構造と連続体の力学基礎

II 37

Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった

9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton (.a) k w(t) m mg k u(t) 9. f (t) k w(t) = f (t) + mg k w(t) = m d w(t) dt m ẅ(t) + k w(t) = mg + f (t) (9.) Einstein 3 F = ma, E = mb, E = mc E = ma F = ma mg 373

374 9 w(t) u(t) w(t) mg k (9.) ü = ẅ (9.) m ü(t) + k u(t) = f (t) (9.3) (9.) u(t) 9. (9.3) f (t) 9. g θ(t) M(t) = mg l sin (θ (t)) mg l θ(t) l θ(t) m 9. θ(t) (.4) J = m l Newton (.3) mg l θ(t) = m l θ(t) l θ(t) + g θ(t) = (9.4) (9.3) [(l, g) (m, k)] (9.3) () (9.3) f (t) t = u() = u, u() = v (9.5a, b) (9.3) m m k ω k m (9.6) ü(t) + ω u(t) = (9.7)

9.. 375 u(t) = exp (µ t) (9.7) µ µ + ω = µ = ±i ω u(t) = A e iω t + B e iω t A B Euler (9.6) u(t) = A sin (ω t) + B cos (ω t) = C sin (ω t ζ), C A + B, ζ tan ( B A ) (9.8a, b, c) C A B C ζ (9.5) ζ sine sin(ω t) t = A B u() = B = u, u() = ω A = v A = v ω, B = u 9.3 T s s ω rad π T = π ω u(t) = v ω sin (ω t) + u cos (ω t) (9.9) (9.) (9.6) ω (9.6) (9.3) (9.4) u(t) u O v T C 9.3 t ω = g l (9.) l? (3) (9.3) u(t) m ü u dt + k u u dt = f (t) u dt + const. K m ü u dt = m { } ( u) dt = m d ( u) = m ( u) f f T = ω /s Hz rad π

376 9 U k u u dt = k u du = k u du = k u K + U = f (t) u dt + const., m ( u) + k u = f (t) u dt + const. (9.) K U (9.9) (9.) K = m ω u sin ωt = k u sin ωt, U = k u cos ωt K + U = const. K U 9 K max = U max K = U (9.3) µ 9.4 m l l x v(x, t) u(t) = A sin (ωt ζ) x v(x, t) u(t) 9.4 v(x, t) = x l u(t) = A x l sin (ωt ζ) K = m ( u) + l µ ( v) dx = ( ω A m + µ ) l cos (ωt ζ) 3 (9.3) ω A U = k u = k A sin (ωt ζ) ( m + µ l 3 ) = k A ω = k m + µ l 3 µ l (4) (9.) f (t) w(t) g y(t) w(t) m ÿ(t) = mg ÿ(t) = g ( )

9.. 377 g (9.5) y() = y, ẏ() = v ( ) w(t) = g t + c t + c c = v, c = y y(t) = g t + v t + y u(t) ( ) y(t) g u(t) = u t u() =, u() = u t x(t) u(t) t = (x(), y()) = (, y ) u v t (x(t), y(t)) y(t) = ( x(t) g v ) + y + v u g g t = v g x = u v g y = y + v g (9.4) θ = θ = const. (θ = ωt + const.) θ v θ r θ = rω 3 u = v θ e θ, e r = e x cos θ + e y sin θ, e θ = e x sin θ + e y cos θ e d (d = x, y, r, θ) e x e y u = v θ ė θ = v θ θ ( e x cos θ + e y sin θ ) = v θ θ e r e r a a = v θ θ = v θ ω (9.4) F = ma = m v θ ω = m r ω = m v θ r (3.64a) 9-. 3

378 9 9.. () 4 () Kelvin- Voigt 9.5 k c k u c u m f (t) 9.5 f (t) 5 c c u(t) f (t) k u(t) c u(t) = m ü(t) m ü(t) + c u(t) + k u(t) = f (t) (9.5) ω k m, β c m ω (9.6a, b) f (t) (9.5) ü(t) + β ω u(t) + ω u(t) = (9.7) c β 6 (9.7) exp (µ t) µ ( ) µ + β ω µ + ω = µ = ω β ± β (9.8a, b) β 4 km 5 6 [7] h ω β

9.. 379 u(t) u(t) O u(t) t O 9.6 u n t u n+ j O t t 9.7. β > 9.6. β = u = (A + Bt) exp ( ωt) 9.6 (9.6b) β = c cr m ω = m k 3. β < 7 β β =..5 µ = β ω ± i ω d, ω d ω β (9.9a, b) u(t) = exp ( β ω t) (A sin ω d t + B cos ω d t) = [ C e β ω t] sin (ω d t ζ) (9.) C 9.7 β (9.9b) ω d T T = π ω d = π ω β (9.) 7 β <

