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 ALE 11. 12/20 3

. 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 (1) 4

, e i,, J =detf. F x i e i e j X j (2) C F T F (3) B F F T (4) E 1 (C I) 2 (5) Π JF 1 T (6) S JF 1 T F T (7) 5

1, W. S ij = W (8) E ij E = 1 2 (C I) S ij =2 W C ij (9) W C., W C. I C trc (10) II C 1 { (trc) 2 tr(c 2 ) } (11) 2 III C det C (12) S ij =2 ( W I C + W II C + W ) III C I C C ij II C C ij III C C ij (13) 6

2 I C = δ ij C ij (14) II C = I C δ ij C ij C ij (15) III C = III C (C 1 ) ij C ij (16) S ij =2 {( W + W ) I C δ ij W C ij + W } III C (C 1 ) ij I C II C 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 (17) (18) T B. 7

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 (19) (20) 8

Mooney-Rivlin 1 W Mooney-Rivlin. W M c 1 (I C 3) + c 2 (II C 3) (21), 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 (22), C ij = δ ij, p 2c 1 +4c 2. T ij = S ij = 0 (23) S ij = pδ ij +(2c 1 +4c 2 )δ ij (24), W M. W M R c 1 (ĨC 3) + c 2 (ĨI C 3) (25) Ĩ C I C III C 1 3 (26) ĨI C II C III C 2 3 (27) 9

Mooney-Rivlin 2 ĨC, ĨI C (reduced invariants). W M R 2 Piola-Kirchhoff W M R I C = W M R ĨC ĨC I C = c 1 III 1 3 C (28) W M R II C = W M R ĨI C W M R III C = W M R ĨC ĨI C II C = c 2 III 2 3 C (29) ĨC III C + W M 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 (30) { S ij = p(c 1 ) ij +2 (c 1 + c 2 I C )δ ij c 2 C ij + ( 13 c 1I C 23 ) } c 2II C (C 1 ) ij (31), p. T ij = S ij = 0 (32) S ij = pδ ij (33) 10

Mooney-Rivlin 3,.,. F. F = J 1 3F (34), F Flory, det F =1. Cauchy-Green C. C = F T F (35) C 1, 2, ĨC =3, ĨI C =3. 11

Mooney-Rivlin, - S. Mooney-Rivlin c 1, c 2., I C, II C 2, 3. 1 0.8 Stress[MPa] 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 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 (36). 12

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 (37) 13

1, c 1,c 2.. 1 1 1 1/ l x 2 l 1/ l x 3 x 1 3: 14

, F, B, II B l 0 0 F = 0 1/ l 0 0 0 1/ (38) l l 2 0 0 B = FF T = 0 1/l 0 (39) 0 0 1/l 1/l 2 0 0 B 1 = 0 l 0 (40) 0 0 l II B =2l + 1 l 2 (41) 15

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 16

2 Cauchy { W H T kl = pδ kl +2 R (2l + 1 H } II B l )+ W R III B δ kl + W H R I B l 2 0 0 0 1/l 0 W R H II B 0 0 1/l 1/l 2 0 0 0 l 0 0 0 l x 1, T 22 = T 33 =0 { 1 WR H p =2 l I B T 11 =2 { (l 2 1 l ) W H R I B +(l + 1 l 2) W H R II B + W R H } III } B +(l 1 l 2) W H R II B (42) (43) l =1+ε ε 2 T 11 =6(c 1 + c 2 )ε (44) 6(c 1 + c 2 ) E. 17

3,. u x 2 1 1 x 1 x 3 1 4: F, B,I B,II B 18

1 u 0 F = 0 1 0 (45) 0 0 1 1+u 2 u 0 B = u 1 0 (46) 0 0 1 1 u 0 B 1 = u 1+u 2 0 (47) 0 0 1 I B =trb =3+u 2 (48) II B = 1 2 { (trb) 2 tr(b 2 ) } =3+u 2 (49) 19

4 Cauchy { W H T kl = pδ kl +2 R (3 + u 2 )+ W H } R II B III B δ kl + W H R I B 1+u 2 u 0 u 1 0 W R H II B 0 0 1 1 u 0 u 1+u 2 0 0 0 1 T 33 =0 { W H p =2 R I B +(2+u 2 ) W H R II B + W R H } III B (50) ( W H T 12 = T 21 =2u R + W H ) R I B II B u 2, (51) T 12 = T 21 =2(c 1 + c 2 )u (52), u., 2(c 1 + c 2 ) G. 20

