~nabe/lecture/index.html 2

Size: px
Start display at page:

Download "~nabe/lecture/index.html 2"

Transcription

1

2 ~nabe/lecture/index.html 2

3 1. 10/ / / / / / / / / / / / / /17 ALE /24 ALE 2 3

4 total Lagrange updated Lagrange 1 v V T : δa (L) dv = S : δe dv = v δa ij T ij dv = ] [{δu (n) } T [B] T {S} dω = e Ω e e [B] = [ ] B (k) = V v t w ds + t w ds + v V v ρg w dv ρg w dv {δu} T [B] T {T } dv [ ]] [{δu (n) } T [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω e [ ] [B (1) ] [B (n) ] N (k) x 1 N (k) x 2 N (k) x 3 N (k) x 2 N (k) x 1 N (k) x 3 N (k) x 3 N (k) x 2 N (k) x 1 [B (n) ] u 1 N (n) X 2 X 2 + ( 1+ u 1 X 1 ) N (n) X 1 u 2 X 1 N (n) X 1 u 3 X 1 N (n) X 1 ( ) u 1 N (n) X 2 X 2 1+ u 1 N (n) u 3 N (n) X 2 X 2 ( X 2 X ) 2 u 1 N (n) u 2 N (n) X 3 X 3 X 3 X 3 1+ u 3 N (n) ( ) ( ) X 3 X 3 1+ u 1 N (n) X 1 X 2 1+ u 2 N (n) X 2 X 1 + u 2 N (n) u 3 N (n) X 1 X 2 X 2 X 1 + u 3 N (n) ( ) ( ) X 1 X 2 1+ u 2 N (n) X 2 X 3 1+ u 3 N (n) X 3 X 2 + u 3 N (n) X 2 X 3 u 1 N (n) X 3 X 2 + u 1 N (n) u 2 N (n) X 2 X 3 X 3 X 2 + ( 1+ u 1 X 1 ) N (n) X 3 + u 1 X 3 N (n) X 1 u 2 X 1 N (n) X 3 + u 2 X 3 N (n) X 1 u 3 X 1 N (n) X 3 + ( 1+ u 3 X 3 ) N (n) X 1 4

5 δa : Ṡ t (t)+ 1 ( ) δf t (t) T L + L T δf t (t) : T dv = δṙ v 2 ( ) S ij δe ij dv = Ṡ ij δe ij + S ij δėijdω Ω = e Ω (δa ij S t (t) ij + δf ki T ij L kj )dv ([B] T [ ) D] [B]+[G] v = {δu} T δe ij Ṡ ij dω + [ { } T δu (n) v Ω Ω δf ki S ij F kj dω updated dv { u} Ω e ( [B] T [D][B]+[A] ) dω Ṡ t (t) ij = C ijkl D kl { } ] u (n) Ṡt(t) Truesdell Kirchoff Oldroyd Total S ij = C ijkl E kl C ijkl (Ṡij, Ėkl ) Ṡ ij = C ijkl Ė kl 5

6 total Lagrange updated Lagrange 2 Ṡ t (t) ij = C ijkl D kl S ij = C ijkl E kl Ṡ ij = C ijkl Ė kl Ṡ 0 (t) =J 0 (t)f 0 (t) 1 Ṡ t (t)f 0 (t) T Ė 0 (t) =F 0 (t) T DF 0 (t) C pqrs = 1 J F pif qj F rk F sl C ijkl 6

7 . F dx u X x dx 1: X, x : u : (= x X) F : C : Cauchy Green B : Cauchy Green E : Green-Lagrange T : Cauchy Π : 1 Piola Kirchhoff S : 2 Piola Kirchhoff F x i X j e i e j C F T F B F F T E 1 (C I) 2 Π JF 1 T S JF 1 T F T, e i,, J =detf. 7

8 1, W. S ij = W E ij E = 1 (C I) 2 S ij =2 W C ij W C., W C. S ij =2 I C trc II C 1 { (trc) 2 tr(c 2 ) } 2 III C det C ( W I C + W II C + W ) III C I C C ij II C C ij III C C ij 8

9 2 {( W S ij =2 + W ) I C I C II C I C = δ ij C ij II C = I C δ ij C ij C ij III C = III C (C 1 ) ij C ij δ ij W C ij + W } III C (C 1 ) ij II C III C S C. Cauchy T kl = 2 {( W II B + W ) III B J II B III B δ kl + W B kl W } III B (B 1 ) kl I B II B T B. 9

10 3,,.,,.,,., ( )., III C = III B =1,J =1 {( W T kl = pδ ij +2 II B + W ) δ kl + W B kl W } (B 1 ) kl II B III B I B II B, p. 2 Piola-Kirchhoff. S ij = p(c 1 ) ij +2 {( W + W ) I C δ ij W C ij + W } (C 1 ) ij I C II C II C III C 10

11 Mooney-Rivlin 1 W Mooney-Rivlin. W M c 1 (I C 3) + c 2 (II C 3), c 1, c 2. Mooney-Rivlin, 2 Piola-Kirchhoff. } S ij = p(c 1 ) ij +2 {(c 1 + c 2 I C )δ ij c 2 C ij, C ij = δ ij T ij = S ij =0 S ij = pδ ij +(2c 1 +4c 2 )δ ij, p 2c 1 +4c 2., W M. W M R c 1 (ĨC 3) + c 2 (ĨI C 3) Ĩ C I C III C 1 3 ĨI C II C III C

