2002 11 21 1

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1. 10/10 2. 10/17 3. 10/24 4. 10/31 5. 11/ 7 6. 11/14 7. 11/21 8. 11/28 11/21 9. 12/ 5 10. 12/12 11. 12/19 skyline 12. 1/ 9 ALE 13. 1/16 14. 1/23 3

, A [B]. [B] A Ω, Ω Ω, Ω Ω D. t, ρg, u V., ρ, g, V. A t Ω D ρg Ω 1: 4

. [B] t, g, u V. [1 ] (Cauchy 1 ) [2 ] x T + ρg =0 X (S F T) + ρ 0 g =0 u = u on Ω D T T n = t on Ω Ω D ( S F T ) T N = t [3 ] [4 ] [1], [2] [4] [3] [4] 5

1 T T, u U. Ť T, ǔ U,. ( x Ť + ρg) ǔ dv =0 v,., s t,s u t, u s, s = s t + s u. Ť :(ǔ x )dv = t ǔ ds + n Ť u ds + v s t s u, s u w =0 w W. ǔ U, w W ǔ + w U. Cauchy T,. v v ρg ǔdv T :(ǔ x )dv = t ǔ ds + n T u ds + ρg ǔdv v s t s u v T : {(ǔ + w) x } dv = t (ǔ + w)ds + n T u ds + ρg (ǔ + w)dv s t s u v 6

2 Cauchy T,. v T :(ǔ x )dv = t ǔ ds + n T u ds + ρg ǔdv v s t s u v T : {(ǔ + w) x } dv = t (ǔ + w)ds + n T u ds + ρg (ǔ + w)dv s t s u v. T :(w x )dv = t w ds + v s t. T : δa (L) dv = v v t w ds + v v ρg wdv ρg w dv δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i S : δe dv = t w ds + ρg w dv V V V 7

2 v T : δa L dv V dv = V v S : δedv 1 J dv T = 1 J F S F T S ij = JFim 1 T mnfjn 1 δe = F T δa L F (F T δa L F ) ij = F ki δa Lkl F lj = x { ( k 1 δuk + δu )} l xl X i 2 x l x k X j = 1 ( xk δu k x l + x ) k δu l x l 2 X i x l X j X i x k X j = 1 ( xk δu k + x ) l δu l 2 X i X j X i X j = 1 {( ) u k δuk δ ki + δu ( )} l u k δ li 2 X i X j X j X i = 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 = δe ij 8

V S : δedv = (F T δa L F ):(JF 1 T F T ) 1 v J dv = J(F ki δa Lkl F lj )(Fim 1 T mnfjn 1 ) 1 v J dv = δ km δ ln A Lkl T mn dv v = T : δa L dv v 9

T ij δe ij dv T ij δe ij = {δe}{t } = {δu} T [B] T [D][B]{u} {δe} =[B]{δu} {T } =[D]{E} =[D][B]{u} S ij δe ij dv S ij δe ij = {δe}{s} = {δu} T [B]{S} {δe} {δu} u {S} 10

{E} {E} u E = 1 ( ui + u j + u ) k u k e i e j 2 X j X i X i X j δe = 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 11

Newton-Raphson 1,. find u h V h such that δu ht (Q(u h ) F )=0 δu h V h, V h V, V h V. Q, F,., Q(u) =F,. F {F k } 0=F 1 < F 2 < < F n = F. Newton-Raphson, Q(u k )=F k u k u k 1. 12

Newton-Raphson 2 u k 1, u k F k F K k 1 K k 2 Q k 2 Q k Q k 1 F k 1 u k 1 u k 2 u k 0 = u k 1 u k 1 u k 2 u k $mbu$ u k 0 = u k 1 Q k 0 = Q(uk 0 ) K k 1 = Q u. K. u=u k 0 13

, K k 1 uk 1 = F k Q k 0 u k 1 = u k 0 + u k 1 u k 1, uk 0 uk 1 uk. 14

Newton-Raphson 3 F k F K k 1 K k 2 Q k 2 Q k Q k 1 F k 1 u k 1 u k 2 u k 0 = u k 1 u k 1 u k 2 u k $mbu$, u k i 1 u k i. Q k i 1 = Q(uk i 1 ) K k i = Q u u=u k i 1 K k i uk i = F k Q k i 1 u k i = uk i 1 + uk i u k i 1 uk i u k, u k i u k, F k Q k i 0. 15

, F k Q k i =0,. 16

v T : δa (L) dv = v t w ds + v ρg w dv δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i t 0 S : δe dv = t w ds + ρg w dv V V updated Lagrange Total Lagrange Total Lagrange V 17

Ω.. Ω = e Ω e,,. dω = dω Ω e Ω e ds = ds Ω e Ω e u N (i), u i., u (i) i u i = N (n) u (n) i, (n). X i = N (n) X (n) i 18

1 δr.. δr = Ω δu k t k ds + Ω ρ 0 δu k g k dω,. δr = = e Ω {δu} = {δu 1,δu 2,δu 3 } T {t} = {t 1,t 2,t 3 } T {g} = {g 1,g 2,g 3 } T {δu} T {t} ds + ρ 0 {δu} T {g} dω [ Ω ] {δu} T {t} ds + Ω e ρ 0 {δu} T {g} dω Ω e N (i) 0 0 [N i ]= 0 N (i) 0 0 0 N (i) [N] =[[N 1 ][N 2 ] [N n ]] (1), 3 3n [N]. 19