. 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 (53) ( ) S F T T N = t (54) C = F T F (55) S ij = p(c 1 ) ij +2 W C ij (56) III C = 1 (57) (53), (54), (55), (56) ( ) (57)., W, (53) (57). 21

W Φ. Φ = W dω t u ds ρ 0 g u dω (58) Ω Ω λ Lagrange, Φ. Φ = Φ + λg(iii C )dω (59) Ω g, g(iii C ),III C =1 g =0, =1. III C, 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 ( λ g ) δc ij dω ) δc ij + δλg dω C ij t δu ds Ω Ω Ω Ω t δu ds ρ 0 g δu dω Ω ρ 0 g δu dω + δλ g dω = 0 (60) Ω (60),. (60),. 22

Lagrange 1 (59) Lagrange λ, (56) p., (60). δc ij = δf ki F kj + F ki δf kj C, W/ C ij, g/ C ij i, j. (60) 1 ( W + λ g ) δc ij dω Ω C ij C ( ij W = + λ g ) (δf ki F kj + F ki δf kj )dω Ω C ij C ( ij W = 2 + λ g ) δf ki F kj dω ( ) (61) C ij C ij Ω, u = x X δu = δx δf ki = δx k X i = δu k X i (62) ( ) = Ω { ( δu k W F kj 2 + λ g )} dω (63) X i C ij C ij 23

Lagrange 2 X i { ( W 2 = X i + λ g C ij { 2 C ij ) F kj δu k } ( W C ij + λ g C ij ) } { ( W F kj δu k + 2 + λ g ) } δuk F kj (64) C ij C ij X i ( ) = Ω X i { ( W 2 + λ g ) } F kj δu k dω C ij C ij Ω X i { ( W 2 + λ g ) } F kj δu k dω (65) C ij C ij (65) 1 V divb dv = n b ds (66) S ( ) = Ω (60) Ω [ X i ( W n i {2 + λ g ) } F kj δu k ds C ij C ij Ω { ( W 2 + λ g ) } ] F kj + ρ 0 g k δu k dω C ij C ij + Ω X i { ( W 2 + λ g ) } F kj δu k dω (67) C ij C ij { ( W [n i 2 + λ g ) } ] F kj t k δu k ds + δλ g(iii C )dω = 0 (68) C ij C ij Ω 24

Lagrange 3 (68) δu V, δλ Q, (69), (70), (71). { ( W 2 + λ g ) } F kj + ρ 0 g k = 0 (69) X i C ij C ( ij W n i {2 + λ g ) } F kj t k = 0 (70) C ij C ij g(iii C ) = 0 (71), (69) (53), (70) (54), (71) (57). ( W S ij =2 + λ g ) (72) C ij C ij g III C, (16) g = g III C C ij III C C ij = g ( III ) C C 1 (73) III ij C (72). (56) S ij =2 W C ij +2λ g III C III C ( C 1 ) ij (74) p = 2λ (75), λ. 25

, (53) (57),. find (u,λ) (V, Q) such that ( W + λ g ) δc ij dω = t k δu k ds + ρ 0 g k δu k dω Ω C ij C ij Ω Ω (76) δλg dω = 0 (77) Ω for (δu,δλ) (V, Q), λ = 1 2 p 26

Newton-Raphson,. Ω.. Ω = e Ω e (78),,. dω = dω (79) Ω e Ω e ds = ds (80) Ω e Ω e u N (i), u i. u i = N (n) u (n) i (81), u (i) i, (n). Lagrange λ M (m), λ., λ (m). λ = M (m) λ (m) (82) 27

Ω ( W C ij + λ g C ij ) δc ij dω = δλg dω =0 Ω Ω t k δu k ds + Ω ρ 0 g k δu k dω Ω δe ij S ij dω = δr (83) 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 (84),. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T (85) {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (86) 28