12 Mooney-Rivlin 2 ĨC, ĨI C (reduced invariants). W M R 2 Piola-Kirchhoff W M R I C = WM R ĨC ĨC = c 1 III C I C 1 3 W M R II C = WM R ĨI C W M R III C = WM R ĨC S ij = p(c 1 ) ij +2 ĨI C = c 2 III 2 3 C II C ĨC III C + WM R ĨI C ĨI C = 1 III C 3 c 1I C III 4 3 C 2 3 c 2II C III 5 3 C { (c 1 + c 2 I C )δ ij c 2 C ij + ( 13 c 1I C 23 ) } c 2II C (C 1 ) ij T ij = S ij =0 S ij = pδ ij, p. 12

13 Mooney-Rivlin 3,.,. F. F = J 1 3F, F Flory, det F =1. Cauchy-Green C. C = F T F C 1, 2, ĨC =3, ĨI C =3. 13

14 Mooney-Rivlin, - S. Mooney-Rivlin c 1, c 2., I C, II C 2, Stress[MPa] Strain 2: - W H = c 1 (I C 3) + c 2 (II C 3) + c 3 (I C 3) 2 + c 4 (I C 3)(II C 3) + c 5 (II C 3) 2 + c 6 (I C 3) 3 + c 7 (I C 3) 2 (II C 3) + c 8 (I C 3)(II C 3) 2 + c 9 (II C 3) 3. 14

15 Mooney-Rivlin 2, W H W M, p. WR H = c 1 (ĨC 3) + c 2 (ĨI C 3) + c 3 (ĨC 3) 2 + c 4 (ĨC 3)(ĨI C 3) + c 5 (ĨI C 3) 2 + c 6 (ĨC 3) 3 + c 7 (ĨC 3) 2 (ĨI C 3) + c 8 (ĨC 3)(ĨI C 3) 2 + c 9 (ĨI C 3) 3 15

16 1, c 1,c / l x 2 l 1/ l x 3 x 1 3:, F, B, II B l 0 0 F = 0 1/ l / l l B = FF T = 0 1/l /l 1/l B 1 = 0 l l II B =2l + 1 l 2 16

17 2 W W H R W H R I B W H R II B = WH R ĨB = III 1 3 B = WH R ĨI B = III 2 3 B ĨB I { B ) c 1 +2c 3 (ĨB 3 ) + c 4 (ĨI B 3 2 ) +3c 6 (ĨB 3) +2c7 (ĨB 3)(ĨI B 3 ĨI B II B { c 2 + c 4 (ĨB 3 ) ) +2c 5 (ĨI B 3 2 ) +c 7 (ĨB 3) +2c8 (ĨB 3)(ĨI B 3 ) } 2 + c 8 (ĨI B 3 ) } 2 +3c 9 (ĨI B 3 W H R III B = WH R ĨB = 1 3 I BIII 4 3 B ĨB + WH R III B ĨI B 2 3 II BIII 5 3 B ĨI B III B { c 1 +2c 3 (ĨB 3 ) ) + c 4 (ĨI B 3 2 ) +3c 6 (ĨB 3) +2c7 (ĨB 3)(ĨI B 3 { ) ) c 2 + c 4 (ĨB 3 +2c 5 (ĨI B 3 2 ) +c 7 (ĨB 3) +2c8 (ĨB 3)(ĨI B 3 ) } 2 + c 8 (ĨI B 3 ) } 2 +3c 9 (ĨI B 3 17

18 2 Cauchy { W H T kl = pδ kl +2 R (2l + 1 } II B l )+ WH R III B + WH R I B l /l 0 WH R II B 0 0 1/l δ kl 1/l l l x 1, T 22 = T 33 =0 { 1 WR H p =2 +(l + 1 } R + WH R l I B l 2) WH II B III { B T 11 =2 (l 2 1 l ) WH R +(l 1 } R I B l 2) WH II B l =1+ε ε 2 T 11 =6(c 1 + c 2 )ε 6(c 1 + c 2 ) E. 18

19 3,. u x x 1 x 3 1 4: F, B,I B,II B 1 u 0 F = u 2 u 0 B = u u 0 B 1 = u 1+u I B =trb =3+u 2 II B = 1 { (trb) 2 tr(b 2 ) } =3+u

20 4 Cauchy { } W H T kl = pδ kl +2 R (3 + u 2 )+ WH R δ kl II B III B 1+u 2 u 0 1 u 0 + WH R u 1 0 WH R u 1+u 2 0 I B II B T 33 =0 { W H p =2 R +(2+u 2 ) WH R I B II B } + WH R III B ( W H T 12 = T 21 =2u R I B + WH R II B ) u 2, T 12 = T 21 =2(c 1 + c 2 )u, u.,2(c 1 +c 2 ) G. 20

21 . A Ω, Ω Ω, Ω D Ω. t, ρ 0 g, u V p Q. V, Q,.. find (u,p) (V, Q) such that X (S F ) T + ρ 0 g =0 (1) ( ) S F T T N = t (2) C = F T F (3) S ij = p(c 1 ) ij +2 W C ij (4) III C =1 (5) (1), (2), (3), (4) ( ) (5)., W, (1) (5). 21

22 W Φ. Φ = W dω t u ds ρ 0 g u dω (6) Ω Ω λ Lagrange, Φ. Φ = Φ + λg(iii C )dω (7), g(iii C ),III C =1 g =0,. Ω Ω g III C =1, Lagrange Q., u V, λ Q δu V, δλ Q. δ Φ = = Ω Ω W δc ij dω + C ij ( W + λ g C ij C ij Ω ) δc ij + δλg C ij ( λ g ) δc ij dω Ω dω t δu ds Ω Ω t δu ds ρ 0 g δu dω Ω ρ 0 g δu dω + δλg dω =0 Ω (8) (8),. (8),. 22

23 , (1) (5),. Ω find (u,λ) (V, Q) such that ( W + λ g ) δc ij dω = C ij C ij Ω Ω t k δu k ds + Ω ρ 0 g k δu k dω (9) δλg dω = 0 (10) for (δu,δλ) (V, Q), λ = 1 2 p 23