{δu (n) } {δu (n) } = 2 { } T δu (1) 1 δu(1) 2 δu(1) 3 δu (n) 1 δu(n) 2 δu(n) 3,, δr = e } {δu} =[N] {δu (n) [ { } T [ ]] δu (n) [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω e 20

1 δe ij S ij dω = δr δe ij, S ij i, j, Ω δe ij S ij = δe 11 S 11 + δe 22 S 22 + 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,. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T, δe ij S ij dω = {δe} T {S} dω Ω Ω = {δe} T {S} dω = δr e Ω e 21

2 δe ij, δe ij = 1 2 ( δui + δu j + δu k u k + u ) k δu k X j X i X i X j X i X j [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 } T,. {δe} =[Z 1 ] { } δu X (2) u i X j u i = N(n) u (n) i X j X j 22

3 δu i X j. { } 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) δu i = N(n) δu (n) i X j X j X 2 N (1) X 3 N (1) X 1 N (1) X 2 N (1) X 3 N (n) X 1 N (n) X 2 N (n) X 3 N (n) X 1 N (n) 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 9 3n [Z 2 ], { } δu =[Z 2 ]{δu (n) } X 23

4 [B] [B] [Z 1 ][Z 2 ] {δe} =[B]{δu (n) }. [B (n) ] u 1 X 2 N (n) X 2 + ( ) 1+ u 1 N (n) u 2 N (n) u 3 N (n) X 1 X 1 X 1 X 1 X 1 X 1 ( ) u 1 N (n) X 2 X 2 1+ u 1 N (n) u 3 X 2 X 2 N (n) X 2 X ( ) 2 u 1 N (n) u 2 N (n) X 3 X 3 X 3 X 3 1+ u 3 X 3 ( ) ( ) 1+ u 1 N (n) X 1 X 2 1+ u 2 N (n) X 2 u 1 X 3 N (n) X 2 + u 1 X 2 N (n) X 3 u 2 X 3 N (n) X 2 + ( ) ( ) 1+ u 2 N (n) X 2 X 3 1+ u 3 N (n) X 3 N (n) X 3 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 1 N (n) X 1 X 3 + u 1 N (n) u 2 N (n) X 3 X 1 X 1 X 3 + u 2 N (n) u 3 N (n) X 3 X 1 X 1 X 3 + ( ) 1+ u 3 X 3 X 2 + u 3 N (n) X 2 X 3 N (n) X 1 6 3 [B (n) ], [ ] [B] = [B (1) ] [B (n) ]. 24

,. e Ω e {δe}{s} dω = e ] [{δu (n) } T [B] T {S} dω Ω e 25

total Lagrange ] [{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,, Q = F = u = [B] T {S} dω Ω e [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω } e {u (n) [ T δuh (Q(u h ) F ) ] =0, e find u h V h such that [ T δuh (Q(u h ) F ) ] =0 e for δu h V h, Newton-Raphson. 26

1 Newton-Raphson, K = Q u, dq dt = Q du u dt = K u,,. ( Ω S ij δe ij dv ) = Ω Ṡ ij δe ij + S ij δėijdω S ij = C ijkl E kl C ijkl Ṡ ij = C ijkl Ė kl Hooke S ij = λ(tre ij )δ ij +2µE ij C ijkl = λδ kl δ ij +2µδ ki δ jl 27

2 Ṡ ij δe ij + S ij δėijdω Ω = C ijkl Ė kl δe ij + S ij δėijdω Ω 1 ( = C ijkl Ė kl δe ij + S ij δfki F kj + F Ω 2 ) ki δf kj dω ( ) = C ijkl Ė kl δe ij + S ij δfki F kj dω Ω S ij, Ė kl k, l S ij = C ij 11 Ė 11 + C ij 22 Ė 22 + C ij 33 Ė 33 + 1 2 (C ij 12 + C ij 21 )2Ė12 + 1 2 (C ij 23 + C ij 32 )2Ė23 + 1 2 (C ij 31 + C ij 13 )2Ė31 C ij kl 1 2 (C ij kl + C ij lk ) 28

, S. S 11 S 22 S 33 S 12 S 23 S 31 = C 11 11 C11 22 C11 33 C11 12 C11 23 C11 31 C 22 11 C22 22 C22 33 C22 12 C22 23 C22 31 C 33 11 C33 22 C33 33 C33 12 C33 23 C33 31 C 12 11 C12 22 C12 33 C12 12 C12 23 C12 31 C 23 11 C23 22 C23 33 C23 12 C23 23 C23 31 C 31 11 C31 22 C31 33 C31 12 C31 23 C31 31 Ė 11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31 29

3 C ijkl 6 6 [D]. C ijkl, ij, kl, [D].,, {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T {Ė} = {Ė11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31 } T, δe ij S ij dω Ω = δe ij C ijkl Ė kl dω Ω T = {δe} [D]{Ė} dω Ω { } { } T u (n) u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3, {Ė} { =[B] u (n)}. Ė ij = 1 2 ( ui + u j + u k u k + u ) k u k X j X i X i X j X i X j 30

