構造と連続体の力学基礎

Transcription

1 ? finite deformation infinitesimal deformation large deformation 1 [129] B Bernoulli-Euler [26] 1975 Northwestern Nemat-Nasser Continuum Mechanics 1980 [73] What is the physical meaning? 583

2 C argument : A = B C = BC A i j = B ik C k j, D = S:Q = SQ Di jkl = S i jmn Q mnkl s = u u s = u j v j, u = A u = w B ui = A i j v j = w j B ji, w = u u wi = e i jk u j v k, t = A:B t = A i j B i j, A = D:B Ai j = D i jkl B kl, A = u w A i j = v i w j, Q = A B Qi jkl = A i j B kl A = B C A = BC, Ai j = B ik C k j t t t = 0 3 t = t 3 2 g 2 g 3 t = 0 G 3 0 g 1 1 X G 2 0 G 1 0 G 3 xx, t G 2 G g i i = 1, 2, 3 G I t I = 1, 2, 3 G I 0 = g i I = i 12.1 G I 4 G I 3 updated Lagrange p.621 updated Lagrange total Lagrange total Euler 4 t = t

3 C t = 0 X = X I g I X g I = g i, X I = X i I = i t = t xx, t = x i X, t g i X t t x Xx x u u = U I X g I = u i x g i x X, U I X x i X X I, u i x x i X I x, I = i 12.1a, b, c dx = dx I g I, dx = dx i g i 2 dx = F ij X dx J g i, F ij X x ix X J = x i,j 12.2a, b F 12.1 F ij = X I + U I,J = δ IJ + U I,J, I = i 12.3 δ IJ Kronecker G I dx dx = dx j g j = x j,i dx I g j = F ji g j dx I = G I dx I G I = F ji g j 12.4 G I ds ds 2 = dx dx = F ij dx J g i FiK dx K g i = FiJ F ik dx J dx K u = U I X g I = U I X g I C 6 7 kinematics dynamics statics

4 CX ds 2 = C IJ dx I dx J, C IJ X F ki F kj = x k,i x k,j, C = F t F, t C F F, CIJ = C JI 12.6a, b, c, d, e CX Cauchy-Green 12.5 ds 2 = dx dx = δ JK dx J dx K 12.7 Green EX 2 E IJ dx I dx J ds 2 ds EX 1 2 C I, E IJX = 1 2 C IJ δ IJ = 1 2 F ki F kj δ IJ, E IJ = E JI 12.9a, b, c I 2 Kronecker E IJ = 1 GI G J g 2 i g j Green EX 12.3 E IJ = 1 { } 1 { } δki + U K,I δkj + U K,J δij = δij + U I,J + U J,I + U K,I U K,J δ IJ 2 2 = 1 UI,J + U J,I + U K,I U K,J X x Lagrange E 11 = 1 / 2 G 1 2 g1 2 2 Green elongation elongation dx dx dx dx dx elongation ij = F ij δ ij = u i,j = U I,J Kronecker 9 10 E IJ = 1 UI 2 J + U J I + U K I U K J C

5 dv dv G 1 G 2 G 3 dv, dv dx 1 dx 2 dx dv dv F j1f k2 F l3 g j g k g l = F j1 F k2 F l3 g j e ikl g i = F j1 F k2 F l3 δ ji e ikl = F j1 F k2 F l3 e jkl = det F ij e i jk C.15 dv dv = J det F xi ij = det X J = 1 6 e i jk e IJK F ii F jj F kk J Jacobian ρ 0, ρ ρ 0 dv = ρ dv J = ρ 0 ρ J x = fx, t, F ij = x i X j X = f 1 x, t, F 1 I j = X I x j 12.15a, b 11 F 3 3 F 1 I j = 1 2J e IMN e jmn F Mm F Nn 1/ 2 e IMN e jmn F Mm F Nn F 3 C E R R R 1 = R t, det R = a, b F = R U, F ij = R ik U KJ, R ik R jk = δ i j, R im R in = δ MN, U KJ = U JK 12.17a, b, c, d, e 11??

6 UX F R U 12.6 C C IJ = F ki F kj = R km U MI R kl U LJ = U MI U LJ R km R kl = U MI U LJ δ ML = U LI U LJ, C = F t F = U t t t U, C = F F = U U 12.18a 12.18b, c C 12.9 Green E U N I Λ I Λ [ ] t, [ ] U = N Λ N Λ 0 Λ 2 0, 12.19a, b 0 0 Λ 3 1 t N N 1 N 2 N 3, N = N 12.19c, d N I [ ] U C = F t F = U t U = { N [ Λ ] t N t } { N [ Λ ] N t } = N [ Λ 2 ] N t [ Λ ] [ Λ ] 2 Λ 2 I C N N C Λ 2 N U Λ N U = C 1/ 2, C = U a, b U = C = 3 Λ N N N N N, N=1 3 Λ 2 N NN N N, N=1 U = N [ Λ ] N t, 12.22a, b C = N [ Λ 2 ] N t 12.22c, d C b U

7 F = R U R = F U 1 Green E = 3 N=1 1 Λ 2 2 N 1 N N N N, E = N [ 1 2 Λ 2 1 ] N t 12.23a, b U C Green E E 11 E E 12 G 1 G 2 E E11 1, E E E a, b 4 Euler F = u R, F ij = v ik R kj, v ik = v ki 12.25a, b, c 12.6 C C IJ = F ki F kj = v kl R li v km R mj = R li v kl v km R mj = R li b lm R mj, b lm v kl v km = v lk v mk 12.26a, b R ii C IJ R jj = R ii R li b lm R mj R jj = δ il b lm δ m j = b i j b 12.7 ds ds 2 = dx dx = X I, j dx j g I X J,k dx k g J = X I, j X I,k dx j dx k = F ji F ki 1 dx j dx k F ji F ki F ji F ki = v jl R li vkm R mi = R li v jl v km R mi = v jl v km δ lm = v jl v kl b jk b F F t = u u t, b jk F ji F ki = v jl v kl, ds 2 = b jk 1 dx j dx k, b jk = b k j 12.28a, b, c, d bx CX Cauchy-Green ux UX u U

8 [ ] t, u = Λ n n n n n, v = n Λ n 12.29a, b b = Λ n = Λ N, n=1 3 Λ 2 n nn n n, n=1 n b = n [ Λ 2 ] n t n 1 n 2 n 3, 12.29c, d n 1 = n t 12.30a, b, c n n N N n n = R N N, n n i = R ij N N J, n = R N 12.31a, b, c n n N N F = R U = v R 12.8 Green ex v = R U R t, vi j = R ik U KL R jl 12.32a, b 2 e i j dx i dx j ds 2 ds X J,k = δ Jk u j,k x e i j x = 1 1 δi j b i j = ui, j + u j,i u k,i u k, j ex Almansi x Euler Euler Euler Euler Lagrange Λ i x 1 -x 2 cos α sin α 0 Λ Λ 1 cos α Λ 2 sin α 0 xi F = = x i,j = sin α cos α 0 0 Λ X 2 0 = Λ 1 sin α Λ 2 cos α 0 J Λ Λ b u C, U B, V b u n n

9 Λ 3 Λ 1 U R Λ 2 α 2 R 1 v 12 v 22 u v v 21 R = cos α sin α 0 sin α cos α , U = Λ Λ Λ a, b U Λ N = Λ N N 1 = 0, N 2 = 1, N 3 = a, b, c x i G I G I cos α sin α 0 n 1 = sin α, n 2 = cos α, n 3 = a, b, c Λ I n n Λ 1 cos v = 2 α + Λ 2 sin 2 α Λ 1 Λ 2 sin α cos α 0 Λ 1 Λ 2 sin α cos α Λ 1 sin 2 α + Λ 2 cos 2 α Λ g i u Green Λ 1 cos α 1 Λ 2 sin α 0 UI,J = Λ 1 sin α Λ 2 cos α Λ E 11 = 1 Λ , E 22 = 1 Λ , E 33 = 1 Λ a, b, c

10 b Lagrange N x i G I G I Lagrange B.2 Bernoulli-Euler Green Green 5? 13 C Green E γ 2 a a cos γ a sin γ 0 FiJ = UIJ = vi j = a sin γ a cos γ Λ 3 2 γ a a γ 1 O a γ Λ 3 x 3 R = I U a cos γ Λ a sin γ 0 det a sin γ a cos γ Λ 0 = Λ 3 Λ { a cos γ Λ 2 a 2 sin 2 γ } Λ 3 Λ = 0 Λ 1 = a cos γ + sin γ, Λ 2 = a cos γ sin γ, Λ 3 = Λ a, b, c N 1 = 1 1 1, N 2 = , N 3 = 0 0 1, n n = N N 12.41a, b, c, d 13

11 x 1 45 Λ 1 45 Λ 2 Λ 1, Λ 2 Λ γ : ϵ E ϵ E a γ = % 2 a = 2 ϵ E = 1 a = 1 / 2 ϵ E = 0.5 ϵ L ϵ L lna ϵ L = 1 a = e 2.72, ϵ L = 1 a = 1 / e 1 / 2.72 = % 1 / % ϵ E ± ln 2 = ±63.3% 2 log 2 a 100% 100% Green E E X U I, E L X ln U 12.44a, b E E X elongation U 3 E E X = ΛN 1 N N N N, N=1 E E = N [ Λ 1 ] N t 12.45a, b Biot E L X 3 E L X = ln Λ N N N N N, N=1 E L = N [ ln Λ ] N t 12.46a, b x 1 -x 2 E11 E = EE 22 = a cos γ 1, EE 12 = a sin γ, 12.47a, b E11 L = EL 22 = 1 2 ln { a 2 cos 2 γ sin 2 γ } = 1 2 ln Λ 1 Λ 2, E L 12 = 1 cos γ + sin γ 2 ln 12.47c, d cos γ sin γ E12 E a EE 11 γ E L 12

12 E E 11 1 U U a O lnu 11 lna O lnu 11 a 12.4? b E11 L Lagrange 12.9 Green E E 11 = E 22 = 1 a 2 1, E 12 = a 2 sin γ cos γ 12.48a, b 2 Lagrange E 11 = 1 a = a = 1 / 2 E 11 = 3 / 8 = a 2 E 12 a 12.2, U: g i R g i E E : g i R g i E L : Green E: G I G I Lagrange Green

13 a 1 lna 3 11 U 11 lnu γ = a E E 12.4 b E L 100% : 12.5 γ π 3 U 12 E a = 1 γ < π / 4 γ ± π / O lnu E 12 U ? γ ± π / γ Green γ ± π / 4 2 a a = 1 γ a Λ 3 Green a γ π : x 1 Λ 1 ln Λ 1 Ė L 11 = ln Λ 1 t = Λ 1 Λ 1 Λ 1 L 0 L Ė11 L = L L 0 L 0 L = L L Lagrange updated Lagrange

14 p updated Lagrange Green 1 Green Ė IJ = 1 ḞkI F kj + F ki Ḟ kj 2 u u = ẋ i g i = V I X g I = v i x g i Ḟ ki = ẋ k,i = V K,I = v k,i = v k, j F ji Ė IJ = d i j F ii F jj, d i j 1 vi, j + v j,i 2 d 1 F 11 = Λ Green E 11 E 11 = 1 / 2 Λ Ė IJ = d i j F ii F jj Ė 11 = Λ 1 Λ 1 = d 11 Λ 1 Λ 1 Ė 11 Λ 2 1 d 11 = Λ 1 Λ 1 Ė L 11 Green updated Lagrange Lagrange σx σx Cauchy σx σ IJ X = C IJKL E L KL X, σx = C : EL X C

15 Nemat-Nasser No offense! theorem of Kelvin and Tait : : X dx K x dx k X gk K dxk dx K g K k dxk = g K k xk L dx L = A + 1 dx K a A + 1 g K k gk G k x k L x k L = g kl g M l gml + U M L a g K k xk L = g K k gkl g M l gml + U M L = g KM g ML + U M L = g K L + U K L g ML g M g L a g K L + U K L dx L = dx K + U K L dx L = A + 1 dx K U K L A δ K L dx L = 0 b δ K L Kronecker b A det U K L A δ K L = 0 U K L 3 A 3? Euler 14 X ux VX, t = V I X, t g I x ix, t t 14 updated Lagrange g I = ẋ i x, t g I = ẋ i x, t g i = v i x, t g i = ux, t 12.49

16 Ḟ ij = ẋ i,j = v i,j = v i,k x k,j, Ḟ = l F, l ik v i,k 12.50a, b, c lx l = Ḟ F 1, l = Ḟ F 1 = v 12.51a, b 12.9 Green Ė IJ = 1 2 ḞkI F kj + F ki Ḟ kj = 1 2 l km F mi F kj + F ki l km F mj = 1 2 l km F mi F kj + l mk F mi F kj = 1 2 l mk + l km F mi F kj Ė IJ = d mk F mi F kj, d mk 1 2 l mk + l km = 1 vm,k + v k,m, 12.52a, b 2 Ė = F t t 1 d F, Ė = F d F, d = l + l t 1 { t }, d = l + l 12.52c, d, e, f 2 2 dx 3.6 x d 11 d 11 = ẋ 1 ẋ 1 = x 1 x 1 x 1 x 1 = {ln x 1 } R Ṙ = ω R R, Ṙ ij = ω R ik R kj, Ṙ = ω R R 12.53a, b, c ω R x U u N I n i Ṅ I = Ω L N I, ṅ i = ω E n i, Ṅ I J ṅ i j = Ω L JK NI K, Ṅ = Ω L N, 12.54a, b, c = ω E jk ni k, ṅ = ω E n 12.54d, e, f Ω L X ω E x 15 [73] ṅ = Ṙ N + R Ṅ = ω R R N + R Ω L N = ω E n = ω E R N 15 N I K NJ K = δij, n i j n = δ k k i j δ IJ δ i j Kronechker Ω L MN = ΩL NM, ωe mn = ωnm E

17 N t R t ω E = ω R + R Ω L R t, ω E i j = ω R i j + R im R jn Ω L MN 12.55a, b ω ω 1 2 u ω i 1 2 e i jk j v k = ei jk j v k + e ik j k v j = ei jk vk, j v j,k 2 i j, j k, i k w i j li j l ji = vi, j v j,i, w = 2 2 { l l t } 12.56a, b wx l i j = d i j + w i j, l = d + w, l = d + w 12.57a, b, c d w l J = 1/ 6 e i jk e IJK F ii F jj F kk = 1 t 6 e i jk e IJK ḞiI F jj F kk + F ii Ḟ jj F kk + F ii F jj Ḟ kk = 1 ei jk e IJK v i,m F mi F jj F kk + = 1 vi,m e i jk e m jk J + = 1 vi,m 2δ im J = 1 6 2vi,i J + 2v i,i J + 2v i,i J = J v i,i J = J d kk dv dv = dv dv d kk ρ 0 = 0 = ρ J + ρ J = ρ J + ρ d kk J ρ + ρ d kk = J = 0 d kk = ax, t ax, t ux, t = a i x, t g i 12.61

18 X X a x x xx, t a i x, t = v i x, t = v i xx, t, t = v ix, t t + v ix, t x j x j t = v ix, t t + v i, j x, t v j x, t v j t x j 2 Navier-Stokes Euler Lagrange Euler x X Euler 3.164a Lagrange : a R ω R = α n n ω E = α ω R Ω L = N N x 1, x 2 α Ω L F Λ 1 cos 2 α + Λ 2 sin 2 α α + Λ 1 sin α cos α Λ 2 sin α cos α 0 Λ 1 Λ 2 Λ 1 Λ 2 l = α + Λ 1 sin α cos α Λ 2 Λ 1 sin α cos α sin 2 α + Λ 2 cos 2 α 0 Λ 1 Λ 2 Λ 1 Λ 2 Λ Λ 3

19 Λ 1 cos 2 α + Λ 2 Λ sin 2 1 α sin α cos α Λ 2 sin α cos α 0 Λ 1 Λ 2 Λ 1 Λ 2 Λ d = 1 sin α cos α Λ 2 Λ 1 sin α cos α sin 2 α + Λ 2 cos 2 α 0 Λ 1 Λ 2 Λ 1 Λ 2 Λ Λ 3 d = cos α sin α 0 sin α cos α Λ Λ 1 Λ Λ 2 Λ Λ 3 cos α sin α 0 sin α cos α = n [ ln Λ ] n t Euler updated Lagrange Euler w = α ω R, ω E α : 12.3 ω R = 0, ω E = 0, Ω L = F l = d = ȧ γ γ tan 2γ a γ cos 2γ cos 2γ ȧ γ tan 2γ 0 a Λ 3 γ d = Λ Λ 1 Λ Λ 2 Λ Λ 3 Λ = n [ ln Λ ] n t

20 : R = 2 Ω L = 0 cos α sin α 0 sin α cos α , N = cos γ sin γ 0 sin γ cos γ , [ Λ ] = Λ Λ Λ 3 γ Λ I α x 3 Λ 1 cos U = 2 γ + Λ 2 sin 2 γ Λ 1 Λ 2 sin γ cos γ Λ 1 Λ 2 sin γ cos γ Λ 1 sin 2 γ + Λ 2 cos 2 γ, cosγ + α sinγ + α n = sinγ + α cosγ + α ω R = α , ω E = γ + α , Ω L = γ Ω L ω R F = Λ 1 cos γ cosγ + α + Λ 2 sin γ sinγ + α Λ 1 sin γ cosγ + α Λ 2 cos γ sinγ + α Λ 1 cos γ sinγ + α Λ 2 sin γ cosγ + α Λ 1 sin γ sinγ + α + Λ 2 cos γ cosγ + α [ t γ Λ d = n ln Λ ] 2 1 Λ2 2 sin 2γ + α cos 2γ + α n 2Λ 1 Λ 2 cos 2γ + α sin 2γ + α = n [ ln Λ ] n t γ Λ 2 1 Λ t n 2Λ 1 Λ n w = ω R Λ 1 Λ 2 2 2Λ 1 Λ 2 Ω L 2 γ Λ 1 Λ 2 2 Λ 1 Λ 2 γ 2 Λ ϵ 1, Λ ϵ 2 Λ 1 Λ 2 2 ϵ 1 ϵ 2 2 2Λ 1 Λ R ω R 6? Ḟ = Ṙ U + R U = ω R F + R U

21 l = ω R + R U U 1 R t 12.22b U U U 1 = Ω L N [ Λ ] N t [ ] + N Λ N t [ ] t + N Λ N Ω L t [ ] 1 t N N Λ = Ω L + N [ ] Λ t N + U Ω L t 1 U Λ l = ω R + R Ω L R t [ ] Λ t + n n F Ω L F 1 Λ d = d L + d EV, d = d L + d EV 12.64a, b 1 d L d L n [ ] Λ t t n = n [ ln Λ ] n Λ d L 2 d EV d EV 1 [ F Ω L F 1 + F Ω L F 1 t] Euler d EV 1 [ v ω E ω R v 1 + v ω E ω R v 1 t] tr = d kk = J J, tr 1 = tr d L = Λ 1 Λ 1 + Λ 2 Λ 2 + Λ 3 = Λ 1 Λ 2 Λ 3 Λ 3 Λ 1 Λ 2 Λ 3 J = J 2 tr 2 = tr d EV = 0 1 d L 2 d EV E L ĖL = Ω L E L + N [ ln Λ ] N t + E L Ω L t d L w = ω R + { R Ω L R t 1 2 [ F Ω L F 1 + F 1 t Ω L F t ] } 12.69

22 Ω L w = ω E 1 [ F Ω L F 1 F Ω L F 1 t] w = ω E 1 [ v ω E ω R v 1 v ω E ω R v 1 t] ω R ω E F = F e F p e p 16 x 0 Xt = 0, x n xt, F = F e m F p j F e 4 Fp 3 Fe 2 Fe 1, x 0 t = 0 x 1 t = t x k t = k t x k+1 t = k + 1 t x n 1 t = n 1 t x n t = n t ξt + t ξ α X N = ξ α x m x m X N, x m X N = 1 k=n x k i k x k 1 j k, i n = m, j 1 = N, j k = i k 1 16 Asaro[4] Asaro

23 k n 1 ξ = ξx, t + t = ξ xx, t, t + t ξ α X J = ξ α x k x k X J, ξ α,j = ξ α,k x k,j t = t + t ξ α,j = ξ α,β ξ β,k x k,j + ξ α,l ẋ l,k x k,j 17 l αη t + t l αη ξ α,j X J,η = ξ α,β ξ β,k x k,j X J,η + ξ α,l ẋ l,k x k,j X J,η = ξ α,η + ξ α,k ẋ k,l ξ 1 η,l 12.4 updated Lagrange 2 ẋ k,l t = t t = t + t 2 ξ α,k ξ 1 η,l t ξ α,k ξ 1 η,l Kronecker ξ α,k δ αk, ξ 1 η,l δ ηl l αη ξ α,η + ẋ α,η [127] 2 1 t = t + t ẋ M α,η ξ α,k ẋ k,l ξ 1 η,l l αη = ξ α,η + ẋ M α,η [4, 72] 12.6 dx = d e x + d p x, wx = w e x + w p x 12.72a, b d ξ α,η ẋα,η M 12.6 α γ 17

24 ξ α,η ẋα,η M ϵ ϵx = ϵ e x + ϵ p x, ϵx = ϵ e X + ϵ p X 12.73a, b a 0 0 C T C 0 p a a s T F s T as T a 0 = 1 + α T α p a F s p a a s T = 1 + p K s K s p F s a = Fp s FT s a = 1 + p K 1 + α T s 0 p F a a 0 = 1 + p K K F = F s 3 p K = a a 0 = α T 1 K 1 + α T K s 1 + α T K 1 K s K α T Ṫ ṗ ė s T ė s T = α Ṫ

25 ė s p ė s p = ṗ K s ė s ė s ȧ a = ės T + ės p = α Ṫ + ė ė ȧ a = ṗ K ė = ė s ṗ K s ṗ = K K s K s K α Ṫ ȧ a = K s K s K α Ṫ a K s = exp a 0 K s K α T Taylor a a 0 = 1 + K s K s K α T α ,000 C α T 0.1 T + a K K s < a 0 K a a 0 ν = α T

26 ν = 1 / 3 a a 0 4, p 3µ Cauchy nominal Newton 18 Euler Lagrange Euler t = t da d f t t = t i g i = d f da g 2 da g 2 g 1 g 1 M t = t t = 0 t = 0 m σ 21 S N m d f t = t t = m σ, t i = m j σ ji d f i = m j σ ji x da Cauchy σx da d f

27 Lagrange 12.8 da d f d f Cauchy d f i = M J SJi N X da nominal S N X 20 M da da da dx X = e IJK dx J X K g I = da M I g I e IJK C.15 da da dx x = e i jk dx j x k g i = da m i g i da m i g i = e i jk x j,j x k,k dx J X K g i = e i jk x j,j x k,k dx J X K δim g m = ei jk x j,j x k,k dx J X K xi,l X L,m gm δ im Jacobian da m i g i = e i jk x i,l x j,j x k,k XL,m dx J X K g m = e LJK J X L,m dx J X K g m = J X L,m e LJK dx J X K g m = J X L,m da M L g m = J X L,i da M L g i m i da = J X L,i M L da, M I da = 1 J x k,i m k da 12.81a, b d f i = m j σ ji da = J X L, j M L da σ ji = M L J XL, j σ ji da = ML S N Li da S N I j = J X I,i σ i j, σ i j = 1 J x i,i S N I j, SN = J F 1 σ, σ = 1 J F SN, 12.82a, b, c, d S N I j = ρ 0 ρ X I,i σ i j, σ i j = ρ ρ 0 x i,i S N I j 12.82e, f [62] nominal 1 Piola-Kirchhoff [73] 1980 [73] nominal [38] 1 Piola-Kirchhoff [97, 98] nominal 1 Piola-Kirchhoff

28 σ = σ i j g i g j, S N = S N I j G I g j 12.83a, b 21 2 Cauchy 3.22 x σ + ρ π = ρ u, σ ji, j + ρ π i = ρ v i 12.84a, b πx ux m u i = m j σ ji = t i tx Cauchy d dt + d dt σ i j = σ ji = + e { } d 1 dt 2 ρ u u + ρ e dv = ρ π u dv + t u ds + ρ ḣ dv v v s v s m q ds ḣx qx ρv = const. 2 v ρ {v i v i + ė} dv = v ρ π i v i dv + t i v i ds s π Gauss t σ ji v i, j ρ ė dv = 0 v ρ ė = σ ji v i, j = 1 2 σ ji v i, j + σ i j v j,i ė = 1 ρ σ i j d ji = ẇ nominal

