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 =0), : dx = F 1 dx (4). dx dx,. R, U F F = R U (5). U (right stretch tensor). (right ploar decomposition)., R, V F = V R (6). (left ploar decomposition), V (left stretch tensor). 3

2 90 F = R U (7) 0 1 0 0 1 0 2 0 0 2 0 0 = 1 0 0 0 1 0 (8) 0 } 0 {{ 1 } 0 } 0 {{ 1 0 0 1 }}{{} F R U 90 2 F = V R (9) 0 1 0 1 0 0 0 1 0 2 0 0 = 0 2 0 1 0 0 (10) 0 } 0 {{ 1 } 0 0 1 0 }{{}} 0 {{ 1 } F V R X 2 0 X 1 4

3 Cauchy-Green Cauchy-Green C = F T F = U R T R U = U 2 (11) B = F F T = V R R T V = V 2 (12) F 0 (τ) =F t (τ) F 0 (t) (13) t F t (τ) τ t 0 F 0 (t) F 0 (τ) dx dx X x O 5

1 dx 1 dx 2 dx 1 dx 2 =(F dx 1 ) (F dx 2 ) dx 1 dx 2 = dx 1 (C I) dx 2 (14) dx 1 dx 2 dx 1 dx 2 = dx 1 dx 2 (F 1 dx 1 ) (F 1 dx 2 ) = dx 1 (I B 1 ) dx 2 (15) t t 0 dx 2 u dx 2 dx 1 X dx 1 x O 6

2 dx 1 dx 2 dx 1 dx 2 =(F dx 1 ) (F dx 2 ) dx 1 dx 2 = dx 1 (C I) dx 2 (16) Green-Lagrange E Almange A dx 1 dx 2 dx 1 dx 2 = dx 1 dx 2 (F 1 dx 1 ) (F 1 dx 2 ) = dx 1 (I B 1 ) dx 2 (17) E = 1 (C I) (18) 2 A = 1 2 (I B 1 ) (19) 7

3 1 F = l L 0 0 0 1 0 (20) 0 0 1 dx 1 dx 1 dx 1 dx 1 = l L dx 1 l L dx 1 dx 1 dx 1 ( ) l 2 = dx 1 L 1 dx 2 1 (21) Green-Lagrange E Almange A dx 1 dx 1 dx 1 dx 1 = dx 1 dx 1 L l dx 1 L l dx 1 ) = dx 1 (1 L2 dx 1 (22) l 2 E = 1 ( ) l 2 2 L 1 2 (23) A = 1 ) (1 L2 2 l 2 (24) 8

E = 1 ( ) ( ) l 2 2 L 1 = 1 (u + L) 2 L 2 u 2 2 L 2 L A = 1 ) ( ) (1 L2 = 1 (u + L) 2 L 2 u 2 l 2 2 l 2 l u L (25) (26) 9

3 E = E ij e i e j = 1 { ui + u j + u } k u k e i e j 2 X j X i X i X i A = A ij e i e j = 1 { ui + u j + u } k u k e i e j 2 x j x i x i x i E L = E Lij e i e j = 1 { ui + u } j e i e j (27) 2 X j X i E A E = 1 (C I) 2 = 1 2 (F T F I) = 1 2 F T (I F T F 1 )F = 1 2 F T (I B 1 )F = F T AF (28) 10

4 F = RU (29) l L 0 0 l cos θ sin θ 0 L U = 0 h H 0 cos θ h H sin θ 0 l R = sin θ cosθ 0 F = L sin θ h H cos θ 0 h 0 0 0 0 1 0 0 1 H l 2 l L 0 0 2 2 h C = 0 2 0 H E = 1 L 1 0 0 2 h 0 2 1 0 2 2 H (30) 2 h 0 0 2 h H 0 0 2 2 H 1 2 X 2,x 2 h h l L θ H H X 1,x 1 11

