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 (6) δs 2 = δx i δx i s ij = 1 ( ui + u ) j (7) 2 x i 1 r A r r = r + u(r, t) (1) u 1 B B r + δr r = r + δr + u(r + δr, t) (2) 1 1: u i / s ij r ij = 1 2 ( ui u ) j x i u i δx i δx j = (s ij + r ij )δx i δx j (8) = s ij δx i δx j (9) 1
r ij δx i δx j = s ij 2 2.1 (1) r r ρ (x, y, z ) (x, y, z) = ρ (1) (x, y, z ) (x, y, z) V V = (x, y, z ) (x, y, z) = Det [ δ ij + u ] i (11) 1 + u i x i = 1 + div u (12) δ ij Det[ ] u i / 1 V V = div u (13) V div u = 3 x f f = kx (14) 2 z δθ A (x yδθ, y + xδθ, z) u = ( yδθ, xδθ, ) r xy = r yx = δθ k t ij = C ijkl s kl (15) C ijkl 4 4 3 4 = 81 i j k l {ij} 6 6 6 = 36 {ij} {kl} 6 + (36 6)/2 = 21 2 λ µ C ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ) (16) 2 t ij = C ijkl s kl = [λδ ij δ kl + µ(δ ik δ jl + δ il δ jk )]s kl = λδ ij s kk + µ(s ij + s ji ) = λδ ij div u + 2µs ij (17) 2
3.1 t ij + K i = (18) K i (7) (17) t ij = λ div u + µ 2 u i x i x 2 + µ 2 u j j x i = µ u i + (λ + µ) div u x i (19) µ u + (λ + µ)grad div u + K = (2) = 2 x 2 j = 2 x 2 + 2 y 2 + 2 z 2 (21) u rot rot u = grad div u u (22) u (λ + 2µ)grad div u µrot rot u + K = (23) (2) (23) 3.2 f i t ij n j = f i (24) n j 3.3 U = 1 2 kx2 (25) ɛ el E el = V ɛ el dv (26) ɛ el = ɛ el (s ij ) s ij 2 ɛ el (s ij ) = ɛ + ɛ el s ij s ij + 1 2 ɛ el 2 s ij s kl s ij s kl (27) ( s ij = s ij = ɛ el = (28) s ij ɛ ɛ el (s ij ) = 1 2 ɛ el 2 s ij s kl s ij s kl (29) δɛ el = 2 ɛ el s ij s kl s kl δs ij (3) f i = t ij (31) 3
δu i t ij δw = δu i dv (32) V δu i t ij = (δu it ij ) (δu i) t ij (33) 1 (δu i ) δw = δu i t ij n j ds t ij dv (34) S t ij 2 (δu i ) t ij dv = δs ij t ij dv (35) V 1 Λ V V ρ Λ x i + t ij = (36) δu i δw = V ρ Λ δu i dv = ρδλdv = δu x i V (37) δu δλ = Λ(r + δu) Λ(r) = Λ x i δu i (38) (35) E el δe el = t ij δs ij dv (39) V (3) t ij = 2 ɛ el s ij s kl s kl (4) C ijkl = 2 ɛ el s ij s kl (41) {ij} {kl} ɛ el = 1 2 C ijkls ij s kl (42) 4 a b c 4.1 2 F z z δc s zz = δc/c s xx = δa/a δc 2: 4