29 d ẇ ė = ẇ + ḣ 1 ρ q i,i = 1 ρ σ i j d ji + ḣ 1 ρ q i,i Cauchy σ d Lagrange { } d 1 dt V 2 ρ 0 V V + ρ 0 e dv = ρ 0 π V dv + T V ds V S T π πx X nominal M nominal M S N = T, M J S N Ji = T i 12.91a, b Gauss { V i ρ0 V i SJi,J N ρ } 0 π i dv + ρ0 ė SJi N V i,j dv = 0 V Lagrange nominal V X S N + ρ 0 π = ρ 0 V, S N Ji,J + ρ 0 π i = ρ 0 V i 12.92a, b ẇ = 1 ρ 0 S N I j V j,i = 1 ρ 0 S N I j v j,i nominal S N d Lagrange v j,i Cauchy X Cauchy nominal x i,i S N I j = x j,i S N Ii nominal S N x T ds + x ρ 0 π dv = 0 1 T = e i jk x j M L S N Lk ds + V S S V V V e i jk x j ρ 0 π k dv Gauss = e i jk x j S N Lk dv + e i jk x j ρ 0 π k dv X L

30 = V e i jk x j,l S N Lk + x S N Lk j dv + X L V e i jk x j ρ 0 π k dv = e i jk x j,l SLi N dv = V V 1 2 e i jk x j,l SLi N e i jk x j,l SLi N 1 dv = V 2 e i jk x j,l SLi N x i,l SL N j dv Kirchhoff Kirchhoff τ K x ẇ = 1 ρ σ i j d ji = 1 ρ 0 τ K i j d ji τ K i j x ρ 0 ρ σ i j Lagrange Ė IJ = d i j F ii F jj, d i j = X I,i X J, j Ė IJ 12.97a, b ẇ = 1 ρ σ i j d ji = 1 ρ σ i j X I,i X J, j Ė IJ = 1 ρ0 ρ 0 ρ σ i j X I,i X J, j Ė IJ SX ẇ = 1 ρ 0 S IJ Ė JI SX 2 Piola-Kirchhoff SX = S IJ G I G J, S IJ ρ 0 ρ X I,i X J, j σ i j, σ i j = ρ ρ 0 x i,i x j,j S IJ 12.99a, b, c G I C.4 S IJ 2 Piola-Kirchhoff nominal 2 Piola-Kirchhoff Green nominal Cauchy S N I j = x j,j S IJ, S IJ = S N I j X J, j a, b

31 Piola-Kirchhoff T R T R ij = x i,k S KJ, S KJ = X K,i T R ij a, b σ i j = ρ ρ 0 x i,i x j,j S IJ = ρ ρ 0 x i,i S N I j = ρ ρ 0 x i,i x j,j S IJ = ρ ρ 0 T R ij x j,j nominal 2 Piola-Kirchhoff nominal xi,k S JK,J + ρ 0 π i = ρ 0 V i πx g i 2 Piola-Kirchhoff G K x i,k 2 2 Piola-Kirchhoff Green B.2 Bernoulli-Euler 2 Piola-Kirchhoff 2 Piola-Kirchhoff Green B.21 S 11 δe 11 dv = σ δe dv = 2 Piola-Kirchhoff δ Green dv V V V B.22 2 Piola-Kirchhoff σ = g S 11 B.19 Green e = E 11 1 = g 1 = ϵ x 3 κ σ Cauchy S 11 G 1 G 1 = g S 11 G 1 = σ G 1 G 1 S 11 σ g 1 G 1 G 1 = g e Green E xx 12.24a 12.44a E E σ = E e, 2 Piola-Kirchhoff = Green

32 B.27 d cos θx sin θx NX dx sin θx cos θx VX + px qx = 0 0 N θ V G K p q g i θ x i,k X p q X 3 Biot 12.44a Biot E E [73] a Ė IJ = 1 2 U KI U KJ + U KI U KJ a Ė E = U S U Ė E ẇ = 1 ρ 0 S IJ Ė JI = 1 ρ S JK U KI + U JK S KI Ė E JI T 1 2 S U + U S, T IJ 1 2 S IK U KJ + U IK S KJ a, b TX Biot ẇ = 1 ρ σ i j d ji = 1 ρ 0 τ K i j d ji = 1 ρ 0 S N I j v j,i = 1 ρ 0 S IJ Ė JI = 1 ρ 0 T IJ Ė E JI T nonsym IJ U IK S KJ nominal S N I j = T nonsym IK nominal S N Biot Biot R jk

33 σ 12 σ 11 R G 2 T12 R G 1 1 R S N 12 R G 2 S 12 S 11R T11 R S N 11 G 1 t = t t = t t = t t = t g 2 g g 2 2 g 2 t = 0 g 1 t = 0 g 2 1 g 1 2 t = 0 g 1 t = 0 g 1 g 1 g 1 g : Cauchy nominal 2 Piola-Kirchhoff Cauchy σ = σ i j g i g j : R 1 Piola-Kirchhoff T R = T R ij g i G J : R ρ 0 Kirchhoff ρ T R 11 T R 12 nominal S N = S N I j G I g j : R 1 Piola-Kirchhoff updated Lagrange 2 Piola-Kirchhoff S = S IJ G I G J : R Cauchy S 11 S 12 Lagrange updated Lagrange Biot T T IJ G I G J G J : Λ i i = 1, 2, 3 x 1 -x 2 α G 1 R G 1 R

34 Λ 3 Λ 2 t = 0 Λ 1 R σ 12 t = t α σ 11 σ 21 σ 22 Λ 2 Λ 3 R Λ 3 Λ 1 σ 12 σ σ 21 σ 22 α G 1 2 Piola- Kirchhoff G 1 R = R G 1 G 1 = S 11 G 1 S 11 = R G 1 G 1 Λ 1 = G 1 S 11 X = R Λ 1, S IJ = a, b R 2 Piola-Kirchhoff S 11 Biot T 11 = R, T IJ = 0 Lagrange nominal g i S N 11 X = R cos α, S N 12 X = R sin α, S N 21 X = 0, S N 22 X = a, b, c, d R 2 Piola-Kirchhoff Cauchy Cauchy 2 σ 11 Λ 2 Λ 3 cos α + σ 21 Λ 2 Λ 3 sin α = R cos α, σ 12 Λ 2 Λ 3 cos α + σ 22 Λ 2 Λ 3 sin α = R sin α a Cauchy R Λ 2 Λ 3 2 Piola-Kirchhoff 22 σ 11 Λ 3 Λ 1 sin α σ 21 Λ 3 Λ 1 cos α = 0, σ 12 Λ 3 Λ 1 sin α σ 22 Λ 3 Λ 1 cos α = 0 σ 21 = σ 11 tan α, σ 22 = σ 12 tan α = σ 11 tan 2 α 22 [27] R

35 a Cauchy σ 11 x = R cos2 α R sin α cos α, σ 12 x =, σ 22 x = R sin2 α a, b, c Λ 2 Λ 3 Λ 2 Λ 3 Λ 2 Λ 3 R Λ 2 Λ 3 Cauchy G I 3.41 G I 1 Piola-Kirchhoff = ρ 0 dv 0 = ρ dv ρ 0 1 = ρ Λ 1 Λ 2 Λ 3 ρ 1 = ρ 0 Λ 1 Λ 2 Λ 3 b Cauchy Kirchhoff τ K 11 = ρ 0 ρ σ 11 = Λ 1 Λ 2 Λ 3 σ 11 = R Λ 1 cos 2 α, τ K 12 = R Λ 1 sin α cos α, τ K 22 = R Λ 1 sin 2 α a, b, c g 1 Kirchhoff f 1 f 1 = τ K 11 g 1 + τ K 12 g 2 G 1 G 1 τk 11 G 2 G 2 τk 12 f 1 = τ K 11 g 1 + τ K 12 g 2 = τ K 11 G 1 G 1 + τk 12 G 2 G 2 τ K 11 = τk 11 cos α + τk 12 sin α = R Λ 1 cos α, τ K 12 = τk 12 cos α τk 11 sin α = 0 1 Piola-Kirchhoff f 1 = τ K 11 g 1 + τ K 12 g 2 = τ K 11 G 1 G 1 + τk 12 G 2 G 2 = T R 11 G 1 + T R 12 G 2 T R 11 = τk 11 G 1 = τk 11 Λ 1 = R cos α, T R 12 = τk 12 Λ 2 = 0 g 2 τ K 21 = τk 21 cos α + τk 22 sin α = R Λ 1 sin α, τ K 22 = τk 22 cos α τk 21 sin α = 0 T R 21 = τk 21 G 1 = τk 21 Λ 1 = R sin α, T R 22 = τk 22 Λ 2 = 0 T11 R x = R cos α, T 12 R x = 0, T 21 R x = R sin α, T 22 R x = a, b, c, d nominal

36 xi = X J cos α sin α 0 Λ x i,j = sin α cos α Λ 1 cos α Λ 2 sin α 0 0 Λ 2 0 = Λ 1 sin α Λ 2 cos α Λ Λ 3 Cauchy 2 Piola-Kirchhoff Cauchy b σ 11 = ρ 1 x 1,1 x 1,1 S 11 = Λ 2 1 ρ 0 Λ 1 Λ 2 Λ cos2 α R = R cos2 α 3 Λ 1 Λ 2 Λ 3 σ 12 = ρ 1 x 1,1 x 2,1 S 11 = Λ 2 1 ρ 0 Λ 1 Λ 2 Λ sin α cos α R R sin α cos α = 3 Λ 1 Λ 2 Λ 3 σ 22 = ρ 1 x 2,1 x 2,1 S 11 = Λ 2 1 ρ 0 Λ 1 Λ 2 Λ sin2 α R = R sin2 α, 3 Λ 1 Λ 2 Λ Piola-Kirchhoff Piola-Kirchhoff T R 11 = x 1,1 S 11 = R Λ 1 Λ 1 cos α = R cos α, T R 21 = x 2,1 S 11 = R sin α nominal S N 11 = S 11 x 1,1 = R Λ 1 cos α = R cos α, S N 12 Λ = S 11 x 2,1 = R sin α Piola-Kirchhoff Lüders nominal 23 nominal stress Piola-Kirchhoff 1 Piola-Kirchhoff nominal [73] 24 engineering stress conventional stress

37 σx σx = Cx : { dx d p x } 11 σ R R ω R Lagrange 2 Piola-Kirchhoff Ṡ 2 Piola-Kirchhoff Lagrange 12.4 updated Lagrange Euler Cauchy Kirchhoff Cauchy σ 0 ω 25 t = 0 Cauchy 3.41 σ 11 = σ 0 cos 2 ωt, σ 22 = σ 0 sin 2 ωt, σ 12 = 1 2 σ 0 sin 2ωt a, b, c σ 11 = ω σ 0 sin 2ωt, σ 22 = ω σ 0 sin 2ωt, σ 12 = ω σ 0 cos 2ωt a, b, c Cauchy 25 [73] Nemat-Nasser 1

38 x i t = t 2 σ 0 ωt x i t = x it dx i O dx i t + δt = dx i t + δt dv i t = δ i j + δt v i, j dx j t dx i t dx i t = dx it = dx i t + δt dx i t + δt = δ i j + δt v i, j dx j t + δt dx i t + δt = δ i j δt v i, j dx j t + δt t = t + δt Cauchy σ i j t + δt = σ i j t + δt σ i j t Cauchy 2 σ i j t + δt = δ ik δt v i,k δ jl δt v j,l σkl t + δt δt 1 σ i j t + δt = σ i jt + δt [ ] σ i j v i,k σ k j v j,k σ ki t = t σ i j t = σ i jt Cauchy σ i j σ i j lim t + δt σ i j t = σ i j v i,k σ k j v j,k σ ki = σ i j l ik σ k j l jk σ ki δt 0 δt v 1,1 = 0, v 1,2 = ω, v 2,1 = ω, v 2,2 = a, b, c, d σ 11 = 0, σ 22 = 0, σ 12 = 0 σ σ Oldroyd

39 Jaumann l d w σ i j σ i j w ik σ k j w jk σ ki Cauchy Jaumann Jaumann 60 [58] σ i j d kk dik σ k j + d jk σ ki Kirchhoff Jaumann τ K τ K i j σ i j + σ i j d kk [62] Cauchy σ d updated Lagrange Lagrange Lagrange Lagrange updated Lagrange Lagrange F I lim F = I, 0 t lim J = 1, 0 t lim ρ 0 = ρ etc a, b, c 0 t lim 0 t Lagrange E L d L updated Lagrange lim E L = 0, 0 t lim N = n 0 t lim ĖL = n [ ln Λ ] n t = d L t Green E lim Ė = d 0 t

40 updated Lagrange Green Lagrange Ė updated lagrange d d f Prandtl-Reuss d p i j = λ pr σ i j d p i j = λ f σ i j Lagrange Green 2 Piola Kirchhoff X I,k x k,j = δ IJ 0 = X I,k x k,j = ẊI,k x k,j + X I,k ẋ k,j = Ẋ I,k + X I,m v m,k xk,j Ẋ I, j = X I,k v k, j Piola-Kirchhoff Ṡ IJ = J v k,k σ i j X I,i X J, j J X I,k v k,i X J, j σ i j J X J,k v k, j X I, j σ i j + J X I,i X J, j σ i j updated Lagrange ṡ i j lim 0 t Ṡ IJ = σ i j + v k,k σ i j v i,k σ k j v j,k σ ik ṡ 11 = 0, ṡ 22 = 0, ṡ 12 = 0 ṡ Truesdell σ σi σ i j ṡ i j = σ i j + v k,k σ i j v i,k σ k j v j,k σ ki = j + σ i j d kk d ik σ k j d jk σ ki Kirchhoff Jaumann Oldroyd σ i j = σ i j + σ i j d kk Truesdell Green-Naghdi Cauchy Jaumann w ω R Green-Naghdi updated Lagrange Cauchy Jaumann

41 σ σ σ σ w i j d kk d i j Truesdell Truesdell { }} { { }} { σ i j = σ i j w ik σ k j w jk σ ki σ i j d kk d ik σ k j d jk σ ki } {{ }} {{ } σ i j Truesdell 2 Piola-Kirchhoff updated Lagrange 4 Cauchy 3 Cauchy Jaumann Cauchy S = S IJ G I G J G I G I Cauchy 5, 6 G I Biot Ḟ ij = Ṙ ik U KJ + R ik lim Ḟ ij = l i j = lim Ṙ ik + lim U KJ 0 t 0 t 0 t lim Ṙ ij = lim ω R i j t 0 t 2 Biot Ṫ IJ = 1 2 lim w i j = lim ω R i j 0 t 0 t lim 0 t U KJ = l i j w i j = d i j ṠIK U KJ + U IK Ṡ KJ + S IK U KJ + U IK S KJ U KJ σ i j ṫ i j lim Ṫ IJ = σ i j + 1 σik d k j + d ik σ k j 0 t 2 = σ i j + v k,k σ i j 1 σik d k j + d ik σ k j = τ K 2 i j 1 σik d k j + d ik σ k j

42 ± σ i j d kk ± dik σ k j + d jk σ ki [126] convected [73] nominal Ṡ N I j = J v k,k σ m j X I,m J X I,m v m,k σ k j + J X I,m σ m j updated Lagrange nominal ṅ + d ik σ k j + d jk σ ki convected σ i j Oldroyd σ i j +2 d ik σ k j + d jk σ ki + d ik σ k j + d jk σ ki Cauchy Jaumann σ i j ṅ i j lim 0 t Ṡ N IJ = σ i j + σ i j d kk v i,k σ k j = σ i j + σ i j d kk d ik σ k j + w jk σ ki σ i j d kk Truesdell σ i j Lagrange +σ i j d kk Kirchhoff Jaumann τ K i j 2nd Piola- Kirchhoff S IJ Total Lagrange updated Lagrange Lagrange nominal updated Lagrange Ṡ N Ji,J + ρ 0 π i = Ṡ N lim Ji,J + ρ 0 π i = 0 ṅ ji, j x + ρ π i x = t ṅ m j ṅ ji = ṫ i, v i = Cauchy σ ji, j + ρ π i = nominal Cauchy Cauchy σ ji, j + σ ji, j d kk v j,k σ ki, j + ρ π i = 0 σ ji, j v j,k σ ki, j + ρ π i + σ ji, j d kk = 0 27 Newton-Raphson

43 Cauchy σ ji, j v j,k σ ki, j + ρ π i ρ π i d kk = [150] Cauchy Cauchy σ i j t + t = σ i j t + σ i j t Cauchy σ i j t t = t Cauchy t = t updated Lagrange nominal σ i j t + ṅ i j t t = t t = t + t t = t nominal t = t + t S N i j t + t; t = σ i jt + ṅ i j t nominal t Cauchy σ i j t + t = ρt + t; t ρt x i,k t + t S N ρt + t; t k j t + t; t = ρt δik + v i,k σi j t + ṅ i j t x i,k x i,k S N k j = δ ik + v i,k σk j + σ k j + σ k j d ll v k,l σ l j = σi j + σ i j + σ i j d kk ρt + t; t ρt = det x i, j t + t 1 = 1 + dkk 1 = 1 d kk σ i j t + t = 1 d kk σ i j + σ i j + σ i j d kk = σi j t + σ i j t σ i j t + t = 1 det δ mn + v m,n t { δ ik + v i,k t } { σk j t + ṅ k j t } Cauchy t ξ I t t x i t + t σ t ji x = σ JIξ + σ JI ξ 28 σ i j t t

44 t t = t + t I t ξ σ t ji, j = σ JI, j + σ JI, j = σ JI,K ξ K, j + σ JI, j = σ JI,K δk j v K, j + σji, j = σ JI, j σ JI,K v K, j + σ JI, j t + t σ t ji, j + ρ t π t i = 0 = σ JI, j σ JI,K v K, j + σ JI, j + ρ t π t i t 0 σ JI, j σ JI,K v K, j + ρ t π t i ρ π I = 0 σ ji, j σ ji,k v k, j + ρ π i = ρ π i = ρ π i + ρ π i = ρ π i d kk + ρ π i σ ji, j v j,k σ ki, j + ρ π i ρ π i d kk = Kirchhoff t = t + t Kirchhoff Cauchy τi K j t + t = ρ 0 ρt + t σ i jt + t = ρ 0 ρt { σi j t + σ i j t } ρt ρt + t τ i K j t = ρ { 0 σi j t + σ i j t d kk t } ρt ρt ρt + t = 1 + d kkt 1 τi K j t + t = ρ 0 ρt {1 + d kkt} { σ i j t + σ i j t } = ρ { 0 σi j t + σ i j t d kk t + σ i j t } = τi K j ρt t + τk i j t ?

45 R ln Λ 1 d 2 σ ϵ 4 C σ = C : ϵ R = Rln Λ 1 ln Λ 1 4 C Oi C Ei i = 1, 2, C σ = C O1 : ϵ + C E1 : ϵ 2 + C O2 : ϵ 3 + σ = C σ, ϵ : ϵ = = f σ, ϵ ϵ m ϵ m = 3 ϵ m k nk n k, ϵ = k=1 3 ϵ k n k n k 2 ϵ k n k C Ei C Ei 1 Cauchy σ 1 E E 1 C k=1 σ 1 x = C E E 1 x = C Λ 1 1 R = Λ 2 Λ 3 σ 1 = C Λ 2 Λ 3 Λ 1 1 Λ 2 Λ 3 2 R ln Λ 1 2 Piola-Kirchhoff Green { 1 S 1 X = C E1 L X = C ln {Λ IX}, S 1 = C E 1 X = C Λ } R = Λ 1 S 1 = C Λ 1 ln Λ 1, R = C Λ 1 { 1 2 Λ } R ln Λ 1

46 S N = C:E L R C 0 S = C:E R C MR S = C:E lnλ O 0.08 σ = C:E L lnλ O 0.1 σ = C:E E τ K = C:E L σ = C:E L a b ±10% ±10% Lagrange a Lagrange 2 Piola-Kirchhoff Green b 2 Piola-Kirchhoff Green Truesdell O R C 0 lnλ 1 Jaumann Cauchy Kirchhoff ±10% % updated Lagrange σx σx = Cx : dx σx = Cσ, x : dx σx = Cσ, x, X : { dx d p x } C σ ±10%

47 updated-lagrange Truesdell b ϕ ϕ = 1 ρ σ i j d ji = 1 ρ 0 S IJ Ė JI ρ 0, ρ ẇ 29 S IJ = ρ 0 ϕ E IJ ρ 0 ϕ E 2 C S = C E 30 curve-fitting Green 1 ϕe 31 B.2 Bernoulli-Euler 2 Piola-Kirchhoff Green Young E ρ 0 ϕ 1 2 E e2, e E 11 1 = ϵ + x 3 κ σ g S 11 = {ρ 0 ϕe} e = E e ϕe 1 2ρ 0 E e 2 S 11 = {ρ 0 ϕe} E 11 = E e 1 + e a, b, c Green Bernoulli-Euler S 11 = {ρ 0 ϕ E 11 } E 11 = E e 1 + e ρ 0 ϕ E 11 E E E 11 ϕe 1 2ρ 0 E e a, b, c Green 2 29 ė Hooke C Saint Venant-Kirchhoff [12] Saint 31 Argonne Einstein Fermi Fano Fano Fermi X-Day PhD Fano Fermi Why don t you...? Fano Fermi Fermi Why don t you...? 1 Fano PhD Fermi Fano Fermi Fermi journal X-Day Fano 1

48 Mooney-Rivlin ϕ = µ 2 I µ I µ µ I 1, I 2 1, 2 U U = Λ Λ Λ 1 Λ 2, I 1 = U IJ U IJ, I 2 = 1 2 nominal { I 2 1 U KI U KJ U LI U LJ } a, b, c σ i j = C i jkl d kl C i jkl 3.49b C 32 C i jkl = c 0 δ i j δ kl + c 1 1/ 2 δik δ jl + δ il δ jk + c2 σ i j δ kl + c 3 δ i j σ kl + c 4 σik δ jl + δ ik σ jl + c 5 σ im σ m j δ kl + c 6 σ i j σ kl + c 7 δ i j σ km σ ml + c 8 σim σ mk δ jl + σ jm σ ml δ ik + c 9 σ im σ m j σ kl + c 10 σ i j σ km σ ml + c 11 σ im σ m j σ kn σ nl c 0 c 11 [127, 162] ? 1 σ ϵ σ = C : ϵ σ 2 Piola-Kirchhoff S Kirchhoff τ K Cauchy σ nominal S N ϵ Green E E E E L ẇ = 1 ρ σ i j d ji = 1 ρ 0 τ K i j d ji = 1 ρ 0 S N I j v j,i = 1 ρ 0 S IJ Ė JI = 1 ρ 0 T IJ Ė E JI Jaumann [32, 59, 60, 67]

49 Piola-Kirchhoff Green Kirchhoff Cauchy σ S ϵ E, σ τ K, σ, S N ϵ E E, E L 33 ϕ 2 3 σ 11 σ 22 σ 33 C 0 C 1 C 1 = C 1 C 0 C 1 C 1 C 1 C 0 ϵ 11 ϵ 22 ϵ x 1 R Λ 1 = Λ L σ 22 = 0, σ 33 = 0, ϵ 22 = ϵ 33, Λ 2 = Λ 3 = Λ T ϵ 22 = C 1 C 0 + C 1 ϵ 11, σ 11 = C 0 C 0 + C 1 2 C 2 1 C 0 + C 1 ϵ α σ 11 S 11 = R Λ L, σ 11 σ 11 = R, σ Λ 2 11 τ11 K = R Λ L, σ 11 S N 11 = R T Green ϵ a Λ 2 L 1, Λ 2 T = 1 C 1 C 0 + C 1 Λ 2 L 1 ϵ 11 Λ L 1, Λ T = 1 C 1 C 0 + C 1 Λ L b ϵ 11 lnλ L, ln Λ T = C 1 C 0 + C 1 ln Λ L a b C 1/ C0 = 1 / 2 Hooke Poisson ν = 1 / 3 nominal 2 Piola-Kirchhoff Cauchy Kirchhoff 33 nominal nominal Lagrange

