85. 8 The Stress Around a Dislocation I. Screw dis location D i sp l acement l=0un,=0 b 2丌2兀 d Strain:a =0 0 a dx
§5.8 The Stress Around a Dislocation ◆ Ⅰ. Screw dislocation: Displacement: Strain: ( ) 2 2 0 0 1 x y t g b b u u u x y z − = = = = 0 d d 0 d d 0 d d = = = = = = z u y u x u z z y y x x
=0 Oy a b M b-- Ox dz 2T( +y-) x y2 Oy az y Stresses:o、=o,=o,=0 Gb y Gb x 2nx2+ x-+
( 1 2) = 0 + = = x u y ux y xy z 2 ( ) 2 2 z x y u b y x u x xz + = − + = z 2 ( ) 2 2 z x y u b x y u y yz + = + = Stresses: 0 x = y = z = ( ) 2 ( ) 2 0 z 2 2 z 2 2 x y G b x x y G b y x y x y + = + = = −
I. Edge dislocation Gb y3x-+y 2(1-v)(x2+y Gb y(x2 y-2x(1-)(x2+y 女 z=(0x+o1) Gb x =0 2m(1-v)(x2+
◆ Ⅱ.Edge dislocation 2 2 2 2 2 ( ) (3 ) 2 (1 ) x y Gb y x y x + + − = − 2 2 2 2 2 ( ) ( ) 2 (1 ) x y Gb y x y y + − − = ( ) z = x + y 0 ( ) ( ) 2 (1 ) 2 2 2 z z 2 2 = = + − − xy = x y x y Gb x x y
fy=0,ox=0y=02=0 Gb Gb 2m(1-U)x2z(1-U)r For either edge or screw dislocation shear stress on slip plane along SD G edge dislocation n b screw dislocation n
if y=0 , 0 x = y = z = r Gb x Gb xy 2 (1 ) 2 (1 ) − = − = For either edge or screw dislocation shear stress on slip plane along S.D. − = = screw dislocatio n 2 edge dislocatio n 2 (1 ) s G G r b
35.9 Elastic Energy and Line Tension of dislocation I M tot W core elast s misfit P
§5.9 Elastic Energy and Line Tension of Dislocation ◆ Ⅰ. Wtot = Wcore+ Welast = Wmisfit+ We l P
1. We of an edge dislocation Model: see fiqure consider an area element x-x+dx which displaced from b’→b+db then M We=S x,( x)db IGb 1 dxd b 2(1-y)x Glb--In(R)=aGb? +dx 4m(1-y)
1. We of an edge dislocation. Model: see figure. consider an area element x→x+dx which displaced from b'→ b'+db'. then Gb l r Glb R x b x lGb W l x b R r e xy 2 2 b 0 ln( ) 4 (1 ) d d 1 2 (1 ) ( d )d o = − = − = =
2. W of a screw dislocation Glb R We=4丌r 3.W of a mixed dislocation bsin a b a G(bsin a) 1 G(b cos a)LlIn( R mIX 4m(1+y) 4丌
2. We of a screw dislocation: ln( ) 4 2 e r Glb R W = 3. We of a mixed dislocation: l bcos bsin b ]ln( ) 4 ( cos ) 4 (1 ) ( sin ) [ 2 2 mix r G b l G b l R W + + =
◆‖l. Line tension line tension ≈d dl @Gb2
◆ Ⅱ.Line tension e 2 d d line tension Gb l W T = =
Examples and Discussions
Examples and Discussions
Exercise
Exercise