all.dvi

29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F

30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan γ =tan = Δδ (4.4) 2 Δz Δδ X 2 Δz γ X 1 4.1: 4.2.,,.,.,.

4.2. 31 4.2:. t 0, X, X t x, X t 0 t u. u = x X (4.5) t 0 X u x t O 4.3:,,. 1 1 X 1, (1, 0) (a, 0) (X 1,X 2 ) (x 1,x 2 ) u 1,u 2. (b), (c). x 1 = X 1 + ax 1 (4.6) x 2 = X 2 (4.7) u 1 = ax 1 (4.8) u 2 =0 (4.9)

32 4 Green-Lagrange x 1 X 2 u 1 X 2 X 2 X 1 X 1 (a) a X1 x 2 X 2 u 2 X 2 X 1 X 1 (b) (c) 4.4: x u 1 1 X 1, (0, 1) (a, 1) (X 1,X 2 ) (x 1,x 2 ) u 1,u 2. (b), (c). x 1 = X 1 + ax 2 (4.10) x 2 = X 2 (4.11) u 1 = ax 2 (4.12) u 2 = 0 (4.13) x 1 X 2 u 1 X 2 X 2 X 1 X 1 X1 x 2 X 2 u 2 X 2 a (a) X 1 X 1 (b) (c) 4.5: x u

4.2. 33 1 1 X 1 X 2, (0, 1) (a, b), (X 1,X 2 ) (x 1,x 2 ) u 1,u 2. (b), (c). x 1 = X 1 + ax 2 (4.14) x 2 = X 2 bx 2 (4.15) u 1 = ax 2 (4.16) u 2 = bx 2 (4.17) x 1 X 2 u 1 X 2 X 2 b X 1 X 1 a X1 x 2 X 2 u 2 X 2 (a) X 1 X 1 (b) (c) 4.6: x u.,, t 0. {(X 1,X 2 ) 0 X 1 1, 0 X 2 1}., X 1,X 2..,,,,..

34 4 Green-Lagrange 4.3,,,,. M 4.7:,,,.,.,. 4.8:,, 2,,.,., 2. dx, 2 X 1 = X (4.18) X 2 = X + dx (4.19)

4.3. 35,. x 1 = X 1 + u(x 1 )=X + u(x) (4.20) x 2 = X 2 + u(x 2 )=X + dx + u(x + dx) (4.21) X + dx + u(x)+ u dx j X j (4.22) dx = x 2 x 1 2 dx = dx + u X j dx j (4.23) dx dx = dx i dx i (4.24) ( = dx i + u )( i dx j dx i + u ) i dx k (4.25) X j X k, u i X j = dx i dx i + u i X j dx j dx i + u i X k dx k dx i + u i X j u i X k dx j dx k (4.26) u i X k 2, dx dx dx i dx i + u i dx j dx i + u i dx k dx i (4.27) X j X k ( ui = dx dx + + u ) j dx i dx j (4.28) X j X i ε ij. ε ij = 1 ( ui + u ) j 2 X j X i (4.29) 2. dx dx dx dx =2ε ij dx i dx j (4.30) 4.1. u 1 = ax 1, u 2 =0 ε 11 = 1 ( u1 + u ) 1 2 X 1 X 1 = 1 (a + a) =a 2

36 4 Green-Lagrange u 1 = ax 1, u 2 =0 ε 12 = 1 ( u1 + u ) 3 2 X 2 X 1 = 1 2 (a) =a 2 γ γ =2ε 12 (4.31) 4.2. 1, u X,,., +, 1., ε 12 = ε 21. u 1 = ax 2 (4.32) u 2 = bx 2 (4.33) ε 11 = 1 (0 + 0) = 0 (4.34) 2 ε 12 = 1 2 (a +0)= a 2 ε 21 = 1 2 (0 + a) = a 2 (4.35) (4.36) ε 22 = 1 ( b b) =b (4.37) 2, 2, u 1 = ax1 2 (4.38) u 2 = 0 (4.39) ε 11 = 1 2 (2aX 1 +2aX 1 )=2aX 1 (4.40) ε 12 = 1 (0 + 0) = 0 2 (4.41) ε 21 = 1 (0 + 0) = 0 2 (4.42) ε 22 = 1 (0 + 0) = 0 2 (4.43)., ε ij, t 0 X.,., ε ij., u X, u 1 = a 1,u 2 = a 2, u i X j =0 ε ij =0.

4.3. 37. X 1,X 2 θ, { } [ ]{ } { } x 1 cos θ sin θ X 1 X 1 cos θ X 2 sin θ = = (4.44) x 2 sin θ cos θ X 2 X 1 sin θ + X 2 cos θ u i = x i X i { } { } u 1 X 1 (cos θ 1) X 2 sin θ = X 1 sin θ + X 2 (cos θ 1) u 2 [ ] cos θ 1 0 [ε] = 0 cosθ 1 (4.45) (4.46), 2,,.,,. θ 0 cos θ 1,. [ε] = [ ] [ cos θ 1 0 0 cosθ 1 0 0 0 0 ] (4.47),,., = Δt (4.48).,, Δt,. n x X u θ u X 4.9:

38 4 Green-Lagrange θ cos θ 1, sin θ θ (4.49), X 1. X 1 (X 1,X 2 ). n,.,, x = X + u = X + θn X (4.50) u = θn X (4.51) u 1 = θ(n 2 X 3 n 3 X 2 ) (4.52) u 2 = θ(n 3 X 1 n 1 X 3 ) (4.53) u 3 = θ(n 1 X 2 n 2 X 1 ) (4.54) u 1 u 1 u 1 =0, = n 3, = n 2 X 1 X 2 X 3 (4.55) u 2 u 2 u 2 = n 3, =0, = n 1 X 1 X 2 X 3 (4.56) u 3 u 3 u 3 = n 2, = n 1, = 0 X 1 X 2 X 3 (4.57) u 1 X 2 = u 2 X 1, u 2 X 3 = u 3 X 2, u 3 X 1 = u 1 X 3 (4.58) ε ij = 0 (4.59), ε ij =0.,, ε ij =0. 4.4 Green-Lagrange, θ ε ij =0..,