50 R R 2 C 0 S = C:E C 0 2 S = C:E 1 S N = C:E E τ K = C:E E σ = C:E E 0.5 O Λ L 1 1 S N = C:E L τ K = C:E L lnλ L O σ = C:E L 1 a b R C O σ = C:E E 0.4 τ K = C:E E Λ L 1 lnλ L O σ = C:E L 0.5 τ K = C:E L R C a b Cauchy Kirchhoff 2 Piola-Kirchhoff Green Staint Venant-Kirchhoff [12] Λ L = 1 R Σ Cauchy Kirchhoff 2 Piola-Kirchhoff ±100%

51 S = C:E 3 R C 0 σ = C:E E σ = C:E E 6 3 R C 0 MR S = C:E 0.5 O 0.5 Λ 1 L O 0.5 lnλ L 3 6 MR+ σ = C:E L MR MR+ 3 6 σ = C:E L a b C 1 = C 0/ 2 σ L σ T σ T = C C 0 0 / 2 C 0 / 2 C 0 / 2 C C 0 0 / 2 C 0 / 2 C 0 / 2 C 0 ϵ L ϵ T ϵ T p + p p Hooke σ i j = 2G ϵ i j, σ kk = K ϵ kk σ i j = 2G ϵ i j, σ kk = 3p Poisson ν = 1 / 2 ν = 1 C 1/ C0 = λ/ λ+2µ = 1 / 2 ϵ kk = 0 C 1 = 0 Λ 1 Λ 2 Λ 3 = 1 p 1 Λ 2 = Λ 3 = Λ T = 1 ΛL, σ T = 0 p Λ L σ S ϵ E Cauchy Kirchhoff σ σ ϵ E E E L σ S ϵ E R C 0 = 3 4 Λ 3 L 1, σ σ ϵ E E R C 0 = Λ 3/2 L, ϵ E L R C 0 = 9 4 lnλ L Λ L Mooney-Rivlin ϕ = C 0 2 I C 1 I

52 R C 0 MR+ MR σ = C:E E R C O Λ L 1 MR 0.5 O 0.5 lnλ L 3 3 σ = C:E L σ = C:E E σ = C:E L MR+ 6 6 a b Cauchy I 1, I 2 C 1, 2 I 1 = F kj F kj, I 2 = 1 { I F ki F kj F li F lj }, F = Λ Λ Λ 1 Λ a, b, c nominal SI N j = ϕ { } p X I, j = C 0 x j,i 2C 1 xk,j x k,j x j,i x j,k x l,k x l,i + p XI, j x j,i p Cauchy σ i j = x i,k S N K j = C 0 x j,i x i,i 2C 1 { xk,j x k,j x j,i x i,i x j,k x l,k x l,i x i,i } + p δi j p Λ 2, Λ 3 σ 11 = R Λ L = 1 C 0 + 2C 1 Λ 2L Λ 1ΛL, Λ 2 = Λ 3 = 1 L ΛL Λ L = 1 C 1 C 0 = MR C 1 Poisson ν = 1 / 7 C 1 C 0 = 1 8 MR+ 2 Piola-Kirchhoff Mooney-Rivlin Cauchy 12.20

53 σx σx = Cσ, x, X : dx Cauchy Jaumann σ = C : d σ 11 C 0 C 1 C 1 d11 L σ 22 = C 1 C 0 C 1 d22 L σ 33 C 1 C 1 C 0 d L d L 33 n = N = I, d L = [ ln Λ ] I Cauchy σ 11 C 0 C 1 C 1 ln Λ 1 σ 22 = C 1 C 0 C 1 ln Λ 2 C 1 C 1 C 0 ln Λ 3 σ 33 σ = C : E L Cauchy ϕ ρ0 ϕ Ṡ IJ = = 2 ρ 0 ϕ Ė KL E IJ E IJ E KL updated Lagrange updated Lagrange Truesdell σ 2 ρ 0 ϕ i j = C i jkl ρ, σ d kl, C i jkl ρ, σ lim a, b 0 t E IJ E KL C C σ = C : d Truesdell Oldroyd σ = C : d

54 Oldroyd O R C 0 Truesdell Jaumann Cauchy Kirchhoff a C 1 / C0 = 1 / 4 C 1 / C0 = 1 / 2 C 1 / C0 = 2 / 3 lnλ L ν = 1 3 C 1 C 0 = 1 2 Oldroyd O Σ C 0 hyper Jaumann Cauchy Kirchhoff lnλ L Truesdell b Cauchy Kirchhoff Jaumann τ K = C : d Cauchy Jaumann σ 11 = α + β σ 11 {lnλ L } σ 11 α + β σ 11 = Λ L Λ L σ 11 = R Λ 2 T = α β Λ β L 1, ln Λ T = C 1 C 0 + C 1 ln Λ L α C 0 C 0 + C 1 2C 2 1 C 0 + C 1, β C 0 C 1 : C 0 + C 1 Kirchhoff Jaumann C 0 + 3C 1 : C 0 + C 1 Truesdell 2 : Oldroyd C 1/ C0 = 1 / Truesdell C 1/ C0 = 1 / 4, 2 / 3 Poisson ν = 0.2, 0.4 C 1/ C0 = 1 / 2 Oldroyd Truesdell Truesdell 4 Cauchy

55 R C 0 Truesdell 2 R C O Λ L 1 C 1 / C0 = O 0.5 Truesdell lnλ L C 1 / C0 = 0 2 C 1 / C0 = 1 / 2 2 C 1 / C0 = 1 / 2 Jaumann Jaumann a b Jaumann Kirchhoff Jaumann Oldroyd Truesdell a b Σ σ 11 Cauchy Jaumann Cauchy hyper S = C:E Saint Venant-Kirchhoff Λ 2 L = 1 { } 3 + 2ν 9 + 8ν ν Λ 2 L = 1 + ν ν Σ Λ T ν = Λ L = 2 Truesdell Poisson σ = C : d + p I σ 11 C 0 C 1 C 1 σ 22 = C 1 C 0 C 1 σ 33 C 1 C 1 C 0 d 11 d 22 d 33 p + p, d 22 = d 33 = 1 2 d 11 p 1 σ 22 = σ 33 = 0 p σ 11 Truesdell Cauchy Jaumann σ σ R = 3 C 0 C 1 Λ 2 L 4 Λ 1 L σ σ R = 3 2 Λ L C 0 C 1 lnλ L

56 Λ 2 T = 1 / ΛL C 1 / C0 = 1 / 2 Poisson ν = Hooke σ i j = 2G ϵ i j, σ kk = K ϵ kk σ i j = 2G ϵ i j, σ kk = 3p ν = 1 / 2 C 1/ C0 = 0 Truesdell Cauchy Jaumann Cauchy Jaumann [58] Truesdell x 1 γ S N 21 C 66 S = C:E Truesdell u 1 = X 2 tan γ tan γ F = 0 1, 1 1 tan γ F = 0 1 O 0.2 γ S N π J = 1 γ Jaumann O 0 tan γ l = / 2 tan γ d = 1/ 2 tan γ 0, w = 0 1/ 2 tan γ 1 / 2 tan γ nominal S N 11 = σ 11 σ 12 tan γ, S N 21 = σ 12, S N 12 = σ 12 σ 22 tan γ, S N 22 = σ 22 S N 21 S N 12 S N 22 tan γ = 0 Jaumann σ 11 = σ 11 σ 12 tan γ, σ 22 = σ 22 + σ 12 tan γ, σ 12 = σ 12 + σ 11 σ tan γ 34 Cauchy Jaumann Kirchhoff Jaumann

57 d d 12 = d 21 σ 11 = 0, σ 22 = 0, σ 12 = C 66 d 12 = 1 2 C 66 tan γ C G σ 11 = σ 12 tan γ, σ 22 = σ 12 tan γ, σ 12 = 1 { C 66 σ 11 σ 22 } tan γ 2 σ 11 + σ 22 = 0 σ 11 + σ 22 = 0 σ 12 = C 66 σ 11 σ 22 σ 11 σ 22 4 σ σ12 σ11 σ = 1 C 66 C 66 σ 12 = 1 C 66 2 sin θ, σ 11 σ 22 = 1 cos θ C 66 σ 12 σ12 = 1 1 σ 11 σ 22 tan γ C 66 2 C cos θ tan γ = 1 2 sin θ tan γ = θ θ = tan γ σ 12 = 1 C 66 2 sin tan γ, σ 11 = σ 22 = 1 {1 cos tan γ} C 66 C 66 2 S N 21 = 1 C 66 2 sin tan γ, S N 12 = 1 [ ] sin tan γ + tan γ {1 cos tan γ}, C 66 2 S N 11 = 1 [ ] S N 1 cos tan γ tan γ sin tan γ, 22 = 1 {1 cos tan γ} C 66 2 C γ / 2 γ π / 2 sine S N 22 Jaumann γ 45 Truesdell Oldroyd σ 11 = σ 11 2 σ 12 tan γ, σ 22 = σ 22, σ 12 = σ 12 σ 22 tan γ

58 d { } 1 σ 11 = 2 σ 12 tan γ, σ 22 = 0, σ 12 = 2 C 66 + σ 22 tan γ σ 12 C 66 = 1 2 tan γ, σ 11 C 66 = 1 2 tan2 γ, σ 22 C 66 = 0 S N 21 C 66 = S N 12 C 66 = 1 2 tan γ, S N 11 C 66 = S N 22 C 66 = γ S N Piola-Kirchhoff S 11 = σ 11 2 σ 12 tan γ = 1 2 C 66 tan 2 γ, S 22 = σ 22 = 0, S 12 = σ 12 = 1 2 C 66 tan γ Green 12.9 E 11 = 0, E 22 = 1 2 tan2 γ, E 12 = 1 2 tan γ 2 Piola-Kirchhoff Green Truesdell S 11 C 11 C 66 0 E 11 S 22 = C E C 66 E 12 S 12 2 Piola-Kirchhoff Green S = C:E S N 11 = 1 C1 + C 66 tan 2 γ, S N 22 2 = 1 2 C 0 tan 2 γ, S N 12 = 1 2 C 66 tan γ, S N 21 = 1 2 C 66 tan γ C 0 tan 3 γ C 0/ C 66 = 2 Hooke Poisson ν = 1/ γ π / ±100% 3 Truesdell Truesdell Truesdell 2 Piola-Kirchhoff

59 B.3 Timoshenko γ V = Gk t A γ + N γ 1 + ϵ N ϵ k t [17] A G γ N V Engesser [95] { } 1 V N 1 + ϵ γ γ 1 + ϵ ϵ = Gk 2 t A γ + EA γ 1 + ϵ ϵ 2 2 updated Lagrange γ = 0 ϵ 1 V N γ = Gk t A γ Timoshenko σ 11 σ 13 v 1 = v 1 x 1, x 3 v 3 = v 3 x Cauchy Jaumann σ 13 = σ 13 σ 11 w 31 = 2µ d Truesdell σ 13 = σ 13 σ 11 v 3,1 = 2µ d Oldroyd σ 13 = σ 13 σ 11 v 3,1 σ 13 v 1,1 = 2µ d 13 ϵ 1 ϵ v 1,

60 nominal ṅ ji, j + ρ π i = 0 v ṅ δv i ji, j + ρ π i dv = δvi, j ṅ ji ρ δv i π i dv δv i ṫ i ds = 0 v v s nominal ṅ ji = F jikl v k,l 1 δuv δuv δv i, j F jikl v k,l dv v F jikl = F lki j updated Lagrange Uv updated Lagrange Uv Cauchy Jaumann C i jkl = µ δ ik δ jl + δ il δ jk + λ δi j δ kl nominal F i jkl = C i jkl + σ i j δ kl σ li δ jk 1 2 σ ki δ jl 1 2 σ l j δ ik 1 2 σ k j δ il F i jkl F lk ji Uv Kirchhoff Jaumann Truesdell F i jkl = C i jkl + σ li δ jk F i jkl = F lk ji Uv Kirchhoff Jaumann Truesdell

61 Prandtl-Reuss Mises J dx = d e x + d p x, wx = w e x + w p x a, b Hooke ρ0 ϕ Ṡ IJ = = 2 ρ 0 ϕ Ė KL E IJ E IJ E KL d e 2 Piola-Kirchhoff Green Lagrange updated Lagrange updated Lagrange Truesdell σ i j = C i jkl ρ, σ d e kl, 2 C ρ 0 ϕ i jkl ρ, σ lim 0 t E IJ E KL a, b C Hooke 3.49 σ i j = C i jkl d e kl, C i jkl = µ δ ik δ jl + δ il δ jk + K 2µ δ i j δ kl a, b 3 µ K 3.59 di e j = D i jkl σ kl, D i jkl = 1 4 µ 1 1 δik δ jl + δ il δ jk K 1 δ i j δ kl a, b 2 µ Truesdell Cauchy Jaumann di e j = D i jkl σ kl

62 Cauchy Jaumann Truesdell Mises f σ τ y ϵ p, ϵ p σ 11.23a d p i j = λ f σ i j f = a λ = 1 H = λ σ i j 2 σ 2 d p i j dp i j dt a, b f σ i j σ i j = σ i j 2 H σ σ i j, H τ yϵ p ϵ p a, b Mises Cauchy Cauchy d p i j = 1 H σ i j σ kl 4 σ 2 σ kl Cauchy Jaumann Jaumann σ kl w k j σ jl Cauchy σ kl w k j σ jl = 1 2 σ kl w k j σ jl + σ jl w 1 jk σ kl = σ 2 kl w k j σ jl σ jl w k j σ kl = = a Jaumann d p i j = 1 H σ i j σ kl 4 σ 2 σ kl d i j = D i jkl + 1 σ i j σ kl σ H 4 σ 2 kl σ i j = C i jkl µ2 µ + H σ i j σ kl σ 2 d kl : λ = 0 f < a : λ = 0 f = 0 σ i j σ i j < 0 : λ = 0 f = 0 σ i j σ i j = 0 : λ > 0 f = 0 σ i j σ i j > b c d

63 [75] f g f σ FI 1, p, ϵ p, g σ + GI a, b σ 11.23a I p ρ 0 ρ dp kk dt, ϵp a d p i j = λ σ i j 2 σ + β δ i j 2 d p i j dp i j dt a, b, β = βi 1 GI 1 I a, b f = σ σ i j σ i j F I 1 İ 1 F p p F ϵ p ϵ p = σ i j 2 σ σ i j F σ kk = ρ 0 I 1 ρ λ λ = 1 σ kl H 2 σ F δ kl I 1 F p dp kk + F ϵ p σ kl, H 3 ρ 0 ρ 2 d p i j dp i j F p β + F ϵ p a, b H d p i j = 1 σ { i j H 2 σ + β δ σ } kl i j 2 σ + α δ kl σ kl, α = αi 1, p, ϵ p FI 1, p, ϵ p a, b I Cauchy Jaumann d p i j = 1 σ { i j H 2 σ + β δ σ } kl i j 2 σ + α δ kl σ kl Hooke d i j = D i jkl σkl + 1 σ i j σ H 2 σ + β δ kl i j 2 σ + α δ kl σ kl µ σ i j µ σ σ + 3K β δ kl i j σ + 3K α δ kl σ i j = C i jkl d kl d kl H + µ + 9K α β Cauchy Jaumann

64 [82] d p i j = λ g + 1 σ i j σ i j 2 h σ 2 σ i j σ kl σ kl d p i j = 1 H σ i j 2 σ + β δ i j { σ } kl 2 σ + α δ kl σ kl h 1 σ i j 1 2 σ 2 σ i j σ kl σ kl Hooke d i j = 1 σ i j µ 3 3K 1 2µ + 1 σ i j H 2 σ + β δ i j δ i j σkk { σ } kl 2 σ + α δ kl σ kl h 1 σ i j 1 2 σ 2 σ i j σ kl σ kl Cauchy Jaumann 4 Asaro [3, 5] Cauchy Kirchhoff Jaumann σ i j + σ i j d e kk = C i jkl d e kl n α s α γ α γ α α σ i j σ i j w e ik σ k j w e jk σ ki Truesdell Cauchy Burgers n α α γ α γ α 36 s α

65 α v p i, j = γα s α i n α j, α = 1, 2,, N α N d p i j = p α i j γα, α w p i j = ω α i j γα a, b α N p α i j 1 2 s α i n α j + nα i s α j α, ω α i j 1 2 s α i n α j nα i s α j a, b α w p ṡ α i = w e i j sα j, ṅα i = w e i j nα j a, b s α n α α τ α y τ α σ i j s α i n α j = τ α y σ 12 τ 0 y ϵ 11 γ = O 2 s 3 s 4 s2 s τ α y Schmid σ 11 τ 0 y τ α = h αβ γ β β h αβ β α α β τ α = σ i j s α i n α j + σ ṡα i n α j + σ sα i ṅ α j τ α = σ i j s α i n α j + σ we ik sα k nα j + σ sα i w e jk nα k = sα i n α j σi j w e ik σ k j w e jk σ ki

66 τ α = σ i j s α i n α j Prandtl-Reuss Cauchy σ i j p α i j = β h αβ γ β σ i j p α i j = τα y α : γ α = 0 σ i j p α i j < τα y a : γ α = 0 σ i j p α i j = τα y σ i j p α i j < h αβ γ β b : γ α 0 σ i j p α i j = τα y σ i j p α i j = h αβ γ β β β c σ i j p α i j = pα i j Ci jkl d kl σ i j d kk C i jkl d p kl + σ i j dkk p a σ i j p α i j = pα i j Ci jkl d kl σ i j d kk β p α i j Ci jkl p β kl σ i j p β kk γ β = h αβ γ β M αβ N αβ γ α = M αβ p β i j Ci jkl σ i j δ kl dkl β β M αβ N αβ 1, N αβ h αβ + p α i j Ci jkl σ i j δ kl p β kl σ i j + σ i j d kk = C i jkl d kl α Ci jkl p α kl σ i j p α kk + ωα ik σ k j + ω α jk σ ki γ α σ i j = C i jkl d kl σ i j d kk α β Ci jkl p α kl σ i j p α kk + ωα ik σ k j + ω α jk σ ki M αβ p β mn Cmnpq σ mn δ pq dpq

67 Asaro [51] α = τ α y τ α y = τ 0 y = const., γ 0 τ0 y µ H µ x Lüders [40] ν vi, j = gi ν j g v i, j v i, j ν j ṅ ji = ṅ nominal ṅ ji = F jikl v k,l impotent eigenstrain? 38 [51]

68 ν j F jikl ν l gk = g det ν j F jikl ν l = [40] Drucker σ i j ϵ p i j 0 ϵ p i j Cep i jkl ϵ kl 0, σ i j = C ep i jkl ϵ kl ϵ i j ϵ p i j = 1 2 gi ν j + g j ν i g j νi C ep i jkl ν l gk 0 Drucker g 2 det ν i C ep i jkl ν l F F, σ, ϵ p, C ep Prandtl-Reuss 2 [104] Cauchy Jaumann Truesdell Hooke C i jkl = µ δ ik δ jl + δ il δ jk + λ δi j δ kl µ λ Lamé Cauchy Jaumann σ i j = 2µ d i j + λ δ i j d kk µ2 µ + H σ i j σ kl σ 2 d kl + w ik σ k j + w jk σ ki

69 σ 0 µ H/ µ = 10 2 σ 0 µ H/ µ = θ H/ µ = H/ µ = 0 θ a Cauchy Jaumann b Truesdell H τ y ϵ p H τ yϵ p ϵ p, ϵ p 2 d p i j dp ji dt a, b Truesdell σ i j = 2µ d i j + λ δ i j d kk µ2 σ i j σ kl d µ + H σ 2 kl µ + µ + H σ i j d µ σ i j kk σ lm vl,k σ km + v m,k σ kl + vi,k σ k j + v j,k σ ki σ i j d kk 2µ + H σ F x 3 d p 33 = 0 σ 33 = 0 σ 33 = 1 2 σ 11 + σ x 1 σ 11 = σ 0 1

70 σ cr µ σ cr µ convected convected 10 1 Jaumann 10 1 Jaumann Truesdell Truesdell θ H/ µ θ H/ µ Truesdell convected Truesdell Jaumann convected 35 Jaumann H/ µ H/ µ a b convected Truesdell Cauchy Jaumann [50] updated Lagrange 1

71 nominal ṅ ṅ ji, j + π i = 0 π ν j ṅ ji = ṫ i ν ṫ Euler updated Lagrange δv i, j ṅ ji dv δv i π i dv δv i ṫ i ds = v v v s u σ i j t + t = σ i j t + σ i j t updated Lagrange pt p i t + t = p i t + v i t s 2 { } v v 1,1 v 2,2 v 3,3 v 3,2 v 2,3 v 1,3 v 3,1 v 2,1 v 1,2 t nominal { } ṅ ṅ 11 ṅ 22 ṅ 33 ṅ 23 ṅ 32 ṅ 31 ṅ 13 ṅ 12 ṅ 21 t Internal Virtual Work δv i, j ṅ ji dv = v v { } t { } δ v ṅ dv { } { } { } v = S v, v t, v 1 v 2 v a, b S x x x 3 x x x x 2 0 x 1 x t c

72 IVW = v { } t δ v S t { } ṅ dv ṅ i j = F i jkl v k,l { } ṅ = F { } v { } t t IVW = δ v S v F S { } v dv F 3 : Cauchy { } σ σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 t { } σ = G { } v G 6 9 Cauchy Jaumann : Cauchy Jaumann σ i j = C i jkl d kl d C Hooke C i jkl = µ δ ik δ jl + δ il δ jk + λ δi j δ kl µ λ Lamé Cauchy σ i j = 2µ d i j + λ δ i j d kk + w ik σ k j + w jk σ ki w G λ + 2µ λ λ 0 0 G λ λ + 2µ λ σ 23 σ 23 λ λ λ + 2µ σ 23 σ µ 1 2 σ σ 22 µ σ σ σ σ σ σ 13

73 σ 13 σ 13 σ 12 σ σ 12 σ 12 σ 13 σ σ σ 12 µ 1 2 σ σ 33 µ σ σ σ σ σ σ σ σ 23 µ 1 2 σ σ 11 µ σ σ Hooke nominal ṅ i j = 2µ d i j + λ δ i j d kk + σ i j d kk + w jk σ ki d ik σ k j F λ + 2µ λ + σ 11 λ + σ λ + σ 22 λ + 2µ λ + σ 22 σ 23 0 λ + σ 33 λ + σ 33 λ + 2µ 0 σ 23 F σ 23 0 σ 23 µ 1 2 σ σ 22 µ 1 2 σ σ 22 σ 23 σ 23 0 µ 1 2 σ σ 22 µ σ σ 22 σ 13 σ σ σ σ 13 σ 13 2 σ σ 12 0 σ 12 σ σ σ 13 σ 12 0 σ σ σ 13 0 σ 13 σ σ 12 σ σ 12 2 σ σ σ σ σ σ σ 13 µ 1 2 σ σ 33 µ 1 2 σ σ σ σ µ 1 2 σ σ 33 µ σ σ σ σ σ σ 23 µ 1 2 σ σ 11 µ 1 2 σ σ σ σ 23 µ 1 2 σ σ 11 µ σ σ Hooke Cauchy Jaumann Truesdell : Truesdell σ i j = C i jkl d kl Cauchy σ i j = 2µ d i j + λ δ i j d kk σ i j d kk + v i,k σ k j + v j,k σ ki

74 G G λ + 2µ + σ 11 λ σ 11 λ σ σ σ 12 λ σ 22 λ + 2µ + σ 22 λ σ σ σ 12 0 λ σ 33 λ σ 33 λ + 2µ + σ 33 2σ σ σ µ + σ 22 µ + σ 33 0 σ 12 σ σ 13 0 σ 12 0 µ + σ 33 µ + σ 11 0 σ σ 12 0 σ 13 σ 23 0 µ + σ 11 µ + σ 22 Hooke nominal ṅ i j = { µ } δ ik δ jl + δ il δ jk + λ δi j δ kl + σ li δ jk vk,l = F i jkl v k,l F i jkl = F lk ji i j lk F F λ + 2µ + σ 11 λ λ 0 0 σ σ 12 λ λ + 2µ + σ 22 λ 0 σ σ 12 0 λ λ λ + 2µ + σ 33 σ σ σ 23 µ + σ 22 µ 0 σ σ 23 0 µ µ + σ σ 13 0 σ µ + σ 33 µ 0 σ σ 13 σ 12 0 µ µ + σ σ σ µ + σ 11 µ σ σ 23 0 µ µ + σ 22 Hooke Kirchhoff Jaumann updated Lagrange { } t { } { } t { } { } t { } δπt δ v Ft v dv δ v π dv δ v ṫ ds = 0 F vt Πt 1 2 vt { } t v Ft vt { } { } t { } { } t { } v dv v π dv v ṫ ds vt st 2 Piola-Kirchhoff updated Lagrange Truesdell vt st st