4, 1 δe ij Ṡ ij dω = Ω e. [ { δu (n) } T Ω e [B] T [D][B]dΩ { } ] u (n) 31

, δf ki S ij 5 F kj S 11 S 12 S 13 δf ki S ij F kj = {δf 11 δf 12 δf 13 } S 21 S 22 S 23 S 31 S 32 S 33 S 11 S 12 S 13 +{δf 21 δf 22 δf 23 } S 21 S 22 S 23 S 31 S 32 S 33 S 11 S 12 S 13 +{δf 31 δf 32 δf 33 } 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 32

, [σ] = S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 [σ] [Σ] = [σ] [σ] {δf} = {δf 11 δf 12 δf 13 δf 21 δf 22 δf 23 δf 31 δf 32 δf 33 } T { F } = { F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T, δf ki S ij F kj = {δf} T [Σ]{ F} 33

6, δf ij = δx i X j = δu i X j F ij = ẋ i X j = u i X j, [Z 2 ] { } δu } {δf} = =[Z 2 ] {δu (n) X { F { } } =[Z 2 ] u (n), δf ki S ij F kj = { } T { } δu (n) [Z2 ] T [Σ][Z 2 ] u (n) 34

T [Z 2 ] T [Σ][Z 2 ] [A ij ]= { 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] = [A 21 ]......... [A n1 ]...... [A nn ] N (j) X 1 N (j) X 2 N (j) X 3 1 1 1, [Z 2 ] T [Σ][Z 2 ]=[A] 35

7, 2 δf ki S ij F kj dω = Ω e. [ { δu (n) } T Ω e [A]dΩ { } ] u (n) δe ij Ṡ ij dω + δf ki S ij F kj dω Ω Ω = [ { } T ( δu (n) [B] T [D][B]+[A] ) dω e Ω e { } ] u (n). ( [B] T [D][B]+[A] ) dω Ω e e 36

v Updated Lagrange T : δa (L) dv = δv t w ds + v ρg w dv δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i Updated Lagrane v δa ij T ij dv = δr, δr, A ij, Almange. δa ij = 1 ( δui + δu ) j 2 x j x i T ij 37

1 δa ij T ij., δa ij T ij. δa ij T ij = δa 11 T 11 + δa 12 T 12 + δa 13 T 13 + δa 21 T 21 + δa 22 T 22 + δa 23 T 23 + δa 31 T 31 + δa 32 T 32 + δa 33 T 33 = δa 11 T 11 + δa 22 T 22 + δa 33 T 33 +2δA 12 T 12 +2δA 23 T 23 +2δA 31 T 31, δa ij,t ij i, j., {δa} T = {δa 11 δa 22 δa 33 2δA 12 2δA 23 2δA 31 } T {T } = {T 11 T 22 T 33 T 12 T 23 T 31 },. v δa ij T ij dv = v {δa} T {T } dv 38

2, δa ij. δa ij., δa 11 = δu 1 x 1 δa 22 = δu 2 x 2 δa 33 = δu 3 x 3 2δA 12 = δu 1 x 2 + δu 2 x 1 2δA 23 = δu 2 x 3 + δu 3 x 2 2δA 31 = δu 3 x 1 + δu 1 x 3 u i = N (k) u (k) i u i = N(k) x j x j, δu i u (k) i δu i = N (k) δu (k) i δu i x j = N(k) δu (k) i x j 39

., {δu} =,. { } T δu (1) 1 δu (1) 2 δu (1) 3 δu (2) 1 δu (2) 2 δu (2) 3 δu (n) 1 δu (n) 2 δu (n) 3 {δa} =[B][δu], [B] [B (k) ],. [ ] B = ] [B (k) = [[B (1) ][B (2) ] N (k) x 1 N (k) x 2 N (k) x 3 N (k) x 2 N (k) x 1 N (k) x 3 [B (n) ]] N (k) x 3 N (k) x 2 N (k) x 1,. δa ij T ij dv = {δu} T [B] T {T } dv Q,. v Q = v v 40 [B] T [T ]dv

, V, δa ij δa ij = 1 ( δui + δu ) n 2 X j X i 41

. 1, Newton-Raphson, Q u K. K = Q u, u, Q = Q t = Q u u t = Q u u,,.,. T : δa (L) dv = t w ds + ρg w dv v δv v 42

2 F, J. F = x i X j e i e j J =detf, 2 Piola-Kirchhoff S. T = 1 J F S F T S Green-Lagrange E δe. E = 1 ( ui + u j + u ) k u k e i e j 2 X j X i X i X j δe = 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 δa δe. δe = F T δa F,. δe : S dv = (F T δa F ):(JF 1 T F T ) 1 v v J dv = δa : T dv v 43