4.4. Green-Lagrange 39 dx, 2 X 1 = X, X 2 = X + dx,. x 1 = X 1 + u(x 1 )=X + u(x) (4.60) x 2 = X 2 + u(x 2 )=X + dx + u(x + dx) (4.61) X + dx + u(x)+ u dx j X j (4.62) dx = x 2 x 1 dx = dx + u X j dx j (4.63), 2 2 u i X j. u i X k dx dx = dx i dx i (4.64) ( = dx i + u )( i dx j dx i + u ) i dx k (4.65) X j X k = dx i dx i + u i X j dx j dx i + u i X k dx k dx i + u i X j u i X k dx j dx k (4.66).,, (4.66) 2,3,4 ( ui dx dx = dx dx + + u j + u ) k u k dx i dx j (4.67) X j X i X i X j ε ij E ij. E ij = 1 ( ui + u j + u ) k u k 2 X j X i X i X j (4.68) dx dx dx dx =2E ij dx i dx j (4.69), dx dx dx dx 2ε ij dx i dx j (4.70). E ij =0, ε ij..

40 4 Green-Lagrange { u 1 u 2 } { = X 1 (cos θ 1) X 2 sin θ X 1 sin θ + X 2 (cos θ 1), (4.68), E ij. E 11 = 1 ( u1 + u 1 + u ) k u k 2 X 1 X 1 X 1 X 1 = 1 ( u1 + u 1 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 2 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 } (4.71) (4.72) (4.73) = 1 2 { (cos θ 1) + (cos θ 1) + (cos θ 1) 2 +sin 2 θ +0 } (4.74) = 0 (4.75) E 22 = 1 ( u2 + u 2 + u ) k u k 2 X 2 X 2 X 2 X 2 = 1 ( u2 + u 2 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 (4.76) (4.77) = 1 2 { (cos θ 1) + (cos θ 1) + ( sin θ) 2 +(cosθ 1) 2 +0 } (4.78) = 0 (4.79) E 12 = 1 ( u1 + u 2 + u ) k u k 2 X 2 X 1 X 1 X 2 = 1 ( u1 + u 2 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 2 X 2 X 1 X 1 X 2 X 1 X 2 X 1 X 2 (4.80) (4.81) = 1 { sin θ +sinθ +(cosθ 1)( sin θ) sin θ(cos θ 1) + 0} 2 (4.82) = 0 (4.83) E 21 = E 12. 2,. Green-Lagrange. 4.5 Green- Lagrange.

4.5. 41, u 1 = ax 1, u 2 =0, u 3 = 0 (4.84)., 2, 2.,,, 10. u 1 = ax 1, u 2 = νax 1,u 3 = νax 1 (4.85). ν Poisson. ε 11,E 11 ε 11 = 1 (a + a) =a 2 (4.86) E 11 = 1 ( ) a + a + a 2 = 1 ( ) 2a + a 2 2 2 (4.87)., 1 L 1 l. u = l L ε 11 = u L E 11 = 1 ( ) l 2 2 L 1 2 (4.88) (4.89).. l =1.01L, E 11 =0.01005,ε 11 =0.01,. l =1.1L, E 11 =0.105,ε 11 =0.1. l =2L, E 11 =1.5,ε 11 =1.,,. 2,. u 1 X 2 = γ 0 (4.90) 0. u 1 = γx 2 (4.91) 0.,.. u 1 / X 2 = γ.

42 4 Green-Lagrange X 2 X 1 4.10: [ ] ε = 1 0 γ 2 γ 0 (4.92),, Green-Lagrange E 22 0. [ ] E = 1 2 (F T F I) = 1 0 γ, (4.93) 2 γ γ 2 4.6 Green-Lagrange 1 1 X 1 a θ Green-Lagrange. X 2 θ 1 a X 1 4.11: (X 1,X 2 ) (x 1,x 2 ) { } { } x 1 (1 + a)x 1 = X 2 x 2 (4.94)

4.6. Green-Lagrange 43 (x 1,x 2) θ (x 1,x 2 ) { } [ ]{ } { } { } x 1 cos θ sin θ x 1 x 1 = = cos θ x 2 sin θ (1 + a)x 1 cos θ X 2 sin θ x 2 sin θ cos θ x 2 x 1 sin θ + x 2 cos θ = (1 + a)x 1 sin θ + X 2 cos θ (4.95) u 1,u 2 { } { } u 1 X 1 ((1 + a)cosθ 1) X 2 sin θ = (4.96) X 1 (1 + a)sinθ + X 2 (cos θ 1) u 2. Green-Lagrange E 11 = 1 ( u1 + u 1 + u ) k u k (4.97) 2 X 1 X 1 X 1 X 1 = 1 ( u1 + u 1 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 (4.98) 2 X 1 X 1 X 1 X 1 X 1 X 1 X 1 X 1 = 1 { ((1 + a)cosθ 1) + ((1 + a)cosθ 1) + ((1 + a)cosθ 1) 2 +(1+a)sin 2 θ +0 } 2 (4.99) = 1 2 (2a + a2 ) (4.100) E 22 = 1 ( u2 + u 2 + u ) k u k 2 X 2 X 2 X 2 X 2 = 1 ( u2 + u 2 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 (4.101) (4.102) = 1 2 { (cos θ 1) + (cos θ 1) + ( sin θ) 2 +(cosθ 1) 2 +0 } (4.103) = 0 (4.104) E 12 = 1 ( u1 + u 2 + u ) k u k 2 X 2 X 1 X 1 X 2 = 1 ( u1 + u 2 + u 1 u 1 + u 2 u 2 + u ) 3 u 3 2 X 2 X 1 X 1 X 2 X 1 X 2 X 1 X 2 (4.105) (4.106) = 1 { sin θ +(1+a)sinθ + ((1 + a)cosθ 1)( sin θ)+(1+a)sinθ(cos θ 1) + 0} 2 (4.107) = 0 (4.108)

44 4 Green-Lagrange, Green-Lagrange.,., Green-Lagrange,. Green-Lagrange.