75 { } v = Nx 1, x 2, x 3 { } ṗ { } Nx 1, x 2, x 3 ṗ a { } σ = G S N { } ṗ = G B { } ṗ, B S N a, b t t k = N S F S N dv = B v v t F B dv Cauchy Jaumann Truesdell b B Lagrange { } t { } ḣ N π dv, v { } f s N t { } ṫ ds a, b k { } { } ṗ = ḣ { } + f { } p { } t+ t { } t { } t p = p + ṗ { } t+ t { } t { } t, { } t+ t { h = h + ḣ f = f } t { } t + f a, b Cauchy { } σ σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 t { } t+ t { } t { } t σ = σ + σ

76 : 6 1 Prandtle-Reuss { } σ t σ 11 σ 22 σ σ 23 σ 31 σ { } s σ 11 σ 22 σ 33 σ 23 σ 23 σ 13 σ 13 σ 12 σ 12 t / 3 1/ 3 1/ / 3 2/ 3 1/ { } σ = R { } σ = 1/ 3 1/ 3 2/ { } σ / 3 1/ 3 1/ / 3 2/ 3 1/ / 3 1/ 3 2/ { } s = T { } σ = σ σ i j σ i j = 1 { } t { } s s 2 { } σ ϵ p 2 d p i j dp i j = 1 2H σ σ kl σ kl = 1 2H σ σ kl σ kl = 1 2H σ { } t { } s ṡ { } { } { } { } ṡ = T σ = T G S N ṗ = T G B ṗ B f σ τ y = τ y τ y = τ 0 y + H ϵ p H

77 : { } σ = G { } G p v G p nominal { } ṅ = F F p { } F p v Cauchy Jaumann : σ i j = 2µ d i j + λ δ i j d kk + w ik σ k j + w jk σ ki µ2 µ + H G p µ 2 G p µ + H σ 2 nominal ṅ i j = 2µ d i j + λ δ i j d kk + σ i j d kk + w jk σ ki d ik σ k j µ2 µ + H F p µ 2 F p µ + H σ 2 Cauchy σ i j σ kl d σ 2 kl { σ } { s } t σ i j σ kl d σ 2 kl { } { } t s s Truesdell : P Q Q1 Q 2 Q 3 Q 4 Q 5 Q 6 Q 7 Q 8 Q 9 Q 1 σ 11 σ 11 + σ σ2 13, Q 2 σ σ 22 σ 22 + σ2 23, Q 3 σ σ σ 33 σ 33,12.248a, b, c Q 4 σ 12 σ 13 + σ 23 σ 22 + σ 23 σ 33, Q 5 σ 12 σ 13 + σ 23 σ 22 + σ 23 σ 33, d, e Q 6 σ 13 σ 11 + σ 12 σ 23 + σ 13 σ 33, Q 7 σ 13 σ 11 + σ 12 σ 23 + σ 13 σ 33, f, g Q 8 σ 12 σ 11 + σ 12 σ 22 + σ 13 σ 23, Q 9 σ 12 σ 11 + σ 12 σ 22 + σ 13 σ h, i Cauchy σ i j = 2µ d i j + λ δ i j d kk σ i j d kk + v i,k σ k j + v j,k σ ki µ2 µ + H σ i j σ kl σ 2 d kl µ 2 µ + H σ i j σ kl σ 2 vk,m σ ml + v l,m σ mk + µ µ + H σ i j d kk

78 G p µ 2 G p µ + H σ 2 { } { } t σ µ { } s + σ µ { } Q σ P µ + H σ 2 µ + H nominal ṅ i j = 2µ d i j + λ d kk + σ ki v j,k σ i j σ kl µ2 d µ + H σ 2 kl F p µ 2 µ + H σ i j σ kl σ 2 vk,m σ ml + v l,m σ mk + µ µ + H σ i j d kk µ 2 F p µ + H σ 2 { } { } t µ { } µ { } s s + s Q s P µ + H σ 2 µ + H Truesdell Cauchy Jaumann MN m E = 10 MN/m 2, ν = a, b x 2 x 3 x 1 A B C 0 λ + 2µ = 15 MN/m 2, C 1 λ = 7.5 MN/m 2, µ = 3.75 MN/m a, b, c 1 m x 2 x 1, x 3 A x 1 B x R x 2 1 m model-steps Λ L = 0.5 Λ L = m Jaumann J J J Truesdell T T T Truesdell Cauchy Jaumann

79 ν = 1 3 C 1 = 1 C R C 0 Truesdell Truesdell Σ C 0 lnλ L lnλ L O O Jaumann 1 Jaumann ν = 1 3 C 1 C 0 = 1 2 x 2 R x 2 Cauchy Σ Λ L = 0.5, Cauchy Cauchy Cauchy Jaumann x 1 3 h = 16 cm l = cm x 2 t 5 mm l = 20 cm x 1 40 x ,985 15,360 1,034 3,840 O x 3 l x 1 = 0, x 2 = 0, x 3 = 0 x 2 = 0 x 2 x 1 = 0, x 2 = t x 1 x 3 x 1 = 0 x 1 x 1 = l x 1 P Young Poisson Timoshenko k t A.7b F LU h x 1

80 x 1 = l P λ A 2 I l 1 A, B 4 J Cauchy Jaumann K Kirchhoff Jaumann T Truesdell O Oldroyd Euler Timoshenko A B.5 2 B.59 B 1 B.56 A B Truesdell Oldroyd 0.3 P cr EA Pl 2 EI B 2 A Jaumann Euler λ Truesdell : K : J : T : O w l A B B 2 Jaumann A Euler λ = Jaumann x 1 = 0 x 3 x 1 = l x mm l = 20 cm λ 4 P x 2 = 0, x 3 = h / 2 w Cauchy Jaumann Truesdell B 39 l h 25%

81 A B.6.2 LU Truesdell B Cauchy Jaumann A [111] 40 Kirchhoff Jaumann Truesdell F Cauchy Jaumann Oldroyd Jaumann Asaro Kirchhoff Jaumann [3, 5] 2 Piola-Kirchhoff 2 Truesdell Truesdell Cauchy Jaumann : h = 16 cm 16 l = 20 cm 20 t = 1 cm 1,034 3,840 : h = 8 cm 32 l = 20 cm 40 t = 5 mm 2,034 7,680 E = 200 GN/m 2, ν = 1 3, τ y = 140 MN/m 2, H = 0.25 MN/m a, b, c, d τ y H x 1 = 0 x 1 = l 43 x 2 = 0 x 2 = t 40 1 PhD 41 Femap Copyright c 2012 Siemens Product Lifecycle Management Software Inc., Version x 1 = 0, x 3 = h / 2 x 3

82 : x 2 : σ 22 σ : x 1 = l x 2 = t, x 3 = h x 1 x 2 20% : x 1 = l x 2 = t, x 3 = 0 x 1 x 2 20% : x 1 = l / 2 x 2 = t, x 3 = 0 x 1 x 2 20% 3 σ ϵ σ P ht, ϵ U l a, b P x 1 U x 1 = l x 1 LU nd 44 Truesdell : x 1 U ϵ = 0.4% 2 h nd

83 12.7. 非線形挙動の数値的予測の例 665 p p p a ϵ = 0.331%, ϵ max = 0.416%, b ϵ = 0.356%, ϵ max = 0.477%, c ϵ = 0.381%, ϵ max = 0.519%, d ϵ = 0.389%, p ϵ min = 0.411%, nd = 0 p ϵ min = 0.455%, nd = 0 p e ϵ = 0.306%, ϵ max = 0.367%, f ϵ = 0.356%, p ϵ min = 0.362%, nd = 0 p p ϵ max = 0.472%, ϵ min = 0.457%, nd = 1 p ϵ min = 0.507%, nd = 2 p p ϵ max = 1.29%, ϵ min = 0.507%, nd = 2 p p g ϵ = 0.364%, ϵ max = 0.485%, h ϵ = 0.389%, ϵ max = 0.718%, p ϵ min = 0.474%, nd = 2 p ϵ min = 0.510%, nd = 4 図 平面ひずみ状態で公称応力のピーク前後 a ϵ = 0.620%, p p ϵ max = 4.46%, ϵ min = 0.509%, nd = 0 e ϵ = 0.795%, p p ϵ max = 6.20%, ϵ min = 0.532%, nd = 1 b ϵ = 1.67%, p p ϵ max = 25.3%, ϵ min = 0.512%, nd = 0 f ϵ = 1.83%, p p ϵ max = 27.6%, ϵ min = 0.565%, nd = 0 c ϵ = 2.50%, p p ϵ max = 39.6%, ϵ min = 0.512%, nd = 0 g ϵ = 2.50%, p p ϵ max = 40.0%, ϵ min = 0.565%, nd = 0 d ϵ = 2.50%, p p ϵ max = 39.6%, ϵ min = 0.512%, nd = 0 h ϵ = 2.50%, p p ϵ max = 40.0%, ϵ min = 0.565%, nd = 0 図 平面ひずみ状態で公称応力のピーク後 ものであり いわゆる式 で定義されるような支配方程式の楕円性の喪失といった微視的かつ孤立した 局所変形ではない またその周期も増分ステップ毎にかなり変化するが ある程度変形が大きくなると接線剛 性行列の対角項はすべて正に戻り いくつかの巨視的な変形集中帯が孤立するような傾向を示す 最終的に ϵ = 2.5% に達したあたりでは このような 45 度方向のせん断の変形集中帯の発達に伴った絞りが顕著になった 解析的な研究 [40] でも ピーク前後に周期的な変形が先行したあとに式 で定義された変形の局所化が 生じると指摘されている 解析的研究の偉大さと重要さを再認識させる数値結果である なお相当塑性ひずみ が非常に大きな値になるのは用いた硬化係数 H が非常に小さいからである これに対し 亜弾性に Cauchy 応力の Jaumann 速度を用いた結果を図 に示した まず図 f に生 じた変形の周期が その直前の図 e や前述の Truesdell 応力速度を用いた図 h の周期と異なるの は興味深い また四角板の場合に接線剛性行列の対角項に現れる負の要素数 nd は Truesdell 応力速度を用い た場合は 1 ないし 2 であったのに対し Jaumann 速度を用いた場合は 1 から 3 の間になっていた また細長板 では 前者の nd が 1 から 4 の間であったのに対し 後者は 1 から 7 までにわたり Jaumann 速度を用いた場 合には比較的短い周期の応力分布が発生していることを示唆している 用いる応力速度の違いだけがその原因 であるとは断定できないが ここで対象とした辺長比が整数ではない領域に現れていることを念頭に置き 以 下のように推測している 前節までの比較の結果から Truesdell 応力速度を用いた構成則は Jaumann 速度を

84 a ϵ = 0.389%, ϵ p max = 0.545%, b ϵ min p = 0.518%, nd = 3 p ϵ = 0.398%, ϵ max = 0.725%, p c ϵ = 2.50%, ϵ max = 32.8%, p d ϵ = 2.50%, ϵ max = 32.8%, ϵ min p = 0.518%, nd = 2 ϵ min p = 0.518%, nd = 0 ϵ min p = 0.518%, nd = 0 e ϵ = 0.389%, ϵ p max = 0.541%, f ϵ min p = 0.523%, nd = 5 p ϵ = 0.423%, ϵ max = 0.659%, p g ϵ = 2.50%, ϵ max = 42.7%, p h ϵ = 2.50%, ϵ max = 42.7%, ϵ min p = 0.563%, nd = 3 ϵ min p = 0.575%, nd = 0 ϵ min p = 0.575%, nd = Cauchy Jaumann 300 MN/m MN/m 2 σ σ Jaumann 100 Jaumann O 1 2 ϵ % O a b ϵ % Jaumann σ y = 2τ y = 280 MN/m 2 Truesdell Cauchy Jaumann ϵ = 1% Truesdell ϵ = 2% g Jaumann g x 1 2l l Truesdell

85 a ϵ = 0.164%, ϵ p max = 0.110%, b ϵ min p = %, nd = 0 p ϵ = 0.248%, ϵ max = 0.292%, p c ϵ = 0.331%, ϵ max = 2.62%, p d ϵ = 2.50%, ϵ max = 71.0%, ϵ min p = 0.254%, nd = 0 ϵ min p = 0.303%, nd = 0 ϵ min p = 0.303%, nd = a ϵ = 0.165%, ϵ p max = 0.111%, b ϵ min p = %, nd = 0 p ϵ = 0.281%, ϵ max = 0.399%, p c ϵ = 0.373%, ϵ max = 2.24%, p d ϵ = 2.50%, ϵ max = 59.8%, ϵ min p = 0.300%, nd = 0 ϵ min p = 0.300%, nd = 1 ϵ min p = 0.300%, nd = a ϵ = 0.141%, ϵ p max =0.0635%, b ϵ min p = %, nd = 0 p ϵ = 0.250%, ϵ max = 0.334%, c ϵ min p = 0.210%, nd = 0 p ϵ = 0.291%, ϵ max = 0.788%, p d ϵ = 2.50%, ϵ max = 73.0%, ϵ min p = 0.214%, nd = 0 ϵ min p = 0.214%, nd = ϵ 0.4% nd = 1 nd = 0 x , ϵ = 2.5% : x h nd = 1 ϵ = 2.5% , Jaumann g d Truesdell Truesdell Jaumann

86 第 12 章 有限変形理論を直感で噛み砕く 668 a p p ϵ= 0.322%, ϵ max =0.401%, ϵ min = 0.393%, nd = 0 b p p ϵ= 0.331%, ϵ max =0.469%, ϵ min = 0.393%, nd = 0 p c p p ϵ= 0.372%, ϵ max =1.08%, ϵ min = 0.393%, nd = 0 p p ϵ= 2.50%, ϵ max =38.1%, p ϵ min = 0.393%, nd = 0 d p ϵ= 0.322%, ϵ max =0.400%, f ϵ= 0.331%, ϵ max =0.445%, g ϵ= 0.372%, ϵ max = 4.96%, h ϵ = 2.50%, ϵ p = 58.5%, max p p p ϵ min = 0.394%, nd = 1 ϵ min = 0.394%, nd = 1 ϵ min = 0.396%, nd = 0 ϵ min = 0.429%, nd = 0 e p 図 平面ひずみ状態の圧縮 300 MN/m2 300 σ MN/m2 σ 平面ひずみ 圧縮 平面ひずみ 圧縮 Truesdell Jaumann 細長 Truesdell Jaumann O 1 a 四角板の場合 O 2 ϵ % 1 b 細長板の場合 2 ϵ % 図 平面ひずみ状態の圧縮 p p p p ϵ= 0.322%, ϵ max =0.401%, b ϵ= 0.339%, ϵ max =0.435%, c ϵ= 0.347%, ϵ max =0.475%, d ϵ= 0.372%, ϵ max =0.669%, p p p p ϵ min = 0.395%, nd = 1 ϵ min = 0.428%, nd = 2 ϵ min = 0.434%, nd = 3 ϵ min = 0.434%, nd = 2 a p p = 4.02%, f ϵ = 0.940%, ϵ max = 20.6%, g ϵ = 1.58%, e ϵ = 0.555%, ϵ max p ϵ min = 0.434%, nd = 2 p ϵ min = 0.434%, nd = 1 p p ϵ max = 20.8%, ϵ min = 0.439%, nd = 0 h ϵ = 2.50%, p p ϵ max = 24.9%, ϵ min = 0.439%, nd = 0 図 Cauchy 応力の Jaumann 速度を用いた場合の平面ひずみ状態の圧縮 ϵ = 2.5% に至った状態では Truesdell 応力速度を用いた場合は板の寸法に関連した周期を持つ変形パターン になっているが Jaumann 速度を用いた場合には短い周期のパターンも含む状態になった これも 用いる応 力速度の違いだけが原因であるかどうかは判断できないが 巨視的な挙動がほぼ同じであっても 構成モデル の違いによって微視的な変形パターン等には違いが生じることを示している

87 : ϵ = 5% 45 Jaumann Truesdell ϵ = 15% Jaumann ϵ = 20% a ϵ = 2.33%, ϵ max p = 3.80%, p b ϵ = 5.00%, ϵ max = 8.26%, ϵ min p = 3.79%, nd = 1 ϵ min p = 8.26%, nd = x 3 a ϵ = 2.25%, ϵ max p = 3.65%, p b ϵ = 4.25%, ϵ max = 7.39%, p c ϵ = 5.00%, ϵ max = 12.2%, p d ϵ = 5.00%, ϵ max = 12.2%, ϵ min p = 3.65%, nd = 2 ϵ min p = 6.68%, nd = 0 ϵ min p = 6.68%, nd = 0 ϵ min p = 6.68%, nd = Truesdell a ϵ = 2.25%, ϵ max p = 3.65%, p b ϵ = 4.25%, ϵ max = 8.36%, p c ϵ = 5.00%, ϵ max = 20.8%, p d ϵ = 5.00%, ϵ max = 20.8%, ϵ min p = 3.64%, nd = 2 ϵ min p = 6.65%, nd = 0 ϵ min p = 6.65%, nd = 0 ϵ min p = 6.65%, nd = Cauchy Jaumann 12.44, x 1 ϵ = 5% Poisson σ y = 3 τ y = 242 MN/m 2 Truesdell ϵ = 5% x 1 l 300 σ MN/m 2 Jaumann O 2 4 ϵ % Jaumann x 1 2l ϵ 4% 2 Truesdell 45

88 a ϵ = 0.150%, ϵ max p = 0.144%, p b ϵ = 5.00%, ϵ max = 23.4%, ϵ min p = %, nd = 2, ϵ min p = 3.31%, nd = 0, c ϵ=0.132%, ϵ max p =0.0372%, p d ϵ = 5.00%, ϵ max = 43.3%, ϵ min p = %, nd = 2, ϵ min p = %, nd = 1, e ϵ=0.133%, ϵ max p =0.0893%, p f ϵ = 5.00%, ϵ max = 41.7%, ϵ min p = 0.00%, nd = 0, ϵ min p = 0.00%, nd = 0, c ϵ = 5% σ MN/m σ ave 85 MN/m 2 75 MN/m 2 x x 3 3 Lüders? α, β ϕ Hill x 1 -x 3 d 11 cos 2 ϕ + 2d 13 sin ϕ cos ϕ + d 13 sin 2 ϕ = 0, d 11 sin 2 ϕ 2d 13 sin ϕ cos ϕ + d 13 cos 2 ϕ = a, b d p 11 cos2 ϕ + 2d p 13 sin ϕ cos ϕ + dp 13 sin2 ϕ = 0, d p 11 sin2 ϕ 2d p 13 sin ϕ cos ϕ + dp 13 cos2 ϕ = a, b Prandtl-Reuss σ 11 cos2 ϕ + 2σ 13 sin ϕ cos ϕ + σ 33 sin2 ϕ = 0, σ 11 sin2 ϕ 2σ 13 sin ϕ cos ϕ + σ 33 cos2 ϕ = a, b

89 α, β [39] Hill x 1 -x 3 : tan ϕ ± 2, ± 1, x 1 -x 2 : tan ϕ 0, ± a, b 2 x 1 -x 3 : tan ϕ ±1, x 1 -x 2 : tan ϕ = a, b x 1 -x 3 x 1 ±45 x 1 -x 3 Hill α, β x 1 ± ±45 ±45 Asaro[4] 46 tan ϕ ± 3, ± 1 ϕ ±60, ± tan ϕ ± 2, ± 1 ϕ ±54.7, ± Hill [39] Lüders ϕ Hill Lüders Asaro [4] : 100 H = 25 MN/m 2 - ϵ = 2.5% c Young 1 / 1000 H = 250 MN/m 2 - H = 25 MN/m 2 ϵ = 2.5% H = 250 MN/m Lüders 3? 46 kink band coarse slip band CSB macroscopic slip band MSB

90 a w= , ϵ max p =0.0331%, p b w = 0.025, ϵ ϵ min p max = 2.86%, = 0.00% ϵ min p = 0.00% c w = 0.250, ϵ max p = 45.6%, p d w = , ϵ max = ϵ min p = % 45.7%, ϵ min p = % e w = 0.250, ϵ max p = 78.1%, ϵp min = % x 3 σ y = 2 τ y w W h, p P P y, P y M y l, M y h2 t 6 σ y a, b, c, d W x 1 = l, x 3 = h / 2 x 2 = 0, t 2 x 3 P 2 x 3 w σ a w = 0.25 w = 0.25 w = e Truesdell Jaumann 1 p O p 1 p 1 H = 250 MN/m w

91 H = 250 MN/m 2 H = 0.25 MN/m 2 W = 0.05 mm H = 0.25 MN/m 2 W = mm H = 0.25 MN/m 2 W = mm O 0.5 w p a w = 0.500, ϵ max p = 25.0%, p b w = 0.500, ϵ max = 68.8%, ϵ min p = 0.145% ϵ min p = 0.819% H = 250 MN/m a W = mm, ϵ max p = p b W = mm, ϵ max = 163%, ϵ min p = % 230%, ϵ min p = % H = 0.25 MN/m 2 ; w = W 0.05 mm W 12.53? 1 p 0.5 H = 250 MN/m 2 p 10 5 O / h x 1 x 2 x 2 x 3 x 3 x 1 w u 2 l, t, h / 2 h O / h w = 0.5 ; ϵ p max = 19.2%, ϵ p min = % x 1 = l, x 2 = t, x 3 = h / 2 1 x 3 x 1 -x 3 47?

