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1 72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson
2 9.1. Hooke 7 Young Poisson G 2(1 + ν) (9.4) Hooke ( Hooke ) ε 11 σ 11 ν σ 22 + σ ε 22 σ 22 ν σ + σ 11 ε σ ν σ 11 + σ 22 γ 12 σ 12 G γ 2 σ 2 G γ 1 σ 1 G (9.5) ν σ 11 (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 11 ν σ 22 (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 22 ν σ (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε σ 12 1+ν ε 12 σ 21 σ 2 1+ν ε 2 σ 2 σ 1 1+ν ε 1 σ 1 (9.6) γ ij 2ε ij σ 11 σ 0 (9.5)
3 74 9 Hooke ε 11 σ 11 ν σ 22 + σ ε 22 σ 22 ν σ + σ 11 ε σ ν σ 11 + σ 22 γ 12 σ 12 G 0 γ 2 σ 2 G 0 γ 1 σ 1 G 0 σ ν σ ν σ (9.6) ν σ 11 (1 + ν)(1 2ν) ( σ ν σ ν σ )+ σ 1+ν σ ν σ 22 (1 + ν)(1 2ν) ( σ ν σ ν σ ) 1+ν ν σ 0 ν σ (1 + ν)(1 2ν) ( σ ν σ ν σ ) 1+ν ν σ 0 σ 12 1+ν 00 σ 2 1+ν 00 σ 1 1+ν (9.5) (9.6) ε 11,ε 22,ε ε 11 + ε 22 + ε 1 (σ 11 + σ 22 + σ ) 2ν (σ 11 + σ 22 + σ ) 1 2ν (σ 11 + σ 22 + σ ) (9.7) σ 22 + σ (9.5) ε 11 1 σ 11 ν { } 1 2ν (ε 11 + ε 22 + ε ) σ 11 (9.8) ε 11 1+ν σ 11 + ν 1 2ν (ε 11 + ε 22 + ε ) σ 11 { ε 11 + ν } 1+ν 1 2ν (ε 11 + ε 22 + ε ) ν (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 11 (9.9)
4 9.2. Hooke 75 (9.6), Cauchy T ij, σ ij, 9..,,, Cauchy σ ij, σ ij ε ij Hooke. 9.. Hooke,, 4.,., Cauchy Green-Lagrange,. A F. cauchy T 11 F/A 0. θ.. Green-Lagrange, [ ] [ ][ ] T11 T 12 cos θ sin θ ][ T11 T12 cos θ sin θ (9.10) T 21 T 22 sin θ cos θ T 21 T22 sin θ cos θ, Cauchy Green-Lagrange,. T f() (9.11),. Green-Lagrange,., Cauchy Cauchy 2Piola-Kirchoff (??, ) Hooke 45 [ ] [ ] σ 0 0 σ (9.12) 0 σ σ 0 σ 22 σ σ 11 σ σ 11 0 σ 12 σ σ 21 σ σ :
5 76 9 Hooke Hooke ε 11 Hooke ε 11 σ 11 ν σ 22 + σ ε 22 σ 22 ν σ + σ 11 ε σ ν σ 11 + σ 22 γ 12 σ 12 G γ 2 σ 2 G γ 1 σ 1 G (9.1) ε 11 σ 11 ν σ 22 σ ν σ 1+ν σ (9.14) ε 22 σ 22 ν σ 11 σ + ν σ 1+ν σ (9.15) ε 12 ( 1 ) 2 γ 12 0 (9.16) ε 11 0 (9.17) ε 22 0 (9.18) ε 12 ( 1 ) 2 γ 12 σ 12 2G σ (9.19) 2G ε (ε 11 ε 22 ) (9.20) 1 ( 1+ν 2 σ 1+ν ) σ (9.21) 1+ν σ (9.22) σ 2G 1+ν G σ (9.2) 2(1 + ν) (9.24)
6 9.. Hooke Hooke,,, Hooke. (9.9),,.,,,.,.,,.,.,,,. a i σ ij a 1 ε ij + a 2 ε 2 ij + a ε ij + a 4 ε 4 ij + (9.25), a 0 δ ij ε ij 0 σ ij 0, σ ij a 0 δ ij. ε ij I 1 ε ii, I {(ε ii) 2 ε ij ε ji }, I detε, Cayley-Hamilton ε ij I 1ε 2 ij I 2ve ij + I δ ij (9.26), 2,. σ ij φ 0 (I 1,I 2,I )δ ij + φ 1 (I 1,I 2,I )ε ij + φ 2 (I 1,I 2,I )ε 2 ij (9.27) δ ij, ε ij 0 φ 0 (I 1,I 2,I )0.,, φ 0,φ 1,φ 2 λ, μ φ 0 λi 1, φ 1 μ, φ 2 0. σ ij λi 1 δ ij + με ij (9.28). λ, μ Lamé., Hooke. (9.6),,.