38 9 β..3.3.5 ω d ω ω ω d, T π ω (9.) 9.7 n (n + j) j = 3 u n, u n+ j j = 8 u n C exp ( βωt) π jβ = = exp ( jβωt) = exp u n+ j C exp { βω (t + jt)} (9.3) β β j = δ ln u n u n+ j = β = π j β β (9.4) δ δ + 4π j (9.5) β β δ π j 9.8 β 5 T =.485 s (9.) ü ( m / s ).5.5 4 6 9.8 (9.6) ω =.95 rad / s n j (9.4) (9.6) β =.385 9.9 R C L E(t) Q(t) I(t) E(t) R L C t (s) L Q(t) + R Q(t) + C Q(t) = E(t), I(t) = Q(t) (9.7a, b) 9.9 8 β

9.. 38 (3) (9.7) 9.5 ü = d u dt = d u du du dt = ud u du u d u du + ω u = u d u + ω u du = ( u) + ω u = const. ( u ω ) + u = const. (9.8) 9.. (3) u u u u ω 9. F = µ mg µ 9 u > k u(t) F = m ü(t) m ü(t) + k u(t) + F = u < k u(t) + F = m ü(t) m ü(t) + k u(t) F = k F u(t) m 9. m ü(t) + k u + Sgn ( u) F = (9.9) Sgn + u Sgn ( u) = u < (9.3) (9.5) 9. (9.9) 3 F (9.9) { u d u + ω u + Sgn ( u) F } du = { k ω u d u + u + Sgn ( u) F } { d u + Sgn ( u) F } = k k 9 (9.46) Heaviside m ü(t) + k u + H( u) F H( u) F =

38 9 t = 5 u(t) F/ k u(t) F/ k 5 u(t) ω F / k 5 O 5 O 4 ωt π 5 5 9. ( u ω ) { + u + Sgn ( u) F } = const. (9.3) k 9. ( ±F / k, ) 9 k u < F = µ mg (9.) (9.9) u(t) (9.) 3 Sgn ( u) F u dt = F u dt = F du = F ( ) m ( u) + k u + µ mg u dt = f (t) u dt + const. 9..3 ()? m k c f (t) (9.5) m (9.7) ü(t) + β ω u(t) + ω u(t) = m f (t) (9.3) µ s (> µ ) k u µ s mg u µ s mg k

9.. 383 Fourier f (t) = f sin pt (9.3) ü(t) + β ω u(t) + ω u(t) = g sin pt, g f m, f (t) = f sin pt (9.33a, b, c) u p (t) u(t) = exp ( β ω t) (A sin ω d t + B cos ω d t) + u p (t) ( ) u p (t) sin pt, cos pt u p (t) = C sin pt + D cos pt (9.33a) sine, cosine sin pt ( p C β ω p D + ω C g ) + cos pt ( p D + β ω p C + ω D ) = ω p β ω p C β ω p ω p D = C D = ω p β ω p ( ω p ) + 4 β ω p β ω p ω p g ( p = ω { ω) ( p ) } ( p + 4 β ω ω) β p = ω g β = M d.5 5 { ( p ω O.8 π α π.5. β =...8 β = 3 3 9. g = g ω p ( ω p ) + 4 β ω p β ω p f ( p k ω) ) } ( p + 4 β ω) β p ω p ω u p (t) = { ( p ω f k ) } ( p + 4 β ω [{ ( p } ) sin pt β ω) p ] ω cos pt (9.34) u st f k u p (t) = u st M d sin (p t α), M d β p, tan α ω { ( p ) } ( ( p ) p (9.35a, b, c) + 4 β ω ω) ω

384 9 u st f M d α 9.3 t = M d α 9. β = p = ω M d M d p = ω β u =, v = ( ) (9.34) A u st = B u st = β β B u st { ( p ω β p ω { ( p ) } ( p + 4 β ω ω) p ω d ) } + 4 β ( p ω { ( p } ), ω) u(t) u st 5 O 5 M d 4 p =.9 β =.5 ω 9.3 ωt A B p =.9 β =.5 9.3 ω ω d (9.35b) M d p M d 9% 5 m k 3 ω p m k 9-. 9.7 3. 9. 3 c

9.. 385 4. u =, v = f (t) = f sin pt 9.3?? β =. p.,.5,.97,.,.5 ω ±M d u > u st M d 5. β f (t) = f sin ωt p = ω (9.35b) M d u p (t) sin pt = sin ωt 6. (9.9) u() = u >, u() = v > τ ωt x u F/ k +, y v ω F / k, τ, < τ tan y x < π, τ n = τ n + π (n ) τ n τ < τ n+ (n ) u(τ) = F/ n (x cos τ + y sin τ) + k x + y ( )n+ () 9.4 w(t) u(t) m ü(t) + c { u(t) ẇ(t)} + k {u(t) w(t)} = (9.36) u(t) m k c w(t) 9.4 Newton ü(t) v(t) u(t) w(t) (9.37) (9.36) m v(t) + c v(t) + k v(t) = m ẅ(t) (9.38) (9.5) v(t) f (t) = m ẅ(t) = ( ) ( ) (9.39) (9.39)