2 (83), δe ij S ij dω = {δe} T {S} dω Ω Ω = {δe} T {S} dω = δr e Ω e δe ij = 1 ( δui + δu j + δu k u k + u ) k δu k 2 X j X i X i X j X i X j, (87),. [Z 1 ] 1+ u 1 u X 1 0 0 2 u X 1 0 0 3 X 1 0 0 u 0 1 X 2 0 0 1+ u 2 u X 2 0 0 3 X 2 0 u 0 0 1 u X 3 0 0 2 X 3 0 0 1+ u 3 X 3 u 1 X 2 1+ u 1 X 1 0 1+ u 2 u 2 u X 2 X 1 0 3 u 3 X 2 X 1 0 u 0 1 u 1 u X 3 X 2 0 2 X 3 1+ u 2 X 2 0 1+ u 3 u 3 X 3 X 2 u 1 X 3 0 1+ u 1 u 2 u X 1 X 3 0 2 X 1 1+ u 3 u X 3 0 3 X 1 } { δu X { δu1 X 1 δu 1 X 2 δu 1 X 3 δu 2 X 1 δu 2 X 2 δu 2 X 3 δu 3 X 1 δu 3 X 2 δu 3 X 3 {δe} =[Z 1 ] { } δu X (88) } T (89) (90) 29

3 { } δu X δu 1 X 1 δu 1 X 2 δu 1 X 3 δu 2 X 1 δu 2 X 2 δu 2 X 3 δu 3 X 1 δu 3 X 2 δu 3 X 3 = N (1) X 1 N (1) X 2 N (1) X 3 N (1) X 1 N (1) u i X j = N(n) X j u (n) i (91) N (n) X 1 N (n) X 2 N (n) X 3 N (n) X 1 X 2 N (n) N (1) X 3 N (1) X 1 N (1) X 2 N (1) X 3 X 2 N (n) X 3 N (n) X 1 N (n) X 2 N (n) X 3 δu (1) 1 δu (1) 2 δu (1) 3. δu (n) 1 δu (n) 2 δu (n) 3 (92). 9 3n [Z 2 ], { } δu =[Z 2 ]{δu (n) } (93) X, [B] [B] [Z 1 ][Z 2 ] (94) 30

{δe} =[B]{δu (n) } (95) 31

4 [B] [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 (96) 6 3 [B (n) ], [ ] [B] = [B (1) ] [B (n) ]., e Ω e {δe}{s} dω = e ] [{δu (n) } T [B] T {S} dω Ω e. (??), (98), (76) ] [{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 (97) (98) (99). 32

, (77). {M} = {M (1) M (2) M (m) } T (100) {δλ (m) } = {δλ (1) δλ (2) δλ (m) } T (101), (77). δλgdω = δλg dω (102) Ω e Ω e = ] [{δλ (m) } T {M}g dω = 0 (103) e Ω e 33

, {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 (104), (99), (103). [ ] ] [{δu (n) δλ (m) } T [B] T {S} dω e Ω e {M}g = [ [ ] [{δu (n) δλ (m) } T [N] e Ωe T {t} ds + 0 Ω e [ ] ]] ρ 0 [N] T {g} dω 0,, (105) Q = F = u = [ ] [B] T {S} dω (105) Ω e {M}g [ ] [N] T [ ] {t} ρ 0 [N] T {g} ds + dω (106) Ω e 0 Ω e 0 } {u (n) λ (m) (107) [ T δuh (Q(u h ) F ) ] = 0 (108)., e 34

find u h V h such that [ T δuh (Q(u h ) F ) ] = 0 (109) e for δu h V h, Newton-Raphson. 35

Newton-Raphson, K = Q u, dq dt = Q du u dt = K u (110), (76), (77),.,. (76) Ω {( W + λ g ) ( W δc ij + + λ g ) C 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 (111) C ij 36

, (77) δλ ġ dω = Ω Ω δλ g C kl Ċ kl dω (112) 37

(111). 2 Ċ kl =2Ėkl (113). ( 2 W 2 ) g D ij kl =4 + λ (114) C ij C kl C ij C kl, (72), (??), (114) (111) ( δe ij D ij kl Ė kl + δf ki S ij F kj + δe ij 2 g ) λ dω (115) C ij. Ω S ij D ij kl Ė kl (116). Sij, δe ij i, j, δe ij D ij kl Ė kl = δe ij Sij = {δe 11 δe 22 δe 33 δ2e 12 δ2e 23 δ2e 31 } { } T S11 S22 S33 S12 S23 S31 (117) S ij, Ė kl k, l S ij = D ij 11 Ė 11 + D ij 22 Ė 22 + D ij 33 Ė 33 + 1 2 (D ij 12 + D ij 21 )2Ė12 + 1 2 (D ij 23 + D ij 32 )2Ė23 + 1 2 (D ij 31 + D ij 13 )2Ė31 (118) 38

