転位の応力場について
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- よしたか もちやま
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1 y.koyama
2 f ( F ( F( f( xp( i d f( F( xp( i ( π xp( i d δ ( F( δ ( f( δ ( xp( i f ( δ ( F( f ( xp( i d xp( i d F( f( xp( i d ' F( xp( i xp( i ' ' d ' F( xp{ i( } d ' ' ' d ' ' F( ' xp{ i( } d ' ' ' F( ' δ ( ' F( ' ign u ( ( l u ( U ( xp( i l ( l ( xp( i δ ( ( l l l
3 u u l δ l ( i{ U l ( Ul ( }xp( i l A A A Est ijijd ijijd ijijd A A A 0 Cijl ( l l l ( ij ij ij d Cijllij d Cijllijd δ δ A A A Cijl{ δl ( l l }{ δij ( ij ij } d Cijll d ij Cijld l ij Cijl ( ij ij ( l l d Cijl ( l l ijd Cijl ij ld δ δ δ A A A Cijllij d Cijllij d Cijl ( ij ij ( l l d Cijl l ( l δijd Cijl ij ld δ δ A A A Cijllij d Cijllij d Cijl ( ij ij ( l l Cijl ( l l ijd Cijl ij ld δ δ δ A A A Cijllij d Cijllij d E st ij Cijl ( l δ l l d 0 Cijlld Cijl ld Cijlld 0 δ Cijlld Cijlld 0 Cijll Cijl ld 0 Cijl l ld 0 l l ( d l ( l ign
4 δ ( ( δ ( ( 0, 0 l l l l ij, j Cijl Cijl j j j j C ijl δ l ( l ( 0 j j Cijl { ( ( }xp( ( xp( 0 j U l Ul i ijl i ( π ( π Cijl j{ U l ( Ul ( } icijlj l ( xp( i 0 (π Cijl j{ U l ( U l( } icijlj l( 0 CijljU l ( CijljUl ( icijl j l ( ij pq l mn ij pq i i l mn ( U( ( η ( l mn j ( i ij pq lq C U ( C U ( ic pql q l pql q l pqmn q mn ( C U ( C U ( pql q l pql q l C U ( ic ( pql q l pqmn q mn G ( C pl pql q Ul ( C U ( ic ( pql q l pqmn q mn U ( ig ( C ( l pl pqmn q mn
5 δ l ( i{ U l ( Ul ( }xp( i { Gp ( ql Gpl ( q} Cpqmnmn ( xp( i ( π ( δ ( l l l { G ( G ( } C ( xp( i ( π l p q l pl q pqmn mn l ( { p ( nqnl pl ( nqn} Cpqmnmn ( xp( i Ω n Ω n ( π Ω pl ( n G pl ( Ω ( n G nn Ω n nn pl pl ( q ( q pl q G pl { δ ( λ µ ( λ µ } δ ( λ µ ( λ µ n n ( ( λ µ µ ( λ µ µ pl p l pl p l 6 δ pl ( λ µ ( λ µ nn p l Ω pl ( n ( λ µ µ ( C { ( ( } ij ijl l l Cijl l ( { ( ( } ( xp( ( Ω p n nqnl Ωpl n nqn Cpqmn mn i l ( π Cijl { Ω p ( nqnl Ωpl ( nqn } Cpqmnmn ( xp( i C ( ( ijl l l n n (π { } l ( ( ( ( l l l l ( { Ω p ( nqnl Ωpl ( nqn} Cpqmnmn ( xp( i ( l n n ( π { } { p ( nn q l pl ( nn q } Cpqmnmn ( xp( i ( ( l l Ω n Ω n ( π 4