386 9 4 w(t) = w sin pt (9.4) β =.5 M w (9.38) m v(t) + c v(t) + k v(t) = m p w sin pt 5.8 f = m p w (9.35) ( p ) v p (t) = w M w sin (p t α), M w = Md (9.4a, b) ω α (9.35c) (9.4b) M w 9.5 O. 3 9.5 p ω p ω vp w α π p ω M d, α v p w ( p / ω ) sin (pt) vp / ω ẅ k vp = m ẅ 5 v(t) v(t) m k p p ω 8 vp (t) w(t) m k p ω vp (t) / ω ẅ(t) (3) m t = τ f (t) ϵ t = τ ϵ / t = τ + ϵ / τ+ϵ/ τ ϵ/ m ü dt = τ+ϵ/ τ ϵ/ ( m d u = m u τ + ϵ ) ( m u τ ϵ ) 4 Fourier 5 (p ω) (M w ) α (p ω) (M w ) α π

9.. 387 ϵ τ+ϵ/ τ ϵ/ k u dt, τ+ϵ/ τ ϵ/ c u dt = τ+ϵ/ τ ϵ/ c du = c {u(τ + ϵ/) u(τ ϵ/)} ( m u τ + ϵ ) ( m u τ ϵ ) = τ+ϵ/ τ ϵ/ f (t) dt (9.4) t = τ f (t) t = τ f (t) 9.6 f (t) (4.57) Dirac δ(x a) c ψ(a) b < a < c ψ(x) δ(x a) dx = b a < b c < a (9.43) f (t) = δ (t τ) (9.4) = τ+ϵ/ τ ϵ/ f (t) dt = O f (t) ϵ τ+ϵ/ τ ϵ 9.6 τ ϵ/ δ (t τ) dt = t = τ ϵ (9.4) m u(τ) = τ+ϵ/ τ ϵ/ f (t) dt = τ+ϵ/ τ ϵ/ δ (t τ) dt = u(τ) = m / m t = τ t τ t m ü(t) + c u(t) + k u(t) =, u(τ) =, u(τ) = m (9.44a, b, c) t < τ u(t) δ(t τ) [ ] (9.44a) u(t) = exp ( β ω t) (A cos ω d t + B sin ω d t) (9.44b) (9.44c) = exp ( β ω τ) (A cos ω d τ + B sin ω d τ), m = exp ( β ω τ) ω d ( A sin ω d τ + B cos ω d τ) β ω exp ( β ω τ) (A cos ω d τ + B sin ω d τ) cos ω d τ sin ω d τ sin ω d τ cos ω d τ A B = exp (β ω τ) mω d

388 9 A B = exp (β ω τ) cos ω d τ sin ω d τ sin ω d τ cos ω d τ m ω d = exp (β ω τ) m ω d sin ω d τ cos ω d τ t < τ u(t) = u i (t; τ) = u i (t τ) exp { β ω (t τ)} sin {ω d (t τ)} t > τ m ω d (9.45) τ (t τ) u i (t τ) [ ][ ] [ ][ ] [ ] (9.45) t τ H(t τ) =, t < τ, t > τ δ(t τ) d = dh(t τ) dt (9.46) H(t τ) O τ t 9.7 Heaviside Heaviside δ(t τ) (9.43) Dirac d = (.) Heaviside t > β u i (t τ) = m ω d exp { β ω (t τ)} H(t τ) sin {ω d (t τ)} (9.47) u i (t τ) = H(t τ) sin {ω (t τ)} (9.48) m ω (9.44b) (9.44c) t = τ (4) 9.8 F u(t) = F u i (t τ) 3 u(t) = F n u i (t τ n ) n=

9.. 389 t F m ω d exp { β ω (t τ )} H(t τ ) sin {ω d (t τ )} Heaviside t < τ t Heaviside 9.8 f (t) τ Riemann τ f (τ) f (t) O f (t) f (t) F F F F 3 t t O τ τ τ τ 3 τ τ f (τ) u i (t τ) t = u(t) = τ f (τ) u i (t τ) τ= O τ 9.8 t Heaviside τ 6 τ dτ f (t) u(t) dτ f (τ) u i (t τ) u(t) = u i (t τ) f (τ) dτ (9.49) Duhamel t = u i (t τ) t = τ f (τ) Green 9.3 f sin pt u(t) u st = ω ω p sin pt p ω ω ω p sin ωt, u st f k (9.49) f (τ) = f sin pτ (9.48) u(t) = f m ω sin pτ H(t τ) sin ω(t τ) dτ = f m ω t sin pτ sin ω(t τ) dτ Heaviside t > τ τ t u(t) = f ( t t ) sin ωt sin pτ cos ωτ dτ cos ωt sin pτ sin ωτ dτ m ω t τ! 6 Riemann

39 9 (5) Duhamel f (t) Northwestern Olmstead (98 Differential Equations of Mathematical Physics m ü + c u + k u = f (t), u() =, u() = (a) t = τ u i (t τ) m ü i + c u i + k u i = δ(t τ), u i () =, u i () = (b) δ(t τ) (9.43) Dirac u(t) u i (t τ) (b) t = τ u i (t τ ) m ü i c u i + k u i = δ(t τ ), u i ( ) =, u i ( ) = (c) t = (a) u i u i (t τ ) (m ü + c u + k u) dt = u i (t τ ) f (t) dt = ( u i m u u i m u + u i c u ) + u ( m ü i c u i + k u ) dt (a) (a) 3 (c) (c) 3 (c) = u ( m ü i c u i + k u ) dt = u δ(t τ ) dt = u(τ ) (9.43) (d) (d) = u(τ ) = (d) = t τ u(t) = u i (τ t) f (τ ) dτ Close but no cigar! u i (t τ ) f (t) dt u i (t τ ) (m ü i + c u i + k u i ) dt = (d) (e) u i (t τ ) δ(t τ) dt ( f ) (a) (a) 3 (c) (c) 3 (c) = ( u i m u i u i m u i + u ) i c u i + ( u i m ü i c u i + k u ) dt = u i (t τ) δ(t τ ) dt = u i (τ τ)