3 C ij kl 1 2 (D ij kl + D ij lk ) (119), S. S 11 C 11 11 C 11 22 C 11 33 C 11 12 C 11 23 C 11 31 S 22 C 22 11 C 22 22 C 22 33 C 22 12 C 22 23 C 22 31 S 33 = C 33 11 C 33 22 C 33 33 C 33 12 C 33 23 C 33 31 S 12 C 12 11 C 12 22 C 12 33 C 12 12 C 12 23 C 12 31 S 23 C 23 11 C 23 22 C 23 33 C 23 12 C 23 23 C 23 31 S 31 C 31 11 C 31 22 C 31 33 C 31 12 C 31 23 C 31 31 Ė 11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31 (120) C ijkl 6 6 [D 1 ].C ijkl, ij, kl, [D 1 ].,, { S} = { } T S11 S22 S33 S12 S23 S31 (121) {Ė} = { T Ė 11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31} (122), δe ij D ijkl Ė kl dω = δe ij Sij dω Ω Ω = {δe} T [D 1 ]{Ė} dω Ω = {δe} T [D 1 ]{Ė} dω (123) e Ω e 39

4 { } u (n), (95), (124). { } T u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3 (124) {Ė} { } =[B] u (n) (125), (115) 1 δe ij D ijkl Ė kl dω = Ω e. [ { δu (n) } T Ω e [B] T [D 1 ][B]dΩ { } ] u (n) (126) 40

, δf ki S ij 1 F kj = {δf 11 δf 12 δf 13 } [σ] = +{δf 21 δf 22 δf 23 } +{δf 31 δf 32 δf 33 } S 11 S 12 S 13 S 21 S 22 S 23 S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 (127) (128) S 31 S 32 S 33 [σ] [Σ] = [σ] (129) [σ] {δf} = {F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T (130) { F } = { F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T (131) 41

, δf ki S ij F kj = {δf} T [Σ]{ F } (132) 42

2,, (93) δf ij = δx i = δu i X j X j (133) F ij = ẋ i = u i X j X j (134) { } δu } {δf} = =[Z 2 ] {δu (n) X { F { } } =[Z 2 ] u (n) (135) (136), δf ki S ij F kj = { } T { } δu (n) [Z2 ] T [Σ][Z 2 ] u (n) (137), [A ij ]= [A] = { N (i) X 1 N (i) X 2 N (i) } S 11 S 12 S 13 S X 21 S 22 S 23 3 S 31 S 22 S 33 [A 11 ] [A 12 ]... [A 1n ] [A 21 ]......... [A n1 ]...... [A nn ] N (j) X 1 N (j) X 2 N (j) X 3 1 1 1 (138) (139) 43

3. [Z 2 ] T [Σ][Z 2 ]=[A] (140), (115) 2 δf ki S ij F kj dω = Ω e. [ { δu (n) } T Ω e [A]dΩ { } ] u (n) (141) 44

1 { λ } { (m) λ(1) λ(2) λ } T (m) (142) { {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 (143), (115) 3 δe ij 2 g λ dω = C ij. Ω e = e {δe} T {D 2 }[M] Ω e [ { } T δu (n) { λ(m)} dω Ω e [B] T {D 2 }[M]dΩ { } ] λ(m) (144) 45

( 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) (145) Ω e. 46

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} (146) δλ g Ċ kl dω = [ { } T δλ (m) [M] T {D 2 } T [B]dΩ Ω C kl e Ω e. { } ] u (n) (147) 47

, { } (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) (148) [K 1 ] [B] T [D 1 ][B]+[A] (149) [H] [B] T {D 2 }[M] (150) (145), (147) [ { } [ ] δu (n) δλ (m) [K 1 ] [H] dω [H] T 0 e Ω e, [K] =. Ω e { ] u (n) λ(m)} (151) [ ] [K 1 ] [H] dω (152) [H] T 0 48