92 x x 3 W x 3 P σ y = 3τ y x 2 = t / 2 x 1 -x w = 0.5 p = p 10 P a w= , ϵ p max =0.0848% b w=0.0333, ϵp max =1.23% ; ϵ p min=0% 65 kn 1 / H = 250 MN/m 2, w = 0.5, H = 250 MN/m 2 ϵ max p = 11.1%, ϵ min p = % 12.56

93 IV 675

94 移設後の高麗橋と桜島 気仙沼湾横断橋 仮称 2017 年 3 月ヤードにて

95 A Timoshenko A.1 A ux Bernoulli-Euler z x A u z x, z wx γx n 3.199b z B w x ϑx u x x, z Timoshenko Bernoulli-Euler A.1 Timoshenko ϵ xz x, z = γx A.1 z 2 k t A.1 x, z 4.3 u x x, z = ux + z ϑx, u z x, z = wx A.2a, b ϑx A A.2 2 ϵ xz x, z = ϑx + w x = γx ϑ = w + γ A.3 677

96 678 A. TIMOSHENKO x 4.4 θ ϑ w γ ϵ xx x, y, z = u + z ϑ = u + z γ w A.4 1 Hooke 3.183a Hooke 3.46 σ xx = E ϵ xx = E { u + z γ w }, σ xz = 2G ϵ xz = G γ A.5a, b E G Young x Nx σ xx da = EA u, Mx A A z σ xx da = EI γ w = EI ϑ, Vx A σ xz da = Gk t A γ A.6a, b, c A.6c k t 1 γx Poisson [17] k t = 61 + ν 7 + 6ν, k ν t = ν A.7a, b [17] / 2 A A.2 A.3 A.4 δu x = δu + z δϑ, δu z = δw, δϵ xx = δu + z δϑ, 2δϵ xz = δϑ + δw Bernoulli-Euler l { σ xx δϵ xx + 2σ xz δϵ xz dv = σ xx δu + z δϑ + σ xz δϑ + δw } da dx V 0 A A.6 = l { N δu + M δϑ + V δϑ + δw } dx = [N δu + M δϑ + V δw] l l { N δu + M V δϑ + V δw } dx 0 = n i [N δu + M δϑ + V δw] x=0,l 0 l 0 { N δu + M V δϑ + V δw } dx A.8

97 A Bernoulli-Euler w ϑ l 0 p δu + m δϑ + q δw dx, [F i δu + C i δϑ + S i δw] A.9a, b x=0,l A.8 A.9 Timoshenko [n i N F i δu + n i V S i δw + n i M C i δϑ] l 0 x=0,l [ N + p δu + M V + m δϑ + V + q δw ] dx = 0 A.10 N + p = 0, V + q = 0, M V + m = 0 A.11a, b, c m { } { } { } u = ui n i N = F i, w = wi n i V = S i, ϑ = ϑi n i M = C i A.12a, b, c n i 4.26 u i Bernoulli-Euler 4.25 ϑ ϑ γ w 2 A.1.3 wx A.6 A.11 A.12 A.3 γ ϑ EI w + q α t q = 0 A.13 A.12 { w = w i n i EI w α t q } = S i, A.14a w α t w q { + α t = ϑ i n i EI w α t q } = C i A.14b EI α t EI Gk t A = l2 α t, α t E Gλ t 2, λ t l I/k ta A.15a, b, c λ t Timoshenko k t Poisson λ t G α t 0 α t 0

98 680 A. TIMOSHENKO 4.88 wx = P 12EI x3 + Pl2 16EI x + P 2Gk t A x A.16 γx γx = P 2Gk t A A Bernoulli-Euler q wx = q 24EI x x l x 2 lx l 2 q + x l x 2Gk t A A.18 γx = q Gk t A w shear x x l 2 A.19 w shear x = x 0 γξ dξ A.18 2 A.20 P wx = P 12EI x3 + Pl 16EI x2 + P 2Gk t A x A.21 γx 3 w shear x q wx = q 24EI x2 l x 2 q + x l x 2Gk t A A.22 1 Bernoulli-Euler 2 w shear x wx = 1 l 3 x2 3l 2x + 12α t x l 2x l x l α t A.23 γ = 12α t l α t, V = 12EI l α t = 12EI l α t α t A.24a, b α t 12α t/ 1+12α t A.2 A.2.1 A.11 0 = l 0 { δw V + q + δϑ M V } dx

99 A A.12 = l { V δϑ + w + M δϑ } l dx q δw dx [ ] S 1 δw 1 + C 1 δϑ 1 + S 2 δw 2 + C 2 δϑ A.3 A.5 l { Gkt Aγ δγ + EIϑ δϑ } l dx q δw dx [ ] S 1 δw 1 + C 1 δϑ 1 + S 2 δw 2 + C 2 δϑ 2 = A.25 A.1.2 A.3 l 0 { Gkt Aγ δγ + EI w + γ δ w + γ } dx A.26 A A.25 γ ϑ 1 A.3 w 2 γ, w w i, ϑ i A.26 2 γ γ 1 2 γ 1 γx = w 2 w 1 l + ϑ 1 + ϑ 2, wx = 1 x xx l w 1 + ϑ 1 + x 2 l 2l l w xl x 2 + ϑ 2 2l A.26 Gk t A l Symm. Gk ta 2 EI l + Gk tal 4 Gk ta l Gk t A 2 Gk t A l Gk ta 2 EI l + Gk tal 4 Gk t A 2 EI l + Gk tal 4 A.27 Timoshenko Bernoulli-Euler G A.27 G 5.25 A.27 [44] 2 w A.13 4 w 3

100 682 A. TIMOSHENKO γ γx = γ 0, A.28a wx = w 1 ψ 1 x w ψ 2 x + w 2 ψ 3 x w 2 ψ 4x = w 1 ψ 1 + ϑ 1 γ 0 ψ 2 + w 2 ψ 3 + ϑ 2 γ 0 ψ 4 = w 1 ψ 1 x + ϑ 1 ψ 2 x + w 2 ψ 3 x + ϑ 2 ψ 4 x + γ 0 ψ 5 x A.28b ψ n 5.22 ψ 5 x ψ 2 x ψ 4 x = x 3 x2 l + 2 x3 l 2 γ A.26 2 γ A.28 w γ 0 2 γ 1 A.28 A.25 S 1 C 1 S 2 C q 1 q 2 q 3 q 4 q 5 = k b h h t h 5 k b 5.23a 5.25 q i i= b h 5 h h h 1 h 2 h 3 h 4 t, q 5 h n EI l 0 l 0 ψ n ψ 5 dx n = 1, 2, 3, 4, h 5 Gk t Al + w 1 ϑ 1 w 2 ϑ 2 γ 0 q ψ 5 dx, A.29a, b l 0 EI ψ 5 2 dx A.29c, d q 5 5 γ 0 A.12 γ γ 0 5 γ 0 5 γ 0 = q 5 h t u h 5 u w 1 ϑ 1 w 2 ϑ 2 t 4 S 1 C 1 S 2 C 2 + q 1 q 2 q 3 q 4 = k u + h γ 0 = k u + h q 5 h t u h 5

101 A S 1 w 1 C 1 S 2 C 2 + q T = k t ϑ 1 w 2 ϑ 2 A.30 { } q T q T i, q T i q i q 5 h i, A.31a, b h l 3 l 2 l 3 l 2 k t k b h ht h 5 = EI α t Symm α t l α t l 2 l 12 6 l 3 l α t l A.31c A.15b α t G α t 0 α t = 0 A.31c 5.25 A.23 A.24 12α t/ 1+12α t A.30 A.13 A.14 A.30 G MEMORANDUM H

102 684 A. TIMOSHENKO : 1 m A.1 r Gesundheit! God bless you!?

103 B B.1 B dp 0 = dx i e i B.1 [26] C G i dp = dp j e j = G i δ ji + u j dx i e j = dx i G i x i δ ji + u j e j x i e i e i G i 3.10 ds 2 ds 0 2 = G i G j e i e j dxi dx j = G i G j δ i j dxi dx j 2 E i j dx i dx j B.2 B.3 E i j E i j 1 2 Gi G j δ i j B.4 E Green B.2 G i B.4 E i j 1 2 ui x j + u j x i + u k x i u k x j B.5 685

104 686 B E 11 E 11 = 1 2 G1 2 1 = 1 2 { 1 + ϵ } ϵ E 12 E 12 = 1 2 G 1 G 2 cos G 1 G 2 = ϵ ϵ 22 sin ϵ 12 + ϵ 21 ϵ B.1.2 Green 5.64 S ji δe i j dv X i δu i dv F i δū i ds = 0 B.6 V V S S ji 2 Piola-Kirchhoff G j G i S ji = S ji G i i B.7 1, 2, 3 x, y, z B.1.3 Timoshenko [43, 49] A A z e x x B sine cosine A.2 u x x, z = ux + z sin ϑx, B.8a e z B n G x ϑ Λx, z Γx, z u z x, z = wx + z {cos ϑx 1} B.8b ϑx G z ϑx B.1 Timoshenko 1 G x x Λx, z {ϑx Γx, z} Γ tan Λ 0 x = tan {ϑx Γ 0 x} = w x 1 + u B.9 x 1 -

105 B.2. BERNOULLI-EULER 687 x 0 Λ 0 x, Γ 0 x Λx, z Γx, z z = 0 B.9 B.8 B.5 E xx = 1 2 g 1, 2 E xz = g sin Γ = g 0 sin Γ 0 = γ B.10a, b g G x 2 g = 1 + ϵ + z κ 2 + γ 2, g 0 gz = 0 = 1 + ϵ 2 + γ 2 = 1 + u 2 + w 2, B.11a, b ϵ g 0 cos Γ 0 1, γ g 0 sin Γ 0, κ ϑ B.11c, d, e B.9 B.11 u = 1 + ϵ cos ϑ + γ sin ϑ 1, w = 1 + ϵ sin ϑ + γ cos ϑ B.12a, b cos Λ = 1 + ϵ + zκ cos ϑ γ + zκ sin ϑ, sin Λ = B.13a, b g g cos Γ = 1 + ϵ + z κ g, sin Γ = γ g B.14a, b B.2 Bernoulli-Euler B.2.1 Bernoulli-Euler Γ 0 ϑ θ B.8 u x x, z = ux + z sin θx, u z x, z = wx + z {cos θx 1}, Λx, z = Λ 0 x = θx B.15a, b, c θx B.9 tan θx = w x 1 + u x B.16 θ u, w B.11 g = 1 + ϵ + z κ, ϵ = g0 1, κ = θ = 1 g 0 { 1 + u w w u } B.17a, b, c B.16 B.17 u = 1 + ϵ cos θ 1, w = 1 + ϵ sin θ B.18a, b

106 688 B. B.4 B.10a B.11a g G x 2 e G x e G x gx = g 1 B.19 B.17a e = ϵ + z κ B.20 Bernoulli-Euler! B.17c x ξ, ζ ζ = ζξ ξ s 2 ξ = x + ux, ζ = wx B.17c κ = θ x = dθx dx ξ, ζ κ = κ dθs ds κ dθξ dξ 1 + = 1 + u 1 d 2 ζ 2 dζ dξ dξ = u κ 1 d 2 ζ 2 dζ dξ = ζ 2 d2 ds 2 dξ 1 = 2 dζ 1 + dξ x 12 x X x 3/2 d 2 ζ dξ 2 B.2.2 B.10 B.6 E xz B.19 S xx δe xx dv = g Sxx δe dv V V B.21 E xx G x B.7 x S xx g σ g S xx B.22

107 B.2. BERNOULLI-EULER 689 σ δe dv V B.23 B.20 δe = δϵ + z δκ B.23 N δϵ + M δκ dx B.24 x N A σ da, M A z σ da B.25a, b B.16 B.17 δϵ = cos θ δu sin θ δw, δκ = δθ B.16 δθ = 1 g0 cos θ δw + sin θ δu x p δu + q δw dx F δu + S δw + C δθ x=0,l B.26 B.24 Euler N cos θ + M g0 sin θ + p = 0, B.27a N sin θ + M g0 cos θ + q = 0 B.27b u = n i N cos θ + M sin θ = F i, B.28a g0 w = n i N sin θ + M cos θ = S i, B.28b g0 θ = n i M = C i B.28c n i 4.26 B.2.3 B.23 σ e σ = E e = E ϵ + z κ B.29

108 690 B. E Young N = EA ϵ, M = EI κ B.30a, b B.17 ϵ, κ u, w B.2.4 P cr 6.23 u = P P x, w = 0, θ = 0, ϵ = EA EA, N = P, M = 0 6 ux := Px + ux, wx := wx etc. B.31a, b EA P cr [43, 49] [95] P cr l 2 = λ2 π 2 1 EI 2 1 λ λ B.32 P cr l 2 EI = π2 4 B b - B.32 π B.32 π2 4 Timoshenko B.32 Taylor 2 B.33 B.2.5 Elastica Bernoulli- Euler b c Π B.26 1 Π 2 E e2 dv p u + q w dx F u + S w + C θ V x x=0,l B.34

109 B.2. BERNOULLI-EULER 691 B.30 Π Euler B.27 B Elastica 6.86 ϵ = Lagrange P B.34 Π V 1 2 E e2 dv p u + q w dx F u + S w + C θ P ϵ dx x x=0,l x 1 δπ = {N + P δϵ + M δκ} dx + ϵ δp dx p δu + q δw dx F δu + S δw + C δθ = 0 x=0,l x Euler {N + P cos θ + M g0 sin θ} + p = 0, x x { N + P sin θ + M g0 cos θ} + q = 0, ϵ = 0 B.35 { } u = n i N + P cos θ + M sin θ = F i, g0 { } w = n i N + P sin θ + M cos θ = S i, g0 θ = n i M = C i Euler B.30a g 0 = 1, N 0 B.36a, b P cos θ + M sin θ + p = 0, B.37a P sin θ + M cos θ + q = 0, 1 + u 2 + w 2 = 0 B.37b, c u = n i P cos θ + M sin θ = F i, B.38a w = n i P sin θ + M cos θ = S i, B.38b θ = n i M = C i B.38c N P P sin θ = w, cos θ = 1 + u, κ θ = w 1 + u = u w B.39a, b, c

110 692 B. Lagrange P ϵ = 0 ϵ kk = 0 u k x k = 0 B.40 p Hooke Navier-Stokes µ 2 u i x j x j + p x i + X i = 0 B.41 p Hooke p Hooke ν 1 / 2 K B.3 Timoshenko B.3.1 Timoshenko B.19 e B.10 B.6 S xx δe xx + 2 S xz δe xz dv = V V g Sxx δe + S xz δγ dv G z B.7 z S xz B.22 x S xx g σ g S xx, τ S xz B.42a, b σ δe + τ δγ dv B.43 V B.11a δe = cos Γ δ ϵ + z κ + sin Γ δγ B.43 N δϵ + M δκ + V δγ dx B.44 N σ cos Γ da, M A x A z σ cos Γ da, V τ + σ sin Γ da B.45a, b, c A

111 B.3. TIMOSHENKO 693 B.44 B.45 σ σ cos Γ τ σ σ sin Γ B.45 B.9 B.11 δϵ = cos ϑ δu sin ϑ δw g 0 sin Γ 0 δϑ, δκ = δϑ, δγ = sin ϑ δu + cos ϑ δw + g 0 cos Γ 0 δϑ Bernoulli-Euler B.26 B.44 Euler N cos ϑ + V sin ϑ + p = 0, B.46a N sin ϑ + V cos ϑ + q = 0, M g 0 V cos Γ 0 N sin Γ 0 = 0 B.46b B.46c B.11 B.46c M 1 + ϵ V + γ N = 0 B.47 u = n i N cos ϑ + V sin ϑ = F i, B.48a w = n i N sin ϑ + V cos ϑ = S i, B.48b ϑ = n i M = C i B.48c n i 4.26 B.3.2 B.43 σ e τ γ σ = E e, τ = G γ B.49a, b E G Young B.11 B.45 Bernoulli-Euler B.30 B.14 cos Γ 1, sin Γ γ 1 + ϵ B.50a, b

112 694 B. B.50b ϵ 1 B.49 B.50 B.45 1 N = EA ϵ, M = EI κ, V = Gk t A γ + N γ 1 + ϵ B.51a, b, c k t A.7 B.51c 2 B.45c B.51a B.51c V = Gk t A γ B.52 2 B u = 1 + ϵ cos ϑ + γ sin ϑ 1, w = 1 + ϵ sin ϑ + γ cos ϑ, B.53a, b ϑ = M EI, ϵ = N EA, γ = V Gk t A + N B.53c, d, e 1 + ϵ N cos ϑ + V sin ϑ + p = 0, B.54a N sin ϑ + V cos ϑ + q = 0, M 1 + ϵ V + γ N = 0 B.54b B.54c ϵ ϵ ϵ ϵ 1 Elastica B.53a B.53b B.53e B.54c ϵ γ = V Gk t A + N, M V + γ N = 0 B.55a, b 1 2 B.53e B.55a B.52 B.3.4 Timoshenko u = P P x, w = 0, ϑ = 0, ϵ =, γ = 0, N = P, V = 0, M = 0 EA EA

113 B Bernoulli-Euler P cr = ζ 1 β 2 ζ 2 1 β 2 B.56 + α t ζ π 2 ζ P cr l 2, β 1 EI λ B.57a, b β λ α t A.15 ζ = π 2 /4 1 + α t π 2 /4 B.58 Engesser B.51c B.52 2 ζ = 1 1 π 2 β 2 α t 2 β 2 α t B.59 ζ 5 B.59 B.32 B.56 E = 47 Gk t 15 2 [95] αt π ζ = 2 1 B.60 2 α t 2.5 B.33 B.58 Engesser B.60 mod-engesser 5 λ 10 B.2 Engesser Engesser B.59 [95] B.56 1 [43] B.51c B.56 B.2 10 mm 160 mm Poisson ν = 1 / 3 B.4 - B B.11b B.17b ϵ u w 2 B.61 2

114 696 B. 6.31b B.17c κ w B.62 B.30 B.24 p N δϵ + M δκ dx q δw dx { F δu + S δw + C δ w } x=0,l x x B.63 B.63 N = 0, B.64a N w + M + q = 0 B.64b 6.26 u = n i N = F i, B.65a w = n i N w + M = S i, B.65b w = n i M = C i B.65c B.64a B.65a N = P = const EI w P w + q = B Bernoulli-Euer B.63 B.61 B.62 B { } F i = { } K i j u j + { }{ } K i jk u j u k + K i jkl { u j }{ u k }{ u l } B.66

115 B B.1 K i j 11 A/l 2 14 A/l 2 44 A/l I/l I/l I/l I/l I/l I/l I/l I/l I/l I/l 4 B.2 K i jk : Ki jk = A K 2l 2 i jk 122 6/ / / / / / / / / / / / / / / / / / / /15 B.3 K i jkl : Ki jkl = A K 2l 2 i jkl / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /35 { u i } { } u 1 /l w 1 /l w 1 u 2 /l w 2 /l w 2 t, F i F 1/ El 2 S 1 / El 2 C 1 / El 3 F 2 / El 2 S 2 / El 2 C 2 / El 3 t B.67a, b K i j B.1 B.3 K i j 5.39 B.66 K i jk { u j }{ u k } K g i j { u j } K g i j { } u m

116 698 B. 0 c/ El 2 a/ El 2 0 c/ El 2 b/ El 2 6N 0 / 5El 2 N 0 / 10El 2 c/ El 2 6N 0 / 5El 2 N 0 / 10El 2 K g i j = 1 2 2N 0 / 15El 2 a/ El 2 N 0 / 10El 2 N 0 / 30El 2 0 c/ El 2 b/ El 2 B.68 6N 0 / 5El 2 N 0 / 10El 2 Symm. 2N 0 / 15El 2 u2 u 1 N 0 EA l b EA 30, a EA 30 w 1 4w 2 3 w 2 w 1 l 4w 1 + w 2 3 w 2 w 1, B.69a, b l, c EA w 1 10 w 2 12 w 2 w 1 B.69c, d l N 0 0 c/ 2El 2 a/ 2El 2 0 c/ 2El 2 b/ 2El 2 K g i j = 0 6N 0 / 5El 2 N 0 / 10El 2 0 6N 0 / 5El 2 N 0 / 10El 2 0 N 0/ 10El 2 2N 0 / 15El 2 0 N 0 / 10El 2 N 0 / 30El 2 0 c/ 2El 2 a/ 2El 2 0 c/ 2El 2 b/ 2El 2 B N 0/ 5El 2 N 0 / 10El 2 0 6N 0 / 5El 2 N 0 / 10El 2 0 N 0/ 10El 2 N 0 / 30El 2 0 N 0 / 10El 2 2N 0 / 15El 2 1, 4 B P N 0 B u 1, u 2 w w 2 B B.61 B.63 N δϵ + M δκ dx = N δu + Nw δw + M δκ dx x x 1 N N EAu 2 N N = P EAu δu Pw δw + EIw δw dx B.71 x Timoshenko Timoshenko [33] Bernoulli-Euler P l 0 EI ϑ δϑ P w δw + Gk t A γ δγ dx B.72

117 B A w 3 γ γ { } f t { } S 1 l 2 / EI C 1 l/ EI S 2 l 2 / EI C 2 l/ EI, u w 1/ l ϑ w 1 2 / l ϑ 2 t B.73a, b { } { } f = K l Pl2 K nl u EI l K nl K l = Symm α t α t α t, K nl = / 5 1/ / 5 1/ 10 2/ / 10 1/ 30 2 B / 5 1/ 10 Symm. 2/ B.75a, b α t, , 2 2α t + 12α 2 t B.76a, b, c B.5 B Vlasov x s n r 0 t r 0/ t r, ψ r = r 0 n, ds = r 0 dψ z-x E zx = 0, E zz = 0 B.77a, b S nx = 0, S sn = 0, S nn = 0, Sss = 0, s S sx ds = 0 B.78a, b, c, d, e Hooke E nx = 0, E sn = 0, E nn = 0, E ss = 0, s E sx ds = 0 B.79a, b, c, d, e u = u 0 + ξ sin ψ + η cos ψ r cos ψ sin θ + ū cos θ, v = ξ cos ψ η sin ψ, B.80a, b w = w 0 + r cos ψ + ξ sin ψ + η cos ψ r cos ψ cos θ ū sin θ B.80c

118 700 B. s, n, x ξ, η, u θ tan θ = w u 0 B.81 x B.79 ξ = ξ ψ, x n cos α 1 sin α, η = η ψ, x + n cos α 1 cos α 1, ū = n sin α 1, B.82a, b, c ξ = η, tan αψ, x = ξ + η r 0 + ξ η, tan α 1ψ, x = ψ η cos α ξ sin α g0 + θ B.82d, e, f ξ sin ψ + η cos ψ r cos ψ g u w 0 2 B.83 α α 1 ξ = r 0 f x Φ 1 ψ, η = r 0 f x Φ 2 ψ B.84a, b Φ 2 cos 2ψ B.79 Φ 1 Φ 1 = 1 2 sin 2ψ, Φ 2 = cos 2ψ B.85a, b f tan α sin α f ϕ, tan α 1 sin α 1 r 0 f Φ 2, cos α 1, cos α 1 1, ϕ Φ 1 + Φ 2 B.86a, b, c, d, e ux, r, ψ = u 0 x + Z sin θx n r 0 f x Φ 2 cos θx, wx, r, ψ = w 0 x + r cos ψ + Z cos θx + n r 0 f x Φ 2 sin θx, B.87a B.87b ξx, r, ψ = w 0 x sin ψ + f x r 0 Φ 1 n ϕ + Z sin ψ {cos θx 1} + n r 0 f x Φ 2 sin ψ sin θx, B.87c ηx, r, ψ = w 0 x cos ψ + r 0 f x Φ 2 + Z cos ψ {cos θx 1} + n r 0 f x Φ 2 cos ψ sin θx B.87d Z z x z + Z = ξ sin ψ + η cos ψ r cos ψ = r cos ψ + f Z f, Z f r 0 Φ 1 n ϕ sin ψ + r 0 Φ 2 cos ψ B.88a, b u 0, w 0 B.84 Φ i i = 1, 2 f x

119 B B.5.2 Euler f P ss + N f + M f M sx m Z f cos θ m X f sin θ M Z f sin θ + MX f cos θ = 0, B.89a N cos θ + M XZ sin θ + p x + { sin θ m ZZ sin θ + m XZ cos θ f M Z f cos θ f M X f sin θ } = 0, B.89b N sin θ + M XZ cos θ + p z + { cos θ m ZZ sin θ + m XZ cos θ f M Z f cos θ f M X f sin θ } = 0 B.89c f = or B.90a M sx M f + M Z f sin θ M X f cos θ = n i { M Z f cos θ 1 + M X f sin θ + M s f + M n f }, f = or M f = n i M f cos θ M f f sin θ, B.90b u 0 = or B.90c N cos θ + M XZ sin θ + sin θ m ZZ sin θ + m XZ cos θ f M Z f cos θ f M X f sin θ = n i N, w 0 = or B.90d N sin θ + M XZ cos θ + cos θ m ZZ sin θ + m XZ cos θ f M Z f cos θ f M X f sin θ = n i V, θ = or M XZ = n i M XZ cos θ M ZZ sin θ f M f sin θ f M f f cos θ B.90e N σ da, M XZ σ Z da, N f σ θ Z f da, A A A P ss S ss n r ϕ da, M sx S sx {r 0 Φ 1 n A A M f A σ n r 0 Φ 2 da, B.91a, b, c, d ϕ + r 0 r Φ 2 } da B.91e, f X x, X z F s, F n, F x p z X z da, p x X x da, m zz X z z da, m xz X x z da, m Z f X z Z f da, A A A A A m X f X x Z f da, m ZZ X z Z da = m zz + f m Z f, m XZ X x Z da = m xz + f m Xd, A A A M Z f X z n r 0 Φ 2 da, M X f X x n r 0 Φ 2 da, B.92 A A N F x da, M xz F x z da, M X f F x Z f da, M XZ F x Z da = M xz + f M X f, A A A A M f F x n r 0 Φ 2 da, V F s sin ψ + F n cos ψ da, A A M zz F s sin ψ + F n cos ψ z da, M Z f F s sin ψ + F n cos ψ Z f da, A A M ZZ F s sin ψ + F n cos ψ Z da = M zz + f M Z f, M f f F s sin ψ + F n cos ψ n r 0 Φ 2 da, A A M s f F s r 0 Φ 1 n ϕ da, M n f F n r 0 Φ 2 da. A A

120 702 B. B.5.3 f e e ϵ + Z θ n r 0 Φ 2 f, ϵ u { u w 2 } 0 σ = Ee, S ss = E E ss, E ss = f n r 0 ϕ, S sx = 2G E sx, 2E sx = r 0 f Φ 1 n f ϕ + r 0 r Φ 2 B.93a, b B.94a, b, c, d, e E Young, G N = EA M XZ = EI ZZ θ, P ss = EA 1 f, M sx = GI s f, u N f = EI 1 θ 2, M f = EI 2 f, { u w 2 } 0, A I ZZ A 1 A I s A I 1 I 2 A A da = 2π r 0 t, B.95a, b Z 2 da = π r 3 0 t f 3π r3 0 t, B.95c, d A 2 n r ϕ 2 3π t 3 da =, B.95e, f 4r 0 { r 0 Φ 1 n A ϕ + r 0 r Φ } 2 π 2 da = 4 r3 0 t, B.95g, h Z f Z da = 3π r3 0 t 4 n r 0 Φ 2 2 da = π r3 0 t f 5π r3 0 t, B.95i, j 8 B.95k, l f 4 f f f f w 3 B.5.4 x B.89a κ2 r0 4 f = 6 κ2 r0 4 t 2 t 2 B.96 κ θ k κ r2 0 M, m B.97a, b t π E r 0 t 2 B.95 m = k 1 3 f 2 B.98 m M κ = 1 ρ k B.3 ρ 1 Brazier 2 M