7 78 9 Hooke 9.4 I 1 ε 11 + ε 22 + ε. l 0. ε ε 22 0 (9.29) 0 0 ε, v,,, v {l 0 (1 + ε 11 )}{l 0 (1 + ε 22 )}{l 0 (1 + ε )} (9.0) l0{1+(ε 11 + ε 22 + ε )+(ε 11 ε 22 + ε 22 ε + ε ε 11 )+ε 11 ε 22 ε } (9.1) l0 {1+(ε 11 + ε 22 + ε )} (9.2), J V l 0 J v V l 0 {1+(ε 11 + ε 22 + ε )} l o ε 11 + ε 22 + ε (9.), J I 1. ε kk. ε ij. ε ij 1 ε kkδ ij + (ε ij 1 ) ε iiδ ij (9.4) ε ε 11 ε 12 ε kk 1 ε 11 ε kk ε 12 ε 1 ε ε 21 ε 22 ε 2 kk + ε 21 ε 22 ε kk ε 2 (9.5) ε ε 1 ε 2 ε kk ε 1 ε 2 ε ε kk 2 0,. Hooke, (9.28). σ ij λε kk δ ij +2με ij (9.6) ( λε kk δ ij +2μ ε ij + 1 ) ε kk (9.7) (λ + 2 ) μ ε kk δ ij +2με ij (9.8) κε kk δ ij +2με ij (9.9) κ λ + 2 μ, μ G.
8 Young, Poisson ν κ, G., σ 11 0 σ ij 0 ε 11 σ 11. σ 11 Young κ, G. 9Gκ κ + G ε 11 (9.40) 9Gκ κ + G (9.41) 9.4., (9.9). (9.9), ε ii σ ii., σ ii κε ii δ ii +2G (ε ii 1 ) ε iiδ ii (9.42) κε ii (9.4) ε ii 1 κ σ ii (9.44), (9.9), σ ij 1 σ iiδ ij +2G ( ε ij 1 ) 9κ σ iiδ ij (9.45), ε ij 1 9κ σ iiδ ij + 1 (σ ij 1 ) 2G σ iiδ ij (9.46) σ 11 0 σ ij 0.,. ε 11 σ 11. ε κ σ ( σ 11 1 ) 2G σ 11 (9.47) ( 1 G + 1 ) σ 11 κ + G 9κ 9Gκ σ 11 (9.48) σ 11 9Gκ κ + G ε 11 (9.49) ν,, ε 22 σ 11, Poisson ν ε 22 /ε 11,. ν κ 2G 2(κ + G) (9.50)
9 80 9 Hooke 9.5. ε 22, σ 11. ε 22 1 ( 0 1 ) 2G σ 11 ( 1 6G + 1 9κ Poisson ν ε 22 /ε 11, ν ε 22 κ 2G ε 11 18Gκ κ 2G 2(κ + G) + 1 9κ σ 11 (9.51) ) κ +2G σ 11 18Gκ σ 11 (9.52) 9Gκ κ + G (9.5) (9.54) (9.41) (9.50) κ, ν,ν.. κ (1 2ν), G 2(1 + ν) 9.6. G, κ,ν. κ, ν κ 2 G κ (9.55) (κ + G) 9Gκ (9.56) ( 9G)κ G (9.57) κ G (9G ) (9.58) 2(κ + G)ν κ 2G (9.59) (6ν )κ 2G(1 + ν) (9.60) κ 2G(1 + ν) (1 2ν) G 2G(1 + ν) (9G ) (1 2ν) 2(1 + ν) (9G ) (1 2ν) (1 2ν) G 2(1 + ν) G 1 ( ) (1 2ν) 2(1 + ν) + 2(1 + ν) (9.61) (9.62) (9.6) (9.64) (9.65) (9.66)
10 κ 2G(1+ν) (1 2ν) G 2(1+ν) κ,ν. 2(1 + ν) κ (1 2ν) 2(1 + ν) (9.67) (1 2ν) Hooke,ν. (9.9) σ ij (κ 2 ) G ε ii δ ij +2Gε ij (9.68) ( (1 2ν) 2 ) ε ii δ ij + 2 2(1 + ν) 2(1 + ν) ε ij (9.69) ν (1 2ν)(1 + ν) ε iiδ ij + (1 + ν) ε ij (9.70), (9.6). 9.5,,., X 2 0 X 2 0., ε ij 9 ε 22, ε 12, ε 21, ε 2, ε 2 0 δε ij δε 22 δε 12 δε 21 δε 2 δε 2 0 X X 2 X 1
11 82 9 Hooke x 0 ε 1 ( u + u ) 0 2 x x ε 1 1 ( u1 + u ) 0, ε 1 1 ( u + u ) x x 1 2 x 1 x ε 2 1 ( u2 + u ) 0, ε 2 1 ( u + u ) 2 0 (9.71) 2 x x 2 2 x 2 x 0 ν σ 11 (1 + ν)(1 2ν) (ε 11 + ε 22 )+ 1+ν ε 11 ν σ 22 (1 + ν)(1 2ν) (ε 11 + ε 22 )+ 1+ν ε 22 ν σ (1 + ν)(1 2ν) (ε 11 + ε 22 ) σ 12 1+ν ε 12 σ 21 σ 2 σ 2 0 σ 1 σ 1 0 (9.72) ε 0 σ 0 σ,,., X 2 e 2 t 2 0 t 2 σ 21 e 1 + σ 22 e 2 + σ 2 e (9.7) σ 21,σ 22,σ 2 0 Cauchy σ 12,σ 2 0 X X 2 X 1