Mooney-Rivlin 1 α Mooney-Rivlin W S = W H + α 2 W V (III C ) 2 (153) W V (III C ) III C III C =1 W V W V =2(J 1), III C 1 =0 WV III C =1 α 49

Mooney-Rivlin 2. 1 1 1 1+δ x 2 1+ε 1+δ x 3 x 1, F, B, II B ε, δ l + ε 0 0 F = 0 1+δ 0 0 0 1+δ l +2ε 0 0 B = FF T = 0 1+2δ 0 0 0 1+2δ l 2ε 0 0 B 1 = 0 1 2δ 0 0 0 1 2δ (154) (155) (156) 50

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 (157) W S I C = WS ĨC (158) ĨC I C { = c 1 +2c 3 (ĨC 3) + c 4 (ĨI C 3) (159) +3c 6 (ĨC 3) 2 +2c 7 (ĨC 3)(ĨI C 3) + c 8 (ĨI C 3) 2} III 1/3 C (160) W S II C = WS ĨI C = ĨI C II C (161) { c 2 + c 4 (ĨC 3) + 2c 5 (ĨI C 3) (162) +c 7 (ĨC 3) 2 +2c 8 (Ĩ 3)(ĨI 3) + 3c 9(ĨI 3)2} III 2/3 C (163) 51

W S = WH III C ĨC ĨC + WH ĨI C I C ĨI + αw V WV (164) 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 (165) = 1 3 2 { c 2 + c 4 (ĨC 3) + 2c 5 3 (ĨI C 3+c 7 (ĨC 3) 2 (166) } +2c 8 (ĨC 3)(ĨI C 3) + 3c 9 (ĨI C 3) 2 II C III 5/3 (167) + αw V WV III (168) 52

Mooney-Rivlin 4 Ĩ C = I C III 1/3 C =(3+2ε +4δ) (1 23 ε 43 ) δ (169) = 3 (170) ĨI C = II C III 2/3 C =(3+4ε +8δ) (1 43 ε 83 ) δ (171) = 3 (172) W V W V = III C 1 αw V W V = α(iii 1) III (173) = α(2ε +4δ) (174) W V =2(J 1) αw V W V III = α2(j 1) 1 J (175) = α2(ε +2δ)(1 ε 2δ) (176) = α(2ε +4δ) (177) αw V W V III 53 = α(2ε +4δ) (178)

W S = c 1 (1 2 I C 3 ε 4 δ) 3 (179) W S = c 2 (1 4 II C 3 ε 8 δ) 3 (180) W S = (c 1 +2c 2 )(1 2ε 4δ)+2α(ε +2δ) III C (181) T kl T 22 = T 33 =0 δ = 3α (c 1 + c 2 ) 6α +(c 1 + c 2 ) ε (182) ν E α E =6(c 1 + c 2 ) ν = 3α (c 1 + c 2 ) 6α +(c 1 + c 2 ) (183) T 11 = 36(c 1 + c 2 )α 6α +(c 1 + c 2 ) ε (184) E = 36(c 1 + c 2 )α 6α +(c 1 + c 2 ) (185) 54

κ = E 3(1 2ν) =4α (186) 55

1 (R ) Φ= WdΩ+ α (W V ) 2 R (187) Ω 2 Ω α selective/reduced integration V Q αw V Q λ αw V λ P Ω ( W V λ ) δλdω =0 δλ Q (188) α P (αw V )=λ (189) 56

Q 57

2 P Φ = WdΩ+ α (PW V ) 2 R (190) Ω 2 Ω U V δ Φ W = δc ij dω+α (PW V )P (δw V )dω δr = 0 (191) Ω C ij Ω u λ Q Ω ( W V λ ) δλdω =0 δλ Q (192) α 58

3 u V,λ Q α Ω (PW V )P (δw V )dω = Ω λδw V dω δu V (193) δ Φ = Ω W δc ij dω+ λδw V dω δr = 0 (194) C ij Ω ( W V λ ) δλdω = 0 (195) α Ω W V III ( W + λ W V ) δc ij dω=δr (196) Ω C ij C ij ( W V λ ) δλdω = 0 (197) α Ω 59