45 5 Cauchy.,,,,., σ.,, 3,,. 5.1?,,,,,,,.,,,. 5.1:,,.,,,

46 5 Cauchy.,,,,,.,,,,,.,,,,,,.,.,,,,.,.,, (a) (b),,,.,,,,,. (a) (b) (c) 5.2: 5.2 Cauchy.,.

5.2. Cauchy 47.,,,.,, F., Q,.,,.,,., Newton,. F F F Q F Q 5.3:, n. n., ds df n ( 5.2). df n n df(n), df n., t n. t n = df n (5.1) ds

48 5 Cauchy t l n mds df n 5.4:,,,,.. t n n.,,,. x 1. x 2 x 3 x 1 e 1, t 1 = F /A., x 1 x 2, x 3 e 3, t 3 =0. : n = e 1 t 1 =1 e 1 +0 e 2 +0 e 3 : n = e 2 t 2 =0 e 1 +0 e 2 +0 e 3 : n = e 3 t 3 =0 e 1 +0 e 2 +0 e 3 (5.2) x 3 x 2 F A x 1 n t 1 = F /A n t 3 =0 5.5:

5.3. Cauchy 4 Cauchy 49, n e 1, e 2, e 3, t 1, t 2, t 3. t 1, t 2, t 3 T ij. t 1 = T 11 e 1 + T 12 e 2 + T 13 e 3 (5.3) t 2 = T 21 e 1 + T 22 e 2 + T 23 e 3 (5.4) t 3 = T 31 e 1 + T 32 e 2 + T 33 e 3 (5.5) t i = T ij e j (5.6) T ij,, T 11,T 22,T 33, T 12,T 21,T 23,T 32,T 31,T 13. 5.1. 5.0,,, (5.2)., 0,,, 2. : n = e 1 t 1 =0 e 1 +1 e 2 +0 e 3 : n = e 2 t 2 =0 e 1 +0 e 2 +0 e 3? : n = e 3 t 3 =0 e 1 +0 e 2 +0 e 3? (5.7) T ij e i e j T = T ij e i e j (5.8) Cauchy. Cauchy. Cauchy 2,,,.,. 5.3 Cauchy 4 Cauchy,,, Cauchy t n = T T n (5.9)

50 5 Cauchy. (5.15), T t i = T ij e j (5.10). (5.10), (5.15). 4 (Cauchy 4 ),. ABC n, t n. t n ABC,. Δs, t n ds = t n Δs (5.11) ABC., e 1, e 2, e 3 ABC t 1, t 2, t 3., OCB, OAC, OBA, OCB, OAC, OBA Δs 1, Δs 2, Δs 3 t 1 ds = t 1 Δs 1, t 2 ds = t 2 Δs 2, t 3 ds = t 3 Δs 3 (5.12) OCB OAC OBA x 3 x 2 C B t n t 1 Δs Δs 1 O Δs 3 Δs 2 A x 1 t 2 t 3 ρ, Δv, a, g, Cauchy 4. ρ(a g)δv = t n Δs t 1 Δs 1 t 2 Δs 2 t 3 Δs 3 (5.13)

5.3. Cauchy 4 Cauchy 51,., 4, 4. Δs, Δv/Δs 0. 5.3. Δs i /Δs = n i (5.14), n = n i e i Δs 1 t n = t 1 Δs + t Δs 2 2 Δs + t Δs 3 3 Δs = t i n i = T ij e j n i = T ij (e j e i )n k e k = T T n (5.15) Cauchy., T T n t n. Cauchy. 5.2. (5.15), (e i e j ) e k = δ jk e i t n = t i n i = T ij e j n i = T ij e j δ ik n k = T ij (e j e i )n k e k = T T n 5.3. A, B, C {a, 0, 0}, {0,b,0},{0, 0,c} CA = a, CB = b n a, b a 0 bc a b = 0 b = ca, n = 1 bc ca (5.16) c c ab b2 c 2 + c 2 a 2 + a 2 b 2 ab Δs 1 = 1 2 bc, Δs 2 = 1 2 ca, Δs 3 = 1 2 ab, Δs = 1 2 a b, Δs = 1 2 b2 c 2 + c 2 a 2 + a 2 b 2 (5.17) n 1 = Δs 1 Δs, n 2 = Δs 2 Δs, n 3 = Δs 3 Δs (5.18)

52 5 Cauchy,, n a, b 2 ΔS(= 2Δs. a, b θ, a b sin θ ΔS = a b sin θ = a b 1 cos 2 θ (5.19) { } 2 a b = a b 1 = a a b 2 b 2 (a b) 2 (5.20) = (a 2 + c 2 )(b 2 + c 2 ) c 4 = a 2 b 2 + c 2 a 2 + b 2 c 2 (5.21) 5.4,.. T 11 T 12 T 13 T 21 T 22 T 23 (5.22) T 31 T 32 T 33,, e i e j,,,,.. T 1, T 2, T 3, ē i, T 11 T 12 T T 13 1 0 0 T 21 T 22 T 23 = 0 T2 0 T 31 T 32 T 33 0 0 T3 (5.23),, T ij e i e j = T i ē i ē i (5.24) = T 1 ē 1 ē 1 + T 2 ē 2 ē 2 + T 3 ē 3 ē 3 (5.25),. T 11 ε x ε 11 T 22 ε y = ε 22 T 12 γ xy 2ε 12 (5.26)

5.5., 53,. {ε x,ε y,γ xy },...,,,,. 5.5,, 9.,.., σ y,, σ y.., σ ij σ = ( ) 1 3 2 2 σ ij σ ij (5.27) σ ij = σ ij 1 3 σ kkδ ij (5.28) = σ ij 1 3 (σ 11 + σ 22 + σ 33 )δ ij (5.29),,,.,.