9.. 39 ( f ) u i (τ τ ) ( f ) u i (τ τ) = u i (τ τ ) (g) (3.44) (4.65) (g) (e) u(t) = u i (t τ ) f (τ ) dτ τ u(t) = Duhamel u i (t τ) f (τ) dτ (9.5) (6) [8] t t u u(t t), u u(t), u + u(t + t) u(t) u st 5 O 5 ω t =.5 ω t = 4 ωt t u(t) = ( u + u ), ü(t) = { u + u t t t 9.9 u } u = ( u u + u +) (9.5a, b) t ( t) t t ± t t (9.5) u + [ u + f (t) = ( τ + β ) k τ m ( u u + u +) + c ( u + u +) + ku = f (t) ( t) t ( ) ( u τ τ β ) ] u, τ ωt, ω τ k m (9.5a, b, c) t t t u u t u + u( t) t = u() = u() u t = u t =

39 9 start ω β t N f (t)/ k n = u := u(), u := u() u( t) = Eq.(9.53) t = n t u + := Eq.(9.5a) u, ü := Eq.(9.5) t u, u, ü u := u, u := u + n = n + no n = N yes end 9. ẅ(t) m/ s 4 u(t) w(t) 3 ü(t) m/ s t (s) t (s) 9. u (9.5) u + u = u t u + ( t) t = 3 t = ü ü() = m { f () c u() k u()} u( t) = } { τ u() τ ( + β τ) u() ω + f () τ k (9.53) t = u 9.3 9.9 τ = ω t =.5 ω t = 9. 9. 9.8 t s / π [8] 9.9 τ 7 ω t =

9.. 393 dt t 9-3 7. f sin pt Duhamel 9.3 8. 9. 9. Duhamel u st f k u(t) u st ωt β =., ωt =, ωt = 3 ωt Heaviside f O f (t) t t t 9. f (t) = f {H(t t ) H(t t )} Duhamel (9.47) Heaviside. 9. Duhamel. t = τ v(τ) = (7) 9 9.6 ϵ δt 9. t =, t = δt f = F δt t = F δt δt 9.3 β =. ωδt =. ωδt =. m =, kg h = 5 m F = m hg = 9.9 kn s ωδt =..5 s f = MN δt =.8 s ω (8) exp (ipt) f (t) = f exp (ipt) = f (cos pt + i sin pt) (9.54)

394 9 mω d u(t) F.8 Impulse mω d u(t) F ωδt =.5 ωδt =. ωδt =.5.4 O ωδt =. ωδt =. β =..4.8..6 ωt O Impulse ωδt =. 4 8 ωδt =. β =. ωt 9.3 u(t) = ϕ exp (ipt) ϕ (9.5) m ω = k ( p + i β ω p + ω ) ϕ exp (ipt) = f m exp (ipt) ϕ = f H(ip), H(ip) m ( ω p ) + i β ω p = k { ( p } + i β ω) ( p ω) (9.55a, b) u(t) = f H(ip) exp (ipt) = H(ip) f (t) (9.56) H(ip) 9..3 () H(ip) α H(ip) = H(ip) exp ( iα), α arg (H(ip)) (9.57a, b) H(ip) = k β p, tan α = ω { ( p ) } ( ( p ) p + 4 β ω ω) ω (9.35b) M d H(ip) H(ip) = M d k u(t) = u st M d exp {i (pt α)} (9.58) α (9.35c) 9..3 () w(t) = w exp (ipt)

9.. 395 ϕ = w p ( ω p ) + i β ω p v(t) = w H w (ip) exp (ipt), H w (ip) ( p ω) { ( p } + i β ω) p ω (9.59a, b) H w (ip) = M w α = arg (H w (ip)) = arg (H(ip)) f = (9.56) u(t) = H(ip) exp (ipt) ( ) Duhamel t = f (τ) = exp (ipτ) Duhamel u(t) = u i (t τ) exp (ipτ) dτ s = t τ ds = dτ, s : = u i (s) exp {ip (t s)} ds = exp (ipt) ( ) H(ip) = u i (s) exp ( ips) ds u i (t) exp ( ipt) dt H(ip) = F {u i (t)} (9.6) H(ip) u i Fourier (9.64b) F Fourier 9..4 () (9.6) M d = k F(u i ) 9..4 () Fourier t m k

