~nabe/lecture/index.html 2

2001 12 13 1

http://www.sml.k.u-tokyo.ac.jp/ ~nabe/lecture/index.html nabe@sml.k.u-tokyo.ac.jp 2

1. 10/ 4 2. 10/11 3. 10/18 1 4. 10/25 2 5. 11/ 1 6. 11/ 8 7. 11/15 8. 11/22 9. 11/29 10. 12/ 6 1 11. 12/13 2 12. 12/20 13. 1/10 14. 1/17 ALE 1 15. 1/24 ALE 2 3

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

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

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

. 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

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

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

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

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 2 3 11

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

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

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. 14

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

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

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

2 Cauchy { W H T kl = pδ kl +2 R (2l + 1 } II B l )+ WH R III B + WH R I B l 2 0 0 0 1/l 0 WH R II B 0 0 1/l δ kl 1/l 2 0 0 0 l 0 0 0 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

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

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 0 0 1 0 0 1 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

. 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

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

, (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

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

( 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

[B] [B (n) ] u 1 N (n) X 2 X 2 + 4 ( 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

, (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

, { {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

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

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

( 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

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

, { (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

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

Mooney-Rivlin 2. 1 1 1 1+δ x 2 1+ε 1+δ x 3 x 1, F, B, II B 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 ε, δ 35

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) 2 2 3 } +2c 8 (ĨC 3)(ĨI C 3) + 3c 9 (ĨI C 3) 2 II C III 5/3 + αw V WV III 36

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

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

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

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

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

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

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

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

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

, 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

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

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

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

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 2 2 21 + σ 22 + σ +σ 2 2 31 + σ 32 + σ 2 13 2 23 2 33, σ ij. σ ij =σ ij 1 3 σ kkδ ij =σ ij 1 3 (σ 11 + σ 22 + σ 33 ) δ ij 51

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

(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

(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

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

, 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

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

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

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

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

von Mises σ = ( )1 3 2 2 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

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

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

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

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

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

,. 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

. 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

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

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

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

. 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

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

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

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 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 (120) Lamé λ, µ E, ν. νe λ = (1 + ν)(1 2ν) E µ = 2(1+ν) (121) (122) 75

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

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

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 ) (135) 78

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 ) 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 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 79