54 6 Gauss Cauchy 1 Gauss,,,.,., Cauchy Gauss, Cauchy 1., Navier-Stokes. Cauchy 1 Newton Navier-Stokes., Cauchy Cauchy 2. 6.1 Gauss Gauss ρ v S ds ds v v ds ( (a) ) ds n v ds = n vds

6.1. Gauss 55 ds n v v n ds n v n v ds v θ ds (a) (b) (c) (d) n (b) n v (c) n v θ v cos θ n =1 n v = n v cos θ n v n vds (d) v n cos θ<0 n v < 0 S S ρn vds (6.1) S v a n ads (6.2) S V V S a S S S V V μ(v ) μ(v )= n ads (6.3) μ(v ) V V = V 1 + V 2 S μ(v )=μ(v 1 + V 2 )=μ(v 1 )+μ(v 2 ) (6.4)

56 6 Gauss Cauchy 1 n 1 n 2 V 1 V 2 dv μ(dv ) dv a diva diva = dμ (6.5) dv V V divadv = V V divadv = Gauss dμ dv = μ(v ) (6.6) dv S n ads (6.7) 6.2 diva dx 1,dx 2,dx 3 P (x 1,x 2,x 3 ), Q 1 (x 1 +dx 1,x 2,x 3 ), Q 2 (x 1,x 2 +dx 2,x 3 ), Q 3 (x 1,x 2,x 3 +dx 3 ) PQ 1,PQ 2,PQ 3 3 dv diva divadv =divadx 1 dx 2 dx 3 (6.8) V n ads a S x 1 P, Q 1 a = a(x 1,x 2,x 3 ), a 1 = a(x 1 + dx 1,x 2,x 3 ) dx 1 a 1 (x 1 + dx 1,x 2,x 3 )=a 1 (x 1,x 2,x 3 )+ a 1 dx 1 x 1 (6.9) a 2 (x 1 + dx 1,x 2,x 3 )=a 2 (x 1,x 2,x 3 )+ a 2 dx 1 x 1 (6.10) a 3 (x 1 + dx 1,x 2,x 3 )=a 3 (x 1,x 2,x 3 )+ a 3 x 1 dx 1 (6.11)

6.2. 57 n P n =( 1, 0, 0), Q 1 n 1 =(1, 0, 0) dx 2 dx 3 n ads S x 1 ( n adx 2 dx 3 + n 1 a 1 dx 2 dx 3 = a 1 dx 2 dx 3 + a 1 + da ) 1 dx 1 dx 2 dx 3 (6.12) dx 1 = da 1 dx 1 dx 1 dx 2 dx 3 (6.13) a a 1 n Q 3 n 1 Q 2 P Q 1 x 2,x 3 ( a1 n ads = + a 2 + a ) 3 dx 1 dx 2 dx 3 (6.14) x 1 x 2 x 3 S V divadv =divadx 1 dx 2 dx 3 (6.15) Gauss n ads = divadv S V ( a1 divadx 1 dx 2 dx 3 = + a 2 + a ) 3 dx 1 dx 2 dx 3 (6.16) x 1 x 2 x 3 diva diva = a 1 x 1 + a 2 x 2 + a 3 x 3 (6.17), Hamilton (nabla) x = x i e i (6.18) x a = a 1 x 1 + a 2 x 2 + a 3 x 3 =diva (6.19) Gauss x adv = n ads (6.20) V S

58 6 Gauss Cauchy 1 6.3 Gauss Gauss (6.20) a 1 + a 2 + a 3 dv = n 1 a 1 + n 2 a 2 + n 3 a 3 ds (6.21) V x 1 x 2 x 3 S a 1,a 2,a 3 T 1j,T 2j,T 3j (j =1, 2, 3) T 1j + T 2j + T 3j dv = n 1 T 1j + n 2 T 2j + n 3 T 3j ds (6.22) V x 1 x 2 x 3 S T ij dv = n i T ij ds (6.23) x i V e j T ij e j dv = n i T ij e j ds (6.24) V x i S e k T ij e i e j dv = n k e k T ij e i e j ds (6.25) V x k S x T dv = n T ds = T T nds (6.26) V (6.21) T j1,t j2,t j3 (j =1, 2, 3) T x dv = T nds (6.27) V (6.26) ( ) (6.27) ( ) S S S S 6.4 Cauchy (6.26) T Cauchy 2 T T n 0 T T nds + b ρadv = 0 (6.28) S V

6.5. Cauchy 1 59 (6.26) (6.28) 1 ( x T + b ρa)dv = 0 (6.29) V ρa = x T + b (6.30) Cauchy 1 ρa j = T ij x i + b j (6.31) j 6.1 (6.30) x = x i e i 1 x T = x i e i T kl e k e l = T il x i e l = T ij x i e j Tji x j e i 6.1 6.5 Cauchy 1 Cauchy 1 Gauss Cauchy 1 Gauss dx 1,dx 2 P (x 1,x 2 ), Q 1 (x 1 + dx 1,x 2 ), Q 2 (x 1,x 2 + dx 2 ) PQ 1,PQ 2 2 x 1 P, Q 1 t 1 n = P n =( 1, 0, 0),Q 1 n 1 =(1, 0, 0) t 1 (x 1,x 2 )= T 11 (x 1,x 2 )e 1 T 12 (x 1,x 2 )e 2 (6.32) t 1 (x 1 + dx 1,x 2 )=T 11 (x 1 + dx 1,x 2 )e 1 + T 12 (x 1 + dx 1,x 2 )e 2 (6.33)

60 6 Gauss Cauchy 1 dx 1 T 11 (x 1 + dx 1,x 2 )=T 11 + T 11 dx 1 x 1 (6.34) T 12 (x 1 + dx 1,x 2 )=T 12 + T 12 dx 1 x 1 (6.35) x 2 T 22 + T 22 x 2 dx 2 T 21 + T 21 x 2 dx 2 T 12 + T 12 x 1 dx 1 T 11 T 11 + T 11 x 1 dx 1 T 12 T 21 T 22 x 1 ( T 11 dx 2 + T 11 + T ) ( 11 dx 1 dx 2 T 21 dx 1 + T 21 + T ) 21 dx 2 dx 1 = 0 (6.36) x 1 x 2 dx 1 dx 2 x 2 T 11 x 1 dx 1 dx 2 + T 21 x 2 dx 2 dx 1 = 0 (6.37) T 11 x 1 + T 21 x 2 = 0 (6.38) T 12 x 1 + T 22 x 2 = 0 (6.39) T ij x i = 0 (6.40)