3 v δe : S dv. Total-Lagrange. δe : Ṡ + 1 ( ) δf T 2 Ḟ + Ḟ T δf : S dv. V 2 Piola-Kirchhoff S t (t). Ṡ = JF 1 S t (t) F T 1, δe : Ṡ =(F T δa F ):(JF 1 S t (t) F T ) = JδA : Ṡ t (t) S t (t) Ṡ, V δe : ṠdV = v δa : Ṡ t (t)dv 44

4 Ḟ δf,., L, 2, Ḟ = L F δf = δf t (t) F L = u i e i e j x j 1 ( δf T 2 Ḟ + F ) T δf : S = 1 ( ) F T δf t (t) T L F + F T L T δf t (t) F : ( JF 1 T F T ) 2 = J 1 ( ) δf t (t) T L + L T δf t (t) : T 2 V 1 ( δf T 2 Ḟ + F ) T 1 ( ) δf : S dv = δf t (t) T L + L T δf t (t) : T dv v 2 updated Lagrange. δa : Ṡ t (t)+ 1 ( ) δf t (t) T L + L T δf t (t) : T dv = δṙ v 2 45

5. { } 1 δa ij Ṡ t (t) ij dv + 2 (δf t(t) ki L kj + L ki δf t (t) kj ) T ij v v dv = δṙ, 1., Ṡt(t) ṠF t(t) F. δa ij T ij. δa ij Ṡ ij = {δa} T { Ṡ } Ṡ = { Ṡ 11 Ṡ 22 Ṡ 33 Ṡ 12 Ṡ 23 Ṡ 31 }, Ṡ ij D ij. Ṡ ij = C ijkl D kl D ij = 1 ( ui + u ) j 2 x j x i Ṡt(t) Truesdell Kirchoff Oldroyd 46

5 Ṡij = C ijkl D kl.,, { Ṡ } = [ C] { } D Ṡ ij = C ij11 D 11 + C ij12 D 12 + C ij13 D 13 + C ij21 D 21 + C ij22 D 22 + C ij23 D 23 + C ij31 D 31 + C ij32 D 32 + C ij33 D 33 = C ij11 D 11 + C ij22 D 22 + C ij33 D 33 + 1 2 (C ij12 + C ij21 )(2D 12 ) + 1 2 (C ij23 + C ij32 )(2D 23 ) + 1 2 (C ij31 + C ij13 )(2D 31 ). C ijkl = 1 2 (C ijkl + C ijlk ) 47

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 2311 C3322 C3333 C2312 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 48

6 δa ij, {D} =[B] { u} { { u} = u (1) 1 u (1) 2 u (1) 3 u (2) 1 u (2) 2 u (2) 3 u (n) 1 u (n) 2 u (n) 3, 1. δa ij Ṡ t (t) ij dv = {δu} T [B] T [ D] [B] { u} dv v v } T 49

7 2. T ij = T ji,. 1 2 (δf kil kj + L ki δf kj ) T ij = 1 2 T ijl ki δf kj + 1 2 T ijδf ki L kj = δf ki T ij L kj., δf ki T ij L kj = {δf} T [Σ] {L} {δf} = {δf 11 δf 12 δf 13 δf 21 δf 22 δf 23 δf 31 δf 32 δf 33 } {L} = {L 11 L 12 L 13 L 21 L 22 L 23 L 31 L 32 L 33 } T 11 T 12 T 13 [T ]= T 21 T 22 T 23 [Σ] = T 31 T 32 T 33 [T ] [0] [0] [0] [T ] [0] [0] [0] [T ] 50

δf ij,l ij. 8 δf ij = δu i x j L ij = u i x j. { } δu {δf} = =[Z] {δu} x, {L} = { } u x =[Z] { u} { } { } δu δu1 δu 1 δu 1 δu 2 δu 2 δu 2 δu 3 δu 3 δu 3 = x x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x { } { } 3 u u1 u 1 u 1 u 2 u 2 u 2 u 3 u 3 u 3 = x x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 51

[ ] Z = N (1) N (2) x 1 x 1 N (1) N (2) x 2 x 2 N (1) N (2) x 3 x 3 N (1) N (2) x 1 x 1 N (1) N (2) x 2 N (1) N (2) x 3 x 3 N (1) N (2) x 1 x 1 N (1) N (2) x 2 x 2 N (1) N (2) x 3 x 3 N (n) x 1 N (n) x 2 N (n) x 3 N (n) x 1 x 2 N (n) x 2 N (n) x 3 N (n) x 1 N (n) x 2 N (n) x 3 52

9, δf ki T ij L kj. δf ki T ij L kj = {δf} T [Σ] {L} [Z] T [Σ] [Z]. ] { N (i) [G ij = x 1 N (i) x 2 [Z] T [Σ] [Z] = = {δu} T [Z] T [Σ] [Z] { u} N (i) x 3 } T 11 T 12 T 13 T 21 T 22 T 23 T 31 T 32 T 33 N (i) x 1 N (i) x 2 N (i) x 3 [G 11 ] [G 1n ].. =[G] [G n1 ] [G nn ] 1 0 0 0 1 0 0 0 1 v δf ki T ij L kj dv = v {δu} [G] { u} dv 53

10. (δa ij S t (t) ij + δf ki T ij L kj )dv v = {δu} ([B] T T [ ) D] [B]+[G] v dv { u} V, δa ij, Ṡ t (t) ij Caushy. 2. 54