Lagrange Ω ( W C ij + λ g C ij 4 W V λ α g(= W V ) Lagrange Q = Q = ) δc ij dω = t k δu k ds + ρ 0 g k δu k dω (198) Ω Ω δλg dω = 0 (199) Ω Ω e [ Ω e [ ] [B] T {S} dω (200) {M}g [B] T {S} {M} ( W V λ/α ) ] dω (201) Lagrange [ ] [K 1 ] [H] [K] = dω (202) [H] T 0 [K] = Ω e Ω e [ ] [K 1 ] [H] dω (203) [H] T [G] 60

α [G] = 1 α [M]T [M] (204) 61

,,,, 62

(Hooke, ) t t t,,,,. 63

B A e e e p e (A), (B),,.,,,. 64

A B e e e p e e e e e p. e = e e + e p (205) σ E. σ = E(e e p ) (206), Hooke. σ ij = C e ijkl(e kl e p kl) (207), σ ij,e ij,e p ij 2 Cauchy,,, C e ijkl 4 Hooke. 65

, Hooke. σ ij = C e ijkl(e kl e p kl) (208),. σ ij = C e ijkl(ė kl e p kl) (209),. σ ij = C ep ijklė kl (210), (flow rule) 66

:. :. :. A B e e e p e 67

3 2, 9, vonmises Tresca B A e e e p e 68

,, 3 A,, B. von Mises. A B e e e p e 69

Mises Mises σ σ ij σ ij σ = ( ) 1 3 2 σ ijσ 2 ij (211) σ ijσ ij =σ 2 2 11 + σ 12 + σ +σ 2 2 21 + σ 22 + σ +σ 2 2 31 + σ 32 + σ 2 13 2 23 2 33 (212) (213) (214) (215), σ ij. σ ij =σ ij 1 3 σ kkδ ij (216) =σ ij 1 3 (σ 11 + σ 22 + σ 33 ) δ ij (217) 70

F = σ σ y (218) σ y F =0 ( ) σ y σ ij σ ij 71

(associated flow rule),, λ Ψ ė p ij = λ Ψ σ ij (219) (associated flow rule), ė p F ij = λ (220) σ ij F = σ σ y (221) 72

(normality rule) σ ij / t ( ) F =0, F =0, ė p ij ė p σ ij ij t 0, F = λ σ ij (222) F σ ij = λ σ ij t (223) ė p ij σ ij = λ F (224) ė p ij σ ij = λ F = 0 (225), σ ij ė p ij 73

von Mises σ = ( ) 1 3 2 2 σ ij σ ij F = σ σ y F = λ σ ij ė p ij A B e p e e e 74

, Hooke. σ ij = C e ijkl(e kl e p kl) (226),. σ ij = C e ijkl(ė kl e p kl) (227),. σ ij = C ep ijklė kl (228) 75

1 F =0 F =0 F = F σ ij = 0 (229) σ ij σ ij = C e ijkl(ė kl ė p ( kl ) ) (230) = C e F ijkl ė kl λ (231) σ kl F/ σ ij F σ ij = F C e ijklė kl F C e F ijkl λ (232) 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 (233) σ kl 76

2 λ σ ij = C e ijkl = σ ij = C e ijkl(ė kl ė p ( kl ) ) (234) = C e F ijkl ė kl λ (235) σ kl ( λ = ( F σ ij C e ijklė kl F σ ij C e ijkl F (236) σ kl ė kl F C e σ ab abcdė cd F F σ C e abcd F σ ab σ kl cd C e ijkl Ce ijcd F σ cd F σ ab C e abkl F σ C e abcd F ab σ cd ) ) (237) ė kl (238) 77

F/ σ ij 3, F = 3 σ ij 2 σ σ ij (239) σ ij = σ ij Ce ijklė kl σ ij Ce ijklσ kl λ = 2 σ 3 (240) ( ) C e ijkl Ce ijcdσ cd σ ab C e abkl σ ab C e ė kl abcdσ cd (241) 78

4 Hooke C e ijkl λ, μ Lamé C e ijkl = λδ ij δ kl +2μδ ik δ jl (242) μ G, σ ij = ( λ = σ klėkl σ C e ijkl 3Gσ ijσ kl σ 2 ) (243) ė kl (244) 79

von Mises σ = ( )1 3 2 σ ijσ 2 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 (245) 80