396 9 Fourier Fourier T T T = Fourier f (t) T Fourier T/ T/ T/ T/ f (t) = a + a n cos npt + b n sin npt, a n = f (t) cos npt dt, b n = f (t) sin npt dt T T n= n= (9.6a, b, c) p p = π (9.56) (9.57) T a u(t) a n H(inp) cos (npt α) + b n H(inp) sin (npt α) n= n= Fourier (np) H(inp) Fourier (np) Fourier f a n, b n np (n =,,, ) ω, β u H Fourier O np H filter O np Fourier np 9.4 9.4 Fourier ω β H (9.6a) Euler 7 cos θ = eiθ + e iθ e iθ + e iθ, sin θ = eiθ e iθ i (9.6a, b) f (t) = a + n= a n ( e inpt + e inpt) + n= b n ( e inpt e inpt) = a ( i + an + b ) n e inpt + i n= (9.6b) (9.6c) n = a n ± b n i = T/ T = T T/ T/ T/ n= ( an b ) n e inpt i f (τ) (cos npτ ± i ) sin npτ dτ = T/ f (τ) { e inpτ + e inpτ ( e inpτ e inpτ)} dτ T T/ f (τ)e inpτ dτ 7 [4] e i π + = TEX Version π e

9.. 397 T ( 5 s).5. T (s).3 F. F 9.5 Fourier f (t) = e inpt T/ f (τ)e inpτ dτ + e inpt T T = n= e inpt T n= T/ T/ T/ f (τ)e inpτ dτ + n= n= e inpt T T/ T/ T/ T/ Fourier f (t) = c n (p) exp (inpt), c n (p) = T n= f (τ)e inpτ dτ f (τ)e inpτ dτ = T/ T/ e inpt T n= T/ T/ f (τ)e inpτ dτ f (τ) exp ( inpτ) dτ (9.63a, b) c n Fourier p = π T, p n = n p (9.63) f (t) = π exp (ip n t) p n= T/ T/ T p, p n p n p f (t) = π f (t) = π exp (ipt) dp F(ip) exp (ipt) dp F (F), F(ip) f (τ) exp ( ip n τ) dτ p n= f (τ) exp ( ipτ) dτ dp f (τ) exp ( ipτ) dτ F( f ) (9.64a, b) (9.64b) f (t) Fourier (9.64a) Fourier / π / π Fourier 9.4 f Fourier Fourier f (t) Fourier 7 nm.5 5 s 3 nm 8.3 5 s 9.5 Fourier s f (t) F(ip) 8 Northwestern school color Northeastern

398 9 () 9 f (t) σ f { f (t) } = π { lim T F(ip) f (t) Fourier F(ip) } dp (9.65) T F(ip) S f (p) lim T T (9.66) f (t). f (t) S f (p). f (t) = sin qt q S f (p) = δ(p q) δ (9.43) Dirac F(ip) Fourier f (t) τ T/ R f (τ) lim f (t) f (t + τ) dt (9.67) T T T/ f (t) τ R f () = σ f Fourier = R f (τ) exp ( ipτ) dτ = lim T T = lim T T = lim T T T/ T/ lim T T/ f (t) f (t + τ) exp ( ipτ) dτ T T/ f (t) f (t + τ) exp (ipt) exp ( ip(t + τ)) dτ f (t) exp (ipt) dt F( ip) F(ip) = lim T f (t + τ) exp ( ip(t + τ)) d(t + τ) F(ip) = S f (p) T S f (p) = F { R f (τ) }, R f (τ) = F { S f (p) } (9.68a, b) Wiener-Khintchine 9 ( f g) (τ) f (t) g(τ t) dt Khinchin Khinchine

9.. 399 (3) Duhamel u(t) Duhamel u(t) = R u (τ) = lim T T T/ T/ R u (τ) lim T T T/ T/ u(t) u(t + τ) dt (9.69) u i (t ξ) f (ξ) dξ { } { } u i (t ξ) f (ξ) dξ u i (t + τ η) f (η) dη dt s = t ξ, ζ = t + τ η T/ { } { = lim u i (s) f (t s) ds T T T/ { T/ = u i (s) ds u i (ζ) dζ lim T T T/ ξ = t s f { T/ s } = u i (s) ds u i (ζ) dζ lim f (ξ) f (ξ + τ + s ζ) dξ = T T T/ s u i (ζ) f (t + τ ζ) dζ } f (t s) f (t + τ ζ) dt Fourier S u (p) = R u (τ) exp ( ipτ) dτ = η = τ + s ζ = = u i (s) exp (ips) ds = H(ip) H( ip) S f (p) } dt u i (s) u i (ζ) R f (τ + s ζ) ds dζ u i (s) u i (ζ) R f (τ + s ζ) exp ( ipτ) ds dζ dτ u i (s) u i (ζ) R f (η) exp { ip (η + ζ s)} ds dζ dη u i (ζ) exp ( ipζ) dζ R f (η) exp ( ipη) dη S u (p) = H(ip) S f (p) (9.7) (9.58) ( Md ) S u (p) = S f (p) k S f (p) S u (p)