11,,. 1 dv =, [J]. v 1 1 1 1 [ ] J = x i = N(k) r j r j x (k) i, [B], N(k) x j. N (k) x 1 N (k) x 2 N (k) x 3 = [ ] 1 J = 1 (det J) dr 1 dr 2 dr 3 x 1 r 1 x 1 r 2 x 1 r 3 x 2 r 1 x 2 r 2 x 2 r 3 x 3 r 1 x 3 r 2 x 3 r 3 ( = N(k) X (k) i r j ) + u (k) i [J] r 1 x 1 r 2 x 1 r 3 x 1 r 1 x 2 r 2 x 2 r 3 x 2 r 1 x 3 r 2 x 3 r 3 x 3 r 1 x 1 r 2 x 1 r 3 x 1 r 1 x 2 r 2 x 2 r 3 x 2 r 1 x 3 r 2 x 3 r 3 x 3 N (k) r 1 N (k) r 2 N (k) r 3 (3) 55

, [J] X 1 X 1 [ ] r 1 X J = 2 X 1 r 2 r 3 r 1 r 2 r 3 X 2 X 2 X 3 r 1 X 3 r 2 X 3 r 3 56

total Lagrange updated Lagrange 1 v V T : δa (L) dv = S : δe dv = V v t w ds + t w ds + V v ρg w dv ρg w dv 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 updated Ω Ω Ṡ t (t) ij = C ijkl D kl Ṡt(t) Truesdell Kirchoff Oldroyd Total S ij = C ijkl E kl C ijkl (Ṡij, Ėkl ) Ṡ ij = C ijkl Ė kl 57

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 58

. F dx u X x dx 2: 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 59

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. 60

1, W. S ij = W E ij E = 1 (C I) 2 S ij =2 W C ij W C., W C. I C trc II C 1 { (trc) 2 tr(c 2 ) } 2 III C det C S ij =2 ( W I C + W II C + W ) III C I C C ij II C C ij III C C ij 61

2 S ij =2 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 {( 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 T B. δ kl + W B kl W } III B (B 1 ) kl I B II B 62

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 63

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 64

Mooney-Rivlin 2 ĨC, ĨI C (reduced invariants). W M R 2 Piola-Kirchhoff W M R I C W M R II C = WM R ĨC = WM R ĨI C W M R III C = WM R ĨC ĨC = c 1 III C I C 1 3 ĨI C = c 2 III C II C ĨC III C + WM R ĨI C 2 3 ĨI C = 1 III C 3 c 1I C III 4 3 C 2 3 c 2II C III 5 3 C { 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 T ij = S ij =0 S ij = pδ ij, p. 65

Mooney-Rivlin 3,.,. F. F = J 1 3 F, F Flory, det F =1. Cauchy-Green C. C = F T F C 1, 2, ĨC =3, ĨI C =3. 66

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 3: - 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. 67

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 68

1, c 1,c 2.. 1 1 1 1/ l x 2 l 1/ l x 3 x 1 4: 69

, 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 70

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 71

2 Cauchy { W H T kl = pδ kl +2 R (2l + 1 } II B l )+ WH R III B δ kl + WH R I B l 2 0 0 0 1/l 0 WH R 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 ) WH R I B +(l + 1 R + WH R l 2) WH II B III } B +(l 1 l 2) WH R II B l =1+ε ε 2 6(c 1 + c 2 ) E. T 11 =6(c 1 + c 2 )ε } 72

3,. u x 2 1 1 x 1 x 3 1 5: F, B,I B,II B 73

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 74

4 Cauchy { } W H T kl = pδ kl +2 R (3 + u 2 )+ WH R II B III B δ kl + WH R I B 1+u 2 u 0 u 1 0 WH R 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 ) WH R II B } + WH R III B ( W H T 12 = T 21 =2u R I B u 2,, u. + WH R II B T 12 = T 21 =2(c 1 + c 2 )u, 2(c 1 + c 2 ) G. ) 75

. 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 (4) ( ) S F T T N = t (5) C = F T F (6) S ij = p(c 1 ) ij +2 W C ij (7) III C =1 (8) (4), (5), (6), (7) ( ) (8)., W, (4) (8). 76

W Φ. Φ = W dω t u ds ρ 0 g u dω (9) Ω Ω λ Lagrange, Φ. Φ = Φ + λg(iii C )dω (10) g, g(iii C ),III C =1 g =0, =1. III C, Lagrange Q. Ω, 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 + δλg dω = 0 (11) Ω (11),. (11),. 77

Lagrange 1 (10) Lagrange λ, (7) p., (11). δc ij = δf ki F kj + F ki δf kj C, W/ C ij, g/ C ij i, j. (11) 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 ( ) (12) C ij C ij Ω, u = x X δu = δx δf ki = δx k X i = δu k X i (13) ( ) = Ω { ( δu k W F kj 2 + λ g )} dω (14) X i C ij C ij 78