1 O x t dx L d + d 1: 2 X X +dx, t x x +dx, ẋ ẋ +dẋ. ẋ, dẋ d, (elocity gradient tensor) L. d = L dx (31), 2 d dx. t grad. L = x (32) 12

2 dx = F dx (33) dx d = Ḟ dx (34), d = Ldx dx = F dx d = L F dx (35),. Ḟ = L F (36) L = Ḟ F 1 (37), J = detf. J J =trl ( = ) i =di x i (38) 13

, L. L = D + W D = 1 ( ) L + L T 2 = 1 ( i + ) j e i e j 2 x j x i W = 1 ( ) L L T 2 = 1 ( i ) j e i e j (39) 2 x j x i D (deformation rate tensor) (stretching tensor), W (spin tensor) (rotation rate tensor). 14

3 D = 1 2 R( UU 1 + U 1 U)R T W = ṘRT + 1 2 R( UU 1 U 1 U)R T Ė = F T DF D = Ȧ + LT A + AL (40) 15

1 ds N dx B n dx B ds dx A F dx A 2:,,,.,, ds, N, ds, n,. n ds =(detf )F T N ds (41) F T n ds =(detf )N ds (42) Nanson (Nanson s formula). 16

2 dv d dx3 dx 2 dx 3 dx 2 dx 1 F dx 1 3:, dx 1,dX 2,dX 3 3 dv 6, dx 1,dx 2,dx 3, 3 d 6. dv d. d =(detf )dv (43) J det F (44) J. 17

m ρ, m = ρ d (45) (principle of conseration of mass), m, ṁ = 0 (46),. d =(detf )dv, J = detf, m = ρj dv (47) ṁ = = = V V V ( ρj + ρ J)dV ( ρ + ρtrl)j dv, ( ρ + ρdi)d (48) ρ + ρdi = 0 (49) 18

1 (body force) ρg, (surface force) t. g, ρg., t., ρ d ( ) ρ d = ρg d + t ds (50). Euler 1 (Euler s first law of motion). a,( ρfd) = ρf d. ρ(a g)d = t ds (51) s s 19

2, (angular momentum). ( ) x ρ d = x ρg d + x t ds (52),, (moment of momentum). Euler 2 (Euler s second law of motion). ( ρfd) = ρf d ẋ = = 0 (53). x ρa d = x ρg d + s s x t ds (54) 20

Cauchy 1 t l n df n m ds 4: ds df n ( 6)., t., t n. ( ) t n = df n ds (55) 21

Cauchy 2 t l n mds df n 5: 2 l, m, t n. t n = T nn n + T nl l + T nm m (56) T 1, 2., (n, l, m) (e 1, e 2, e 3 ), (e 2, e 3, e 1 ), (e 3, e 1, e 2 ), t 1, t 2, t 3,. t i = T ij e j (57) Cauchy T T ij. T = T ij e i e j (58) 22

Cauchy 3 Cauchy t 3 t 1 t n t 2 6: Δ s1,δ s2,δ s3,δ s Eular ( ρ d) = ρg d + s t ds ρ(a g)δ t n Δ s t 1 Δ s1 t 2 Δ s2 t 3 Δ s3 (59) Δ si /Δ s n i t n = t i n i (60) 23

Cauchy 4 t n = t i n i t i = T ij e j Cauchy (Cauchy s formula) : t n = T T n (61). Cauchy, Eular ρ(a g)d = s tds t t n = T T n ρ(a g)d = dit d (62). ρa =dit + ρg (63). ρa i = T ji x j + ρg i (64) Cauchy 1 (Cauchy s first law of motion) (equilibrium equation). 24

Cauchy 5, Euler x ρad = x ρg d + s x t ds t t n = T T n. x ρa d = {x (ρg +dit )+e ijk T ij e k } d (65) ρa =dit + ρg e ijk T ij e k d =0 e ijk T ij = 0 (66). T ij = T ji (67) T T = T (68), Cauchy 2 (Cauchy s second law of motion). Cauchy, 6. 25