24 Newton-Raphson,. Ω.. Ω = e Ω e (11),,. dω = dω (12) Ω e Ω e ds = ds (13) Ω e Ω e u N (i), u i. u i = N (n) u (n) i (14), u (i) i, (n). Lagrange λ M (m), λ λ = M (m) λ (m) (15)., λ (m). 24

25 ( W + λ g ) δc ij dω = Ω C ij C ij δλg dω =0 Ω Ω t k δu k ds + Ω ρ 0 g k δu k dω δe ij S ij dω = δr (16) Ω S ij =2 ( W + λ g ) C ij C ij δe ij = 1 2 δc ij δe ij, S ij i, j, δe ij S ij = δe 11 S 11 + δe 22 S 33 + δe 33 S 33 +2δE 12 S 12 +2δE 23 S 23 +2δE 31 S 31 =(δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 )(S 11 S 22 S 33 S 12 S 23 S 31 ) T (17),. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T (18) {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (19) 25

26 [B] [B (n) ] u 1 N (n) X 2 X ( 1+ u 1 X 1 ) N (n) X 1 u 2 X 1 N (n) X 1 u 3 X 1 N (n) X 1 ( ) u 1 N (n) X 2 X 2 1+ u 1 N (n) u 3 N (n) X 2 X 2 ( X 2 X ) 2 u 1 N (n) u 2 N (n) X 3 X 3 X 3 X 3 1+ u 3 N (n) ( ) ( ) X 3 X 3 1+ u 1 N (n) X 1 X 2 1+ u 2 N (n) X 2 X 1 + u 2 N (n) u 3 N (n) X 1 X 2 X 2 X 1 + u 3 N (n) ( ) ( ) X 1 X 2 1+ u 2 N (n) X 2 X 3 1+ u 3 N (n) X 3 X 2 + u 3 N (n) X 2 X 3 u 1 N (n) X 3 X 2 + u 1 N (n) u 2 N (n) X 2 X 3 X 3 X 2 + ( 1+ u 1 X 1 ) N (n) X 3 + u 1 X 3 N (n) X 1 u 2 X 1 N (n) X 3 + u 2 X 3 N (n) X 1 u 3 X 1 N (n) X 3 + ( 1+ u 3 X 3 ) N (n) X 1 (20) 6 3 [B (n) ], [ ] [B] = [B (1) ] [B (n) ] (21)., {δe}{s} dω = ] [{δu (n) } T [B] T {S} dω e Ω e e Ω e (22). (??), (22), (9) ] [{δu (n) } T [B] T {S} dω = e Ω e e. [ ]] [{δu (n) } T [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω e (23) 26

27 , (10). {M} = {M (1) M (2) M (m) } T (24) {δλ (m) } = {δλ (1) δλ (2) δλ (m) } T (25), (10). δλgdω = δλg dω (26) Ω e Ω e = ] [{δλ (m) } T {M}g dω = 0 (27) e Ω e 27

28 , { {δu (n) δλ (m) } = δu (1) 1 δu (1) 2 δu (1) 3 δu (n) 1 δu (n) 2 δu (n) 3 δλ (1) δλ (m)} T (28), (23), (27). [ ] ] [{δu (n) δλ (m) } T [B] T {S} dω e Ω e {M}g = [ [ ] [{δu (n) δλ (m) } T [N] e Ωe T [ ] ]] {t} ρ 0 [N] T {g} ds + dω 0 Ω e 0, Q = F = u = [ ] [B] T {S} dω (29) Ω e {M}g [ ] [N] T [ ] {t} ρ 0 [N] T {g} ds + dω (30) Ω e 0 Ω e 0 { u (n) λ (m)} (31), (29) [ T δuh (Q(u h ) F ) ] = 0 (32) e., find u h V h such that [ T δuh (Q(u h ) F ) ] = 0 (33) e for δu h V h, Newton-Raphson. 28

29 Newton-Raphson, K = Q u, dq dt = Q du u dt = K u (34), (9), (10),.,. Ω (9) ( W C ij + λ g {( W + λ g ) ) δc ij + C ij C ij C ij [ {( 2 W 2 ) g = + λ Ċ kl + Ω C ij C kl C ij C kl ( W + + λ g ) ( δf ki C ij C ij { ( 2 W 2 ) g = + λ Ċ kl δc ij Ω C ij C kl C ij C kl ( + 2 W +2λ g C ij C ij δċij } dω } g λ δc ij C ij F kj + F ki δf kj ) ] dω ) δf ki F kj + } g λ δc ij dω C ij (35), (10) δλ ġ dω = δλ g Ċ kl dω (36) C kl Ω Ω 29

30 1 { λ } { (m) λ(1) λ(2) λ } T (m) (37) { {D 2 } 2 g 2 g 2 g 2 g 2 g 2 g } T C 11 C 22 C 33 C 12 C 23 C 31 (38), (??) 3 δe ij 2 g { λ dω = {δe} T {D 2 }[M] λ(m)} dω Ω C ij e Ω e = [ { } T { ] δu (n) [B] T {D 2 }[M]dΩ λ(m)} e Ω e (39). 30

31 ( δe ij D ij kl Ė kl + δf ki S ij F kj + δe ij 2 g ) λ dω Ω C ij [ { } T ( δu (n) [B] T [D 1 ][B]+[A] ) { } dω u (n) e Ω e } T { } + {δu ] (n) [B] T {D 2 }[M]dΩ λ(m) (40) Ω e. 31