1, Hooke. Hooke. F Cauchy T (elastic material). T (t) =f(f (t)) (246) f. f(f )=f(q F )=Q f(f ) Q T (247) F, F O,O, O O Q., P f(f )=f(f P ) (248). V. T = f(v ) (249). f(v )=f(q V Q T )=Q f(v ) Q T (250) 81

V, V O,O, O O Q. f(v ) (isotropic tensor function). (250) T, V, T = f(v )=φ 0 I + φ 1 V + φ 2 V 2 (251)., φ i (i =0, 1, 2) V. (representation theorem). (249) V = B 1/2. T = g(b) (252) g(b )=g(q B Q T )=Q g(b) Q T (253), g(b),., B. T = ψ 0 I + ψ 1 B + ψ 2 B 2 (254) = ξ 0 I + ξ 1 B + ξ 1 B 1 (255) Hooke. V I + 1 {u x + x u} (256) 2 82

, E (L), (251) E (L) E (L) = 1 {u x + x u} (257) 2 T =(φ 0 + φ 1 + φ 2 )I +(φ 1 +2φ 2 )E (L) (258) = η 0 I + η 1 E (L) (259), η 0, η 1 E (L). T E (L), Hooke., λ, μ Lamé. T =(λtre (L) )I +2μE (L) (260) 83

, Hooke. 2 T = f(v ), T = g(b), B Almansi A A = 1 (I B) (261) 2, T = h(a) (262). h(a )=h(q A Q T )=Q h(a) Q T (263) A, A O,O, O O Q. h(a),. T = h(a) =ζ 0 I + ζ 1 A + ζ 2 A 2 (264) Hooke T =(λtra)i +2μA (265). A E (L) (266), λ, μ Lamé. 84

3 T =(λtra)i +2μA. Ṫ, Ȧ T = QT Q T (267) Ṫ = QT Q T + QṪQT + QT Q T (268) W, T, A T, Å. T = Ṫ W T + T W (269) Å = Ȧ W A + A W (270) Jaumann T (J) = Ṫ W T + T W (271) Oldroyd T (O) = Ṫ L T T LT (272) Cotter Rivlin T (C) = Ṫ + LT T + T L (273) Green Naghdi T (G) = Ṫ Ω T + T Ω (Ω = Ṙ RT ) (274). T =(λtrå)i +2μÅ (275) 85

,, (, ), F t (τ) R t (τ), U t (τ) I (276),. T (J) T (O) T (C) T (G) (277) Å (J) Å(O) Å(C) Å(G) (278), T (J) = Ṫ W T + T W (279) T (J) = T (O) + D T + T D (280) T (J) = T (C) D T T D (281) T (G) = Ṫ Ω T + T Ω (282) W Ω (283), Å(C) = D, T =(λtrd)i +2μD (284). T Kirchhoff ˆT t (τ) =J t (τ)t (τ) ˆT t (t) =(λtrd)i +2μD (285) 86

., ˆT t (t) (J) = T (J) + T trd (286) ˆT t (t) (O) = T (O) + T trd (287) ˆT t (t) (C) = T (C) + T trd (288) 87

. v v e v p v = v e + v p (289), L D. D = D e + D p (290) σ ij T ij, e p ij D p ij., C ep ijkl. T ij = C ep ijkld kl (291),,, (291), Cauchy Kirchhoff ˆT ij = C ep ijkld kl (292). Kirchhoff, Jaumann. 88

. C ep ijkl = ( C ijkl 3G T T ) ij kl σ 2 (293) T ij, T ij = T ij 1 3 T kk δ ij. pe = T kl D kl σ (294) λ = T kl D kl σ (295) 89

1,,., t t e p ij = t σ ij = = t 0 t 0 0 τė p ij dτ (296) τ σ ij dτ (297) τ C ep ijkl τ ė kl dτ (298) t t σ ij t σ ij = τ C ep ijkl τ ė kl dτ (299) t. t C ep ijkl, (298), (299),. 90

2,. Kirchhoff.,, (305) 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 (300) {τ ˆTτ ij (tr τ D) τ T ij } dτ (301) {τ ˆTτ ij + τ W ik τ ˆTτ kj τ ˆTτ ik τ W kj (tr τ D) τ T ij } {τ ˆTτ ij + τ W ik τ T kj τ T ik τ W kj (tr τ D) τ T ij } dτ (302) dτ (303) { 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 (304) = t T ij + {t C ep ijkl t D kl + t W ik t T kj t T ik t W kj (tr t D) t T ij } Δt (305) 91