Cauchy t.,,. Kirchhoff ˆT t 0 t J ˆT JT (69). 26

1 Piola Kirchihoff, 1 Piola Kirchihoff (first Piola Kirchihoff stress tensor), df n : t df n ds (70) t Π T N (71)., n, ds,, N, ds, Π (nominal stress). t t 0 N ds Π T df n ds n df n 7: 1 Piola Kirchihoff Nanson df n = Π T N ds = 1 J ΠT F T n ds (72) 27

, Cauchy t n (= df n /ds) =T T n T Π. T = 1 J F Π (73) 28

2 Piola Kirchihoff 2 Piola Kirchihoff (second Piola Kirchihoff stress tensor) S, df n F 1 df n :. t F 1 df n ds = F 1 t (74) t S T N (75) t t 0 N S T ds F 1 df n F 1 ds n df n 8: 2 Piola Kirchihoff Nanson df n = F S T N ds = 1 J F ST F T n ds (76) 29

, Cauchy t n (= df n /ds) =T T n T S. T = 1 J F S F T (77) 30

1,.,, (constitutie equation). 31

2 F, Cauchy T. Q, Q T Q T., Q F., T F. T = f(f ) (78) f, (tensor alued tensor function). Q T Q T = f(q F ) (79) Q f(f ) Q T = f(q F ) (80). f. 32

3, 3. 1. (principle of determinism for the stress) :. 2. (principle of local action) : X, X,. 3. (principle of material frame indiffernce or principle of material objectiity) : (referance frame) 1., 2 O O x = c(t)+q(t) x (81) T = Q(t) T Q(t) T (82),. 1 (eent) x t {x,t} (obserer). {x,t}, {x,t }. 33

4 2 O O x = c(t)+q(t) x (83) 2 b = x 2 x 1, b = x 2 x 1 x 2 x 1 = Q(t)(x 2 x 1 ) (84) b = Q(t) b (85) Q(t) K = Q(t) K Q(t) T (86) V, B, A, D, T (87) R, L, W (88) U, C, E, S (89) 34

4 ε ij (u) = 1 ( ui + u ) j 2 X j X i (90) T ij = κ(di u)δ ij +2Gε D ij(u) (91) x 1 x 2 90 0 1 0 F ij = 1 0 0 (92) 0 0 1 E ij = 1 1 1 0 1 1 0 1 1 0 + 1 1 0 2 0 0 0 0 0 0 1 0 0 = 0 1 0 (93) 0 0 0 cos θ 1 sin θ θ 35

4 T = ae (94) T = a E T = QT Q T a = a E = E T = a E QT Q T = ae = T (95) T = aa (96) S = ae (97) 36

, A [B]. [B] A Ω, Ω Ω, Ω Ω D. t, ρg, u V., ρ, g, V. A t Ω D ρg Ω 9: 37

. [B] t, g, u V. [1 ] (Cauchy 1 ) x T + ρg = 0 (98) X (S F T) + ρ 0 g = 0 (99) [2 ] u = u on Ω D (100) T T n = t on Ω Ω D (101) ( S F T ) T N = t (102) [3 ] [4 ] [1], [2] [4] [3] [4] 38

1 T T, u U. Ť T, ǔ U,. ( x Ť + ρg) ǔ d = 0 (103),., s t,s u t, u s, s = s t + s u. Ť :(ǔ x )d = t ǔ ds + n Ť u ds + s t s u, s u w =0 w W. ǔ U, w W ǔ + w U. Cauchy T,. ρg ǔd (104) T :(ǔ x )d = t ǔ ds + n T u ds + ρg ǔd (105) s t s u T : {(ǔ + w) x } d = t (ǔ + w)ds + n T u ds + ρg (ǔ + w)d (106) s t s u 39