32 Ω δλ ġ dω = Ω δλ g C kl Ċ kl dω δλ g Ċ kl = δλ 2 g Ė kl C kl C { } kl T = δλ (m) [M] T {D 2 } T {Ė} { } T = δλ (m) [M] T {D 2 } T [B]{ u} (41) δλ g Ċ kl dω = Ω C kl e [ { δλ (m) } T. Ω e [M] T {D 2 } T [B]dΩ (42) { } ] u (n) 32

33 , { (n) u λ(m) } { u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3 λ (1) λ } T (m) (43) [K 1 ] [B] T [D 1 ][B]+[A] (44) [H] [B] T {D 2 }[M] (45) (40), (42) [ { } [ ] δu (n) δλ (m) [K 1 ] [H] { } ] (n) dω u e Ω e [H] T λ(m) 0 (46), [ ] [K 1 ] [H] [K] = dω (47) [H] T 0. Ω e 33

34 Mooney-Rivlin 1 α Mooney-Rivlin W S = W H + α 2 W V (III C ) 2 W V (III C ) III C III C = 1 W V = 0 WV = 1 W V = III C 2(J 1), III C 1 α 34

35 Mooney-Rivlin δ x 2 1+ε 1+δ x 3 x 1, F, B, II B l + ε 0 0 F = 0 1+δ δ l +2ε 0 0 B = FF T = 0 1+2δ δ l 2ε 0 0 B 1 = 0 1 2δ δ ε, δ 35

36 Mooney-Rivlin 3 Cauchy T kl = 2 {( W II B + W ) III B δ kl + W B kl W } III B (B 1 ) kl J II B III B I B II B W S I C = WS ĨC ĨC I C { = c 1 +2c 3 (ĨC 3) + c 4 (ĨI C 3) +3c 6 (ĨC 3) 2 +2c 7 (ĨC 3)(ĨI C 3) + c 8 (ĨI C 3) 2} III 1/3 C W S II C = WS ĨI C = ĨI C II C { c 2 + c 4 (ĨC 3) + 2c 5 (ĨI C 3) +c 7 (ĨC 3) 2 +2c 8 (Ĩ 3)(ĨI 3) + 3c 9(ĨI 3)2} III 2/3 C W S = WH III C ĨC = 1 3 ĨC + WH ĨI C I C ĨI + αw V WV C II C III C } {c 1 + c 3 (ĨC 3) + c 4 (ĨI C 3) + 3c 6 (Ĩ 3)2 I C III 4/3 { c 2 + c 4 (ĨC 3) + 2c 5 (ĨI C 3+c 7 (ĨC 3) } +2c 8 (ĨC 3)(ĨI C 3) + 3c 9 (ĨI C 3) 2 II C III 5/3 + αw V WV III 36

37 Mooney-Rivlin 4 Ĩ C = I C III 1/3 C =(3+2ε +4δ) (1 23 ε 43 ) δ =3 ĨI C = II C III 2/3 C =(3+4ε +8δ) (1 43 ε 83 ) δ =3 W V W V = III C 1 αw V WV III = α(iii 1) = α(2ε +4δ) W V =2(J 1) αw V WV III = α2(j 1) 1 J = α2(ε +2δ)(1 ε 2δ) = α(2ε +4δ) αw V WV III = α(2ε +4δ) 37

38 W S I C = c 1 (1 2 3 ε 4 3 δ) W S II C = c 2 (1 4 3 ε 8 3 δ) W S III C = (c 1 +2c 2 )(1 2ε 4δ)+2α(ε +2δ) T kl T 22 = T 33 =0 ν δ = 3α (c 1 + c 2 ) 6α +(c 1 + c 2 ) ε ν = 3α (c 1 + c 2 ) 6α +(c 1 + c 2 ) T 11 = 36(c 1 + c 2 )α 6α +(c 1 + c 2 ) ε E E = 36(c 1 + c 2 )α 6α +(c 1 + c 2 ) α E =6(c 1 + c 2 ) E κ = 3(1 2ν) =4α 38

39 1 (R ) Φ= WdΩ+ α (W V ) 2 R Ω 2 Ω α selective/reduced integration V Q αw V Q λ ( W V λ ) δλdω =0 δλ Q Ω α αw V λ P P (αw V )=λ Q 39

40 2 P Φ = WdΩ+ α Ω 2 Ω (PW V ) 2 R U V δ Φ = Ω W δc ij dω+α (PW V )P (δw V )dω δr =0 C ij Ω u Ω ( W V λ ) δλdω =0 α λ Q δλ Q 40

41 3 u V,λ Q α Ω (PW V )P (δw V )dω = Ω λδw V dω δu V δ Φ = Ω W δc ij dω+ λδw V dω δr =0 C ij Ω ( W V λ ) δλdω =0 α Ω W V III ( ) W + λ WV δc ij dω=δr Ω C ij C ij ( W V λ ) δλdω =0 α Ω 41

42 4 Lagrange Ω ( W C ij + λ g C ij ) δc ij dω = δλg dω =0 Ω Ω t k δu k ds + W V λ α g(= W V ) Ω ρ 0 g k δu k dω Lagrange [ ] [B] T {S} Q = dω {M}g Q = Ω e [ Ω e [B] T {S} {M} ( W V λ/α ) ] dω Lagrange [K] = [K] = Ω e Ω e [ ] [K 1 ] [H] dω [H] T 0 [ ] [K 1 ] [H] dω [H] T [G] [G] = 1 α [M]T [M] α 42

43 ,,,, 43

44 (Hooke, ) t t t,,,,. 44

45 B A e e e p e (A), (B),,.,,,. 45

46 A B e e e p e e e e e p. e = e e + e p σ E. σ = E(e e p ), Hooke. σ ij = C e ijkl(e kl e p kl), σ ij,e ij,e p ij 2 Cauchy,,, C e ijkl 4 Hooke. 46