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 (15) 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 (16) C ij C ij (16) 1 V divb dv = n b ds (17) S ( ) = Ω (11) ( 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 Ω X i C ij C ij + Ω X i { ( W 2 + λ g ) } F kj δu k dω (18) C ij C ij { ( W [n i 2 + λ g ) } ] F kj t k δu k ds + δλg(iii C )dω = 0 (19) C ij C ij Ω 79

Lagrange 3 (19) δu V, δλ Q, (20), (21), (22). { ( W 2 + λ g ) } F kj + ρ 0 g k = 0 (20) X i C ij C ( ij W n i {2 + λ g ) } F kj t k = 0 (21) C ij C ij g(iii C ) = 0 (22), (20) (4), (21) (5), (22) (8). ( W S ij =2 + λ g ) (23) C ij C ij g III C, (??) g = g III C C ij III C C ij = g ( III ) C C 1 (24) III ij C (23). (7) S ij =2 W C ij +2λ g III C III C ( C 1 ) ij (25) p = 2λ (26), λ. 80

, (4) (8),. 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 (27) δλg dω = 0 (28) Ω for (δu,δλ) (V, Q), λ = 1 2 p 81

Newton-Raphson,. Ω.. Ω = e Ω e (29),,. dω = dω (30) Ω e Ω e ds = ds (31) Ω e Ω e u N (i), u i., u (i) i u i = N (n) u (n) i (32), (n). Lagrange λ M (m), λ., λ (m). λ = M (m) λ (m) (33) 82

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 (34) 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 (35),. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T (36) {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (37) 83

2 (34), 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, (38),. [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 (39) } T (40) (41) 84

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 (42) 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 (43). 9 3n [Z 2 ], { } δu =[Z 2 ]{δu (n) } (44) X 85

, [B] [B] [Z 1 ][Z 2 ] (45) {δe} =[B]{δu (n) } (46) 86

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 (47) 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. (), (49), (27) ] [{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 (48) (49) (50). 87

, (28). {M} = {M (1) M (2) M (m) } T (51) {δλ (m) } = {δλ (1) δλ (2) δλ (m) } T (52), (28). δλgdω = δλg dω (53) Ω e Ω e = ] [{δλ (m) } T {M}g dω = 0 (54) e Ω e 88

, {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 (55), (50), (54). [ ] ] [{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,, (56) Q = F = u = [ ] [B] T {S} dω (56) Ω e {M}g [ ] [N] T [ ] {t} ρ 0 [N] T {g} ds + dω (57) Ω e 0 Ω e 0 } {u (n) λ (m) (58) [ T δuh (Q(u h ) F ) ] = 0 (59)., e 89

find u h V h such that [ T δuh (Q(u h ) F ) ] = 0 (60) e for δu h V h, Newton-Raphson. 90

Newton-Raphson, K = Q u, dq dt = Q du u dt = K u (61), (27), (28),.,. (27) Ω ( 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 (62) C ij 91

, (28) δλ ġ dω = Ω Ω δλ g C kl Ċ kl dω (63) 92

(62). 2 Ċ kl =2Ėkl (64). ( 2 W 2 ) g D ij kl =4 + λ (65) C ij C kl C ij C kl, (23), (??), (65) (62) ( δe ij D ij kl Ė kl + δf ki S ij F kj + δe ij 2 g ) λ dω (66) C ij. Ω S ij D ij kl Ė kl (67). 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 (68) 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 (69) 93

3 C ij kl 1 2 (D ij kl + D ij lk ) (70), 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 (71) C ijkl 6 6 [D 1 ].C ijkl, ij, kl, [D 1 ].,, { S} = { } T S11 S22 S33 S12 S23 S31 (72) {Ė} = { T Ė 11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31} (73) 94

, Ω δe ij D ijkl Ė kl dω = δe ij Sij dω Ω = {δe} T [D 1 ]{Ė} dω Ω = {δe} T [D 1 ]{Ė} dω (74) e Ω e 95

4 { } u (n), (46), (75). { } T u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3 (75) {Ė} { } =[B] u (n) (76), (66) 1 δe ij D ijkl Ė kl dω = Ω e. [ { δu (n) } T Ω e [B] T [D 1 ][B]dΩ { } ] u (n) (77) 96

1 δf ki S ij F kj = {δf 11 δf 12 δf 13 } +{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 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 (78) 97

, [σ] = S 11 S 12 S 13 S 21 S 22 S 23 (79) S 31 S 32 S 33 [σ] [Σ] = [σ] (80) [σ] {δf} = {F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T (81) { F } = { F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T (82), δf ki S ij F kj = {δf} T [Σ]{ F} (83) 98

2, δf ij = δx i = δu i X j X j (84) F ij = ẋ i = u i X j X j (85), (44) { } δu } {δf} = =[Z 2 ] {δu (n) X { F { } } =[Z 2 ] u (n) (86) (87), δf ki S ij F kj = { } T { } δu (n) [Z2 ] T [Σ][Z 2 ] u (n) (88) 99

, [A ij ]= [A] = { N (i) X 1 N (i) X 2 N (i) } S 11 S 12 S 13 S 21 S 22 S 23 X 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 (89) (90) 100

3. [Z 2 ] T [Σ][Z 2 ]=[A] (91), (66) 2 δf ki S ij F kj dω = Ω e. [ { δu (n) } T Ω e [A]dΩ { } ] u (n) (92) 101

