocations y t t t 2 2 k k x z t Recall the stress field for a screw dislocations; all stress components are zero except: xz Similar expressions for dislocation 2 yz sin 2 r cos 2 r

9 The force exerted y dislocation () onto dislocation (2) is, therefore, given y ( () F t ( () If and ) 0 xz 0 0 yz 0 xz yz ) 0 2σ. 0 2σ 0 F 2 [cosi sinj] 2 2r are of the same sign, the reaction is repulsive If and 2 have opposite signs then the interaction is attractive and the dislocations may form a dipole, or react to form a new dislocation (if the reaction energetically favorale), or if the dislocation also have the same magnitudes and opposite signs they will annihilate. xz yz

10 Dipole 45 0

11 d dl2 2 F self Self-Force C Force at su-segment d d = Force from segment C+ Force from segment BD+ Force from segment B (see Hirth and Lothe, 982, p. 3) B D d

12 Self-Force on a segment in closed loop One can determine the force on an element in a dislocation caused y other segments of the same dislocation line using the Peach-Koehler equation, e.g. consider the piecewise straight-dislocation configurations: y y B dl C D The interaction force on an element dl in from segments BC, CD, DE, and E is given y F dl t dl ( ( BC). dl ) t dl ( ( CD). dl ) y 2 E t dl ( ( DE ). dl ) t dl ( ( E) But dl also interact with segment B to which it elongs to. The method for determining this interaction is given in Hitth and Lothe (982); it yields:. dl ) df s 4 y e dl e 2 n 3 y y y2 e dl

13 C B D D C B B DB C g F F L f L f F cos sin,, B z x B Explicit expression e.g. Self-Force per unit length on a finite segment F g

14 e.g. f C C z C x B z B z C x B x C y B y cos sin verage force per unit length : F g 4L L f C, f DB B, n FC FD Similar expressions are otained for the normal force. These expressions reduce to those given in Hirth and Lothe (982, p.38) for B C Cut-off parameter; numerical parameter which can e adjusted to account for core energy

15 Dislocation-Surface Interaction surfaces (Image dislocation force!) The dislocation field equations otained so far are for dislocations in an infinite or semi-infinite medium. So, when a dislocation is in a finite domain, it will interact with the surfaces of the domain. Consider, for example, a pure screw dislocation near a free surface. The shear stress from the dislocation if it were in an infinite medium would e non-zero at the location of the free surface. n equilirium condition requires there can e no shear stress and normal stress at the free surface. In general, one can use the principle of superposition to solve this prolem. Here, and ONLY for a purse screw dislocation, it can e shown that the stress will e ZERO on the free surface if an IMGE dislocation with equal and opposite sign at the surface as shown elow. r xz r IMGE xz

16 Z, msm3d simulation of Screw dislocation: Interaction with free surface T X, p Y,

17 Therefore, the IMGE dislocation, which in in effect represents the surface attract the dislocation towards the surface, leaving a ledege.. The interaction force is: F x 2 4r Thus, one can conclude that given a straight dislocation interesting the surface, it would curve due to the force from the surface:

18 Generalize: Interaction with External free surfaces: V t a pennyshaped cracks The solution for the stress field of a dislocation segment is known for the case of infinite domain and homogeneous materials, which is used in DD codes. Therefore, the principle of superposition is employed to correct for the actual oundary conditions, for oth finite domain and homogenous materials. t t a- t dislocation s cracks U a V S u S u S * u * * Zi and Diaz de la Ruia, Multiscale Model of Plasticity, Int. J. Plasticity, 8,

19 Dislocation- defect interaction Stacking-Fault Tetrahedra D C B SFT Oserved in quenched metals and metals and alloys of low stacking-fault energy. It consists of a tetahedron of intrinsic staking faults on {} planes with /6<0> type stair-rod dislocations along the edges of the tetrahedron. dislocation loop (of Frank partial dislocation formed y the collapse of vacancies), may dissociate into a low-energy stair-rod dislocation and a Shockly partial on an intersecting slip plane, leading to the formation of a SFT.