47 , Hooke. σ ij = C e ijkl(e kl e p kl),. σ ij = C e ijkl(ė kl e p kl),. σ ij = C ep ijklė kl, (flow rule) 47

48 :. :. :. B A e e e p e 48

49 3 2, 9, vonmises Tresca B A e e e p e 49

50 ,, 3 A,, B. von Mises. B A e e e p e 50

51 Mises Mises σ σ = ( 3 2 σ ijσ ij ) 1 2 σ ij σ ij σ 2 2 ij σ ij =σ 11 + σ 12 + σ +σ σ 22 + σ +σ σ 32 + σ , σ ij. σ ij =σ ij 1 3 σ kkδ ij =σ ij 1 3 (σ 11 + σ 22 + σ 33 ) δ ij 51

52 F = σ σ y σ y F =0 ( ) σ y σ ij σ ij 52

53 (associated flow rule),, λ Ψ ė p ij = λ Ψ σ ij (associated flow rule), ė p ij = λ F σ ij F = σ σ y 53

54 (normality rule) σ ij / t ( ) ė p ij ė p σ ij ij t F = λ σ ij F σ ij = λ σ ij t ė p ij σ ij = λ F F =0, F =0, ė p ij σ ij = λ F =0 0,, σ ij ė p ij 54

55 von Mises σ = ( )1 3 2 σ ijσ 2 ij F = σ σ y F = λ σ ij ė p ij A B e e e p e 55

56 , Hooke. σ ij = C e ijkl(e kl e p kl),. σ ij = C e ijkl(ė kl e p kl),. σ ij = C ep ijklė kl 56

57 1 F =0 F =0 F = F σ ij =0 σ ij σ ij = C e ijkl(ė kl ė p ( kl ) ) = C e F ijkl ė kl λ σ kl F/ σ ij F σ ij = F C e ijklė kl F C e F ijkl λ σ ij σ ij σ ij σ kl =0, λ λ = F σ ij C e ijklė kl F σ ij C e ijkl F σ kl 57

58 2 λ σ ij = C e ijkl(ė kl ė p ( kl ) ) = C e F ijkl ė kl λ σ kl λ = ( σ ij = C e ijkl = ( F σ ij C e ijklė kl F σ ij C e ijkl F σ kl ė kl F C e σ ab abcdė cd F F C σ e abcd F σ ab σ kl cd C e ijkl Ce ijcd F σ cd F σ ab C e abkl F C σ e abcd F ab σ cd ) ) ė kl 58

59 3 F/ σ ij F = 3 σ ij 2 σ σ ij, σ ij = σ ij Ce ijklė kl σ ij Ce ijklσ kl λ = 2 σ 3 ( ) C e ijkl Ce ijcdσ cd σ ab C e abkl σ ab C e abcdσ cd ė kl 59

60 4 Hooke C e ijkl λ, µ Lamé C e ijkl = λδ ij δ kl +2µδ ik δ jl µ G, σ ij = ( λ = σ klėkl σ C e ijkl 3Gσ ij σ kl σ 2 ) ė kl 60

61 von Mises σ = ( ) σ ij σ ij F = σ σ y F = λ σ ij ė p ij ( σ ij = C e ijkl 3Gσ ij σ kl σ 2 ) ė kl 61

62 1, Hooke. Hooke. F Cauchy T (elastic material). T (t) =f(f (t)) (48) f. f(f )=f(q F )=Q f(f ) Q T (49) F, F O,O, O O Q., P f(f )=f(f P ) (50). V. T = f(v ) (51). f(v )=f(q V Q T )=Q f(v ) Q T (52) 62

63 V, V O,O, O O Q. f(v ) (isotropic tensor function). (52) T, V, T = f(v )=φ 0 I + φ 1 V + φ 2 V 2 (53)., φ i (i =0, 1, 2) V. (representation theorem). (51) V = B 1/2. T = g(b) (54) g(b )=g(q B Q T )=Q g(b) Q T (55), g(b),. T = ψ 0 I + ψ 1 B + ψ 2 B 2 (56) = ξ 0 I + ξ 1 B + ξ 1 B 1 (57), B. 63

64 Hooke. V I + 1 {u x + x u} (58) 2, E (L) E (L) = 1 {u x + x u} (59) 2, (53) E (L) T =(φ 0 + φ 1 + φ 2 )I +(φ 1 +2φ 2 )E (L) (60) = η 0 I + η 1 E (L) (61), η 0, η 1 E (L). T E (L), Hooke. T =(λtre (L) )I +2µE (L) (62), λ, µ Lamé. 64

65 2, Hooke. T = f(v ), T = g(b), B Almansi A, A = 1 (I B) (63) 2 T = h(a) (64). h(a )=h(q A Q T )=Q h(a) Q T (65) A, A O,O, O O Q. h(a),. T = h(a) =ζ 0 I + ζ 1 A + ζ 2 A 2 (66) Hooke. T =(λtra)i +2µA (67) A E (L) (68), λ, µ Lamé. 65

66 3 T =(λtra)i +2µA. Ṫ, Ȧ T = QT Q T Ṫ = QT Q T + QṪQT + QT Q T W, T, A T, Å. T = Ṫ W T + T W (69) Å = Ȧ W A + A W (70) Jaumann T (J) = Ṫ W T + T W Oldroyd T (O) = Ṫ L T T LT Cotter Rivlin T (C) = Ṫ + LT T + T L Green Naghdi T (G) = Ṫ Ω T + T Ω (Ω = Ṙ RT ). T =(λtrå)i +2µÅ (71),, (, ), F t (τ) R t (τ), U t (τ) I (72) 66