1 { λ } { (m) λ(1) λ(2) λ } T (m) (93) { {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 (94), (66) 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) (95) 102

( 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) (96) Ω e. 103

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} (97) δλ g Ċ kl dω = [ { } T δλ (m) [M] T {D 2 } T [B]dΩ Ω C kl e Ω e. { } ] u (n) (98) 104

, { } (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) (99) [K 1 ] [B] T [D 1 ][B]+[A] (100) [H] [B] T {D 2 }[M] (101) (96), (98) [ { } [ ] δu (n) δλ (m) [K 1 ] [H] dω [H] T 0 e Ω e, [K] =. Ω e { ] u (n) λ(m)} (102) [ ] [K 1 ] [H] dω (103) [H] T 0 105

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 W V =2(J 1), III C 1 =0 WV III C =1 α 106

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 107

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 108

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 109

Mooney-Rivlin 4 Ĩ C = I C III 1/3 C =3 =(3+2ε +4δ) (1 23 ε 43 ) δ 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) 110

αw V WV III = α(2ε +4δ) 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 δ = 3α (c 1 + c 2 ) 6α +(c 1 + c 2 ) ε ν E 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 = 36(c 1 + c 2 )α 6α +(c 1 + c 2 ) 111

α E =6(c 1 + c 2 ) κ = E 3(1 2ν) =4α 112

1 (R ) Φ= WdΩ+ α Ω 2 Ω (W V ) 2 R α selective/reduced integration V Q αw V Q λ Ω ( W V λ ) δλdω =0 α δλ Q αw V λ P P (αw V )=λ 113

Q 114

2 P Φ = WdΩ+ α (PW V ) 2 R Ω 2 Ω U V δ Φ W = δc ij dω+α (PW V )P (δw V )dω δr =0 Ω C ij Ω u λ Q Ω ( W V λ ) δλdω =0 α δλ Q 115

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 W V III ( ) W + λ WV δc ij dω=δr Ω C ij C ij ( W V λ ) δλdω =0 α Ω Ω 116

Lagrange 4 Ω ( W C ij + λ g C ij ) δc ij dω = δλg dω =0 W V λ α g(= W V ) Ω Ω t k δu k ds + Ω ρ 0 g k δu k dω Lagrange Q = Q = Ω e [ Ω e [ ] [B] T {S} dω {M}g [B] T {S} {M} ( W V λ/α ) ] dω Lagrange [ ] [K 1 ] [H] [K] = dω [H] T 0 [K] = Ω e Ω e [ ] [K 1 ] [H] dω [H] T [G] 117

α [G] = 1 α [M]T [M] 118

,,,, 119

(Hooke, ) t t t,,,,. 120

B A e e e p e (A), (B),,.,,,. 121

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. 122

, 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) 123

:. :. :. A B e e e p e 124

3 2, 9, vonmises Tresca B A e e e p e 125

,, 3 A,, B. von Mises. A B e e e p e 126

Mises Mises σ σ ij σ ij σ = ( ) 1 3 2 2 σ ij σ ij σ ijσ ij =σ 2 11 + σ +σ 2 21 + σ 12 22 +σ 2 31 + σ 32 2 + σ 2 + σ 2 13 2 23 2 2 + σ 33, σ ij. σ ij =σ ij 1 3 σ kkδ ij =σ ij 1 3 (σ 11 + σ 22 + σ 33 ) δ ij 127

F = σ σ y σ y F =0 ( ) σ y σ ij σ ij 128

(associated flow rule),, λ Ψ ė p ij = λ Ψ σ ij (associated flow rule), ė p F ij = λ σ ij F = σ σ y 129

(normality rule) σ ij / t ( ) F =0, F =0, 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 0, ė p ij σ ij = λ F =0, σ ij ė p ij 130

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 131

, 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 132

1 F =0 F =0 F = F σ ij =0 σ ij F/ 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 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 133

2 λ σ ij = C e ijkl = σ ij = C e ijkl(ė kl ė p ( kl ) ) = C e F ijkl ė kl λ σ kl ( λ = ( 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 σ ab C e abcd F σ σ kl cd C e ijkl Ce ijcd F σ cd F σ ab C e abkl F σ ab C e abcd F σ cd ) ) ė kl 134

F/ σ ij 3 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 135

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 136

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 137

1, Hooke. Hooke. F Cauchy T (elastic material). T (t) =f(f (t)) (104) f. f(f )=f(q F )=Q f(f ) Q T (105) F, F O,O, O O Q., P f(f )=f(f P ) (106). V. T = f(v ) (107). f(v )=f(q V Q T )=Q f(v ) Q T (108) 138

V, V O,O, O O Q. f(v ) (isotropic tensor function). (108) T, V, T = f(v )=φ 0 I + φ 1 V + φ 2 V 2 (109)., φ i (i =0, 1, 2) V. (representation theorem). (107) V = B 1/2. T = g(b) (110) g(b )=g(q B Q T )=Q g(b) Q T (111), g(b),. T = ψ 0 I + ψ 1 B + ψ 2 B 2 (112) = ξ 0 I + ξ 1 B + ξ 1 B 1 (113), B. 139

