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1、1 Tri Lab Short Course on Dislocations in Materials Pleasanton CA June 8 10 1998 Lecture on 3D Dislocation Dynamics Numerical Treatment Tri Lab Short Course on Dislocations in Materials Pleasanton CA June 8 10 1998 Lecture on 3D Dislocation Dynamics Numerical Treatment H M Zbib M Rhee numerical para
2、meter which can be adjusted to account for core energy Similar expressions are obtained for the normal force These expressions reduce to those given in Hirth and Lothe 1982 p 138 for e g CAAB bb 17 A B b For pure edge pure screw dislocations this reduces to e g b g F g F force per unit length L n L
3、b Fgl 4 2 L n L b Fgl 41 1 2 18 Bow out of edge and mixed dislocations Examples Prismatic loop 19 Example Stress field Demir Zbib and Hirth 1992 ExactApproximate 20 IV Short Range Interactions IV Short Range Interactions BCC 4 Burgers vectors without regard to slip planes 8 possible distinct reactio
4、ns 4 repulsive 3 attractive and one annihilation When slip planes are considered For the 110 and 112 planes 420 attractive reactions all of which are sessile Baird and Galye 1965 d A short range interaction occurs when the distance d between two dislocations becomes comparable to the size of the cor
5、e annihilation formation of dipoles jogs and junctions Detailed investigation of each possible interaction can become very cumbersome 21 1 Critical distance criterion Essman and Mughrabi 1979 Implicitly takes into account the effect of the local fields arising from all surrounding dislocations Crite
6、ria for determining interaction Rule 1 If c PK F F and PK F or changes sign as the segment is advanced short range interaction is possible provided that local interaction between segment AB and its closest neighbor CD CDAB F is attractive A B C D 2 Force based criterion 22 MPA bL F c 8 102 This capt
7、ures interactions for a distance varying from 10b to 100b Effective applied stress 2 Force based criterion 23 Rule 2 If Rule 1 is satisfied and 1 CDAB and 0 CDAB bb or 1 CDAB and 0 CDAB bb AB and CD will annihilate by either i Glide if they are on the same plane or ii Cross slip if they are on inter
8、secting planes Annihilation Free nodes 24 T h e probability that two attractive dislocations form a jog or a junction is im p licitly determ ined by their interaction forces For exam p le if they w ere to form a junction the interacting segm ents rotate relative to each other to align them selves in
9、to a configuration m ore suitable for junction form ation yielding a reduction in the energy If not they form a jog since it is energetically m ore favorable A B C D 25 Rule 3 If Rule 1 is satisfied and c jnCDAB with 0 CDAB bb then a junction is formed Segments AB and CD are combined to form a new s
10、egment at the junction whose Burgers vector is equal to bb CDAB Junction node Junction 2 2 2 1 2 bbbb 21 b b in b b in A B 3 111110 3 111110 27 3 2 cos b Wv c jgs oc jgs 120 For Ta Rule 5 If c jgsjg the jog moves forward in the direction of average velocities of the two adjacent dislocation segments
11、 Jogs Motion Vacancy or interstitial generating jogs Line tension approximation Vacancy or interstitial formation energy 28 Dipole Rule 6 If Rule 1 is satisfied and 1 CDAB 0 CDAB bb or 1 CDAB and 0 CDAB bb and AD and CD are on parallel planes they would form a dipole provided that RcCDAB Vhh Dipoles
12、 form naturally numerically without a need for a Rule However a rule can be used for numerical efficiency 29 kT V A kT V AP 0 0 exp W exp Cross Slip e a n ab L aLa naaLL b aLL L nLaLaL b W l l l 12 222 14 2 2 2 22 22 2 22 22 2 Model Activation energy bAWG Junction node Example Initial dislocation so
13、urce on 011 with b b MM gsge 3 111 111 view 30 Short range Interaction Summary 31 P Zo Q A Main Computational Cell B C D Cells Grains C or D Cell with N Dislocations P Dislocation in Cell A For screw Dislocations Stress at P from Q is obtained from the potential Z o zz b z x y zxiy Long Range Intera
14、ction 2D V Numerical Issues V Numerical Issues Long Range Interaction Computationally most expensive Use of superdislocation representation 32 cont Multipolar Expansion LeSar et al 1994 3 2 2 z bz z bz z b z oo For N Dislocations z b z b z z b z z iioiioi 2 2 3 o zzif Field of Single Dislocation wit
15、h Burgers Vector bi Monopole Field of Dipoles with an intensity B bzoi 2 2 Quadrupoles 33 Superdislocations 2D 34 Superdislocations 3D 1 3 r 35 3D Superdislocations N i zixix lb L B 1 1 Monopoles N i ziyiy lb L B 1 1 N i ziziz lb L B 1 1 N i cixixixz Zlb L B 1 1 2 Dipoles N i cixiyiyz Zlb L B 1 1 2
16、N i cixizizz Zlb L B 1 1 2 36 Segment length L Meshing If Lij Lmax 200b sub divided If Lij Lmin 50b combine ij Time step Limited by short range interaction and velocity max vbt20 For strain rate 0 1 s 1 to 1000 s 1 811 1010 t Dynamic time step 37 max vbt20 t t M M i i pp 1 1 100010 s st 47 1010 Example Constant Strain Rate p tE Random distribution of dislocation lines and Frank Read Sources Stress increment 38 Example Constant Strain Rate m10862 10 bGPa770 03393 101010 541 fgsge MMPa s Ta 3 450