t ij = (43) t zz = F ab (44) (17) t xx = λ(s xx + s yy + s zz ) + 2µs xx = (45) t yy = λ(s xx + s yy + s zz ) + 2µs yy = (46) t zz = λ(s xx + s yy + s zz ) + 2µs zz = F ab (47) s ij F s xx + s yy + s zz = ab(3λ + 2µ) (48) s zz = λ + µ F µ(3λ + 2µ) ab = λ + µ µ(3λ + 2µ) t zz (49) λ s xx = s yy = t zz 2µ(3λ + 2µ) (5) Y σ t zz = Y s zz, s xx = s yy = σs zz (51) µ(3λ + 2µ) Y = λ + µ λ σ = 2(λ + µ) (52) (53) Y σ λ = (1 + σ)(1 2σ) Y µ = 2(1 + σ) (54) (55) 2 Y δc c = F ab, δa a = σ δc c (56) σ > 4.2 3 p t ij = pδ ij (57) t xx = λ(s xx + s yy + s zz ) + 2µs xx = p (58) t yy = λ(s xx + s yy + s zz ) + 2µs yy = p (59) t zz = λ(s xx + s yy + s zz ) + 2µs zz = p (6) 3p s xx + s yy + s zz = (3λ + 2µ) (61) s xx + s yy + s zz = div u V/V K V 4.3 K V = pv V = p div u K V = λ + 2 3 µ = Y 3(1 2σ) (62) (63) 3 z x 5
= (F 1 ) i + (F 2 ) i (69) y y x t xy = t yx t yx = F y bc, t xy = F x ac (64) F x b = F y a t xy = t yx 4.4 2 Y > K V > G > 1 < σ < 1/2 σ s xy = t xy 2µ (65) 5 u x = αy, u y = βx (66) u x / y = u y / x α = β 3 s xy = α π/2 ± θ θ = 2α G = t xy θ = 1 t xy Y = µ = 2 s xy 2(1 + σ) (67) 5.1 (18) 4 2 S 1 S 2 (18) tij dv = K i dv (68) tij dv = t ij ζ j ds + t ij ζ x jds j S 1 S 2 3: 3 α = β = ζ j S 1 ζ j S 2 (F k ) j S k s F i (s 1 ) + F i (s 2 ) = K i dv (7) 6
s 2 = s 1 + ds df i ds = K i ds (71) S 1 dn k ds + ε kliζ l F i = ε kli x l K i ds (76) F i df i ds = (72) 4: dn k ds + ε kliζ l F i = (77) s d 2 N k ds 2 + ε kli dζ l ds F i = (78) (72) F i (18) ε kli x l l i t ij ε kli x l dv ) = ε kli ( x l t ij ζ j ds + x l t ij ζ jds S 1 S 2 = ε kli x l K i dv (73) S 1 S 1 N k (s) = ε kli x l t ij ζ j ds (74) S 1 S 1 S 2 S 1 x l S 2 x l x l = x l + ζ lds ε kli x l t ij ζ jds S 2 = ε kli (x S l + ζ l ds)t ij ζ (75) jds 2 = ε kli ζ l F i (s + ds)ds + N k (s + ds) 5.2 x y z x y 4 5 R δθ l = Rδθ y y = l + δl = (R y)δθ s xx = δl l = y R (79) t xx = Y s xx = Y y R (8) 4 z 7
F x = Y ydydz (81) R F x = y 5 ξ (x) = dξ dx (86) F i R (82) F i (77) (78) ξ 5.3 5: z N z = t xx ydydz = Y y 2 dydz (82) R I y = y 2 dydz (83) y h z w I y = w h/2 h/2 ydy = wh3 12 (84) y = ξ(x) 1 R = 1 d 2 ξ [1 + ξ (x) 2 ] 3/2 dx 2 (85) 5 z z ξ(x) ξ 1 1 R = d2 ξ dx 2 (87) ζ = (1, ξ, )/ 1 + (ξ ) 2 = (1, ξ, ) (88) I y Y d3 ξ dx 3 + F y dξ dx F x = (89) F x = I y Y d3 ξ dx 3 + F y = (9) ξ(x) = F yx 3 6Y I y + a + bx + cx 2 (91) a b c 8