Hooke., E (L), (109) E (L) V I + 1 {u x + x u} (114) 2 E (L) = 1 {u x + x u} (115) 2 T =(φ 0 + φ 1 + φ 2 )I +(φ 1 +2φ 2 )E (L) (116) = η 0 I + η 1 E (L) (117), η 0, η 1 E (L). T E (L), Hooke., λ, µ Lamé. T =(λtre (L) )I +2µE (L) (118) 140

, Hooke. 2 T = f(v ), T = g(b), B Almansi A A = 1 (I B) (119) 2, T = h(a) (120). h(a )=h(q A Q T )=Q h(a) Q T (121) A, A O,O, O O Q. h(a),. T = h(a) =ζ 0 I + ζ 1 A + ζ 2 A 2 (122) Hooke T =(λtra)i +2µA (123). A E (L) (124), λ, µ Lamé. 141

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 (125) Å = Ȧ W A + A W (126) 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 (127) 142

,, (, ), F t (τ) R t (τ), U t (τ) I (128),. T (J) T (O) T (C) T (G) (129) Å (J) Å(O) Å(C) Å(G) (130), T (J) = Ṫ W T + T W (131) T (J) = T (O) + D T + T D (132) T (J) = T (C) D T T D (133) T (G) = Ṫ Ω T + T Ω (134) W Ω (135), Å(C) = D, T =(λtrd)i +2µD (136). T Kirchhoff ˆT t (τ) =J t (τ)t (τ) ˆT t (t) =(λtrd)i +2µD (137) 143

., ˆT t (t) (J) = T (J) + T trd (138) ˆT t (t) (O) = T (O) + T trd (139) ˆT t (t) (C) = T (C) + T trd (140) 144

. v v e v p v = v e + v p (141), L D. D = D e + D p (142) σ ij T ij, e p ij D p ij., C ep ijkl. T ij = C ep ijkld kl (143),,, (143), Cauchy Kirchhoff ˆT ij = C ep ijkld kl (144). Kirchhoff, Jaumann. 145

. C ep ijkl = ( C ijkl 3G T T ) ij kl σ 2 (145) T ij, T ij = T ij 1 3 T kk δ ij. pe = T kl D kl σ (146) λ = T kl D kl σ (147) 146

1,,., t t e p ij = t σ ij = = t 0 t 0 0 τė p ij dτ (148) τ σ ij dτ (149) τ C ep ijkl τ ė kl dτ (150) t t σ ij t σ ij = τ C ep ijkl τ ė kl dτ (151) t. t C ep ijkl, (150), (151),. 147

2,. Kirchhoff.,, (157) 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 (152) {τ ˆTτ ij (tr τ D) τ T ij } dτ (153) {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 (154) dτ (155) { 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 (156) = 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 } t (157) 148

. V,v, S, s. s t, u, v g.. T Cauchy. x T + ρg = 0 (158) T T n = t (159) u = u (160) D ij = 1 ( ui + u ) j (161) 2 x j x i ˆT ij = C ep ijkl D kl, T ij = t 0 T ij dt (162) 149

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 (163) (164) δa ij T ij dv = {δu} T [B] T {T } dv (165) v Q = [B] T [T ]dv (166) 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} (167) 150

1,. Ṡ ij = C ijkl D kl (168) D kl = D lk, C ijkl = 1 2 (C ijkl + C ijlk ) (169). 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 (170) 151

Hooke 1 Cijkl e = λδ ijδ kl +2µδ ik δ jl (171) C ijkl e = λδ ijδ kl + µ (δ ik δ jl + δ il δ jk ) (172) C e klij = λδ kl δ ij + µ (δ ki δ lj + δ kj δ li ) (173) = λδ ij δ kl + µ (δ ki δ lj + δ il δ jk ) (δ mn = δ nm ) (174) = C e ijkl (175) [ 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) (176) (177) (178) 152

1. C p ijkl = 3G σ ij σ kl σ 2 (179),. A = 3G σ 2 C p ijkl = A σ ij σ kl (180), C p ijkl = 1 2 ( ) C p ijkl + Cp ijlk = C p ijkl (181), C p ijkl C p ijkl = C p klij (182) 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 [ C 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 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 (183) 153

2, Kirchhoff Jaumann, D. t t ˆT (J) = C ep : D (C 4 ) (184) t tṡ = C : D. t tṡ = t t ˆT (J) D T T D (185) = C ep : D D T T D (186) t tṡij = C ep ijkl D kl D ik T kj T ik D kj (187) = C ep ijkl D kl δ il T kj D kl T ik δ jl D kl (188) { = C ep ijkl 1 2 (δ ijt kj + δ ik T lj ) 1 } 2 (T ikδ ij + T il δ jk ) D kl (189). C ep ijkl = Cep ijkl 1 2 (δ ilt kj + δ ik T lj ) 1 2 (T ikδ jl + T lj δ jk ) (190) 154

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 ) (191) 155

4 Kirchhoff Jaumann Truesdell t tṡ t tṡ = t t t tṡ = t t ˆT (J) D T T D (192) = C ep : D D T T D (193) 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 0 156