67 ,. T (J) T (O) T (C) T (G) (73) Å (J) Å(O) Å(C) Å(G) (74), T (J) = Ṫ W T + T W (75) T (J) = T (O) + D T + T D (76) T (J) = T (C) D T T D (77) T (G) = Ṫ Ω T + T Ω (78) W Ω (79), Å(C) = D, T =(λtrd)i +2µD (80). T Kirchhoff ˆT t (τ) =J t (τ)t (τ) ˆT t (t) =(λtrd)i +2µD (81)., ˆT t (t) (J) = T (J) + T trd (82) ˆT t (t) (O) = T (O) + T trd (83) ˆT t (t) (C) = T (C) + T trd (84) 67

68 . v v e v p v = v e + v p (85), L D. D = D e + D p (86) σ ij T ij, e p ij D p ij., C ep ijkl. T ij = C ep ijkld kl (87),,, (87),Cauchy Kirchhoff ˆT ij = C ep ijkld kl (88). 68

69 Kirchhoff, Jaumann.. ( C ep ijkl = C ijkl 3G T T ) ij kl σ 2 (89) T ij, T ij = T ij 1 3 T kk δ ij. pe = T kl D kl σ (90) λ = T kl D kl σ (91) 69

70 1,,., t e p ij = t 0 τė p ij dτ (92) t σ ij = = t 0 t 0 τ σ ij dτ (93) τ C ep ijkl τ ė kl dτ (94) t σ ij t σ ij = t τ C ep ijkl τ ė kl dτ (95) t. t C ep ijkl, (94), (95),. 70

71 2,. Kirchhoff.,, (101) t C ep ijkl, t Cijkl e. t T ij = t T ij + = t T ij + = t T ij + = t T ij + = t T ij + t t t t t t t t t t τ T ij dτ (96) {τ ˆTτ ij (tr τ D) τ T ij } dτ (97) {τ ˆTτ ij + τ W ik τ ˆTτ kj τ ˆTτ ik τ W kj (tr τ D) τ T ij } (98) } {τ ˆTτ ij + τ W τ ik T kj τ T τ ik W kj (tr τ D) τ T ij (99) { τ C ep ijkl τ D kl + τ W ik τ T kj τ T ik τ W kj (tr τ D) τ T ij } dτ (100) = t T ij + {t } C ep ijkl t D kl + t W t ik T kj t T t ik W kj (tr t D) t T ij t (101) dτ dτ 71

72 . V,v, S, s. s t, u, v g.. T Cauchy. x T + ρg = 0 (102) T T n = t (103) u = u (104) D ij = 1 ( ui + u ) j (105) 2 x j x i ˆT ij = C ep ijkl D kl, T ij = t 0 T ij dt (106) 72

73 T : δa (L) dv = t w ds + v δv v ρg w dv (107) δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j (108) 2 x j x i updated Lagrange v δa ij T ij dv = {δu} T [B] T {T } dv (109) v Q = [B] T [T ]dv (110) v (δa ij S t (t) ij + δf ki T ij L kj )dv ([B] T [ ) D] [B]+[G] v = {δu} T v dv { u} (111) 73

74 1,. Ṡ ij = C ijkl D kl (112) D kl = D lk, C ijkl = 1 2 (C ijkl + C ijlk ) (113). D, D ij [ C]. Ṡ 11 Ṡ 22 Ṡ 33 Ṡ 12 Ṡ 23 Ṡ 31 = C 1111 C1122 C1133 C1112 C1123 C1131 C 2211 C2222 C2233 C2212 C2223 C2231 C 3311 C3322 C3333 C3312 C3323 C3331 C 1211 C1222 C1233 C1212 C1223 C1231 C 2311 C2322 C2333 C2312 C2323 C2331 C 3111 C3122 C3133 C3112 C3123 C3131 D 11 D 22 D 33 2D 12 2D 23 2D 31 (114) [ C], Cijkl = C klij,. 74

75 1 Hooke Cijkl e = λδ ijδ kl +2µδ ik δ jl (115) C ijkl e = λδ ijδ kl + µ (δ ik δ jl + δ il δ jk ) (116) C e klij = λδ kl δ ij + µ (δ ki δ lj + δ kj δ li ) (117) = λδ ij δ kl + µ (δ ki δ lj + δ il δ jk ) (δ mn = δ nm ) (118) = C e ijkl (119) [ C e ]= λ +2µ λ λ λ λ+2µ λ λ λ λ+2µ µ µ µ (120) Lamé λ, µ E, ν. νe λ = (1 + ν)(1 2ν) E µ = 2(1+ν) (121) (122) 75

76 1. C p ijkl = 3G σ ij σ kl σ 2 (123), A = 3G σ 2 C p ijkl = A σ ij σ kl (124)., C p ijkl = 1 2 ( ) C p ijkl + Cp ijlk = C p ijkl (125) C p ijkl = C p klij (126), C p ijkl 6 6. A σ 11 σ 11 A σ 11 σ 22 A σ 11 σ 33 A σ 11 σ 12 A σ 11 σ 23 A σ 11 σ 31 A σ 22 σ 11 A σ 22 σ 22 A σ 22 σ 33 A σ 22 σ 12 A σ 22 σ 23 A σ 22 σ 31 [ C p A σ ]= 11 A σ 33 σ 22 A σ 33 σ 33 A σ 33 σ 12 A σ 33 σ 23 A σ 33 σ 31 A σ 12 σ 11 A σ 12 σ 22 A σ 12 σ 33 A σ 12 σ 12 A σ 12 σ 23 A σ 12 σ 31 A σ 23 σ 11 A σ 23 σ 22 A σ 23 σ 33 A σ 23 σ 12 A σ 23 σ 23 A σ 23 σ 31 A σ 31 σ 11 A σ 31 σ 22 A σ 31 σ 33 A σ 31 σ 12 A σ 31 σ 23 A σ 31 σ 31 (127) 76