y g S ρ df y dx = Sρg (92) (9) (76) F y = Sρgx d (93) ξ(x) = Sρgx4 24Y I y + a + bx + cx 2 + dx 3 /6 (94) 5.4 6 ξ =, dξ dx = (95) ξ =, d 2 ξ dx 2 = (96) d 2 ξ dx 2 =, d 3 ξ dx 3 = (97) y F (9) d 2 ξ dx 2 =, d 3 ξ dx 3 = F (98) Y I y (91) x = x = L F a = b = c = F L/(2Y I y ) ξ(x) = F x2 6Y I y (3L x) (99) 6: (a) (b) x = L H F H F = F L3 3Y I y (1) (94) x = x = L a = b = d = SρgL/(Y I y ), c = SρgL 2 /(4Y I y ) ξ(x) = Sρgx2 24Y I y (x 2 4Lx + 6L 2 ) (11) x = L H g H g = SρgL4 8Y I y (12) (84) H F h 3 H g h 2 5.5 x I y Y d3 ξ dx 3 dξ dx F x = (13) 9
I y Y d2 ξ dx 2 F xξ = c (14) c F x > κ = F x /(Y I y ) ξ(x) = A cosh κx + B sinh κx + C (15) x = = L ξ() = A + C =, ξ () = κb = (16) B = ξ (L) = κ 2 A cosh κl = (17) cosh κl A = = C F x < k = F x /(Y I y ) ξ(x) = A cos kx + B sin kx + C (18) x = x = L ξ() = A + C =, ξ () = kb = (19) B = ξ (L) = k 2 A cos kl = (11) k = (2n + 1)π 2L (n =, 1, 2,... ) (111) A 7 ξ = 2 F c = π2 Y I y 4L 2 (112) 7: F x < F c 6 F c = π2 Y I y L 2 (113) ρ Dv i Dt = t ij + K i (114) Dv i Dt = v i t + v j v i v i t = 2 u i t 2 (115) ρ 2 u = µ u + (λ + µ)grad div u + K (116) t2 6.1 u = u(t, x) x 1
u x ρ 2 u x t 2 = µ 2 u x x 2 y z ρ 2 u α t 2 + (λ + µ) 2 u x x 2 (117) = µ 2 u α x 2 (118) α y z 2 U t 2 = c2 2 U x 2 (119) (119) U(t, x) = f(x ct) + g(x + ct) (12) c x 1 x u x u x u y u z λ + 2µ Y (1 σ) c l = = ρ ρ(1 + σ)(1 2σ) c t = µ ρ = Y 2ρ(1 + σ) (121) (122) 1 < σ < 1/2 c l > c t P S 6.2 u = u l + u t (123) 2 2 rot u l =, div u t = (124) (116) 2 u t t 2 c2 t u t + 2 u l t 2 c2 l u l = (125) u l grad div u = rot rot u + u (126) U t = 2 u t t 2 c2 t u t (127) U l = 2 u l t 2 c2 l u l (128) U t + U l = (129) rot U l =, div U t = (13) (129) rot U t =, div U l = (131) U t = U l = 2 u t t 2 = c2 t u t (132) 2 u l t 2 = c2 l u l (133) 2 u l u t 6.3 11
x t xx = Y u x x t yy = t zz = t xy = t yz = t zx = (134) u x ρ 2 u x t 2 = t xx x = Y 2 u x x 2 (135) Y c b = (136) ρ u y y = u z z = σ u x x (137) u y u z ρ 2 u y t 2 = ρ 2 u z t 2 = (138) (137) (134) L D u y u x = u z u x σ D L (139) (135) u x (t, x) = sin(ωt + φ)f(x) (135) f k = ω/c b ω2 c 2 f = d2 f b dx 2 (14) f(x) = A cos kx + B sin kx (141) t xx = t xx df/dx x = x = L df/dx = x = B = x = L A sin kl = A = A k = nπ L (n = 1, 2,... ) (142) k ω n = nπ L c b (143) 6.4 (72) (77) y d/ds = / x F y x = Sρ 2 ξ t 2 (144) S Y I y 3 ξ x 3 + F y = (145) Y I y 4 ξ x 4 = Sρ 2 ξ t 2 (146) x 4 12