2 Cauchy T,. T :(ǔ x )d = t ǔ ds + n T u ds + ρg ǔd (107) s t s u T : {(ǔ + w) x } d = t (ǔ + w)ds + n T u ds + ρg (ǔ + w)d (108) s t s u. T :(w x )d = t w ds + s t. T : δa (L) d = δ t w ds + δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i ρg wd (109) ρg w d (110) (111) S : δe d = t w ds + ρg w dv (112) V δv V 40

3 T : δa L d V dv = V S : δedv (113) 1 d (114) J T = 1 J F S F T S ij = JF 1 im T mnf 1 jn (115) 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 (116) 41

V S : δedv = = = = (F T δa L F ):(JF 1 T F T ) 1 J d J(F ki δa Lkl F lj )(F 1 im T mnf 1 jn ) 1 J d δ km δ ln A Lkl T mn d T : δa L d (117) 42

T ij δe ij dv (118) T ij δe ij = {δe}{t } = {δu} T [B][D][B]{u} (119) {δe} =[B]{δu} (120) {T } =[D]{E} =[D][B]{u} (121) S ij δe ij dv (122) S ij δe ij = {δe}{s} = {δu} T [B]{S} (123) {δe} {δu} u {S} 43

{E} {E} u E = 1 ( ui + u j + u ) k u k e i e j (124) 2 X j X i X i X j δe = 1 ( δui + δu j + δu k u k + u ) k δu k (125) 2 X j X i X i X j X i X j 44

Newton-Raphson 1,. find u h V h such that δu ht (Q(u h ) F )=0 δu h V h (126), V h V, V h V. Q, F,., Q(u) =F (127),. F {F k } 0=F 1 < F 2 < < F n = F (128). Newton-Raphson, Q(u k )=F k (129) u k u k 1. 45

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 Δu k 1 Δu k 2 u k 0 = uk 1 u k 1 u k 2 u k $mbu$. K. u k 0 = u k 1 (130) Q k 0 = Q(u k 0) (131) K k 1 = Q u (132) u=u k 0 46

, K k 1 Δu k 1 = F k Q k 0 (133) u k 1 = uk 0 +Δuk 1 (134) u k 1, uk 0 uk 1 uk. 47

Newton-Raphson 3 F k F K k 1 K k 2 Q k 2 Q k Q k 1 F k 1 Δu k 1 Δu k 2 u k 0 = uk 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 ) (135) K k i = Q (136) u u=u k i 1 K k i Δu k i = F k Q k i 1 (137) u k i = u k i 1 +Δu k i (138) u k i 1 uk i u k, u k i u k, F k Q k i 0. 48

, F k Q k i =0,. 49

T : δa (L) d = t w ds + δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i ρg w d (139) (140) t 0 S : δe dv = t w ds + ρg w dv (141) V V updated Lagrange Total Lagrange Total Lagrange V 50

Ω.. Ω = e Ω e (142),,. dω = dω (143) Ω e Ω e ds = ds (144) Ω e Ω e u N (i), u i., u (i) i u i = N (n) u (n) i (145), (n). X i = N (n) X (n) i (146) 51

δr.. δr = 1 δu k t k ds + Ω Ω ρ 0 δu k g k dω (147) {δu} = {δu 1,δu 2,δu 3 } T (148) {t} = {t 1,t 2,t 3 } T (149) {g} = {g 1,g 2,g 3 } T (150),. δr = = e Ω {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 (151) N (i) 0 0 [N i ]= 0 N (i) 0 (152) 0 0 N (i) [N] =[[N 1 ][N 2 ] [N n ]] (153), 3 3n [N]. 52

{δu (n) } {δu (n) } = 2 { } T δu (1) 1 δu(1) 2 δu(1) 3 δu (n) 1 δu(n) 2 δu(n) 3 (154), } {δu} =[N] {δu (n) (155), δr = e [ { } T [ ]] δu (n) [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω e (156) 53