77 2, Kirchhoff Jaumann, D. t t ˆT (J) = C ep : D (C 4 ) (128) t tṡ = C : D. t tṡ = t t ˆT (J) D T T D (129) = C ep : D D T T D (130) t tṡij = C ep ijkl D kl D ik T kj T ik D kj (131) = C ep ijkl D kl δ il T kj D kl T ik δ jl D kl (132) { = C ep ijkl 1 2 (δ ijt kj + δ ik T lj ) 1 } 2 (T ikδ ij + T il δ jk ) D kl. (133) C ep ijkl = Cep ijkl 1 2 (δ ilt kj + δ ik T lj ) 1 2 (T ikδ jl + T lj δ jk ) (134) 77

78 3 t tṡij t tṡ = C : D. 2T T 21 0 T 31 0 T 22 0 T 21 T T 22 T 21 T 23 T 31 T 12 T (T T 22 ) 2 T T 23 0 T 23 T T (T 22 + T 33 ) 1 2 T 12 T 31 0 T T T (T 11 + T 33 ) (135) 78

79 4 Kirchhoff Jaumann t tṡ = t t ˆT (J) D T T D (136) = C ep : D D T T D (137) Truesdell t tṡ t tṡ = t t ˆT (O) = t t T (O) + (trd)t = t t T (J) D T T D + (trd)t = t t ˆT (J) D T T D t t T (J) (trd)t S 11 = T 11 (D 11 + D 22 + D 33 ) S 22 = T 22 (D 11 + D 22 + D 33 ) S 33 = T 33 (D 11 + D 22 + D 33 ) S 12 = T 12 (D 11 + D 22 + D 33 ) S 23 = T 23 (D 11 + D 22 + D 33 ) S 31 = T 31 (D 11 + D 22 + D 33 )

OHP.dvi

OHP.dvi 7 2010 11 22 1 7 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2010 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/18 3. 10/25 2, 3 4. 11/ 1 5. 11/ 8 6. 11/15 7. 11/22 8. 11/29 9. 12/ 6 skyline 10. 12/13

More information

2002 11 21 1 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2002 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture nabe@sml.k.u-tokyo.ac.jp 2 1. 10/10 2. 10/17 3. 10/24 4. 10/31 5. 11/ 7 6. 11/14

More information

Report98.dvi

Report98.dvi 1 4 1.1.......................... 4 1.1.1.......................... 7 1.1..................... 14 1.1.................. 1 1.1.4........................... 8 1.1.5........................... 6 1.1.6 n...........................

More information

OHP.dvi

OHP.dvi t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F

More information

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan

More information

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 12 12.1? finite deformation infinitesimal deformation large deformation 1 [129] B Bernoulli-Euler [26] 1975 Northwestern Nemat-Nasser Continuum Mechanics 1980 [73] 2 1 2 What is the physical meaning? 583

More information

all.dvi

all.dvi 72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

meiji_resume_1.PDF

meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E

More information

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2 No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

第5章 偏微分方程式の境界値問題

第5章 偏微分方程式の境界値問題 October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ

More information

note1.dvi

note1.dvi (1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

N/m f x x L dl U 1 du = T ds pdv + fdl (2.1)

N/m f x x L dl U 1 du = T ds pdv + fdl (2.1) 23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1) 24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/

More information

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),

More information

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d ) 23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

all.dvi

all.dvi 38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t

More information

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j = 72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................

More information

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0 1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :

More information

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e

More information

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j 6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..

More information

B ver B

B ver B B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................

More information

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t

More information

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................

More information

第10章 アイソパラメトリック要素

第10章 アイソパラメトリック要素 June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1

More information

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)

More information

5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i

5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i i j ij i j ii,, i j ij ij ij (, P P P P θ N θ P P cosθ N F N P cosθ F Psinθ P P F P P θ N P cos θ cos θ cosθ F P sinθ cosθ sinθ cosθ sinθ 5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3

More information

C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq

C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq 7 2003 6 26 ( ) 5 5.1 F K 0 (q 1,,q N,p 1,,p N ) (Q 1,,Q N,P 1,,P N ) Q i Q i (q, p). (1) P i P i (q, p), (2) (p i dq i P i dq i )df. (3) [ ] Q αq + βp, P γq + δp α, β, γ, δ [ ] PdQ pdq (γq + δp)(αdq +

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

73

73 73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ

More information

Auerbach and Kotlikoff(1987) (1987) (1988) 4 (2004) 5 Diamond(1965) Auerbach and Kotlikoff(1987) 1 ( ) ,

Auerbach and Kotlikoff(1987) (1987) (1988) 4 (2004) 5 Diamond(1965) Auerbach and Kotlikoff(1987) 1 ( ) , ,, 2010 8 24 2010 9 14 A B C A (B Negishi(1960) (C) ( 22 3 27 ) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 1 2 3 Auerbach and Kotlikoff(1987) (1987)

More information

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds 127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds

More information

Note.tex 2008/09/19( )

Note.tex 2008/09/19( ) 1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................

More information

(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008)

(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008) ,, 23 4 30 (i) (ii) (i) (ii) Negishi (1960) 2010 (2010) ( ) ( ) (2010) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 16 (2004 ) 2 (A) (B) (C) 3 (1987)

More information

量子力学 問題

量子力学 問題 3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,

More information

2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................

More information

chap10.dvi

chap10.dvi . q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l

More information

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information

K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0 79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

( )

( ) 7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................