121 B B.96 B.98 m = k 1 9 k k 2 B B.3 [79] ξ = k 0 r k2 0 sin 2ψ k2 0 sin 4ψ, B.100a η = k 0 + 2k2 0 cos 2ψ + 3 r k2 0 cos 4ψ + 3, B.100b k 0 1 ν 2 k 2 B.100c B.84 1 B.100 Reissner k 0 f ξ r 0 = 1 2 f sin 2ψ f 2 sin 4ψ, η r 0 = f cos 2ψ f 2 cos 4ψ + 3 B.4 Reissner m max k cr f = k k k 4 32 k 2 B.101 m = k 1 3 f f 2 B.102 B.101 B.102 B.99 B.3 2 B.4 B.6 B method of adjoints [81] 1 B.5

122 704 B. l z 1 N cos ϑ + V sin ϑ l2, z 2 EI z 3 M l EI, z 4 u l, z 5 w l, z 6 ϑ, d d x / l 2 N sin ϑ + V cos ϑ l2, B.103a, b EI B.103c, d, e, f, g ż 1 = q 1, ż 2 = q 2, B.104a B.104b ż 3 = { 1 + β 2 1 α t y1 } y2, B.104c ż 4 = 1 + β 2 y 1 cos z6 + α t β 2 y 2 sin z 6 1, ż 5 = 1 + β 2 y 1 sin z6 + α t β 2 y 2 cos z 6, ż 6 = z 3 B.104d B.104e B.104f α t E k t G, y 1 z 1 cos z 6 z 2 sin z 6, y 2 z 1 sin z 6 + z 2 cos z 6, q 1 p l3 EI, q 2 q l3 EI B.105a, b, c, d, e α t α t 1 ż 3 = y β 2 y α t β 2 y β 2 y 1, B.106a ż 4 = 1 + β 2 y 1 cos z6 + α t β 2 y 2 sin z 6 1, 1 + α t β 2 y β 2 y 1 ż 5 = 1 + β 2 y 1 sin z6 + α t β 2 y 2 cos z α t β 2 y β 2 y 1 B.106b B.106c 3 2 B.104c ż 3 = 1 α t β 2 y 1 y2 B B.106 ż 3 = y α t β 2 y 1, B.108a ż 4 = 1 + β 2 y 1 cos z6 + α t β 2 y 2 sin z α t β 2 y 1 1, B.108b ż 5 = 1 + β 2 y 1 sin z6 + α t β 2 y 2 cos z α t β 2 y 1 B.108c

123 B B [47] [62] [47] L f i F i L 2 / EI S i L 2 / EI C i L/ EI t, di u i/ L w i / L ϑ i t B.109a, b f 1 = T k 1 T t {d 2 d 1 D}, f 2 = T k 2 T t {d 2 d 1 D} B.110a, b T ϑ 1 D cos ϑ 1 sin ϑ 1 0 T sin ϑ 1 cos ϑ , D cos ϑ1 1 ξ sin ϑ 1 ξ t 0, ξ L l B.111a, b, c l Timoshenko B.75 k i = k l i + z 0 k nl i, z 0 L2 = EI { u2 u 1 L cos ϑ 1 1 w2 w 1 cos ϑ 1 + sin ϑ } 1 ξ sin ϑ 1 ξ L ξ β, 2 B.112a, b k l 1 k l 2 ξ β ξ ξ 2 0 6ξ ϕ ξ 0 ξ β ξ 3 6ξ 2 0 Symm ϕ ξ 0, k nl 1, k nl ξ , B.113a, b / ξ ξ , B.113c, d 2/ Symm. ξ 2 0 β I/A L, α E Gk t, ϕ αβ 2 ξ 2, ϕ, , 2 2ϕ + 12ϕ 2 B.114a, b, c, d, e, f

124 706 B. QR 2 EI R I/A = 100 θ = 215 Q R θ O R B.4 B.110 Newton-Raphson k t B.110 d i H 11 H 12 H 13 + S 1 H 14 H 15 H 16 k t = H 21 H 22 H 23 + S 2 H 24 H 25 H + 26 P 1 P 2 cos ϑ 1 sin ϑ 1 g cos ϑ 1 sin ϑ 1 0 B.115 H i H i1 H i2 H i3 H i4 H i5 H i6 = Tk i T t C, S i Qk i T t + Tk i Q t d 2 d 1 D, B.116a, b P i ξtk nl i T t d 2 d 1 D /β 2, B.116c 1 0 sin ϑ 1 /ξ sin ϑ 1 cos ϑ 1 0 C 0 1 cos ϑ 1 /ξ , Q cos ϑ 1 sin ϑ g {u 2 u 1 /L cos ϑ 1 1 /ξ} sin ϑ 1 + {w 2 w 1 /L + sin ϑ 1 /ξ} cos ϑ 1, B.116d, e B.116f n B.110 B.115 n + 1 d 1 d 2 n+1 = d 1 d 2 n + k n t 1 f 1 { Tk 1 T t d 2 d 1 D } n f 2 { Tk 2 T t d 2 d 1 D } n B.117 n d n+1 d n d n+1 <, d d 1 d 2 B.118a, b 2 B.4 [18] B.5 [185] AA BB BB

125 B PL 2 EI A B w L P H PL 2 EI 2 /L O H L = L = 100 I/A B.5 B A w L P /L B.6 B.5 [ d / dx ] Bernoulli- Timoshenko Bernoulli- Timoshenko Euler 2 A 1 B Euler 2 A 1 B u = 1 + ϵ cos ϑ + γ sin ϑ 1, w = 1 + ϵ sin ϑ + γ cos ϑ, ϵ 0 N cos ϑ + V sin ϑ + p = 0, N sin ϑ + V cos ϑ + q = 0 M V 1 + ϵ + Nγ = 0 M V + Nγ = 0 γ = 0 V = Gk t Aγ V = Gk t A + B.32 B.59 B.56 K l +K nl b K l + K nl a K l + K nl b N = EAϵ, M = EIϑ N 1 + ϵ γ γ = 0 V = Gk t Aγ V = Gk t A + N γ B.33 Euler K l +K nl b ϵ = 0 B.60 Engesser K l + K nl a ϵ = 0 B.58 Engesser K l + K nl b ϵ = 0 B , 6.14 [24] [185] B B Bernoulli-Euler ϵ Elastica

126 708 B. 3 [43] - f f 1 f t 2 f = K l + K nl m d m = A, B, K l = K l 1 K l 2 K l 2 t K l 3, Knl m = z m K nl 1 m K nl 2 m K nlt 2 m K nl 3 B.119a, b, c m A B Timoshenko B.5 L K l 1 K l 3 m K nl β ϵ ϵ Ψ Symm. 1 β Symm ϵ 2 6, 1 + ϵ Ψ m 1, K l 2 m K nl 1 0 m ϵ , m 1 + ϵ 1/ 30 + m β ϵ ϵ Ψ ϵ m ϵ 2 m Symm. m K nl ϵ 2/ 15 + m 2, m 1 m ϵ Symm. 1 + ϵ 2/ 15 + m 2 2, B Ψ, Ψ α t 1 + ϵ, 2 m Ψ m = A m 2 m = B, B.121 m 2 2Ψ 24Ψ 2 m = A 2Ψ + 12Ψ 2 m = B, m 3 2Ψ + 48Ψ 2 m = A m 2 m = B, m Ψ 2 m = A 1 m = B z ϵ z 1 β 2 { u2 u 1 L cos ϑ 1 1 w2 w 1 cos ϑ 1 + sin ϑ } 1 sin ϑ 1, ϵ = zβ 2 B.122a, b ξ L ξ Bernoulli-Euler α t = 0 B

127 B P P C 1 F 1 S 1 β H a S 1 F 1 β k C 1 B.7 b H B.7 - B.7.1 : [9] B.7 B.7 a F 1 = P H sin β, S 1 = H cos β, C 1 = l + wl H cos β + wl P H sin β B.123a, b, c tan β x = 0 w0 = 0, w 0 = 0, S 1 = EI w 0 F 1 w 0, C 1 = EI w 0 x = l S 2 = EI w l F 1 w l = H cos β, C 2 = EI w l = 0 B.124a, b β = 0 x = l B.124a S 2 wl = 0 x = 0 wl H B.123 C 1 2 N H P F 1 - F1 P H sin β wx = a + bx + c sin µx + d cos µx, µ EI = B.125a, b EI a d H a + d = 0, b + µc = 0, µ 2 b = H cos β, c sin µl + d cos µl = 0, B.126a, b, c, d EI dµ 2 = Hl EI cos β + µ2 {a + bl + c sin µl + d cos µl} B.126e H µ µ 0 det = 0 det 0 sin µl cos µl = sin µl cos µl sin µl cos µl = sin µl cos µl 0 sin µl cos µl 1 0 sin µl cos µl + 1 B.127

128 710 B. P µ wx = a [ ] sin µl µx cos µl sin {µ l x} sin µl W W wl = a sin µl µl cos µl sin µl B.129 B.128 B.8 3 µl O wx W µl = 6 µl = µl = 4 1 B.8 - wl = 0 B.129 W = 0 sin µl µl cos µl tan µl µl = 0 B.130 x l µ H µ 2 = F 1 EI = P EI H EI sin β H H = 0 cos µl = 0 sin µl = 0 cos µl = 0 sin µl = 0 H 0 H H sin β N µ 2 = P EI H EI sin β P EI B.131 B.123 B µ 0 det = µl cos µl sin µl = sin µl cos µl 0 l 0 1 B.130 B.123 wl = 0 B B.9 1,000 Elastica P cos β cos β P B.7 a P U U ul 16

129 B Pl 2 EI 20 0 β = 10 β = β = 0 β β = 15 β = 0.5 P cos β β + β β = 0.5 U l 0.5 z l O 0.1 β = 0.5 β = 15 l 2 P cos β EI = 26.4 l 2 P cos β EI = β = 0.5 P cos β x l B.9 B Pl 2 EI 0 β = 0.1 c a f b g h β = 15 h 20.8 z l 0.4 a Pl2 EI = 41.5 b 31.1 c N0l 2 EI β = d e β = U l B.11 β = O 0.5 x g 20.7 l d 97.3 f e 97.4 B.12 1 β = ±0.5 β β = 0 B.130 µl 4.493, Pl2 EI β 0 β H β > 0 β 10 β B.10 β = B a g h B.12 B.11 N0 P N a c, f, g, h β = 30 B.13 a h i, j B

130 712 B. 60 Pl 2 EI a h i g f b U l B.13 β = 30 e c β = 30 N0l 2 d EI j β = 30 c 11.6 d 11.6 z l O 0.5 j 0.02 b 28.3 e 11.7 f 23.1 g 34.7 h 52.0 i 12.6 x l a Pl2 EI = 23.2 B.14 B.13 N0 P N H B.131 a j 1 a b j B.14 B.8 µl Pl2 EI 20 i j Pl2 EI 20 a h 16 β = 30 B B.130 B.6 B.15 a d B.16 B.14 n B.6 µ n l P n l 2 EI B.14 a i 1 B.14 h B.16 d 2 cos β P N β B β 90 β > 88 β = 75, 80 β 88 P Pl2 EI β = 75

131 B Pl 2 EI β = 30 z l 0.6 b 24.0 a Pl2 EI = a b c d c U l B.15 β = 30 d O x 0.6 β = 30 l B.16 Pl 2 EI U l 0.4 β = P 75 N 20 Pl 2 EI 10 λ = β P cos β l cos β U l B.17 β 0 β = U l B.18 β N 75 N H N cos β P cos75 = N U l cos β 2 β 88 B.17 B.18 Elastica λ 10,000 B.130 Pl2 EI B.18 10,000

132 714 B. - B.127 e a A 0 1 µ sin µl cos µl 1 0 sin µl cos µl + 1, { } y 0 0 e, b A = { y } B.132a, b, c c d p.243 Alternative Theorem B.132 A t { v } = { 0 } { v } = t { y } t { v } 0 B.132 B.123 wl B.128 wx = e [ sin µl µx cos µl sin {µ l x} ], µl cos µl sin µl a e sin µl wl = 0 : B.7 b F 1 = P H sin β, S 1 = H cos β, C 1 = l H cos β + wl P H sin β B.133a, b, c x = 0 w0 = 0, k w 0 = EI w 0 = C 1, S 1 = EI w 0 F 1 w 0 x = l S 2 = EI w l F 1 w l = H cos β, C 2 = EI w l = 0 B.134a, b B.134a x = 0 B.125 H a + d = 0, k b + µ c + EI µ 2 d = 0, c sin µl + d cos µl = 0, B.135a, b, c dµ 2 = lµ 2 b 2 + µ 2 {a + bl + c sin µl + d cos µl} B.135d α s α s µ µ 2 l det 0 0 sin µl cos µl 1 0 sin µl cos µl + 1 = 0 det α s α s µ µ 2 l 0 sin µl cos µl 0 sin µl cos µl = sin µl cos µl sin µl cos µl = 0 α s k EI α s kl EI B.136

133 B C 1 F 1 S 1 a P e tanx a x l a = 0.25l l a H 4 a = 0.1l b P a = 0 c P B.19 O π Pl 2 x = EI B.20 µ k wx = a { α α s sin µl s sin µl µl cos µl + µl 2 sin µl } W wl W = 0 α s µl cos µl sin µl µl2 sin µl = 0 B.137 α s sin µl = 0 α s B.130 B.137 µ H β = 0 wl B.133 B α s α s µ µ 2 l det = 0 α s 0 0 sin µl cos µl µl cos µl sin µl µl2 sin µl = 0 0 l 0 1 B.137 B.7.2 B.19 a F 1 = P, S 1 = H, C 1 = ep l + a H x = 0 w0 = 0, w 0 = 0, S 1 = EI w 0 + P w 0, C 1 = EI w 0

134 716 B. x = l S 2 = EI w l P w l = H, C 2 = EI w l = 0 a d H e 1 cos µx sin µl + sin µx µx cos µl P µl cos µl wx = e sin µl µl 1 + a, H = l cos µl sin µl µl 1 + a cos µl l l x = l µ 2 P EI B.138a, b sin µl µl cos µl W wl = e sin µl µl 1 + a cos µl l B.139 a = 0 wl = e B.138 B.139 tan µl µl 1 + a = 0 l a = 0 a 0 B.20 a = 0 µl a 0 0 < µl < π / 2 B.7 a l B.7 µl Pl 2 EI a / l = e / a = tan 2.3 = , e / l = , B.21 W wl U ul + a B.7 B.20 Pl 2 EI Pl2 EI e A P P < 0 B.19 b c 1 W B.139 B.22 B.21 - B.21 Pl2 EI 0.5 A W W > 0 B U U = l / 2 B Pl2 EI 20.19

135 B Pl 2 EI U W Pl 2 EI 10 U ul + a W wl B.139 B A B 0 a l W P U U 0 l, W l B.21 e 0.2 W 0.1 l 0.5 B.22 O B.19 a W W > 0 B.21 W > 0 Take care! Take it easy! TV

136 718 B. B E : 1 a-meter un-measurable 2 60

137 C C.1 [26] C.1 g i i = 1, 2 g j P P = P 1 g 1 + P 2 g 2 P 2 g 2 O 2, g 2 α u P 1 + P 2 cos α sin α P 1 g 1 C.1 P P 2 g 2 1, g 1 P 1 g 1 C.1 x i 1 y i 2 P x y P x, P y P x = P 1 + P 2 cos α, P y = P 2 sin α P u W W = u P = u 1 + u 2 cos α P 1 + P 2 cos α + u 2 sin α P 2 sin α W = u P = u 1 + u 2 cos α P 1 + u 2 + u 1 cos α P 2 C.2 C.2 u 1 u 1 + u 2 cos α, u 2 u 2 + u 1 cos α C.3a, b W = 3 u i P i = u i P i i=1 C.4 719

138 720 C u P C.3 g i i = 1, 2 C.1 u = u 1 g 1 + u 2 g 2 C.5 g i g i C.2 W = u 1 P 1 + P 2 cos α + u 2 P 2 + P 1 cos α C.6 P 1 = P 1 + P 2 cos α, P 2 = P 2 + P 1 cos α C.7a, b C.3 P P = P 1 g 1 + P 2 g 2 C.8 g i 1 2 sin α g i 1 = sin α, gm g n = δ m n C.9a, b δ m n Kronecker W = u i g i P j g j = ui P j g i g j = u i P j δ i j = u i P i = u i P i C.10 g i ξ i x i x 1 = ξ 1 + ξ 2 cos α, x 2 = ξ 2 sin α, C.11a, b C.11 g i = x j ξ i i j C.12 g 1 = i 1, g 2 = cos α i 1 + sin α i 2 C.9 g i C.2 g i j g i g j, g i j g i g j C.13a, b

139 C C.9b u i g i = u j g j u i g i g k = u j g j g k u i δ i k = u j g jk u k = u j g jk C.14 e i jk i jk 123 e i jk = e i jk = 1 i jk C C.15 C 1 1 C 1 2 C 1 3 C C 2 1 C 2 2 C 2 3 C 3 1 C 3 2 C 3 3 c c det C = C i 1 C j 2 Ck 3 e i jk = C 1 i C2 j C3 k ei jk, e lmn c = C i l C j m C k n e i jk, c = 1 6 Ci l C j m C k n e i jk e lmn C.16a, b, c u, u w = u u w k = u i v j e i jk = e ki j u i v j C.17a, b ϵ i jk g det g i j, 1 g det g i j C.18a, b ϵ i jk = g e i jk, ϵ i jk = 1 g e i jk C.19a, b Kronecker ϵ i jk ϵ imn = δ j m δ k n δ j n δ k m, ϵ i jk ϵ i jn = 2δ k n, ϵ i jk ϵ i jk = 6 C.20a, b, c g = 1 C.3 u u = u i g i g j ξ j u ξ j = u i g i ξ j = ui ξ j g i + u i g i ξ j

140 722 C. C.12 g i ξ j = g i, j = 2 x k ξ i ξ j i k g i, j = Γ i jk g k = Γ k i j g k C.21 Γ i jk Christoffel C.21 u ξ j = u i g i ξ j = u i, j g i + u i g i, j = u i, j + uk Γ i jk gi = u i j g i C.22 u i j C.4 C.1 g i C.9 u = u i g i u i ξ 1 = r, ξ 2 = θ, ξ 3 = z [26] g r = i 1 cos θ + i 2 sin θ, g θ = i 1 r sin θ + i 2 r cos θ C.23a, b g θ g rr = 1, g θθ = r 2, g rr = 1, g θθ = 1 r 2 C.24a, b, c, d Christoffel Γ rθθ = r, Γ θθr = r, Γ r θθ = r, Γθ rθ = 1 r C.25a, b, c, d σ ji j + f i = 0 σ rr r σ rθ r σ rz r + σrr r + 3σrθ r + σrz r + σrθ θ + σθθ θ + σzθ θ rσ θθ + σrz z + σθz z + σzz z + f r = 0, C.26a + f θ = 0, C.26b + f z = 0 C.26c f r, f θ g r, g θ g r g θ C.23 τ rr σ rr, τ rθ rσ rθ, τ θθ r 2 σ θθ, τ zz σ zz, τ rz σ rz, τ θz rσ θz, q r f r, q θ r f θ, q z f z C.27a, b, c, d, e, f, g, h, i

141 C τ rr r τ rθ r τ rz r + 1 r τ rθ θ + 2τrθ r + τrr τ θθ r + 1 r + τrz r + 1 r τ θθ θ τ zθ θ + τθz z + τrz z + qr = 0, C.28a + q θ = 0, C.28b + τzz z + qz = 0 C.28c u r r = ur r, ur θ = ur θ r uθ, u θ r = uθ r + 1 r uθ, u θ θ = uθ θ + 1 r ur C.29a, b, c, d u v r u r, v θ r u θ C.30a, b u r r = vr r, ur θ = vr θ vθ, u θ r = r v θ r + 1 r 2 vθ, u θ θ = 1 v θ r θ + 1 r vr C.31a, b, c, d ϵ i j u r r = g rr u r r = vr r, u r θ = g rr u r θ = vr θ vθ, { ϵ rr = vr 1 r, 2ϵ rθ = r r u r θ + vθ r 1 r vθ C.23 ε i j u θ r = g θθ u θ r = r vθ r, u θ θ = g θθ u θ θ = r 2 } 1, ϵ θθ = r 2 v θ r θ + 1 r vr g r = i 1 cos θ + i 2 sin θ, g θ 1 = i 1 r sin θ + i 1 2 r 1 v θ r θ + 1 r vr C.32a, b, c, d C.33a, b, c cos θ C.34a, b ε rr = ϵ rr, ε rθ = 1 r ϵ rθ, ε rr = 1 r 2 ϵ θθ C.35a, b, c ε rr = vr r, ε rθ = r u r θ + vθ r 1 r vθ, ε θθ = 1 v θ r θ + 1 r vr C.36a, b, c z ε zz = vz z, ε rz = 1 v r 2 z + vz, ε θz = 1 v θ r 2 z + 1 r v z θ C.37a, b, c σ i j = C i jkl ϵ kl C.38

142 724 C. C τ i j = C i jkl ε kl C.39 [101] headquarter: 1 s quarters StarTrek 1 Close. choice He don t know nothing, you know! 1 tm Paramount Pictures.