T 22 T 21 6.6. Cauchy 2 61 6.6 Cauchy 2 Cauchy 2 Cauchy T = T T (6.41) T 22 + T 22 x 2 dx 2 T 21 + T 21 x 2 dx 2 T 11 T 12 T 12 + T 12 x 1 dx 1 T 11 + T 11 x 1 dx 1 T 21 T22 T 11,T 22 T 12 T 11 T 12 T 11 T 21 T22 T 12 dx 2 dx 1 2 2=T 21dx 1 dx 2 2 2 (6.42) T 12 = T 21 (6.43) Cauchy

103 11 Cauchy x X Cauchy 1 Cauchy x X 2.4 x X 4 Green-Lagrange u X Cauchy 1 x X Cauchy 1,2Piola-Kirchoff 11.1 Cauchy 1 ρa j = T ij x i + ρg j (11.1) T ij ρ m, v ρ = m v (11.2) u. X i dx i dx i dx 1,dX 2,dX 3 V = dx 1 dx 2 dx 3 6 v

104 11 Cauchy x X dx 3 dx 3 dx 2 dx 2 dx 1 dx 1 dx i,dx i dx 1 = x 1 dx 1 + x 1 dx 2 + x 1 dx 3 X 1 X 2 X 3 (11.3) dx 2 = x 2 dx 1 + x 2 dx 2 + x 2 dx 3 X 1 X 2 X 3 (11.4) dx 3 = x 3 dx 1 + x 3 dx 2 + x 3 dx 3 X 1 X 2 X 3 (11.5) dx 1 dx 2 dx = 3 e i x 1 X 1 x 1 X 2 x 1 X 3 x 2 X 1 x 2 X 2 x 2 X 3 x 3 X 1 x 3 X 2 x 3 X 3 dx 1 dx 2 dx 3 (11.6) dx 1 = x 1 dx 1 e 1 + x 1 dx 2 e 2 + x 1 dx 3 e 3 X 1 X 2 X 3 (11.7) dx 2 = x 2 dx 1 e 1 + x 2 dx 2 e 2 + x 2 dx 3 e 3 X 1 X 2 X 3 (11.8) dx 3 = x 3 dx 1 e 1 + x 3 dx 2 e 2 + x 3 dx 3 e 3 X 1 X 2 X 3 (11.9) V = dx 1 dx 2 dx 3 x 1 x X 1 dx 1 x 1 X 2 dx 1 2 X 3 dx x 1 x 1 x 1 3 X x v = 2 x X 1 dx 2 x 1 X 2 dx 2 1 X 2 X 3 x 2 X 3 dx 3 = 2 x 2 x 2 x 3 x X 1 dx 3 x 1 X 2 dx 3 X 1 X 2 X 3 dx 2 X 3 dx 3 x 3 x 3 x 3 1 dx 2 dx 3 = JV (11.10) X 1 X 2 X 3 ρ 0 = Jρ, ρ = 1 J ρ 0 (11.11) 11.1 a, b, c (a b) c a b a 1 b 1 e 1 a b = a 2 b 2 e 2 a 3 b 3 e 3 =(a 2b 3 a 3 b 2 )e 1 +(a 3 b 1 a 1 b 3 )e 2 +(a 1 b 2 a 2 b 1 )e 3 (11.12)

11.2. x X? 105 (a b) c (a b) c =(a 2 b 3 a 3 b 2 )c 1 +(a 3 b 1 a 1 b 3 )c 2 +(a 1 b 2 a 2 b 1 )c 3 (11.13) = a 1 b 2 c 3 + a 2 b 3 c 1 + a 3 b 1 c 2 a 1 b 3 c 2 + a 2 b 1 c 3 + a 3 b 2 c 1 (11.14) a 1 b 1 c 1 = a 2 b 2 c 2 (11.15) a 3 b 3 c 3 11.1 11.2 x X? A det A A ij Cayley-Hamilton det A A kl =(deta)a 1 lk (11.16) 11.2 A ij B ij det A (11.18) A A 1 ij = 1 det A B ij (11.17) (det A)A 1 ij = B ij (11.18) (det A)δ ij = A im B mj (11.19) i = j det A = A im B mi A kl (11.19) 11.2 (11.16) J x i det A = δ ik δ ml B mi = B lk A kl (11.20) =detaa 1 lk (11.21) J = J x i 2 x k X m X l X m x i X l x k (11.22)

106 11 Cauchy x X 11.3 11.3 J x i = J F kl F kl x i = J X l x k 2 x k X m = J X m X l x i = JF 1 F kl lk x k = J X l x i X l x k (11.23) x i X m x k (11.24) X m x i X l X l x k (11.25) ( J 1 x ) i = 0 (11.26) x i X j 11.4 x i ( J 1 x ) i 2 J x i = J X j x i + J 1 2 x i (11.27) X j x i X j = J 1 2 x k X m X l x i + J 1 2 x i X m (11.28) X m X l x i x k X j X m X j x i = J 1 2 ( ) x k Xm x i Xl + J 1 2 x i X m (11.29) X m X l x i X j x k X m X j x i = J 1 2 x k X m X l + J 1 2 x i X m X m X l X j x k X m X j x i (11.30) = J 1 2 x k X l δ mj + J 1 2 x i X m X m X l x k X m X j x i (11.31) = J 1 2 x k X l + J 1 2 x i X m X j X l x k X m X j x i (11.32) = 0 (11.33) 11.4 P A A 0 Cauchy Piola-Kirchhoff Cauchy n ds df t n = df n ds, t n = T T n (11.34)

11.2. x X? 107 ds df n ds ds t = df n ds, t = Π T N (11.35) Π N n n = Π N df n t 0 df n N df n n ds ds t Nauson nds =(detf )F T NdS (11.36) F T N =(detf )NdS (11.37) df n = Π T NdS = 1 J ΠT F T nds (11.38) Cauchy df n /ds = t n = T T n T = 1 J F Π, Π = JF 1 T (11.39) T Π

108 11 Cauchy x X Piola-Kirchhoff Piola-Kirchhoff df n F 1 ds t = F 1 df n ds = F 1 t, t = S T N (11.40) S F df n F 1 S Nanson f n = F S T NdS = 1 J FST F T nds (11.41) Cauchy df n /ds = t n = T T n S T = 1 J F S F T, S = JF 1 TF T (11.42) t 0 N F 1 df n t n df n ds ds F 1

225 20? 20.1, A [B]. [B] A Ω, Ω Ω, Ω Ω D. t, ρg, u V., ρ, g, V. A t Ω D ρg Ω 20.1: 20.2. [B] t, g, u V.