More information

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1

More information

chap9.dvi

chap9.dvi 9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =

More information

1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,

1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t, 1 Gourgoulhon BSSN BSSN ϕ = 1 6 ( D i β i αk) (1) γ ij = 2αĀij 2 3 D k β k γ ij (2) K = e 4ϕ ( Di Di α + 2 D i ϕ D i α ) + α ] [4π(E + S) + ĀijĀij + K2 3 (3) Ā ij = 2 3Āij D k β k 2αĀikĀk j + αāijk +e

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

2 1 1 (1) 1 (2) (3) Lax : (4) Bäcklund : (5) (6) 1.1 d 2 q n dt 2 = e q n 1 q n e q n q n+1 (1.1) 1 m q n n ( ) r n = q n q n 1 r ϕ(r) ϕ (r)

2 1 1 (1) 1 (2) (3) Lax : (4) Bäcklund : (5) (6) 1.1 d 2 q n dt 2 = e q n 1 q n e q n q n+1 (1.1) 1 m q n n ( ) r n = q n q n 1 r ϕ(r) ϕ (r) ( ( (3 Lax : (4 Bäcklud : (5 (6 d q = e q q e q q + ( m q ( r = q q r ϕ(r ϕ (r 0 5 0 q q q + 5 3 4 5 m d q = ϕ (r + ϕ (r + ( Hooke ϕ(r = κr (κ > 0 ( d q = κ(q q + κ(q + q = κ(q + + q q (3 ϕ(r = a b e br

More information

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

More information

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................

More information

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6

More information

( ) ,

( ) , II 2007 4 0. 0 1 0 2 ( ) 0 3 1 2 3 4, - 5 6 7 1 1 1 1 1) 2) 3) 4) ( ) () H 2.79 10 10 He 2.72 10 9 C 1.01 10 7 N 3.13 10 6 O 2.38 10 7 Ne 3.44 10 6 Mg 1.076 10 6 Si 1 10 6 S 5.15 10 5 Ar 1.01 10 5 Fe 9.00

More information

20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................

More information

Microsoft Word - 11問題表紙(選択).docx

Microsoft Word - 11問題表紙(選択).docx A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx

More information

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3 2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q

More information

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, 変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +

More information

数学Ⅱ演習(足助・09夏)

数学Ⅱ演習(足助・09夏) II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w

More information

/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat

/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat / Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiation and the Continuing Failure of the Bilinear Formalism,

More information

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c)   yoshioka/education-09.html pdf 1 2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h

More information

D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η visco

D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η visco post glacial rebound 3.1 Viscosity and Newtonian fluid f i = kx i σ ij e kl ideal fluid (1.9) irreversible process e ij u k strain rate tensor (3.1) v i u i / t e ij v F 23 D v D F v/d F v D F η v D (3.2)

More information

5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0

5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0 5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â = Tr Âe βĥ Tr e βĥ = dγ e βh (p,q) A(p, q) dγ e βh (p,q) (5.2) e βĥ A(p, q) p q Â(t) = Tr Â(t)e βĥ Tr e βĥ = dγ() e βĥ(p(),q())

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

More information

linearal1.dvi

linearal1.dvi 19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352

More information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................

More information

弾性定数の対称性について

弾性定数の対称性について () by T. oyama () ij C ij = () () C, C, C () ij ji ij ijlk ij ij () C C C C C C * C C C C C * * C C C C = * * * C C C * * * * C C * * * * * C () * P (,, ) P (,, ) lij = () P (,, ) P(,, ) (,, ) P (, 00,

More information

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4 [2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =

More information

( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................

More information

,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,

,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 14 5 1 ,,,17,,,194 1 4 ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 1 4 1.1........................................ 4 5.1........................................ 5.........................................

More information

21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........

More information

arxiv: v1(astro-ph.co)

arxiv: v1(astro-ph.co) arxiv:1311.0281v1(astro-ph.co) R µν 1 2 Rg µν + Λg µν = 8πG c 4 T µν Λ f(r) R f(r) Galileon φ(t) Massive Gravity etc... Action S = d 4 x g (L GG + L m ) L GG = K(φ,X) G 3 (φ,x)φ + G 4 (φ,x)r + G 4X (φ)

More information

2003 12 11 1 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2003 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2002 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 9 2. 10/16 3. 10/23 ( ) 4. 10/30 5. 11/ 6

More information

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) 4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7

More information

ver Web

ver Web ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3

More information

τ τ

τ τ 1 1 1.1 1.1.1 τ τ 2 1 1.1.2 1.1 1.1 µ ν M φ ν end ξ µ ν end ψ ψ = µ + ν end φ ν = 1 2 (µφ + ν end) ξ = ν (µ + ν end ) + 1 1.1 3 6.18 a b 1.2 a b 1.1.3 1.1.3.1 f R{A f } A f 1 B R{AB f 1 } COOH A OH B 1.3

More information

4 Mindlin -Reissner 4 δ T T T εσdω= δ ubdω+ δ utd Γ Ω Ω Γ T εσ (1.1) ε σ u b t 3 σ ε. u T T T = = = { σx σ y σ z τxy τ yz τzx} { εx εy εz γ xy γ yz γ

4 Mindlin -Reissner 4 δ T T T εσdω= δ ubdω+ δ utd Γ Ω Ω Γ T εσ (1.1) ε σ u b t 3 σ ε. u T T T = = = { σx σ y σ z τxy τ yz τzx} { εx εy εz γ xy γ yz γ Mindlin -Rissnr δ εσd δ ubd+ δ utd Γ Γ εσ (.) ε σ u b t σ ε. u { σ σ σ z τ τ z τz} { ε ε εz γ γ z γ z} { u u uz} { b b bz} b t { t t tz}. ε u u u u z u u u z u u z ε + + + (.) z z z (.) u u NU (.) N U

More information

講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K

講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K 2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =

More information

i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................

More information

keisoku01.dvi

keisoku01.dvi 2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [ 3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

More information