143 D D.1 1 R 2 γ D.1 U 0 σ2 max E E Young b 2b O σ ρ E σ max σ z/ b D.1 2a σ max D.2 2γ 2 U 0 = 2γ σ max E γ b γ E b / 100 E = 210 GN/m 2, b 2 Å = 0.2 nm, γ 2 J/m 2 σ max GN/m 2 b + z b σ tip σ max E / 10 E /

144 726 D. I II III D.3 Griffith Orowan Inglis D.2 σ max σ tip σ tip a = σ max ρ a ρ σ tip 2σ max a ρ E γ b ρ b σ max E γ 4a 2a 5000 b γ E b / 100 σ max E ρ 0 D.2 D.2.1 D.3 III D.4 x y > 0 y < 0 O y r θ D.4 III x

145 D ϵ zx = 1 2 u 0, v 0, w z = 0 w x 0, ϵ yz = 1 w 2 y 0 Hooke σ zx σ yz σ rz σ θz σ yz y = 0 = σ θz y = 0 = 0, σ yy y = 0 = σ θθ y = 0 = 0 wy = 0 = 0 x w = 0 along θ = 0, C.28 r σ rz r w θ + σ θz θ Hooke µ σ rz = 2µ ϵ rz = µ w r, = 0 along θ = π D.1a, b = 0 σ θz = 2µ ϵ θz = µ 1 r w θ D.2a, b w r w w r r r θ = w = 0 D.3 D.1 w D.3 w A r p f θ θ θ = 0 r r = 0 D.3 p 2 r p 1 f + r p 1 f = 0 θ f θ D.1 f + p 2 f = 0 f = a sin p θ + b cos p θ f 0 = 0, f π = 0 D.4

146 728 D. b = 0, cos p π = 0 f θ sin p θ, p = ± 1 2, ±3 2, f sine w x D.2 D.4 w = A r p sin p θ, σ rz = A µ p r p 1 sin p θ, σ θz = A µ p r p 1 cos p θ a U U 1 µ π a π 0 1 σ 2 2 rz + σθz 2 π r dθ dr = 2 A2 µ p a 2p p 1 U < 0 a 2 0 U p 1 2 U < w = w 0 + A r 1 /2 sin θ / 2 + O r 3 /2, σ rz = 1 2 A µ r 1 /2 sin θ / 2 + O r 1 /2, σ θz = 1 2 A µ r 1 /2 cos θ / 2 + O r 1 /2 r = 0 σ A µ r 1 /2 gθ r 1 / 2 r 0 µ θ A r 1 / 2 A A A = A cr D.5 A cr

147 D D.2.2 I II σ rr r Hooke + 1 r σ rθ θ σ rθ = 2µ ϵ rθ, σ rr = µ + 1 r σ rr σ θθ = 0, σ rθ r + 1 r σ θθ θ + 2 r σ rθ = 0 3 κ κ 1 ϵ θθ + κ κ κ 1 ϵ rr, σ θθ = µ κ 1 ϵ rr + κ + 1 κ 1 ϵ θθ κ ϵ rr = u r r, ϵ θθ = u r r + 1 u θ r θ, ϵ rθ = 1 1 u r 2 r θ + u θ r u θ r r u r = r p f θ, u θ = r p gθ D I 2a 1 D.5 σ rr σ θθ = K f i θ i σ rz g i θ, i=i, II, 2πr σ h i θ θz = K iii 2πr σ rθ K i, i=i, II, III K i K ii K iii = πa σ yy σ xy σ yz pθ qθ σ yy σ yy σ xy 2a σ xy σ yz σ yz y r θ D.5 x D.5 K i = K ic, i=i, II, III K ic σ cr σ cr = K ic πa D.6

148 730 D. D.6 D.6 D.6 TMCP D.3 Dugdale-Barenblatt??? Hankel

149 E E.1 E [139] x V x V δ ξ V V : 2 ux + px = 0 in V, E.1a E.1 u = ū or ν u = f on V E.1b : x = ξ 2 Gx; ξ + δx ξ = 0, with G 0 as x E.2 Gx; ξ E.1 V ν Gx; ξ = gx; ξ on V E.3 gx; ξ E.1.2 E.1a E.2 G 0 = V G { 2 ux + px } dx 731

150 732 E. 1 Gauss 2 0 = G u,kk + p dx = G ν k u,k dx G,k u,k dx + V V V { = G νk u,k G,k ν k u } dx + G,kk u dx + p G dx V E.2 2 { 0 = G νk u,k G,k ν k u } dx δx ξ u dx + p G dx V V V { = G νk u,k G,k ν k u } dx uξ + p G dx V V V V V p G dx uξ = V px Gx; ξ dx + {Gx; ξ ν ux ux ν Gx; ξ} dx V E.1b E.3 uξ = px Gx; ξ dx + {Gx; ξ f x ūx gx; ξ} dx V V E.4 G g p f, ū E.4 u ū E.4 ξ V ūξ ξ V 2 ū ū Gx; ξ as x ξ ξ V E.4 2 [139] 1 2 ūξ = px Gx; ξ dx + {Gx; ξ f x ūx gx; ξ} dx on ξ V V V Cauchy Gx; ξ = Gξ; x, gx; ξ = gξ; x x ξ 1 2 ūx = Gx; ξ pξ dξ + {Gx; ξ f ξ gx; ξ ūξ} dξ on x V E.5 V ū V E.2 E.5 E.4

151 E ū E.5 N x = x i V i = 1, 2,, N ū i ūx i ūx = N ū i ϕ i x i=1 E ūi = 1 2 ūx i = Gx i ; ξ pξ dξ + Gx i ; ξ f ξ dξ V V N j=1 V gx i ; ξ ū j ϕ j ξ dξ E.6 A i j V gx i ; ξ ϕ j ξ dξ, f i V Gx i ; ξ pξ dξ + Gx i ; ξ f ξ dξ V E.6 N j=1 A i j δ i j ū j = f i, i = 1, 2,..., N E.7 ū i E.3 E ūx + gx; ξ ūξ dξ = Gx; ξ pξ dξ V V on x V E.8 f E.1 p f u E.8 p ū ū p ū p 1 ū p H eigen back stress E.8 pξ ūx pξ E Ω pξ in ξ Ω pξ = 0 in ξ Ω E real time real time

152 734 E. E ūx + gx; ξ ūξ dξ = V Ω Gx; ξ pξ dξ on x V E.10 Ω Ω t pξ Ω [139] E.8 Fourier E.1 Machinac, Michigan 1957 m PhD. Medical Doctor Doctor of Music? PhD PhD PhD BS Bullshit MS More Shit PhD Piled High and Deep bullshit chicken

153 F 1 F.1 Northwestern Achenbach Viscoelasticity γ τ σ γ = 1 η σ F.1 η Hooke 3.45 γ = 1 µ σ F.2 Gt µ F.2 F σ0 t = 0 Heaviside σt = σ0 Ht F.3 γt γt = σ0 Jt F.4 735

154 736 F. 1 Jt Gt G g J g O ψt ϕt 1/ η G r t O a b F.1 t Jt Jt = J g + t η + ψt F.5 J g 3 ψt 2 F.1 a t σt + t Ht + t F.4 γt = σ0 Jt + σt + t Jt t F.6 t N N γt = σ0 Jt + σt + n t Jt n t n=1 F.7 N γt = σ0 Jt + t 0+ Jt s dσs = σ0 Jt + t 0+ dσ s Jt s ds ds F ϵ 0 < ϵ 0 2 Kelvin Jt ψt Kelvin 1 9 Kelvin F.1 F.2 σ = µ γ + η γ γ + 1 τ γ = 1 η σ F Kelvin-Voigt

155 F τ τ η µ F.10 t = 0 σ 0 σt = σ 0 Ht F.9 γ0 = 0 γt = 1 µ σ 0 { 1 exp t } τ F.11 F.4 Jt Jt γt σ 0 = 1 µ { 1 exp t } τ F.12 Kelvin µ 0 η 0 Jt = 1 µ 0 + t η µ { 1 exp t } τ F.13 F.5 1, 2 F γ0 t = 0 γt = γ0 Ht F.14 σt σt = γ0 Gt F.15 Gt Gt = G r + ϕt, ϕ0 = G g G r F.16a, b F.16a 1 G r 2 ϕt F.16b G g F.1 b t γt + t Ht + t t t dγ σt = γ0 Gt + Gt s dγs = γ0 Gt + s Gt s ds ds F.17

156 738 F. 1 2 Maxwell Kelvin Maxwell γ = σ µ + σ η τ F.10 σ + 1 τ σ = µ γ τ η µ F.18 F.19 t = 0 γ 0 γt = γ 0 Ht F.20 F.18 σ + 1 τ σ = µ γ 0 δt F.21 δt Dirac Ht δt F.21 t = 0 σt = µ γ 0 exp t Ht τ F.22 F.21 F.22 Ht F.21 F.15 Gt Ht Gt σt γ 0 = µ exp t τ Maxwell µ r Gt = µ r + µ exp t τ F.23 F.24 F.16a 1

157 G 1 G.1 1 G a A x t ux, t x qx, t q x rx, t x = 0 dx A x = a A qx, t A qx + dx, t x A dx rx, t G.1 = c ρ A dx u G.1 t c ρ G.1 = A dx rx, t + A qx, t A qx + dx, t = = c ρ A dx u t dx 0 A rx, t A rx, t qx + dx, t qx, t dx qx, t x = c ρ ux, t t = c ρ A u t Fourier qx, t = κ ux, t x κ G.3 G.2 x κ ux, t + rx, t = c ρ x G.2 G.3 ux, t, 0 < t, 0 < x < a G.4 t 739

158 740 G. 1 x ux, t t = k 2 ux, t rx, t + x 2 c ρ G.5 k k κ c ρ G.6 2 G.5 : t = 0 ux, 0 = f x G.7 f x t = 0 x : x = 0 x = a 3 Dirichlet 1 u0, t = T l t, ua, t = T r t G.8a, b T l t T r t Neumann 2 q0, t = κ u 0, t = 0, x u qa, t = κ a, t = 0 G.9a, b x x q0, t = κ u x 0, t = Q lt, qa, t = κ u x a, t = Q rt Q l t Q r t x Robin 3 Newton κ u x 0, t = h l {u0, t v l }, κ u x a, t = h r {ua, t v r } G.10a, b h l h r, v l, v r Fourier I

159 G G.1.2 Northwestern Olmstead 1980 Differential Equations of Mathematical Physics 1 k = 1 t t 2 vx, t vx, t + = 0, < x <, t > 0 G.11 x 2 t vx, 0 = f x, x f x, t < G.12a, b Erutarepmet vx, t pp G.2 1 Northwestern Dundurs 1980 Elasticity G.2 a T l 0 T l T r = 0 T l = 0 g Hooke T l T l O T l u u a g T r = 0 x x σ xx = E ϵ xx ϵ T, ϵ T x, t = α ux, t G.13a, b O x ϵ T u E Young α G.2 T l g T 0 ϵ T a = g, ϵ T = α T l T l = T 0 g a α G.14 T l G.2 ux, t = T l 1 x ϵ T = α T l 1 x a a G.13 ϵ xx = σ xx E + α T l 1 x a 1!?

160 742 G. 1 x σ xx = p ϵ xx = p E + α T l 1 x a g g = a 0 ϵ xx dx = p a E + a α T l 2 T 1 p = 0 p a E = a α T 1 g = 0 T 1 = 2g 2 a α = 2 T 0 G.15 G.14 T 0 2 T 0 < T l < T 1 = 2 T 0 T l = T 0 0 T 0 Ut g << a u t = k 2 u x 2, pt a E = α a 0 u dx g 0, 0 < x < a, 0 < t G.16a, b u0, t = T 0 + Ut, ua, t = 0, ux, 0 = T 0, U0 = 0, U = T 0 G.17a, b, c, d, e pt

161 G No I 1 p

162 744 G. 1 No quarter letter-size 100 p A4 4? B5 A4 21 cm 25 mm

163 G No NU quarter TTh MWF MTWF T , Th PhD quarter Freshmen 17 Sophomores 28 Juniors 29 Seniors quarter summer session 3 11 quarter take-home exam p I 3 quarter B average 80 quarter NU No offense GP P/N GPA NU university police NU

164 Dundurs Gasper 24 take-home exam Dundurs i? prerequisite

165 747 letter size 1 quarter 6 21 / 30 Asymp. and Pertur. Meth. in Appl. Math. Prof. Davis Continuum Theory of Fracture Prof. Keer Diff. Eqs and Math. Physics I Prof. Olmstead Diff. Eqs and Math. Physics II Prof. Olmstead Diff. Eqs and Math. Physics III Prof. Olmstead Elasticity I Prof. Dundurs Elasticity II Prof. Dundurs Fourier Series and BVP Assist. Prof. Mahar Mechanics of Continua I Prof. Nemat-Nasser Mechanics of Continua II Prof. Nemat-Nasser Mechanics of Fracture Prof. Achenbach Plasticity Prof. Mura Plasticity Prof. Nemat-Nasser Seminar in Micromechanics Prof. Mura Viscoelasticity Prof. Achenbach Wave Propagation Prof. Achenbach Technological Institute Deering Music Library Weber Arch Northwestern Evanston 8 Sheridan Road 6 OHP PhD A [70] Northwestern Evanston

166 748 G m energy, data, memory, mail, main and sale: 1

167 H H.1 E 1 E 2 E H.1 26 GFRP, CFRP Young E 1, E 2 H.1 749

168 750 H. 1 Voigt 2 E 1, E 2 H.2 ϵ σ = 1 f σ 1 + f σ 2, σ 1 = E 1 ϵ, σ 2 = E 2 ϵ H.1a, b, c f Young σ = E ϵ E 1 f E 1 + f E 2 Voigt H.2a, b E 1 E 1 E 2 E 2 σ ϵ ϵ σ H.2 Voigt Reuss Young H.2 σ ϵ = 1 f ϵ 1 + f ϵ 2, ϵ 1 = σ E 1, ϵ 2 = σ E 2 H.3a, b, c Young ϵ = σ 1 f E + f 1 E E 1 E 2 Reuss H.4a, b Voigt H.2b Reuss Poisson? Young H.8 f [70] Micromechanics 2 H.8 Mori-Tanaka Hill self-consistent [37] SC H.3.1 [70] H.2 H.2.1 Eshelby 2 2 H.3 1 Ω Young Poisson E m, ν m E i, ν i m i a 1, a 2, a 3 1 averaging [154] homogenization m eigen

169 H Eshelby [25] σ H.3 ϵ = in Ω H.5 Ω E m Ω E i H.3 Eshelby Ω ϵ = const. σ eigen [70] eigen Ω [25] H H.4 D Ω u ϵ ϵ i j = 1 2 ui, j + u j,i H.6 σ i j = C m i jkl ϵ kl = C m i jkl u k,l in D Ω, σ i j = C i i jkl ϵ kl = C i i jkl u k,l in Ω H.7a, b Hooke C m C i 2 3 H.6 C k i jkl Lamé µ k, λ k = C k i jlk k = m, i C k i jkl = µ k δik δ jl + δ il δ jk + λk δ i j δ kl, k = m, i H.8 δ i j Kronecker Lamé Young Poisson µ k = E k ν k, λ ν k E k k =, k = m, i H.9a, b 1 + ν k 1 2ν k σ ji, j = 0, σ i j = σ ji σ i j, j = 0 H.10a, b, c H.10c H.10b H.10a n j σ ji = f i at x H.11

170 752 H. σ 0, ϵ 0 σ 0, ϵ 0 C i Ω C m ϵ d Ω ϵ d C m = = C i C m ϵ d C mϵ Ω C m H.4 ; C m C i C m eigen ϵ n f Ω u ν σ ν Ω 2 Ω σ d, ϵ d σx = σ 0 + σ d x, ϵx = ϵ 0 + ϵ d x, σ d x 0, ϵ d x 0 as x H.12a, b, c, d σ 0 f 2 eigen H.4 σ 0 = C m : ϵ 0 σ d ϵ d Eshelby H.4 C m Ω ϵ eigen d eigen ϵ H.2.4 eigen e eigen ϵ i j x = e i j x + ϵi j x, ϵ i j 0 in Ω H.13 H.6 Hooke σ i j = C m i jkl e kl = C m i jkl ϵkl ϵkl = C m i jkl uk,l ϵkl H.14 H.6 H.13 C m i jkl = Cm i jlk H.8 m H.10c H.14 u C i jkl u k,l j = C i jkl ϵ kl, j H.15

171 H n j σ ji = 0 at x H.16 H.14 H.16 u n j C i jkl u k,l = n j C i jkl ϵ kl = 0 at x [ ϵ x = 0 ] H.17 H.2.3 Fourier 1 Fourier Fourier Fourier eigen ϵ Fourier Fourier ϵ Fourier ϵi j x = ϵ i jξ exp i ξ x dξ ϵ i jξ = 1 2π 3 u i x = H.18 H.20 H.15 C i jkl u k ξξ l ξ j exp i ξ x dξ = ϵi j x exp i ξ x dx H.19 u i ξ exp i ξ x dξ C i jkl ϵ klξi ξ j exp i ξ x dξ 3 eigen Fourier H.18 H.20 C i jkl ξ l ξ j u k = i C i jkl ϵ kl ξ j H.21 H.15 Fourier u K ik u k = X i H.22 K ik C i jkl ξ l ξ j, X i i C i jkl ϵ kl ξ j Fourier u k = K ik 1 X i = N kiξ Dξ X i H.23 N i j K i j D N i j = 1 2 ϵ ikl ϵ jmn K mk K nl, D = 1 6 ϵ i jk ϵ lmn K il K jm K kn H.24a, b ϵ i jk Fourier u i x = i C jlmn ϵ mnξ ξ l N i j ξ D 1 ξ exp i ξ x dξ H.25

172 754 H. ϵ i j x = 1 { C klmn ϵ 2 mnξ ξ l ξ j N ik ξ + ξ i N jk ξ } D 1 ξ expi ξ x dξ, [ ] σ i j x = C i jkl C pqmn ϵ mnξ ξ q ξ l N kp ξ D 1 ξ expi ξ x dξ ϵkl x H.26 H.27 2 Green H.19 H.25 { 1 u i x = i dξ C jlmn dx ϵ 2π mnx exp i ξ x } ξ 3 l N i j ξ D 1 ξ exp i ξ x G G i j x x 1 2π 3 dξ N i j ξ D 1 ξ exp { i ξ x x } u i x = C jlmn ϵmnx Gi j x x dx x l H.28 H.29 H.30 G eigen G Green Green G C i jkl G km,l j x x + δ im δx x = 0 H.31 H-1 1 δ im Kronecker 2 δ Dirac δx c δx 1 c 1 δx 2 c 2 δx 3 c 3 X C i jkl u k,l j + X i = 0 H.32 H.31 Green G G km x x x x m x x k H.30 ϵ i j x = 1 { C m klmn 2 ϵ mnx 2 G ik x x + 2 G jk x x } x l x i x l x j [ σ i j x = C m i jkl C m pqmn ϵmnx 2 G kp x x ] dx + ϵkl x q x x l Green dx, H.33 H.34 3 Dξ = µ 2 m λ m + 2µ m ξ 6, N i j ξ = µ m ξ 2 { λ m + 2µ m δ i j ξ 2 λ m + µ m ξ i ξ j }, H.35a, b G i j x x = 1 4πµ m δ i j x x 1 16πµ m 1 ν m 2 x i x j x x, ξ 2 ξ k ξ k H.35c, d 3 Green

173 H : x 3 x 2 11 b O eigen x 3 b x 1 b O H.5 b Burgers H.5 ϵ23 = 1 2 b H x 1 δx 2 x 2 x 3 x 1 x 1 H.36 eigen Hx Heaviside Fourier ϵ 23 = b δξ 2 8π 2 i ξ 1 H.37 H.25 u 1 = 0, u 2 = 0 u 3 = i 2 ξ ξ2 2 b ξ 8π 2 2 exp {i ξ 1 x 1 + ξ 2 x 2 } dξ 1 dξ 2 = b x i ξ 1 2π tan 1 2 x 1 2 x 1 x 3 b H.38 3 Eshelby Eshelby Ω eigen H.33 ϵ ϵ i j x = S i jkl x ϵ kl H.39 4 Sx Ω Sx =, x Ω Poisson a i i = 1, 2, 3 S Eshelby [70] S i jkl = α 1 { 1 3 δ i j δ kl + β 2 } 1 δik δ jl + δ il δ jk 3 δ i j δ kl = α A i jkl + β B i jkl H.40 α A B 1 + ν m 3 1 ν m, β 2 4 5ν m 15 1 ν m H.41a, b A i jkl 1 3 δ i j δ kl, B i jkl 1 2 δik δ jl + δ il δ jk 1 3 δ i j δ kl H.42a, b A i jmn B mnkl = 0 H.43

174 756 H. σ 0, ϵ 0 σ 0, ϵ 0 C i Ω C m ϵ C m Ω C m H.6 C m i jkl = 3κ m 1 3 δ i j δ kl + 2µ m { 1 1 δik δ jl + δ il δ jk 2 3 δ i j δ kl κ m Lamé } = 3κ m A i jkl + 2µ m B i jkl H.44 κ m λ m µ m H.45 A B I I i jkl = 1 1 { 1 3 δ } 1 i j δ kl + 1 δik δ jl + δ il δ jk 2 3 δ i j δ kl = 1 A i jkl + 1 B i jkl H.46 4 S = α, β, C m = 3κ m, 2µ m, I = 1, 1 H.47a, b, c H.43 4 S I = α 1, β 1, C m 1 α β S =, 3κ m 2µ m H.48a, b H.2.4 H.6 1 Eshelby [25] Ω Ω eigen ϵ eigen eigen H.4 σ 0, ϵ 0 σ, ϵ σ 0 = C m : ϵ 0 H.49 4

175 H H.10c H.16 Hooke σ 0 i j + σ i jx = C m i jkl ϵ 0 kl + ϵ klx in D Ω, σ 0 i j + σ i jx = C i i jkl ϵ 0 kl + ϵ klx in Ω H.50a, b Eshelby [25] ϵ i j = in Ω H.51 H.4 H.6 2 H.50 σ 0 i j + σ i jx = C m i jkl ϵ 0 kl + ϵ klx ϵkl H.52 eigen ϵi = 0 in D Ω j 0 in Ω H.53 Ω eigen ϵ Ω H.50b H.52 Ω C i i jkl eigen H.39 eigen ϵ 0 kl + ϵ kl = C m i jkl ϵ 0 kl + ϵ kl ϵkl in Ω H.54 ϵ kl = S klmn ϵ mn H.55 H.54 C i i jkl ϵ 0 kl + S klmn ϵmn = C m i jkl ϵ 0 kl + S klmn ϵmn ϵkl { C i i jkl S klmn C m i jkl S klmn I klmn } ϵ mn = C m i jkl Ci i jkl ϵ 0 kl ϵ i j = { C i i jmn S mnkl C m i jmn S mnkl I mnkl } 1 C m klpq C i klpq ϵ 0 pq H.56 eigen eigen H.56 H.55 H.52 H-1 1. G H.31 Dirac 1 Fourier δx x = 1 2π 3 exp { i ξ x x } dξ H µ i = 10 µ 1 / 10 Poisson ν i = m ν m = 0.3 σ σ 12 σ 0 12 σ 0 kk + σ kk σ 0 kk

176 758 H. H.3 H.3.1 H.2 1 H.2 2 H.7 C m C m H.2 C i ϵ ϵ m Hooke H.7 σ m = C m : ϵ m H.58 ϵ m m H.58 1 H.2 Hooke σ i = C i : ϵ i H.59 γ i ϵ i = ϵ m + γ i H.60 i eigen ϵ i σ i = C i : { } ϵ m + γ i = C m : { ϵ m + γ i ϵ } i H.61 Eshelby γ i = S : ϵ i H.62 eigen H.62 H.61 C i : { ϵ m + S : ϵ i } = C m : { ϵ m + S I : ϵ i }

177 H ϵ i = { C m C m C i S } 1 C m C i : ϵ m = { C m C m C i S } 1 C m C i C m 1 : σ m H.63 eigen H.58 H.63 H.56 H.62 H.61 H.58 σ i = C m : { ϵ m + S I : ϵ i } = σ m + C m S I : ϵ i H.64 f VΩ V H.65 V V σ σ 1 σ dv + 1 σ dv = V Ω V V Ω V D Ω V σ VΩ i + σ V m = f σ i + 1 f σ m Voigt ϵ H.66 ϵ f ϵ i + 1 f ϵ m H.67 Reuss H.1 Voigt H.3 Reuss H.58 H.62 H.63 H.67 ϵ = C m 1 : σ m + f S : ϵ i H.68 H.63 H.64 H.66 H.68 σ m, σ i, ϵ i, ϵ m ϵ = [ C m C m C i {S f S I} ] 1 [ C m 1 f C m C i S ] C m 1 : σ H.69 ϵ m 5 C 1 Eq.H.69 C 1 : σ H.70 C [ [ C m C m C i {S f S I} ] 1 [ C m 1 f C m C i S ] C m 1 ] 1 H.71 4 H.48 [74] E m = 2.81 GN/m 2 ν m = E i = 223 GN/m 2 ν i = f =

178 760 H. E E m = 1.72 GN/m 2, 60 GN/m 2 ν m = 0.45 Voigt 40 E i = 70.2 GN/m 2, SC ν i = Reuss 0 Mori-Tanaka f 1 a Young 80 E E m = 3.01 GN/m 2, ν m = E i = 76 GN/m 2, ν i = 0.23 GN/m 2 40 Voigt SC 20 Reuss Mori-Tanaka f ν 0.4 Mori-Tanaka Reuss 0.5 ν 0.4 Reuss Mori-Tanaka SC 0.3 Voigt SC Voigt f f 1 b Poisson H.8 x 3 Eshelby [70] H.71 Voigt C GN/m 2 H.72 H.3.2 [80, 86] H.8 Voigt Reuss Mori-Tanaka Reuss Hill self-consisten [37] SC f Hill self-consistent Hill self-consistent : Hill self-consistent

179 H H.71 µ = µ m + f µ i µ m µ µ + 2 S 1212 µ i µ, κ = κ m + f κ i κ m κ κ S ii j j κ i κ H.73 Eshelby Poisson 2 S 1212 = 2 4 5ν m 15 1 ν m, 1 3 S i ji j = 1 + ν m 3κ 2µ, ν = 3 1 ν m 2 µ + 3κ H.74 µ f = 1, µ m 1 2 S 1212 κ κ m = 1 f S ii j j H.75a, b Poisson ν = 7 + 5ν m 6 f 1 + ν m D 5 {2 3 f 1 ν m } D 7 5ν m 2 6 f 19 56ν m + 45νm f ν m + 49νm 2 H.76 H.77 f 50% f = π / Poisson 0.5 µ i = 0, ν i = 0.5, κ i =, k κ i κ m 0 H.78a, b, c, d H.9 f Poisson f = 0.5 f = 0.6 ν = 0.5, µ = 0, κ = 5k κ m 3 + 2k H.79a, b, c 0.3 κ ν m = 0.3 κ m 0.2 k = 10 5 k = k = 0.02 k = f 0.6 H.9 κ i 0.14 MN/m 2 k 10 6 k 10 5 self-consistent f = 0.5 H-2 3. H.63 H.69 H.8 µ i µ m = 10 Poisson ν m = ν i = A f B 2. B 1 f A