226 20? [[1]] (Cauchy 1 ) x T + ρg = 0 (20.1) X (S F ) T + ρ 0 g = 0 (20.2) [[2]] u = u on Ω D (20.3) T T n = t on Ω Ω D (20.4) ( S F T ) T N = t (20.5) [[3]] [[4]] [1], [2] [4] [3] [4] T T, u U. Ť T, ǔ U,. ( x Ť + ρg) ǔ dv = 0 (20.6) v,., s t,s u t, u s, s = s t + s u. Ť :(ǔ x )dv = t ǔ ds + n Ť u ds + ρg ǔdv (20.7) v s t s u v, s u w =0 w W. ǔ U, w W ǔ + w U. Cauchy T,. T :(ǔ x )dv = t ǔ ds + n T u ds + ρg ǔdv (20.8) v s t s u v T : {(ǔ+w) x }dv = t (ǔ+w)ds+ n T uds+ ρg (ǔ+w)dv (20.9) v s t s u v Cauchy T,. T :(ǔ x )dv = t ǔ ds + n T u ds + ρg ǔdv (20.10) v s t s u v

20.2. 227 T : {(ǔ+w) x }dv = t (ǔ+w)ds+ n T uds+ ρg (ǔ+w)dv (20.11) v s t s u v. T :(w x )dv = t w ds + ρg wdv (20.12) v s t v. T : δa (L) dv = t w ds + v δv v ρg w dv (20.13) δa (L), w W Almange. δa (L)ij = 1 ( wi + w ) j (20.14) 2 x j x i S : δe dv = t w ds + ρg w dv (20.15) δe = F T δa L F V v δv T : δa L dv V dv = V v V S : δedv (20.16) 1 dv (20.17) J T = 1 J F S F T S ij = JF 1 im T mnf 1 jn (20.18) (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 2 X i X j X j = 1 ( δui + δu j + δu k 2 X j X i X i ( )} u k δ li X i u k X j + u k X i δu k X j = δe ij (20.19) )

228 20? V S : δedv = (F T δa L F ):(JF 1 T F T ) 1 v J dv = J(F ki δa Lkl F lj )(F 1 im T mnf 1 jn ) 1 v J dv = δ km δ ln A Lkl T mn dv v = T : δa L dv (20.20) v 20.3 T ij δe ij dv (20.21) T ij δe ij = {δe}{t } = {δu} T [B][D][B]{u} (20.22) {δe} =[B]{δu} (20.23) {T } =[D]{E} =[D][B]{u} (20.24) S ij δe ij dv (20.25) S ij δe ij = {δe}{s} = {δu} T [B]{S} (20.26) {δe} {δu} u {S} {E} {E} u E = 1 ( ui + u j + u ) k u k e i e j (20.27) 2 X j X i X i X j δe = 1 ( δui + δu j + δu k u k + u ) k δu k (20.28) 2 X j X i X i X j X i X j

229 21 Newton-Raphson Cauchy Cauchy 2Piola-Kirchhoff Newton-Raphson 3 2 2 2 Newton-Raphson Newton-Raphson 21.1 1 1 2 3 4 10 y = x 2 y =2, 3,, 10 x>0 a f(x)

230 21 Newton-Raphson f(x) =a (21.1) x a f(x) a a 10 0 <a 1 <a 2 < <a 10 = a a 1 x 1 a 2 x 2 x 9 a 10 (= a) x a a i 1.1 x (i) r r = a f ( x (i)) (21.2) x r =0 x (i) x (i) x r 1 r =0 a = 2 a 2 2 OS 2 2 bit r =0 ɛ (10 8 10 10 ) r <ɛ x (i) x ɛ 15 ɛ =10 100 ɛ =10 6 10 8 ɛ r <ɛ x (i) 1.2 x (i) Δx x (i) k+1 = x k +Δx i (21.3)

21.1. 1 231 k x k k +1 x k+1 a (i) x (i) k+1 = x k +Δx a (i) (21.4) Newton-Raphson f(x) f(x) f(x) 2.Newton-Raphson f (x) Newton-Raphson Newton-Raphson Newton [x, x + dx] Δx [x, x +Δx] f(x) a 1 f(x) =a 1 x 1 f(x) =a 2 x 2 x (1) 2 f (x 1 )Δx (1) 1 = a 2 f(x 1 ) (21.5) Δx (1) 1 1 = f (x 1 ) (a 2 f(x 1 )) (21.6) x (1) 2 = x 1 +Δx (1) 1 (21.7)

232 21 Newton-Raphson a 2 f(x (1) a 2 f(x (1) 2 ) 2 ) 2 ) f(x (2) a 2 f(x 1 ) Δx (2) 2 a 2 f(x (1) 2 ) a 1 Δx (1) 1 a 1 x 1 x (1) x (1) 2 x 1 x (2) 2 2 r (1) 2 r (1) 2 = a 2 f(x (1) 2 ) (21.8) r (1) 2 r x (2) 2 f (x (1) 2 )Δx(2) 2 = a 2 f(x (1) 2 ) (21.9) 1 Δx (2) 2 = f (x (1) 2 ) (a 2 f(x (1) 2 )) (21.10) x (2) 2 = x (1) 2 +Δx (2) 2 (21.11) x (i) 2 x(i+1) 2 f (x (i) 2 )Δx(i+1) 2 = a 2 f(x (i) 2 ) (21.12) 1 Δx (i+1) 2 = f (x (i) 2 ) (a 2 f(x (i) 2 )) (21.13) r (i+1) 2 <ε x (i+1) 2 = x (i) 2 +Δx (i+1) 2 (21.14) r (i+1) 2 = a 2 f(x (i+1) 2 ) (21.15)