12 x σ 22 σ 12 σ 21 σ 2 σ 2 0 Hooke 0 σ 0 σ 11 σ 22 σ σ 12 σ 2 σ 1 ν (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 11 ν (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 22 ν (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 0 1+ν ε 12 σ 21 1+ν ε 2 σ 2 0 ε 2 ε ν ε 1 σ 1 0 ε 1 ε 1 0 (9.74) ν (1 + ν)(1 2ν) (ε 11 + ε 22 + ε )+ 1+ν ε 0 ε ν 1 ν (ε 11 + ε 22 ) (9.75) Hooke σ 11 1 ν ε ν 1 ν ε 2 22 σ 22 ν 1 ν ε ν ε 2 22 σ 12 1+ν ε 12 (9.76)
13 85 11, Cauchy 1, Hooke Young σ 11 0 σ ij 0,, Cauchy., f. F F X 2 X 1,,,, 0.,,.. Cauchy ( ) : σ 11 + σ 12 + σ 1 0 (11.1) X 1 X 2 X σ 21 + σ 22 + σ 2 0 X 1 X 2 X (11.2) σ 1 + σ 2 + σ 0 X 1 X 2 X (11.) u 1 u 2 u 0 at(0, 0, 0) (11.4)
14 86 11 t n σ T n, X 1 L, n (1, 0, 0) f (f,0, 0) X 1 L, n ( 1, 0, 0) f ( f,0, 0), (Hooke ) σ ij σ 1f atx 1 L (11.5) σ ( 1) f atx 1 L (11.6) ν (1 2ν)(1 + ν) ε iiδ ij + (1 + ν) ε ij (11.7) ε ij 1 ( ui + u ) j (11.8) 2 X j X i,,, ε ij 0 (i j)., Hooke, σ ij 0 (i j). 0, Cauchy σ 11 X 1 0, σ 22 X 2 0, σ X 0 (11.9)., C 1,C 2,C,. (9.46), σ 11 C 1, σ 22 C 2, σ C (11.10) σ 11 f, σ 22 0, σ 0 (11.11) σ 11 ε 11 (11.12) u 1 X 1 (11.1), f u 1 X 1 u 1 X 1 f (11.14) u 1 f X 1 + C (11.15)
15 X 1 0 u 1 0, ε 22 ε νε 11 u 2 νf X 2, u 1 f X 1 (11.16) u νf X (11.17) σ 11 ε 11 u 1 X 1 Cauchy 2 2 u 1 X (11.18) u 1 X 1 f at X 1 L, u 1 0 at X 1 0 (11.19).. 0 X X 1 u 0(X 0) L f ε ij 0(i j), σ ij 0(i j) Cauchy X σ 11 σ 22 σ 0, 0, + b 0 (11.20) X 1 X 2 X b, ρ, g, (ρ, g > 0 ) b ρg (11.21)
16 88 11, σ X ρg 0 (11.22), X L σ f, σ ɛ σ 11 0, σ 22 0, σ C + ρgx (11.2) f C ρgl, C f + ρgl (11.24) u X f + ρg (X L) (11.25) u f X + ρg 2 X2 ρglx + c (11.26) X 0 u 0 c 0 u f X + ρg ( ) X 2 2 2LX (11.27). 2 u a (11.28) X 2 du, dx u. d 2 u f(x) (11.29) dx2,.,, [B] u. d2 u dx u(0) u 0, u(a) u a or 2 f(x) (0 <x<a) (11.0) du (a) d (11.1) dx
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5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
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7 2010 11 22 1 7 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2010 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/18 3. 10/25 2, 3 4. 11/ 1 5. 11/ 8 6. 11/15 7. 11/22 8. 11/29 9. 12/ 6 skyline 10. 12/13
1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th
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tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
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II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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gr09.dvi
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2000年度『数学展望 I』講義録
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2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
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