1 δe ij S ij dω = δr (157) 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 (158),. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T (159) {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (160), δe ij S ij dω = {δe} T {S} dω Ω Ω = {δe} T {S} dω = δr e Ω e 54

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 (161) (162) } T (163),. u i X j {δe} =[Z 1 ] { } δu X (164) u i X j = N(n) X j u (n) i (165) 55

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 3 N (1) X 1 N (1) δu i X j = N(n) X j δu (n) i (166) 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 9 3n [Z 2 ], { } δu =[Z 2 ]{δu (n) } (168) X 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 (167) 56

4 [B] [B] [Z 1 ][Z 2 ] (169). {δe} =[B]{δu (n) } (170) [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 + 6 3 [B (n) ], [ ] [B] = [B (1) ] [B (n) ]. 57 ( ) 1+ u 3 X 3 X 2 + u 3 N (n) X 2 X 3 N (n) X 1 (171) (172)

,. e Ω e {δe}{s} dω = e ] [{δu (n) } T [B] T {S} dω Ω e (173) 58

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 (174),, Q = F = u = [B] T {S} dω (175) Ω e [N] T {t} ds + ρ 0 [N] T {g} dω (176) Ω e } Ω e {u (n) (177) [ T δuh (Q(u h ) F ) ] = 0 (178) e, find u h V h such that [ T δuh (Q(u h ) F ) ] = 0 (179) e 59

for δu h V h, Newton-Raphson. 60

1 Newton-Raphson, K = Q u, dq dt = Q du u dt = K u (180),,. ( Ω S ij δe ij dv ) = Ω Ṡ ij δe ij + S ij δėijdω (181) C ijkl Hooke S ij = C ijkl E kl (182) Ṡ ij = C ijkl Ė kl (183) S ij = λ(tre ij )δ ij +2μE ij (184) C ijkl = λδ kl δ ij +2μδ ki δ jl (185) 61

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 (186) 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 (187) C ij kl 1 2 (C ij kl + C ij lk ) (188) 62

, 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 (189) 63

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 (190) T {Ė11 {Ė} = Ė 22 Ė 33 2Ė12 2Ė23 2Ė31} (191), δe ij S ij dω Ω = δe ij C ijkl Ė kl dω Ω T = {δe} [D]{Ė} dω (192) Ω { } { } T u (n) u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3 (193), {Ė} { =[B] u (n)} (194). 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 (195) 64

4, 1 δe ij Ṡ ij dω = Ω e. [ { δu (n) } T Ω e [B] T [D][B]dΩ { } ] u (n) (196) 65

, 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 (197) 66

, [σ] = S 11 S 12 S 13 S 21 S 22 S 23 (198) S 31 S 32 S 33 [σ] [Σ] = [σ] (199) [σ] {δf} = {δf 11 δf 12 δf 13 δf 21 δf 22 δf 23 δf 31 δf 32 δf 33 } T (200) { F } = { F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32 F 33 } T (201), δf ki S ij F kj = {δf} T [Σ]{ F } (202) 67

6, δf ij = δx i = δu i X j X j (203) F ij = ẋ i = u i X j X j (204), [Z 2 ] { } δu } {δf} = =[Z 2 ] {δu (n) X { F { } } =[Z 2 ] u (n) (205) (206), δf ki S ij F kj = { } T { } δu (n) [Z2 ] T [Σ][Z 2 ] u (n) (207) 68

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 21 S 22 S 23 X 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 (208) (209), [Z 2 ] T [Σ][Z 2 ]=[A] (210) 69

7, 2 δf ki S ij F kj dω = Ω e. [ { δu (n) } T Ω e [A]dΩ { } ] u (n) (211) 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) (212). e Ω e ( [B] T [D][B]+[A] ) dω (213) 70