4 9 (9.7) H(ip) S f S u A A: H a B B: H b O p O p 9.6 9.4 Fourier p 9.6 S f (p) S u (p) B (4) (9.38) f (t) ẅ m ẅ [7] ü Duhamel (9.47) u(t) = t u = = f (τ) m ω d exp { β ω (t τ)} H(t τ) sin {ω d (t τ)} dτ f (τ) e βω(t τ) [ βωh(t τ) sin{ω d (t τ)} + δ(t τ) sin{ω d (t τ)} + ω d H(t τ) cos{ω d (t τ)} ] dτ m ω d f (τ) e βω(t τ) H(t τ) [ βω sin{ω d (t τ)} + ω d cos{ω d (t τ)} ] dτ m ω d t ( f (τ) ü = βω e βω(t τ) [ βωh(t τ) sin{ω d (t τ)} + δ(t τ) sin{ω d (t τ)} + ω d H(t τ) cos{ω d (t τ)} ] ) dτ m ω d ( f (τ) + ω d e βω(t τ) [ βωh(t τ) cos{ω d (t τ)} + δ(t τ) cos{ω d (t τ)} ω d H(t τ) sin{ω d (t τ)} ] ) dτ m ω d ( f (τ) = βω e βω(t τ) H(t τ) [ βω sin{ω d (t τ)} + ω d cos{ω d (t τ)} ] ) dτ m ω d ( f (τ) + ω d e βω(t τ) H(t τ) [ βω cos{ω d (t τ)} ω d sin{ω d (t τ)} ] ) dτ + f (t) m ω d m = ( ) β ω ω f (τ) d e βω(t τ) H(t τ) sin{ω d (t τ)} dτ m ω d f (τ) βωω d e βω(t τ) H(t τ) cos{ω d (t τ)} dτ + f (t) m ω d m

9.. 4 S a ω i S a ω j m k u (S a ) max (gal) O (S a ) max p O (S a ) max p c S a ω k S a ω l ẅ O (S a ) max p O (S a ) max p t O T i T j T k T l (T = π / ω ) 9.7 (9.37) v (9.38) f (t) = mẅ ( ω d β ω ) v = ẅe βω(t τ) H(t τ) sin{ω d (t τ)} dτ + βω ẅe βω(t τ) H(t τ) cos{ω d (t τ)} dτ ẅ ω d (9.37) ũ i (t τ) exp { β ω (t τ)} H(t τ) ü(t) = v + ẅ = ω d ũ i (t τ) ẅ(τ) dτ (9.7) β sin{ω β d (t τ)} + β cos{ω d(t τ)} β (9.7) S a (p) = ω d H(ip) S e (p), H(ip) F {ũ i (t)} (9.73a, b) S a ü S e ẅ ẅ ω i (9.73) (S a ) max ω i 9.7 [7] (S a ) max ω T T (S a ) max m m (S a ) max 9. 9.. ()

4 9 [8] Newton 9.8 f (t) c m c m k k u (t) u f (t) u (t) c u f 9.8 f k u c u + k (u u ) + c ( u u ) k u c ( u u ) k (u u ) f f k (u u ) c ( u u ) Newton f k u c u + k (u u ) + c ( u u ) = m ü, f k (u u ) c ( u u ) = m ü m ü (t) + (c + c ) u (t) c u (t) + (k + k ) u (t) k u (t) = f (t), m ü (t) c u (t) + c u (t) k u (t) + k u (t) = f (t) M ü(t) + C u(t) + K u(t) = f(t) (9.74) u f M C K u(t) u (t) u (t) t, f(t) f (t) f (t) t, (9.75a, b) m M m, C c + c c c c, K k + k k k k (9.75c, d, e) (9.74) u (t), u (t) u() = u, u() = u (9.76a, b) u, u (9.75a) 3

9.. 43 () C f M ü(t) + K u(t) = (9.77) u(t) = A exp (iωt), A A A t (9.78a, b) A, A A (9.78a) (9.77) ( K ω M ) A exp (iωt) = t ( K ω M ) A = (9.79) A, A det ( K ω M ) = (9.8) ω (k + k ) m ω k det k k m ω = { (k + k ) m ω } ( k m ω ) k = m m ω 4 {m k + m (k + k )} ω + k k = {( ) ( ) ω 4 k k + + m m ( m m ) ( )} k ω ω = {( ) ( ) ( ) ( )} {( ) ( ) k k m k k k + + ± + + m m m m m m m ( ) ( ) ω k k + = (9.8) m m ( m m ) ( k m )} ( ) ( ) k k 4 m m > ω = {ω { ω + ω + m ω} ± + ω + m ω} 4 ω ω > (9.8) m m m, ω k m, ω k m (9.83a, b, c) ω ω, ω ±ω, ±ω ω, ω ( < ω < ω ) ω = ω, ω = ω

44 9 (9.79) A A A A = d A k = (9.84) A k m ω k k d =, d k m ω = k m ω u (t) = D sin ω t + D cos ω t + D 3 sin ω t + D 4 cos ω t, u (t) = d (D sin ω t + D cos ω t) + d (D 3 sin ω t + D 4 cos ω t) u(t) = ϕ (D sin ω t + D cos ω t) + ϕ (D 3 sin ω t + D 4 cos ω t) (9.85a) u(t) = a ϕ sin (ω t α ) + a ϕ sin (ω t α ), ϕ =, ϕ = (9.85b, c, d) D D 4 a, a, α, α (9.76) ϕ n (n =, ) k = k = k, m = m = m (9.76) u() = u = u t, u() = u = t ( ) d d (9.8) ω 4 3 ω ω + ω 4 =, ω k m ω = 3 5 ω = 5 ω.68 ω, ω = 3 + 5 ω = 5 + ω.68 ω d n (9.84) d = 5.68, d = 5 +.68 3 d d 9.9 ω ω 3 (p.5)