180 762 H. 2 A B 6 µ f Hill self-consistent H Hashin Shtrikman [35] H [113, 182, 196] σ m = C m : ϵ m ϵ p m, σ i = C i : ϵ m + γ i ϵ p i H.80a, b ϵ p H.80a H.80b σ i = C i : { C m 1 : σ m + γ i ϵ p i } H.81 ϵ p i ϵ p i ϵ p m H.82 σ i = C i : { C m 1 : σ m + γ i ϵ p i } = C m : { C m 1 : σ m + γ i ϵ p i + ϵ i } H.83 Eshelby γ i = S : ϵ p i + ϵ i H.84 eigen ϵ ϵ = C 1 : σ + F : ϵ p m + G : ϵ p i H.85 C F, G [113, 182] C F 1 f I, G f I H.86a, b 2 [92] SiC SiC Mises SiC x 1 -x 3 x 2 x 3 60 a 1 = a 2 a 3/ a1 = % E m = 60 GN/m 2, ν m = 0.3, E i = 450 GN/m 2, ν i = 0.2 σ y m = 700 MN/m 2 Mises 6

181 H σ 3 MN/m p = 7 GN/m κ MN/m 2 Hashin-Shtrikman self-consistent κ 1 = 2000 MN/m 2, ν 1 = 0.3 κ 2 = 300 MN/m 2, ν 2 = principal axis O p = 0 GN/m 2 σ 1 p = 7 GN/m 2 σ f 2 H.10 H.11 H.10 p 2 3 Voigt H.8 Hill self-consistent self-consistent 3 2 [48] A B C C A B 2 C κ 1 κ 2 f 1, f 2 κ m {1 f 1 + f 2 } 3 f 1 + f 2 1 κ κ = 2 i=1 2 i=1 f i κ i κ m κ m κ i α f i κ m κ m κ i α H.87 α β Poisson H.41 κ m ν m

182 764 H. κ m κ = f 1 κ 1 + f 2 κ 2 H.88 H.2b Voigt κ m 0 κ = f1 + f 1 2 H.89 κ 1 κ 2 H.4b Reuss Voigt Reuss Voigt Reuss 1 κ m = κ 1, ν m = ν 1 H.87 f 2 1 κ 2 κ κ 1 = 1 κ 1 1 f 1 1 κ 2 α κ 1 H Hashin Shtrikman [35] Hashin Shtrikman 3 2 κ m = κ 2, ν m = ν 2 H-2 3 H.11 Hill self-consistent 1 Hashin Shtrikman H.87 [48] ϵ σ Hashin Shtrikman Hill self-consistent f 2 Hashin Shtrikman 3 Hill self-consistent f 2 Hashin Shtrikman 3 Hill self-consistent Voigt 3 3 [56, 140] σ m = C m : ϵ m, σ i = C i : { ϵ i ϵ p i } H.91a, b ϵ i = ϵ m + γ i σ i = C i : { ϵ m + γ i ϵ p i } H.92 σ i = C m : [ ϵ m + γ i { ϵ p i + ϵ i } ] H.93

183 H eigen γ i = S i : { ϵ p i + ϵ i } H.94 Eshelby eigen ϵ i H.66 H.67 ϵ = C 1 : σ + 2 i=1 f i P i C 1 M i : ϵ p i H.95 C [140] H.95 2 H.12 Young Poisson E 1 = GN/m 2 ν 1 = 0.32 E 2 = GN/m 2 ν 2 = O σ 11 GN/m 2 Mori-Tanaka Ju[52] 3 σ ϵ H.12 34% a 3/ a1 = 1000 a 2 = a 1 Mises f J 2 F ϵ p 1 = 0 ϵ 11 H.96 F ϵ p 1 F ϵ p 1 = 1 3 σy 1 + h 1 ϵeq n H.97 σ y 1 = MN/m2 h 1 = MN/m 2, n 1 = 0.6 ϵ eq 1 ϵ eq 1 2 ϵ p 1 : ϵ p 1 dt H.98 history [52] 3 [56] H.13 E 1 = 3.16 GN/m 2 ν 1 = 0.35 E 2 = 73.1 GN/m 2 ν 2 = 0.18 Mises σ y 1 = MN/m2, h 1 = MN/m 2, n 1 = 0.26

184 766 H. σ 11 GN/m 2 σ 11 GN/m Mori-Tanaka 0.14 Mori-Tanaka Tandon and Weng [93] Doghri and Ouaar [21] Tandon and Weng [93] Doghri and Ouaar [21] ϵ p a 15% b 35% H.13 ϵ p 11 H.1 FRP 2007 m Guess what!, Tell you what and Chances are...:

185 I Fourier I.1 I ut ODE ut = a ut t ut = c exp a t c a a ut ut = A u, ut = v 1 t v 2 t v 3 t t, A = 3 3 I.1a, b, c u0 = α = α 1 α 2 α 3 t I.2 3 ODE u 3 2 I.1.2 ODE a I.1 A λ e A e = λ e I.3 767

186 768 I. FOURIER λ λ i i = 1, 2, 3; λ i λ j i j e i i = 1, 2, 3 i A e i = λ i e i I.4 I.4 e t j e t j A e i = λ i e t j e i j i e t i A e j = λ j e t i e j A e t i A e j = e t j At e i = e t j A e i e t j A e i = λ j e t i e j = λ j e t j e i 0 = λ i λ j e t j e i λ i λ j e t i e j = 0, i j I.5 I.1.3 I.1 I.2 ut = ut = I.6 I.1 3 c i t e i i=1 3 ċ i t e i i=1 I.6 3 ċ i t e i = A I.4 i=1 3 c i t e i = i=1 3 c i t A e i i=1 3 ċ i t e i = i=1 3 c i t λ i e i i=1 n e t n I.5 n n ċ n t = λ n c n t

187 I c n t ODE c n t = k n exp λ n t I.7 k n u ut = 3 k i e i exp λ i t i=1 I.8 I.2 k i I.8 I.2 u0 = 3 k i e i = α i=1 k i e t n I.5 k n k n e t n e n = e t n α k n = et n α e n 2 ut = 3 e t i α e e i 2 i exp λ i t i=1 I.9 : A = , α = A 5 λ λ λ e = 0 5 λ 0 0 det 0 5 λ λ = 5 λ {5 λ 2 4} = 5 λ3 λ7 λ = 0 λ 1 = 5, λ 2 = 3, λ 3 = 7 λ i e 1 = t, e 1 2 = 1, e 2 = t, e 2 2 = 2, e 3 = t, e 3 2 = 2

188 770 I. FOURIER I.6 c i t ċ i t = λ i c i t c 1 t = k 1 exp5t, c 2 t = k 2 exp3t, c 3 t = k 3 exp7t k 1 = et 1 α e 1 = = 6, k 2 = et 2 α e 2 = = , k 3 = et 3 α e 3 = 7 8 = ut = 6 exp5t e exp3t e exp7t e 3 3 ODE I.2 1 I.2.1 G ux, t a x κ ux, t + rx, t = c ρ x ux, t, 0 < t, 0 < x < a I.10 t κ c ρ rx, t t = 0 ux, 0 = u 0 x I.11 0 < x < a 3 u0, t = T l t, ua, t = T r t I.12a, b T l t T r t q0, t = 0, qa, t = 0 u u 0, t = 0, x a, t = 0 I.13a, b, c, d x Newton q0, t = κ u x 0, t = h l u0, t v l, qa, t = κ u x a, t = h r ua, t v r I.14a, b h l h r, v l, v r rx, t 0 κ ρ, c I.10 ux, t = k 2 ux, t x 2 k k κ c ρ I.15 I.16

189 I I.2.2 I.1 ODE e i A ut = A ut 3 u c i t e i i=1 A e n = λ n e n I.15 A k 2 x 2 k d2 dx 2 f nx = λ n f n x ut = A ut ux, t = k d2 ux, t dx 2. A e n = λ n e n k d2 f n x dx 2 u = = λ n f n x 3 c n t e n ux, t = c n t f n x n=1 n I.17 I.11 I.15 0 I.17 u 0, t = 0, ua, t = 0 I.18a, b x k f n x = λ n f n x I.19 x f n x I.18 f n0 = 0, f n a = 0 I.20a, b λ n < 0 f n x I.19 f n x = A n sin ξ n x + B n cos ξ n x, ξ n λ n /k.

190 772 I. FOURIER I.20 ξ A n = 0, A n sin ξ n a + B n cos ξ n a = 0. A n = 0, B n cos ξ n a = 0. 2 B n = 0 0 cos ξ n a = 0 I.21 ξ n a = 2n 1 2 π ξ n = 2n 1π 2a B n 0 I n 1πx 2n 1π f n x = cos, λ n = k, n = 1, 2,, I.23a, b 2a 2a I.19 i f i x = 1 k λ i f i x f j x 0 a a 0 f j f i dx = 1 a k λ i f j f i dx 0 I.20 f j f i a a a f 0 j f i dx = f j f i dx 0 0 a f j f i dx = 1 a 0 k λ i f j f i dx 0 j f i x a f i f j dx = 1 a 0 k λ j f i f j dx = 1 k a λi λ j f i f j dx 0 I.23b λ i λ j i j a 0 f i x f j x dx = 0 f i, f j = 0 i j I.24 I.24

191 I I.2.3 I.1 ODE PDE ux, t = c i t f i x I.25 I.15 i=1 ċ i t f i x = k i=1 i=1 i=1 c i t f i x I.19 k c i t 1 k λ i f i x = c i t λ i f i x c i t ċ i t f i x = c i t λ i f i x i=1 f n x I.24 n ċ n t f n, f n = λ n c n t f n, f n ċ n t = λ n c n t i=1 i=1 I.25 c n t c n t = A n exp λ n t I.26 ux, t 2 2n 1π 2n 1πx ux, t = A n exp t k 2a cos 2a n=1 I.23b λ i A n I.27 I.11 ux, 0 = A n f n x = u 0 x. f i x a a A n f i f n dx = f i u 0 dx A i f i, f i = f i, u 0 A i = f i, u 0 f i, f i n= f n, u 0 2n 1π 2n 1πx ux, t = f n, f n exp t k 2a cos 2a a 2n 1πx f n, u 0 = u 0 x cos dx, f n, f n = 2a 0 n=1 n=1 ux, t = 2 a 2n 1πη u 0 η cos a 2a n=1 0 dη exp a 0 2n 1πx cos 2 dx = a 2a 2 k 2n 1π 2a 2 t cos 2n 1πx 2a I.27 I.28 I.29a, b 0 I.30 I.30

192 774 I. FOURIER u 0 x = u 0 = const. a 2n 1πη f n, u 0 = u 0 cos dη = 2au 0 1 n+1 2a 2n 1π 0 ux, t = 4u 0 1 n+1 2 2n 1π 2n 1πx 2n 1π exp t k 2a cos 2a n=1 40 I.1 k t / a 2 = x = a ux, t u O 0.2 k t a 2 = x 1 a I.1 x = 0 I.2.4 ux, t t = k 2 ux, t x 2 I.31 u0, t = T l, ua, t = T r I.32a, b T l T r I.11 { vx, t ux, t T l + T r T l x } a I.33 vx, t v0, t = u0, t T l = 0, va, t = ua, t T r = 0 I.34a, b { T l + T r T l x } v u a vx, t t = k 2 vx, t x 2 vx, 0 = ˆ f x, { f ˆx f x T l + T r T l x } a vx, t v n x v n x = A n sin ξ n x + B n cos ξ n x, ξ n λ n /k I.35 I.36a, b I.34 B n = 0 A n sin ξ n a = 0 2 sin ξ n a = 0 ξ n a = nπ ξ n = nπ a

193 I nπx nπ 2 v n x = sin, λ n = k I.37a, b a a v vx, t ux, t f ˆ ξ, v n ξ ux, t = v n η, v n η exp λ n t v n x + n=1 { T l + T r T l x } a I.38 I.2.5 ux, t t = k 2 ux, t x 2 + Fx I-1 1. ux, t t = 2 ux, t x 2, 0 < x < 1, 0 < t u0, t = 0, u1, t = 0 ux, 0 = 1, 0 x < 1 / 2 +1, 1/ 2 x 1 t = 0, , , 0.01, < x < 1, 0 < t u t = 2 u x 2 + cos x u0, t = 2, u1, t = 3, ux, 0 = 5 t = 0, , , 0.01, < x < 1, 0 < t u t = 2 u x + sin x, u 0, t = 0, u1, t = 0, ux, 0 = cos 7πx 2 x I.3 1 I üx, t = c 2 u x, t + gx, t I.39 ux, t c gx, t T m c = T m x ux, t x ux, t I.2

194 776 I. FOURIER t 2 ux, 0 = u 0 x, ux, 0 = v 0 x I.40a, b I.2 a u0, t = 0, ua, t = 0 I.41a, b I.2 u 0, t = 0, ua, t = 0 I.42a, b t := c t I.43 üx, t = u x, t + gx, t I.44 I.3.2 gx, t 0 f n x = λ n f n x f n 0 = 0, f n a = 0 λ n < 0 f n x = A n sin ξ n x + B n cos ξ n x, ξ n λ n B n = 0 A n sin ξ n a = 0 2 sin ξ n a = 0 ξ n a = nπ ξ n = nπ nπ 2 nπx a, λ n =, fn x = sin a a I.3.3 ux, t = nπx c n t sin a n=1 I.45 I.44 nπx c n t sin = a n=1 nπ 2 cn t sin a n=1 nπx a

195 I sin jπx 0 a I.24 a jπ 2 c j t = c j t a ODE c j t = A j sin I.43 jπct c j t = A j sin a n=1 jπt jπt + B j cos a a + B j cos jπct a nπx { nπct nπct } ux, t = sin A n sin + B n cos a a a I.46 I.46 I.40 ux, 0 = nπx B n sin = u 0 x, ux, 0 = a n=1 n=1 nπc nπx A n sin = v 0 x a a f j x jπc B j f j x, f j x = u 0 x, f j x, A j f j x, f j x = v 0 x, f j x a A j = a v 0 x, f j x jπc f j x, f j x = 2 jπc a 0 v 0 x sin jπx a dx, B j = u 0x, f j x f j x, f j x = 2 a a 0 u 0 x sin jπx dx a : 2x u 0 0 < x < a a 2 u 0 x = u 0 2 2x, v a 0 = 0 a 2 < x < a ux, t u 0 = 8 π n=1 sin nπ / 2 n 2 nπx nπct sin cos a a ux, t 1 u 0 O I.3 tc a = x a I I.3.4 gx, t 0 I.45 I.39 sin nπx c n t sin = a n=1 n=1 nπc 2 nπx cn t sin + gx, t a a jπx 0 a a jπc 2 c j t f j x, f j x = c j t f j x, f j x + gx, t, f j x a

196 778 I. FOURIER f j x, f j x = a / 2 c j t = p j t jπc 2 c j t + g a j t, g j t 2 a c j t = A j sin ux, t ux, t = nπx { sin A n sin a n=1 a 0 gx, t sin jπct jπct + B j cos + p j t a a nπct a + B n cos nπct a jπx dx a } + p n t I.47 I.47 I.40 ux, 0 = nπx B n + p n 0 sin = u 0 x, ux, 0 = a n=1 n=1 { nπc } nπx A n + ṗ n 0 sin = v 0 x a a f j x B j + p j 0 { jπc } f j x, f j x = u 0 x, f j x, A j + ṗ j 0 f j x, f j x = v 0 x, f j x a A j = a { 2 a jπx } v 0 x sin dx ṗ j 0, B j = 2 jπc a a a I a 0 u 0 x sin 2 ux, t t 2 = 2 ux, t x 2 + x 1 x, 0 < x < 1, 0 < t u0, t = 0, u1, t = 0 ux, 0 = sin x, I.4 jπx dx p j 0 a ux, 0 t = cos x 2 3 = 2 ux, t + 2 ux, t + 2 ux, t +gx, y, z, x 2 y 2 z 2 ux, t t 2 ux, t = 2 ux, t + 2 ux, t t 2 x 2 y ux, t +gx, y, z z 2 2 ux, t + 2 ux, t + 2 ux, t + gx, y, z = 0 I.48 x 2 y 2 z 2 0 < x < a, 0 < y < b 2 u0, y = Ay, ua, y = By, ux, 0 = Cx, ux, b = Dx I.49a, b, c, d ux, y = u 1 x, y + u 2 x, y + u 3 x, y

197 I u 1 u 2 I.48 gx, y 0 u 3 gx, y u 1 x, y 2 u 1 x, y x u 1 x, y y 2 = 0 u 1 0, y = Ay, u 1 a, y = By, u 1 x, 0 = 0, u 1 x, b = 0 u 2 x, y 2 u 2 x, y x u 2 x, y y 2 = 0 u 2 0, y = 0, u 2 a, y = 0, u 2 x, 0 = Cx, u 2 x, b = Dx u 3 x, y 2 u 3 x, y x u 3 x, y y 2 + gx, y = 0 u 3 0, y = 0, u 3 a, y = 0, u 3 x, 0 = 0, u 3 x, b = 0 u 2 x, y u 2 x, y = Xx Yy X x Xx = Y y Yy = µ2 Xx X x + µ 2 Xx = 0, with X0 = 0, Xa = 0 nπ µ n = a, X n x = sin nπx a µ n Yy nπy Yy = c n exp + d n exp nπy a a u 2 x, y u 2 x, y = n=1 { c n exp nπy + d n exp a nπy a } sin nπx a c n d n u 1 x, y u 3 x, y 2 f 3 x, y x f 3 x, y y 2 = λ f 3 x, y

198 780 I. FOURIER u 3 x, y = n=1 m=1 c nm sin nπx sin a nπy nπ 2 nπ 2, λ nm = b a b u 3 x, y c nm c nm = G nm λ nm, G nm a b 0 0 gx, y sin nπx nπy sin dx dy a b I-3 5. u u 0, y = 0, x 2 ux, y x ux, y y 2 = Fx, y a, y = 0, ux, 0 = Cx, ux, b = Dx. x I.5 I ur, θ, t r r ur, θ, t r + 1 r 2 2 ur, θ, t θ 2 + gr, θ, t = ur, θ, t, 0 < r < R 0, 0 < θ < 2π, 0 < t I.50 t k = 1 ur, θ, 0 = u 0 r, θ I.51 u0, θ, t <, ur 0, θ, t = 0 I.52a, b 1 θ ur, 0, t = ur, 2π, t I.53 I vr, θ r r vr, θ r + 1 r 2 2 vr, θ θ 2 = λ vr, θ I.54 vr, θ = Rr Θθ I.55 I.54 r 2 d2 R dr + r dr 2 dr d 2 Θ λr2 R = dθ 2 R Θ must be = µ const.

199 I J 0 x J 1 x J 2 x 0.5 O Y 0 x Y 2 x 4 8 x 12 O x 12 1 I.4 Bessel Y 1 x Θθ d 2 Θ + µθ = 0. dθ2 I.53 2π µ = n 2, Θθ sin nθ cos nθ I.56a, b n 0 Rr r 2 R + rr n 2 + λr 2 R = 0 I.57 r RR 0 = 0, R0 < I.58a, b λ < 0 λ = β 2 r 2 R + r R + β 2 r 2 n 2 R = 0 I.59 n Bessel Bessel Bessel Rr = J n βr Y n βr I.60a, b J n βr n 1 Bessel Y n βr n 2 Bessel I.4 I.58b r = 0 Y n βr Rr = J n βr I.58a RR 0 = J n βr 0 = 0 I.61 I.4 I.1 J 0 β J 1 I.1 1 Bessel m n = 0 J n = 1 J n = 2 J βr β R 0, βr β R 0, βr β R 0 βr β R 0, βr β R 0, βr β R 0

200 782 I. FOURIER β nm vr, θ J n β nm r A n cos nθ + B n sin nθ n = 0, 1, m = 1, 2, I.5.3 ur, θ, t = {A nm t J n β nm r cos nθ + B nm t J n β nm r sin nθ} I.62 n=0 m=1 sine cosine f i, f j θ 2π 0 f i θ f j θ dθ Bessel J n β ni r, J n β n j r r R0 0 r J n β ni r J n β n j r dr A B I.63 I ur, t r r ur, t r + gr = ur, t, r < R 0, 0 < t I.64 t n = 0 ur, 0 = u 0 r I.65 ur, t = A 0m t J 0 β 0m r m=1 I.66 I.54 2 J 0 r r J 0 r = λ J 0 = β 2 0m J 0 I.66 I.64 A 0m t β 2 0m J 0β 0m r + gr = Ȧ 0m t J 0 β 0m r m=1 m=1 {Ȧ0m t + β 2 0m A 0mt } J 0 β 0m r gr = 0 m=1

201 I I.63 A 0 j R0 0 J 0 β 0 j r r dr Ȧ 0 j t + β 2 0 j A 0 jt = G j, G j gr, J 0β 0 j r r J 0 β 0 j r, J 0 β 0 j r r = R0 0 R0 0 grj 0 β 0 j r r dr {J 0 β 0i r} 2 r dr A 0 j t = c j exp β 2 0 j t + G j β 2 0 j I.67 I.66 ur, t = c m exp β 2 0m t + G m J 0β 0m r I.65 ur, 0 = c m + G m J 0β 0m r = u 0 r m=1 Bessel c j + G j = U β 2 j 0 j m=1 R0 0 R0 0 R0 0 β 2 0m J 0 β 0 j r r dr u 0 rj 0 β 0 j r r dr β 2 0m { J0 β 0 j r } 2 r dr c j = U j G j β 2 0 j ur, t = U m G m exp β 2 0m t + G m J 0β 0m r m=1 β 2 0m β 2 0m I.68 t ur, t G m β 2 m=1 0m J 0 β 0m r d 2 ur dr r dur dr + gr = 0 I.69 Northwestern Mahar 1980 Fourier Series and Boundary Value Problems [78] 1 1

202 784 2 Alternative Theorem Northwestern Olmstead 1980 Differential Equations of Mathematical Physics Consider a boundary-value problem with the 2nd linear differential equation L ux = f x, a < x < b, B 1 u = α, B 2 u = β 1 where d 2 L a 0 dx + a d 2 1 dx + a 2, a 0 0 and f x is a locally integrable function. By the definition of a linear functional as g, h gx hx dx the formal adjoint operator L and the corresponding boundary operators B can be defined by Lg, h = g, L h. Then the auxiliary problems are defined as follows: L u h x = 0, B 1 u h = 0, B 2 u h = 0, 2 L u hx = 0, B 1 u h = 0, B 2 u h = 0, 3 L Gx ξ = δx ξ, a < ξ < b, B 1 G = 0, B 2 G = 0 4 where δx ξ is the Dirac delta function. Theorem: If u h x = 0 is the only solution of 2, then u hx = 0 is the only solution of 3, and 1 and 4 have a unique solution respectively. Alternatively, if 2 has a non-trivial solutions, then 3 also will have a non-trivial solutions, and 4 has no solution. Then 1 can have a solution non-unique if and only if for each u hx satisfying 3. b a f x u hx dx = 0 Consider the problem with the integral equation b a kx, ξ uξ dξ λ ux = f x, a x b 1 where kx, ξ and f x are known functions. Then the auxiliary problems are defined as follows: b a b a kx, ξ u h ξ dξ λ u h x = 0, 2 kξ, x u hξ dξ λ u hx = 0 3

203 785 where b a dξ kξ, x is the adjoint integral operator of the integral operator b a dξ kx, ξ. Theorem: If u h x = 0 is the only solution of 2 for a given value of λ, then u hx = 0 is the only solution of 3, and, for sufficiently smooth kx, ξ, 1 has a solution and it is unique. Alternatively, if 2 has a non-trivial solution, then 3 has a non-trivial solution, and 1 has a solution non-unique if and only if for each u hx satisfying 3. b a f x u hx dx = 0 Consider a linear equation A u = f 1 where A is the n n matrix, and u and f are n 1 vectors. Then the auxiliary problems are defined as follows: A u h = 0, 2 where A is the adjoint transposed matrix of A. A u h = 0 3 Theorem: If u h = 0 is the only solution of 2, then u h = 0 is the only solution of 3, and 1 has a solution and it is unique. Alternatively, if 2 has a non-trivial solutions, then 3 has a non-trivial solutions, and 1 can have a solution non-unique if and only if f, u h = 0 for each u h satisfying 3. Caucasian:

204 786 I.1 p.8 E rafel: /5/24