6 u ( U ( xp( i ig ( C ( xp( i l l pl pqmn q mn Est Cijlij ( l ( d Cijl ij ( ij ( ij ( l ( l ( l ( d δ δ C ijl ij ( ij ( δij ( l ( l ( δl ( d C ijl ij ( ij ( l ( ( l d Cijl δij ( l ( l ( d Cijl δij ( l ( d δ Est C ijl ij ( δij ( ij ( C ijl ij ( ij ( δij ( ij ( ( l ign ( ( δ ( H( H H(, ( < 0 H( 0, ( > 0 ign 5
7 ( ( xp( i d ( H( xp( i d δ ( H ( xp{ i( } ddd δ L δ ( LL δ ( i i { H ( xp( i d}{ ( xp( }{ xp( δ i d i d } xp( i d δ ( L F ( ( xp( id L δ δ ( F( xp( i xp( i π π H( xp( i i π H( xp( i d i L H( xp( i d L i 6
8 u ( ig ( C ( xp( i p pqmn q mn i { Gp ( Cpq q ( Gp ( Cpq q ( } xp( i i { Gp ( Cpq q ( } xp( i i { G( C( G( C( } xp( i ( π ( λ µ ( λ µ ( λ µ δ( i µ 4 µ 4 xp( i ( λ µµ ( λ µµ i ( π ( λ µ ( λ µ ( λ µ ( λ µ ( λ µ ( π 4 4 δ ( xp( i ( λ µ ( λ µ δ ( xp( i ( λ µ (π 4 ( λ µ ( ( ( δ( xp( i δ( 4 xp( i π λ µ π δ ( xp{ i( } ( ( xp{ i ( } π tan tan ( π π xp{ i ( } π log( xp{ i ( } π tan xp{ i ( } π ( xp{ i ( } π log( π ( 7
9 u ( ig ( C ( xp( i p pqmn q mn i { Gp ( Cpq q ( } xp( i i { G ( C G ( C } ( xp( i ( λ µ ( λ µ ( λ µ δ ( i µ 4 µ 4 xp( i ( λ µ µ ( λ µ µ i ( π 0 u ( ig ( C ( xp( i p pqmn q mn i { Gp ( Cpq q ( } xp( i i { G ( C G ( C } ( xp( i i 0 ( λ µ ( λ µ δ ( d µ µ xp( i ( λ µ µ ( λ µ µ i ( π 4 4 ign ( ( δ ( H( ( ( xp( i d δ ( H( xp( i d δ ( i 8
10 u ( ig ( C ( xp( i p pqmn q mn i { Gp ( Cpq q ( Gp ( Cpq q ( } xp( i i { Gp ( Cpq q ( } xp( i i { G( C ( G( C( } xp( i ( π ( λ µ ( λ µ ( λ µ δ ( i µ 4 µ 4 xp( i ( λ µ µ ( λ µ µ i ( π ( λ µ ( λ µ ( λ µ i ( λ µ ( λ µ ( π 4 4 δ ( xp( ( λ µ ( λ µ δ ( 4 xp( i ( λ µ ( π ( λ µ δ( xp( i δ( xp( i ( π ( λ µ ( π 4 δ ( xp{ i( } ( ( λ µ ( λ µ δ ( xp{ i( } ( ( π ( λ µ xp{ i( } xp{ i( } ( ( λ µ ( (π ( λ µ π π tan ( π ( λ µ ( π ( λ µ tan π 4 π ( λ µ tan π 4 π ( ν 9
11 u ( ig ( C ( xp( i p pqmn q mn i { Gp ( Cpq q ( } xp( i i { G ( C ( G ( C ( } xp( i ( λ µ ( λ µ ( λ µ δ ( i ( λ µ µ ( λ µ 4 µ 4 xp( i µµ i ( π ( λ µ ( λ µ ( λ µ i ( λ µ ( λ µ ( π 4 4 δ ( xp( ( λ µ ( λ µ ( λ µ ( π δ ( 4 xp( i ( λ µ δ ( xp( i δ ( xp( i ( π ( λ µ ( π 4 δ ( xp{ i( } ( ( π ( λ µ ( λ µ ( ( π δ ( xp{( i } ( λ µ xp{ i( } xp{ i ( } ( ( λ µ ( ( π ( λ µ ( π ( λ µ ( π π log( π log( π log( log( 4π 4 π ( ν ν log( 8π ν 4 π ( ν ( ( ( xp( 0 u igp Cpqmnj mn i λ ν ( λ µ ( λ µ λ µ λ µ λ λ ν λ µ λ µ ( λ µ 0