9.. 45 u u, u u O ϕ ϕ c ω ω c ω t 5 5 u u, u u. O. u u, u u O ω ω t 5 5 ω ω t 5 5 9.9 D D 4 ( ) D + D 4 = u, d D + d D 4 =, ω D + ω D 3 =, d ω D + d ω D 3 = u (t) u = D =, D = d u d d, D 3 =, D 4 = d u d d d d d cos ω t d d d cos ω t, u (t) u = d d d d (cos ω t cos ω t) 9.9 ω / d ω / d u (3) (9.74) M ü(t) + C u(t) + K u(t) = (9.86) u(t) = A exp (iωt), A A A t (9.87a, b) A (9.87a) (9.86) ( K ω M + i ω C ) A exp (iωt) =

46 9 t ( K ω M + i ω C ) A = (9.88) det ( K ω M + i ω C ) = (9.89) (9.89) ω 4 9.. (6) (4) 9.. (6) 9.8 9.3 k (u u ) k (u w), k (u u ) w(t) u (t) u (t) k (u w) k (u u ) 9.3 m ü + k (u w) k (u u ) =, m ü + k (u u ) = v (t) u (t) w(t), v (t) u (t) w(t) (9.9a, b) m v + (k + k ) v k v = m ẅ, m v k v + k v = m ẅ M ü(t) + K u(t) = f w (t), u(t) v (t) v (t) t, f w (t) ẅ(t) m m t (9.9a, b, c) M K (9.75c) (9.75e) sine w(t) = w sin pt (9.9) sin pt u = A sin pt (9.93)

9.. 47 A w A w 5 5 O ω ω ω 3 ω p ω O ω ω ω 3 ω p ω 9.3 (9.9) (9.93) (9.9a) (9.9c) ( K p M ) A = p w m m k + k p k m m A k k p A = p w m m ω + m ω p m ω ω p ω A A = p w (9.83) A = p w ( ω p ) p m ω ω + m ω p A ω = p w ( p ) ω + m ω p ω + ω + m ω p ( p ) p 4 ( ω + ω + m ω ) p + ω ω (9.8) ω p ω, ω ( p ) = ( p ω ) ( p ω A w p = ( ) ( A p ω p ω) w ) ω + m ω p ω + ω + m ω p (9.94) p

48 9 m = m = m, k = k = k ω k/ m ω = 3 5 3 + 5 ω.68 ω, ω = ω.68 ω (9.94) A w A w = ( p ω ) ( ω ( p ω ) ) ( p ω ω ) ( ) ω ω ( p ) ω ) ( p 3 ω 9.3 p p / ω =.4 p / ω = 3.73 9-4. 9.8 f (t) = f sin pt 3. 9.3 G u(t) θ(t) M G J k r k h u = U exp(iωt), θ = Θ exp(iωt) O x x = U / Θ g ω h kh M, ω r kr J, r J M ( ) ωr = 4, ω h ( ) h = 3 r h h x k r G θ(t) u(t) O k h 9.3 4. m k ω = k / m m / m =., k / k =. (9.9)

9.. 49 9.. () 9.33 N ( ) N n (N > n > ) n m n ü n c n u n + (c n + c n+ ) u n c n+ u n+ k n u n + (k n + k n+ ) u n k n+ u n+ = f n M ü(t) + C u(t) + K u(t) = f(t) (9.95) N = C u(t) u (t) u (t). u N (t), f(t) f (t) f (t). f N (t) c + c c c + c 3 c 3...., M, K m Symm. m.... m N m N k + k k k + k 3 k 3...., (9.96a, b, c) (9.96d, e) Symm. c N + c N c N c N Symm. k N + k N k N k N 3 u() = u, u() = u (9.97a, b) u u N () C f u = ϕ exp (iωt) (9.98) ( K ω M ) ϕ = (9.99) c m c n m n k f (t) u (t) k n c n ( u n u n ) k n (u n u n ) f n f n (t) u n (t) c n+ ( u n+ u n ) k n+ (u n+ u n ) 9.33 N ( 3) c N k N m N f N (t) u N (t)

4 9 ϕ det ( K ω M ) = (9.) ω n > [37] ω n > ω n (n =,,, N) ϕ n ( K ω n M ) ϕ n = (9.) (9.) (9.) ϕ n (9.98) N { u(t) = An exp (i ω n t) + B n exp ( i ω n t) } N ϕ n = {A n sin (ω n t) + B n cos (ω n t)} ϕ n (9.) n= n= A n B n A n B n (9.97) N N ω < ω < < ω N, T > T > > T N (9.3a, b) 6.4. (3) exp ( ) (c + c t) exp ( ) (9.3) (3) (9.) (9.95) i j (i j) (9.) K ϕ i = ω i M ϕ i, K ϕ j = ω j M ϕ j ϕ t j ϕt i ϕ t j K ϕ i = ω i ϕt j M ϕ i, ϕ t i K ϕ j = ω j ϕt i M ϕ j ( ) i j ϕ t j M ϕ i = ( ϕ t j M ϕ i) t = ϕ t i M t ϕ j = ϕ t i M ϕ j, ϕ t j K ϕ i = ( ϕ t j K ϕ i) t = ϕ t i K t ϕ j = ϕ t i K ϕ j