Updated Lagrange T : δa (L) d = δ t w ds + δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j 2 x j x i Updated Lagrane ρg w d (214) (215) δa ij T ij d = δr (216), δr, A ij, Almange. δa ij = 1 ( δui + δu ) j (217) 2 x j x i T ij 71

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 (218) = δa 11 T 11 + δa 22 T 22 + δa 33 T 33 (219) +2δA 12 T 12 +2δA 23 T 23 +2δA 31 T 31 (220), δa ij,t ij i, j., {δa} T = {δa 11 δa 22 δa 33 2δA 12 2δA 23 2δA 31 } T (221) {T } = {T 11 T 22 T 33 T 12 T 23 T 31 } (222),. δa ij T ij d = {δa} T {T } d (223) 72

2, δa ij. δa ij. δa 11 = δu 1 x 1 (224) δa 22 = δu 2 x 2 (225) δa 33 = δu 3 x 3 (226) 2δA 12 = δu 1 + δu 2 x 2 x 1 (227) 2δA 23 = δu 2 + δu 3 x 3 x 2 (228) 2δA 31 = δu 3 + δu 1 x 1 x 3 (229), u i = N (k) u (k) i (230) u i = N(k) u (k) i x j x j (231), δu i δu i = N (k) δu (k) i (232) δu i x j = N(k) δu (k) i (233) x j 73

., {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 (234) {δa} =[B][δu] (235), [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)]] (236),. δa ij T ij d = {δu} T [B] T {T } d (238) Q,. Q = 74 N (k) x 3 N (k) x 2 N (k) x 1 (237) [B] T [T ]d (239)

, V, δa ij δa ij = 1 ( δui + δu ) n (240) 2 X j X i 75

. 1, Newton-Raphson, Q u K. K = Q (241) u, u, Q = Q t = Q u u t = Q u (242) u,,.,. T : δa (L) d = t w ds + ρg w d (243) δ 76

2 F, J., 2 Piola-Kirchhoff S. F = x i e i e j X j (244) J =detf (245) T = 1 J F S F T (246) S Green-Lagrange E δe. E = 1 ( ui + u j + u ) k u k e i e j (247) 2 X j X i X i X j δe = 1 ( δui + δu j + δu k u k + u ) k δu k (248) 2 X j X i X i X j X i X j δa δe. δe = F T δa F (249),. δe : S dv = (F T δa F ):(JF 1 T F T ) 1 d (250) J = δa : T d (251) 77

3 δe : S dv. Total-Lagrange. δe : Ṡ + 1 ( ) δf T 2 Ḟ + Ḟ T δf : S dv (252). V 2 Piola-Kirchhoff S t (t) S t (t) Ṡ. 1, Ṡ = JF 1 S t (t) F T (253) δe : Ṡ =(F T δa F ):(JF 1 S t (t) F T ) (254) = JδA : Ṡ t (t) (255), V δe : ṠdV = δa : Ṡ t (t)d (256) 78

4 Ḟ δf,., L, 2, Ḟ = L F (257) δf = δf t (t) F (258) L = u i e i e j (259) 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 (260) 2 V 1 ( δf T 2 Ḟ + F ) T 1 ( ) δf : S dv = δf t (t) T L + L T δf t (t) : T d (261) 2 updated Lagrange. δa : Ṡ t (t)+ 1 ( ) δf t (t) T L + L T δf t (t) : T d = δṙ 2 (262) 79

5. { } 1 δa ij Ṡ t (t) ij d + 2 (δf t(t) ki L kj + L ki δf t (t) kj ) T ij d = δṙ (263), 1., Ṡt(t) ṠF t(t) F. δa ij T ij., Ṡ ij D ij. δa ij Ṡ ij = {δa} T { Ṡ } (264) Ṡ = { Ṡ 11 Ṡ 22 Ṡ 33 Ṡ 12 Ṡ 23 Ṡ 31 } (265) Ṡ ij = C ijkl D kl (266) D ij = 1 ( ui + u ) j 2 x j x i (267) Ṡt(t) Truesdell Kirchoff Oldroyd 80