ξ(t, x) = sin(ωt + φ)x(x) g Y I y d 4 X dx 4 = ω2 SρX (147) ω 2 = Y I y Sρ k4 (148) d 4 X dx 4 k4 X = (149) 4 g = exp αt α 4 = k 4 α = ±k, ±ik X(x) =A cos kx + B sin kx + C cosh kx + D sinh kx (15) x = x = L X() = X () = A + C = B+D = X (L) = X (L) = k 2 [A(cos kl + cosh kl) + B(sin kl + sinh kl)] = (151) k 3 [ A(sin kl sinh kl) + B(cos kl + cosh kl)] = (152) A = B = (cos kl + cosh kl)(cos kl + cosh kl) + (sin kl + sinh kl)(sin kl sinh kl) = (153) cos kl cosh kl + 1 = (154) k (148) (154) kl = 1.8751 cos kl cosh kl 1 = (155) kl = 4.73 7 2 7.1 σ(t) = σ cos ωt (156) 13
6 s(t) = s cos(ωt δ) (157) s σ s = A(ω)σ (158) δ ω σ(t) = σ Re[exp iωt] (159) s(t) = σ Re[ J(ω) exp iωt] (16) J(ω) = A(ω) exp[ iδ(ω)] (161) J 1/ J = κ σ = Y s σ(t) = Re[Ỹ (ω)s exp(iωt)] (162) Ỹ J(t) = 1 2π J(ω) exp(iωt)dω (164) σ(t) = σ h(t) h(t) { 1 t h(t) = t < t s(t) = σ J(t t )dt t = σ J(t )dt = σ J c (t) (165) (166) J c J(t) = δ(t)/k J c = h(t)/k J(t) = γ exp( γt)h(t)/k J c (t) = [1 exp( γt)]h(t)/k (167) 7.2 J(t) s(t) = t J(t t )σ(t )dt (163) 6 t σ 7.3 8 9 14
8: (a) (b) 9: (a) (b) x y f = Γ(ẋ ẏ) = ky (168) y f = Γ(ẋ f/k) (169) J = 1 k + 1 iωγ (17) J(t) = 1 ( 1 2π k + 1 ) e iωt dω iωγ = 1 k δ(t) + 1 (171) Γ h(t) J t < J = ω = J c (t) = t J(t)dt = ( 1 k + t ) h(t) (172) Γ f = kx + Γẋ (173) J = 1 k + iωγ (174) J(t) = 1 1 2π k + iωγ eiωt dω = 1 (175) Γ exp( kt/γ)h(t) J c (t) = 1 exp( kt/γ) h(t) (176) k k i iωγ i 1 3 k = k + 1 1/k 1 + 1/(iωΓ 1 ) = k + iωγ 1k 1 k 1 + iωγ 1 (177) J = 1/ k = k 1 + iωγ 1 k k 1 + iωγ 1 (k + k 1 ) J (ω) = (178) ωγ 1 k 2 1 (k k 1 ) 2 + ω 2 Γ 2 1 (k + k 1 ) 2 (179) 15
s(t) = s exp( γt) cos(ωt + φ) (185) 1: 3 ω = k k 1 /[(k +k 1 )Γ 1 ] = 1/τ J(t) = δ(t) k 1 + k + k 1 (k + k 1 )k τ exp( t/τ)h(t) 7.4 (18) δ 1 w = = 2π/ω 2π/ω = πσ s sin δ δw = σδs (181) σ(t) ds(t) dt dt σ cos ωtωs sin(ωt δ)dt (182) e = 1 2 Re[ σ s] = s σ 2 cos δ (183) 7 Ξ = w 2e = π tan δ (184) tan δ 7 e = s σ /2 s[(2nπ φ)/ω] = log s[(2(n 1)π φ)/ω] = 2γπ/ω (186) E 1 2 s2 = 1 2 s2 exp( 2γt) (187) 1 E = 1 2 s2 exp( 2γt)[1 exp( 4γπ/ω)] (188) Ξ = E 2E = 1 exp( 4γπ/ω) 2γπ/ω = 2 (189) A x x φ tan φ = ξ (x) (19) ζ = (cos φ, sin φ, ) (191) (19) x 1 dφ cos 2 φ dx = d2 ξ dx 2 (192) 16