21.1. 1 233 Newton-Raphson 2 y = x 2 x 1 =1,y 1 =1 x =0,y =0 0 f (0) = 0 Newton-Raphson x 0 f (0) = 0 a 2 =2,x 1 =2 f(x) =x 2, df (x) dx =2x (21.16) df (1) dx =2, 2Δx(1) 1 =2 1 (21.17) Δx (1) 1 = 1 2 (21.18) a 2 =2,x (1) 2 = 3 2 x (1) 2 = x 1 +Δx (1) 1 =1+ 1 2 = 3 2 df ( 3 2 ) dx =3, f(3 2 )=9 4, 3Δx(2) 2 =2 9 4 =1.5 (21.19) (21.20) Δx (2) 2 = 1 12 (21.21) x (2) 2 = x (1) 2 +Δx (2) 2 = 3 2 1 12 = 17 =1.4166 12 (21.22) a 2 =2,x (2) 2 = 17 12 df ( 17 ) 12 = 17 dx 6, f(17 12 )=289 144, 17 6 Δx(3) 2 =2 289 144 (21.23) Δx (3) 2 = 1 408 (21.24) x (3) 2 = x (2) 2 +Δx (3) 2 = 17 12 1 408 = 577 =1.4142156 408 (21.25) a 2 =2,x (3) 2 = 577 408 df ( 577 408 ) dx = 577 204, f(577 408 577 )=577 408 408, 577 577 577 204 Δx(4) 2 =2 408 408 (21.26)

234 21 Newton-Raphson Δx (4) 2 = 1 (21.27) 470832 x (4) 2 = x (3) 2 +Δx (4) 2 = 577 408 1 470832 = 665857 =1.41421356237 (21.28) 470832 4 do k =2,10 a k = k do i = 1, 100 df (i 1) k dx x (i) k Δx (i) k = x (i 1) k = a k f(x (i 1) k +Δx (i) k r (i) k = a k f(x (i) k ) if r (i) k <ε convergence flag = 1 exit loop else convergence flag = 0 end if end if convergence flag = 1 write x else write not converged exit end if end ) Δx (i) k i k bug k iincre(index of increment) i iiter(index of interation) nincre niter 1dim.f f(x) f(x) =cx(c )

21.2. 2 235 df dx = c cδx = a cx k Δx = 1 c (a cx k) (21.29) x (1) k+1 = x k + 1 c (a cx k)= a c (21.30) r (1) k+1 = a ca c = 0 (21.31) a c K f(x) Ku KΔu = f Ku (21.32) u =0 f(x) Q(u) =f (21.33) Q Newton-Raphson u Newton-Raphson 21.2 2 4 r 1,r 2 x 1,x 2 x i = N (1) (r 1,r 2 )x (1) i + N (2) (r 1,r 2 )x (2) i + N (3) (r 1,r 2 )x (3) i + N (4) (r 1,r 2 )x (4) i (21.34) x 1,x 2 r 1,r 2 Newton-Raphson 2 Jacobi dx 1 = x 1 r 1 dr 1 + x 1 r 2 dr 2 (21.35) dx 2 = x 2 r 1 dr 1 + x 2 r 2 dr 2 (21.36)

236 21 Newton-Raphson { } dx 1 dx 2 = [ x1 x 1 r 1 r 2 x 2 x 2 r 1 r 2 ]{ } dr 1 dr 2 (21.37) x 1,x 2 r 1 =0,r 2 =0 (x G 1,xG 2 ) x 1,x 2 (x G 1,xG 2 ) Δx <1> 1 = x 1 x G 1 (21.38) Δx <1> 2 = x 2 x G 2 (21.39) x 1,x 2 (x G 1,xG 2 ) (Δx 1, Δx 2 ) (Δr 1, Δr 2 ) Δr 1, Δr 2 { } Δr 1 = Δr 2 [ x1 x 1 r 1 r 2 x 2 x 2 r 1 r 2 ] 1 { } Δx 1 Δx 2 (21.40) r 1 <1> =0+Δr 1 (21.41) r 2 <1> =0+Δr 1 (21.42) (r <1> 1,r <1> 2 ) (x 1,x 2 ) (x 1 x G 1,x 1 x G 1 ) > (x 1 x <1> 1,x 1 x <1> 1 ) (21.43) r 1,r 2

237 22 22.1 T : δa (L) dv = v v t w ds + v ρg w dv (22.1) δa (L), w W Almange. δa (L)ij = 1 2 ( wi + w ) j x j x i (22.2) t 0 S : δe dv = t w ds + ρg w dv (22.3) V V updated Lagrange Total Lagrange Total Lagrange V 22.2 Ω.. Ω = e Ω e (22.4)

238 22,,. dω = dω (22.5) Ω e Ω e ds = ds (22.6) Ω e Ω e u N (i), u i. u i = N (n) u (n) i (22.7), u (i) i, (n). X i = N (n) X (n) i (22.8) 22.3 1 δr. δr = δu k t k ds +. Ω Ω ρ 0 δu k g k dω (22.9) {δu} = {δu 1,δu 2,δu 3 } T (22.10) {t} = {t 1,t 2,t 3 } T (22.11) {g} = {g 1,g 2,g 3 } T (22.12), δr = {δu} T {t} ds + ρ 0 {δu} T {g} dω Ω Ω = [ ] {δu} T {t} ds + ρ 0 {δu} T {g} dω e Ω e Ω e (22.13).