12 u ( u ( 0 u ( tan π u π π u π π u u u u u u u 0 u u δ δ δ δ δ δ 0, 0, 0 δ u u u δ π 4π u u δ 4π π 0 u u ign ( ( δ ( H( ( ( d ( H ( ddd δ H ( d δ ( d d L L L L ( 0 ij
13 0 4π 4π C ( C ( C ( C ( 0 l l l C ( C ( C ( C ( 0 l l l C ( C ( C ( C( 0 l l l C ( C ( C ( 0 l l l C ( C ( C ( µ ( l l l Cl ( l l C( C( µ ( 0 µ µ ( µ µ µ π µ µ ( µ µ µ π u ( tan π 4 π( ν ν u ( log( 8π ν 4 π( ν u ( 0
14 u ( ( π 4 π( ν ( π 4 π( ν ( u ( π 4 π( ν ( π u ν 4π ν π( ν ( u ν 4π ν π( ν ( u u u u u 0 ( 4 π( ν ( δ δ δ δ δ π 4 π( ν ( 4π ν π( ν ( u ( u ν u u ( ν π 4 π( ν ( 4π ν π( ν ( ν ( π ν 4 π( ν ( ( 8π ν 8 π( ν ( ( 4 π( ν ( δ δ δ δ 0 ( ν δ π 4 π( ν ( 4π ν π( ν ( ( ν ν ( 4π ν 4π ν 4 π( ν ( 4 π( ν ( 4ν ( 4π ν 4 π( ν ( ν π ν ign ( ( δ ( H(
15 ( ( d ( H ( ddd δ H ( d δ ( d d L L L L ( 0 ij π 4 π( ν ( π π λ µ ( 4π ν π( ν ( π λ µ π λ µ ( ( ( λ µ ( ν µ ( λ µ ( 4 π ( ν ( 0 ν µ π ν π λ µ ( λ µ ν λ λ λ µ λ µ ( λ µ λ µ λ µ λ ν ( λ µ λ ( λ µ µ ν λ ( λ µ λ λ µ ( λ µ λ ν ( λ µ λ λ ν λ ( λ µ λ λ µ ( λ µ 4
16 C ( C ( C ( C ( l l l ( λ µ λ λ ( µ C ( C ( C ( C ( l l l ( λ µ λ λ( µ C ( C ( C ( C ( λ( l l l C ( C ( C ( µ ( l l l 0 λµ µν λ( πλ µ π ν λ ( µ µν ( µ π ν π 4 π( ν ( µν µ ( ν µ ( π ν π ν π( ν ( µ µ ( π ν π( ν ( µ ( ( µ ( π( ν ( π( ν ( λ ( µ µν (ν π ν 4π ν π( ν ( µ µ ν µ (ν µ π ν π ν π( ν ( µ µ π ν π( ν ( µ ( µ ( π( ν ( π( ν ( ( µ ( µ µ µ π( ν ( µ ign SSS LL S 5
17 S - S S - S SS S - (S (S - S S - nn S ign β * ji β * ( x nδ( S ji i j δ ( S S S β S ign * ji n n * * ij ( ( βij β ji ( j i i j δ( S x x S x S y x y z S x-z (x<0 z z n (0,,0 δ ( S δ ( yh ( x ( ±,0,0 (0,0, ± ± (,0,0 (0,0, ign (,0,0 ij( ( jni n i j δ( S ( jni n i j δ( y H( x ( ( ( n n δ( y H( x δ( y H( x ( (0,0, ij ij( ( jni n i j δ( S ( jni n i j δ( y H( x ( ( ( n n δ( yh ( δ( yh ( x ij ( x 6
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