ds = 1 + ξ (x) 2 dx cos φ = 1/(1 + tan 2 φ) = 1/[1 + ξ (x) 2 ] dφ ds = 1 d 2 ξ [1 + ξ (x) 2 ] 3/2 dx 2 = 1 R (193) (82) (78) Y I y d 2 φ ds 2 + cos φf y sin φf x = (194) F x = F (F > ) F y = d 2 φ = sin φf (195) ds2 dφ/ds Y I y 2 ( ) 2 dφ cos φf = c (196) ds s = s = φ = s = L dφ/ds = s = L φ c = F cos φ Y Iy s = 2F φ dφ cos φ cos φ (197) φ Y Iy L = 2F φ dφ cos φ cos φ (198) φ < φ 1 cos φ = 1 φ 2 /2 + φ 4 /24 (199) η = φ/φ F 1 dη L = Y I y (1 η2 )[1 φ 2 (1 + η 2 )/12] (2) 1 (1 η2 )[1 φ 2 (1 + η 2 )/12] 1 = (1 η2 ) [1 + φ 2 (1 + η 2 )/24] (21) η = sin θ F π/2 L = dθ[1 + φ 2 (1 + sin 2 θ)/24] Y I y = π (1 + φ 2 ) 2 16 (22) (112) F = 1 + φ 2 F c 16 (23) F > F c cos φ = dx/ds sin φ = dy/ds x = = φ φ Y Iy 2F cos φ ds dφ dφ φ y = sin φ ds dφ dφ Y φ Iy = 2F = (24) cos φdφ cos φ cos φ sin φdφ cos φ cos φ 2Y Iy F ( 1 cos φ cos φ cos φ ) (25) H = y(φ ) 2Y Iy H = 1 cos φ (26) F Y Iy H = F φ = 8L Fc F 1 (27) π F F c 17
B L z z = z = L Θ z θ = Θz/L u x = θy = yzθ L u y = θx = xzθ L (28) (29) div u = u z z = (21) (28) (29) u z x y s xx = s yy = s zz =, s xy = (211) s zx = 1 ( uz 2 x yθ ) L s zy = 1 ( uz 2 y + xθ ) L (212) (213) t xx = t yy = t zz =, t xy = (214) ( uz t zx = µ x yθ L ( uz t zy = µ y + xθ L ) ) (215) (216) 2 u z x 2 + 2 u z y 2 = (217) u z 2 u z 2 η = x + iy f(η) f(η) = φ + iψ u z = Θ L φ (218) t zx = µ Θ ( ) φ L x y t zy = µ Θ ( ) φ L y + x (219) (22) φ x = ψ y, φ y = ψ x (221) t zx = µ Θ ( ) ψ L y y t zy = µ Θ ( ψ ) (222) L x + x χ = ψ/2 (x 2 + y 2 )/4 t zx = 2µ Θ χ L y t zy = 2µ Θ χ L x (223) 2 χ x 2 + 2 χ = 1 (224) y2 n = (n x, n y, ) t zx n x + t zy n y = (225) z (x(s), y(s)) (dx/ds, dy/ds) n x = dy ds, n y = dx ds (226) 18
(225) (223) (226) χ dy y ds + χ x dx ds = dχ ds = (227) χ (224) χ χ = 1 4 (x2 + y 2 ) + c (228) c ψ = 2c φ = a u z z u z = x y 2a 2b χ = A(x 2 y 2 ) 1 4 (x2 + y 2 ) + c (229) A x = a cos ϕ y = b sin ϕ ϕ (229) χ =A(x 2 y 2 ) (x 2 + y 2 )/4 + c =A(a 2 cos 2 ϕ b 2 sin 2 ϕ) (a 2 cos 2 ϕ + b 2 sin ϕ)/4 + c =[A(a 2 + b 2 ) (a 2 b 2 )/4] cos 2 ϕ (A + 1/4)b 2 + c A = a2 b 2 4(a 2 + b 2 ) (23) (231) ψ = 2A(x 2 y 2 ) = 2AIm[iη 2 ] (232) φ = 2ARe[iη 2 ] = 4Axy (233) u z = Θ L a 2 b 2 a 2 xy (234) + b2 z B.1 N z = ( yt zx + xt yz )ds (235) χ N z = 2µ Θ (y χ L y + x χ )ds (236) x y χ y ds = χydx χds (237) x χ x ds = χxdy χds (238) 8 2 χ χydx = χxdy = χ S (239) χ S N z = 2µ Θ [ ( yχdx + xχdy) 2 L = 4µ Θ ( ) χds χ S L ] χds (24) χ = N z = 4µ Θ L χds (241) Θ/L C = 4µ χds (242) 8 19
a C = 2πµ a (a 2 r 2 )rdr = πµa 4 /2 (243) 2