22.4. 1 239 [N i ]=, 3 3n [N]. {δu (n) } {δu (n) } = N (i) 0 0 0 N (i) 0 (22.14) 0 0 N (i) [N] =[[N 1 ][N 2 ] [N n ]] (22.15) { } T δu (1) 1 δu(1) 2 δu(1) 3 δu (n) 1 δu(n) 2 δu(n) 3 (22.16), {δu} =[N] { δu (n)} (22.17), δr = e [ {δu } [ ]] (n) T [N] T {t} ds + ρ 0 [N] T {g} dω Ω e Ω e (22.18) 22.4 1 δe ij S ij dω = δr (22.19) 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 (22.20),. {δe} = {δe 11 δe 22 δe 33 2δE 12 2δE 23 2δE 31 } T (22.21) {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (22.22)

240 22, δe ij S ij dω = {δe} T {S} dω Ω Ω = {δe} T {S} dω = δr e Ω e δe ij, δe ij = 1 2 ( δui X j + δu j X i + δu k X i u k + u ) k δu k X j X i X j (22.23) [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 X 3 0 u 3 X 1 (22.24) { } { } δu δu1 δu 1 δu 1 δu 2 δu 2 δu 2 δu 3 δu 3 δu T 3 (22.25) X X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 2 X 3,. u i X j {δe} =[Z 1 ] { } δu X (22.26) u i = N(n) u (n) i (22.27) X j X j δu i X j δu i X j = N(n) δu (n) i (22.28) X j

22.4. 1 241 { } 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) 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 (22.29). 9 3n [Z 2 ], { } δu =[Z 2 ]{δu (n) } (22.30) X [B]. [B] [Z 1 ][Z 2 ] (22.31) {δe} =[B]{δu (n) } (22.32) [B (n) ] u 1 N (n) X 2 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 N (n) X 3 X 2 + u 1 N (n) u 2 N (n) X 2 X 3 X 3 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 (22.33) N (n) X 1

242 22 6 3 [B (n) ], [B] = [ [B (1) ] [B (n) ] ] (22.34)., e Ω e {δe}{s} dω = e ] [{δu (n) } T [B] T {S} dω Ω e (22.35). 22.5 total Lagrange ] [{δu (n) } T [B] T {S} dω = [ ]] [{δu (n) } T [N] T {t} ds + ρ 0 [N] T {g} dω e Ω e e Ω e Ω e (22.36),, Q = [B] T {S} dω (22.37) Ω e F = [N] T {t} ds + ρ 0 [N] T {g} dω (22.38) Ω e Ω e u = { u (n)} (22.39) [ T δuh (Q(u h ) F ) ] = 0 (22.40) e, find u h V h such that [ T δuh (Q(u h ) F ) ] = 0 (22.41) e for δu h V h, Newton-Raphson.

22.6. 1 243 22.6 1 Newton-Raphson, K = Q u, dq dt = Q du u dt = K u (22.42),,. ( ) S ij δe ij dv = Ṡ ij δe ij + S ij δėijdω (22.43) Ω C ijkl Hooke Ω S ij = C ijkl E kl (22.44) Ṡ ij = C ijkl Ė kl (22.45) S ij = λ(tre ij )δ ij +2μE ij (22.46) C ijkl = λδ kl δ ij +2μδ ki δ jl (22.47) 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 δf ki F kj + F Ω 2 ) ki δf kj dω ) = C ijkl Ė kl δe ij + S ij (δf ki F kj dω Ω S ij, Ė kl k, l (22.48) 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 (22.49)

244 22 C ij kl 1 2 (C ij kl + C ij lk ) (22.50), S. S 11 C 11 11 C11 22 C11 33 C11 12 C11 23 C11 31 S 22 C S 22 11 C22 22 C22 33 C22 12 C22 23 C22 31 33 = C 33 11 C33 22 C33 33 C33 12 C33 23 C33 31 S 12 C 12 11 C12 22 C12 33 C12 12 C12 23 C12 31 S 23 C 23 11 C23 22 C23 33 C23 12 C23 23 C23 31 S 31 C 31 11 C31 22 C31 33 C31 12 C31 23 C31 31 Ė 11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31 (22.51) C ijkl 6 6 [D]. Cijkl, ij, kl, [D].,, {S} = {S 11 S 22 S 33 S 12 S 23 S 31 } T (22.52) {Ė} = {Ė11 Ė 22 Ė 33 2Ė12 2Ė23 2Ė31} T (22.53), δe ij S ij dω Ω = δe ij C ijkl Ė kl dω Ω T = {δe} [D]{Ė} dω (22.54) Ω { } { } T u (n) u (1) 1 u (1) 2 u (1) 3 u (n) 1 u (n) 2 u (n) 3 (22.55), {Ė} =[B] { u (n)} (22.56). Ė ij = 1 ( ui + u j + u k u k + u ) k u k 2 X j X i X i X j X i X j, 1 δe ij Ṡ ij dω = [ {δu } (n) T [B] T [D][B]dΩ { ] u (n)} Ω e Ω e (22.57) (22.58)

22.6. 1 245., δf ki S ij 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 (22.59), [σ] = [Σ] = S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 [σ] [σ] [σ] (22.60) (22.61) {δf} = {δf 11 δf 12 δf 13 δf 21 δf 22 δf 23 δf 31 δf 32 δf 33 } T (22.62) { F } = { F 33 } T (22.63) F 11 F 12 F 13 F 21 F 22 F 23 F 31 F 32, δf ki S ij F kj = {δf} T [Σ]{ F } (22.64)

246 22, δf ij = δx i X j = δu i X j (22.65) F ij = ẋ i X j = u i X j (22.66), [Z 2 ] { } δu {δf} = =[Z 2 ] { δu (n)} (22.67) X { F } =[Z 2 ] { u (n)} (22.68), δf ki S ij F kj = { δu (n)} T [Z2 ] T [Σ][Z 2 ] { u (n)} (22.69) T [Z 2 ] T [Σ][Z 2 ] { N (i) [A ij ]= 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 (22.70) (22.71) [Z 2 ] T [Σ][Z 2 ]=[A] (22.72), 2 δf ki S ij F kj dω = [ {δu } (n) T [A]dΩ { u (n)} ] Ω e Ω e. δe ij Ṡ ij dω + δf ki S ij F kj dω Ω Ω = [ {δu } (n) T ( [B] T [D][B]+[A] ) dω { ] u (n)} e Ω e (22.73) (22.74)

22.6. 1 247. ( [B] T [D][B]+[A] ) dω (22.75) Ω e e