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 (268) = C ij11 D 11 + C ij22 D 22 + C ij33 D 33 + 1 2 (C ij12 + C ij21 )(2D 12 ) (269) + 1 2 (C ij23 + C ij32 )(2D 23 ) (270) + 1 2 (C ij31 + C ij13 )(2D 31 ) (271). C ijkl = 1 2 (C ijkl + C ijlk ) (272) 81

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 (273) 82

6 δa ij, {D} =[B] { u} (274) { } T { 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 (275), 1. δa ij Ṡ t (t) ij d = {δu} T [B] T [ D] [B] { u} d (276) 83

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 (277) = δf ki T ij L kj (278)., δf ki T ij L kj = {δf} T [Σ] {L} (279) {δf} = {δf 11 δf 12 δf 13 δf 21 δf 22 δf 23 δf 31 δf 32 δf 33 } (280) {L} = {L 11 L 12 L 13 L 21 L 22 L 23 L 31 L 32 L 33 } (281) T 11 T 12 T 13 [T ]= T 21 T 22 T 23 (282) [Σ] = T 31 T 32 T 33 [T ] [0] [0] [0] [T ] [0] (283) [0] [0] [T ] 84

δf ij,l ij. 8 δf ij = δu i x j (284) L ij = u i (285) x j. { } δu {δf} = =[Z] {δu} (286) x, {L} = { } u x =[Z] { u} (287) { } { } 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 (288) (289) 85

[ ] 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 (290) 86

9, δf ki T ij L kj. [Z] T [Σ] [Z]. ] { N (i) [G ij = x 1 δf ki T ij L kj = {δf} T [Σ] {L} (291) N (i) x 2 [Z] T [Σ] [Z] = = {δu} T [Z] T [Σ] [Z] { u} (292) 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 1 0 0 0 1 0 (293) 0 0 1 [G 11 ] [G 1n ].. =[G] (294) [G n1 ] [G nn ] δf ki T ij L kj d = {δu} [G] { u} d (295) 87

10. (δa ij S t (t) ij + δf ki T ij L kj )d = {δu} ([B] T T [ ) D] [B]+[G] d { u} (296) V, δa ij, Ṡ t (t) ij Caushy. 2. 88

11,,. 1 d =, [J]. 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 (297) 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 (298) ) + u (k) i (299) [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 (300) (301) 89

, [J] X 1 X 1 [ ] r 1 X J = 2 r 3 X 1 r 1 r 2 r 3 r 2 X 2 X 2 X 3 X 3 X 3 r 1 r 2 r 3 (302) 90

total Lagrange updated Lagrange 1 updated V T : δa (L) d = S : δe dv = δa : Ṡt(t)+ 1 2 ( Ω S ij δe ij dv V t w ds + t w ds + V ρg w d (303) ρg w dv (304) ( ) δf t (t) T L + L T δf t (t) : T d = δṙ (305) ) = Ω Ṡ ij δe ij + S ij δėijdω (306) Ṡ t (t) ij = C ijkl D kl (307) Ṡt(t) Truesdell Kirchoff Oldroyd Total S ij = C ijkl E kl (308) C ijkl (Ṡij, Ėkl ) Ṡ ij = C ijkl Ė kl (309) 91

total Lagrange updated Lagrange 2 Ṡ t (t) ij = C ijkl D kl (310) S ij = C ijkl E kl (311) Ṡ ij = C ijkl Ė kl (312) Ṡ 0 (t) =J 0 (t)f 0 (t) 1 Ṡ t (t)f 0 (t) T (313) Ė 0 (t) =F 0 (t) T DF 0 (t) (314) C pqrs = 1 J F pif qj F rk F sl C ijkl (315) 92

(103) (117) Newton-Raphson 2 (157) (173) (180) (213) (214) (240) (241) (302) (241) (302) (303) (315) 93