. V,v, S, s. s t, u, v g.. T Cauchy. x T + ρg = 0 (306) T T n = t (307) u = u (308) D ij = 1 ( ui + u ) j (309) 2 x j x i ˆT ij = C ep ijkl D kl, T ij = t 0 T ij dt (310) 92

T : δa (L) dv = v δv t w ds + δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i updated Lagrange v v ρg w dv (311) (312) δa ij T ij dv = {δu} T [B] T {T } dv (313) v Q = [B] T [T ]dv (314) 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} (315) 93

1,. Ṡ ij = C ijkl D kl (316) D kl = D lk, C ijkl = 1 2 (C ijkl + C ijlk ) (317). 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 [ C], Cijkl = C klij,. D 11 D 22 D 33 2D 12 2D 23 2D 31 (318) 94

Hooke 1 Cijkl e = λδ ijδ kl +2μδ ik δ jl (319) C ijkl e = λδ ij δ kl + μ (δ ik δ jl + δ il δ jk ) (320) C e klij = λδ klδ ij + μ (δ ki δ lj + δ kj δ li ) (321) = λδ ij δ kl + μ (δ ki δ lj + δ il δ jk ) (δ mn = δ nm ) (322) = C e ijkl (323) [ C e ]= λ +2μ λ λ 0 0 0 λ λ+2μ λ 0 0 0 λ λ λ+2μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ Lamé λ, μ E, ν. νe λ = (1 + ν)(1 2ν) E μ = 2(1+ν) (324) (325) (326) 95

1. C p ijkl = 3G σ ij σ kl σ 2 (327),. A = 3G σ 2 (328) C p ijkl = A σ ij σ kl (329), C p ijkl = 1 2 ( ) C p ijkl + Cp ijlk = C p ijkl (330), C p ijkl C p ijkl = C p klij (331) 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 ]= 33 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 (332) 96

2, Kirchhoff Jaumann, D. t t ˆT (J) = C ep : D (C 4 ) (333) tṡ t = C : D. t tṡ = t t ˆT (J) D T T D (334) = C ep : D D T T D (335) t tṡij = C ep ijkl D kl D ik T kj T ik D kj (336) = C ep ijkl D kl δ il T kj D kl T ik δ jl D kl (337) { = C ep ijkl 1 2 (δ ijt kj + δ ik T lj ) 1 } 2 (T ikδ ij + T il δ jk ) D kl (338). C ep ijkl = Cep ijkl 1 2 (δ ilt kj + δ ik T lj ) 1 2 (T ikδ jl + T lj δ jk ) (339) 97

3 t tṡij t tṡ = C : D. 2T 11 0 0 T 21 0 T 31 0 T 22 0 T 21 T 23 0 0 0 2T 22 T 21 T 23 T 31 T 12 T 12 0 1 2 (T 1 11 + T 22 ) 2 T 31 1 2 T 23 0 T 23 T 23 1 2 T 31 1 2 (T 22 + T 33 ) 1 2 T 12 T 31 0 T 31 1 2 T 23 1 2 T 12 1 2 (T 11 + T 33 ) (340) 98

4 Kirchhoff Jaumann Truesdell t tṡ t tṡ = t t ˆT (J) D T T D (341) = C ep : D D T T D (342) t tṡ = t t ˆT (O) (343) = t t T (O) + (trd)t (344) = t t T (J) D T T D + (trd)t (345) = t t ˆT (J) D T T D (346) t t T (J) (trd)t S 11 = T 11 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 S 22 = T 22 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 S 33 = T 33 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 S 12 = T 12 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 (347) S 23 = T 23 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 S 31 = T 31 (D 11 + D 22 + D 33 ) 1 1 1 0 0 0 99

Mooney-Rivlin 6(c 1 + c +2), 2(c 1 + c 2 ) OHP p.25 Lagrange (152) (203) (245) (293) (305) (306) (347) 100

Program hyper.tar.gz (Linux intel fortran complier ) hyper Run 1.in Run 1 1.in Mooney-Rivlin 101