20 * E (ev) The interaction energy etween two piecewise straight dislocation segments is calculated using relations ased on either Blin s formulas, e.g. equation (4-40) (Hirth and Lothe 982) or Kroner s derivation (Kroner 958, de Wit 967) of the two-loop interaction: W d Self-energy and Staility of Isolated SFT For the calculation of self-energy E self of straight dislocation segments, we follow an empirical form of cut off radius * with choice of where * is the magnitude of Burgers vector of the dislocation segment. In cases of parallel dislocation segments, the formula for W int is replaced y the sum of the self-energy of each segment when separation of two parallel dislocations ecomes less than 2dditionally, a common core reaction distance, 2 =, is assumed in all annihilations or cominations of parallel dislocation segments. Expressions for the self-energy E self of isolated truncated SFTs can e found in the work of Jossang and Hirth (966) as numerical functions. In copper samples, consideration of isolated defects reveals that SFTs are usually more stale than FSLs or /2[0]-type perfect loops on {} planes. For example, E self of a SFT of size 0 is ev per vacancy (NOTE this value seems a little low for the SFT, atomistic MD results with a low SFE give 55 vacancy SFT (~9-0, Ef=26.7 ev or ~0.49 ev/vacancy) while E self for the corresponding FSL and the perfect loop are ev and 0.63 ev, respectively. Due to large contriution of stacking fault energy to the self-energy of complete SFTs, the SFT's can e truncated over a certain size range. Generally, complete SFTs are elieved to e most stale when the size of SFT, L, is less than a critical size L c while truncated SFTs ecome metastale as L increases up to a second critical size L c2, and then ecome stale at L > L c2., Higher stacking fault energy,, leads to further SFT truncations y pulling Shockley partials towards the ase to reduce the faulted area along the tetrahedron faces. Self-energy of a truncated SFT of various sizes is shown in figure 3 as a function of the degree of truncation, t L' L. The truncated SFT is unstale within the range of size up to L c 00, ecomes metastale at L c, and then most stale at L c2 30. In cases of L 0, the energy gap etween complete and slightly truncated SFTs, say SFTs at t 0.8, is within aout 0.0 ev, and therefore, a population of oth complete and partially truncated SFT is expected to e comparale in equilirium at room temperature P(y, z) t C C z [] D O 0.5 B L y[2] L B x[0] L= t

21 SFT versus FS loop! Interaction etween a defect (Frank sessile loop or SFT) and a glide edge dislocation C D B D B D D C Franksessile loop SFT In the DD code the SFT is constructed from six dislocations segments with Stair Rods Dislocations: e.g. Etc.. a a a [ 2] [ 2 ]

22 F(pN) E(eV) Interaction of SFT with Dislocation Force profile ctivation Energy profile g e s 0 o y z x W G Y() Case for a perfect dislocation gliding on a ase plane of SFT (d-plane) U F(pN)

23 * E (ev) * E (ev) Staility of truncated SFT with Dislocation (V) Unlike case for a glide plane at d = z= t d=0.95 P(y, z) C C z [] D O y[2] System energy when dislocation moves continuously on a plane L B Spontaneous truncation is unlikely! L B x[0] Unlike case at various d (glide planes) d=3.5 The figure shows such a relation etween energy curves at each dislocation position on a glide plane and the curve of energy minimum (lowest curves in the figure). Similarly, one can construct a family of curves for the interaction energy etween truncated SFTs and an incident dislocation gliding on a plane parallel to the BCC B face ((a) plane) of the SFT. The reactions take place most likely along an envelope of the aforementioned curves that minimizes each degree of truncation and positions of the approaching dislocations. These (lower) envelopes f can e given as functions in a form of * P, t Ey t f, where P is a dislocation position on each glide plane specified y a variale as P = (0, y, z(y)), and the parameter y * satisfies minimum conditions, Ey, t y 0and 2 y, t y 0 2 E. Figure 8a depicts the group of curves at d = 0.95 in annihilation cases and figure 8 shows such envelopes for various glide planes on (a) with distance d from the side plane BCC B for unlike cases t Isoloated

24 glide PF-force (N/m) R=0, y=5 y Edge dislocation X 6.00E+08 SFT 4.00E+08 FS loop 2.00E E E E+08 Defect: a)loop, ) dissociate into SFT -6.00E E+08 x ()

25 Stress, Pa SFT s versus FS loops 5.00E E+07 FS loops 4.00E E E E E+07.50E+07.00E E+06 SFT s Defect density=.6x E % 0.02% 0.04% 0.06% 0.08% 0.0% Strain Elastic interaction: FS loops produce higher strength than SFT s

26 Interaction etween a defect (Frank sessile loop; defect clusters in irradiated materials) and a glide edge dislocation Interaction is weak 3 / r z d=2r y l Fig. 5. (a) TEM picture showing dislocation pinning y the dispersed hardening mechanism. () DD picture showing the same mechanism at work as captured in the simulations. The dots in the figure represent defect clusters.

27 Flow Stress [MPa] Dislocation interaction with the thermal stress field of SiC spherical particles in an aluminum matrix. (a) Pile-ups of glide dislocation loops forming aove 2 particles (left) and four particles (right). () The effect of particle numer or volume fraction on the flow stress. Khraishi et al 2003 Dislocation Particle Interaction (MMC) particle radius=800, thermal strain=%, height=00, strain rate=2/sec 4par 2par 3par 0par par 5par 6par 6par particle volume fraction [in percents]

28 Internal surfaces: Internal surfaces such as micro-cracks and rigid surfaces around fiers, say, are treated within the dislocation theory framework, wherey each surface is modeled as a pile- up of infinitesimal dislocation loops. Hence defects of these types are all represented as dislocation segments and loops, and there interaction with external free surfaces follows the method discussed aove.

29 Dislocation pile-up at grain oundary