9.. 4 ( ) ϕ t i K ϕ j = ω i ϕt i M ϕ j, ϕ t i K ϕ j = ω j ϕt i M ϕ j ( ) = ( ω i ω j) ϕ t i M ϕ j (9.3) i j ω i ω j ϕ n ϕ t i M ϕ j =, j i (9.4) M I ϕ i ϕ j M j = i ϕ t i M ϕ j =, j i m i, j = i (9.5) m i m i i (9.) (9.5) ϕ t i K ϕ j = ω, j i j ϕt i M ϕ j = m i ω i, j = i (9.6) K K : 4 N u w (u, w) u t w N I (u, w) = u t I w Euclid A 5 B (u, w) A u t A w, A a Symm. a.... a N, (u, w) B u t B w, B b b b N Symm. b b N.... N 4 (5.57) 5 A B b NN

4 9 (4) (9.97) 9.9 N (9.) q n (t) N u(t) = q n (t) ϕ n (9.7) n= q n (t) (9.95) C, f(t) N N q n (t) M ϕ n + q n (t) K ϕ n = n= n= n= n= ϕ t j j (9.5) (9.6) j n N N q n (t) ϕ t j M ϕ n + q n (t) ϕ t j K ϕ n = m j q j (t) + m j ω j q j(t) = q j (t) q j (t) + ω j q j(t) = (9.8) ω j (9.7) N N 6 N q j (t) = A j sin ω j t + B j cos ω j t (9.9) A j, B j (9.97) N q n () ϕ n = u, n= N q n () ϕ n = u n= ϕ n q n () q n () (9.9) ω n A n B n ϕ n N M ϕ j ( j M N) ϕ t j M N q n () ϕ t j M ϕ n = ϕ t j M u, n= N q n () ϕ t j M ϕ n = ϕ t j M u n= (9.5) (9.6) q j () = ϕt j M u m j, q j () = ϕt j M u m j (9.a, b) 6

9.. 43 q j (t) (9.9) B j = ϕt j M u m j, A j = ϕt j M u m j ω j A j, B j j q j (t) q j (t) + ω j q j(t) =, (9.8) q j () = ϕt j M u m j, q j () = ϕt j M u m j (9.) (9.7) q j (t) j N M q j (t) ( j =,,, M N) u(t) M N n= q n (t) ϕ n M 3 m = m = m = m 3, k = k = k = k 3 u = u t, u = t u u u u ϕ ϕ ϕ ϕ 3 t (9.99) k mω k k k mω k k k mω ϕ ϕ ϕ 3 = ( ) u 3 u 4 9.34 3 ω t ω 6 5ω ω4 + 6ω 4 ω ω 6 =, ω i = ξ i ω ω k m ξ ω ω.445, ξ ω ω.47, ξ 3 ω 3 ω.8 3 7 ( ) ϕ =.8, ϕ =.445, ϕ 3 =.47.47.89.555 7 N.3. Scilab roots

44 9 8 3 (9.5) (9.5) m i = µ i m µ m m = +.8 +.47 = 9.96, µ m m =.84, µ 3 m 3 m =.863 (9.) ϕ i q () = u µ, q () =, q () = u µ, q () =, q 3 () = u µ 3, q 3 () = (9.9) A =, B = u µ, A =, B = u µ, A 3 =, B 3 = u µ 3 { u(t) = u ϕ µ cos (ξ ω t) + ϕ µ cos (ξ ω t) + } ϕ µ 3 cos (ξ 3 ω t) 3 9.34 3 3 (5) (9.95) M ü(t) + K u(t) = f(t) (9.) (9.7) N N q n (t) M ϕ n + q n (t) K ϕ n = f(t) n= n= ϕ t j j (9.5) (9.6) j n j = n N N q n (t) ϕ t j M ϕ n + q n (t) ϕ t j K ϕ n = ϕ t j f(t) m j q j (t) + m j ω j q j(t) = f j (t) n= n= f j (t) ϕ t j f(t) (9.) j f(t) ϕ n (9.7) q j (t) + ω j q j(t) = m j f j (t) (9.3) 8 ϕ i

9.. 45 q j (t) r j (t) (9.) q j (t) = A j sin ω j t + B j cos ω j t + r j (t) (9.4) q j () = B j + r j () = ϕt j M u m j, q j () = ω j A j + ṙ j () = ϕt j M u m j A j = ϕt j M u m j ω j ṙ j() ω j, B j = ϕt j M u m j r j () A j, B j (9.49) Duhamel f j (τ) q j (t) = A j sin ω j t + B j cos ω j t + H(t τ) sin { ω j (t τ) } dτ m j ω j Duhamel A j = ϕt j M u, B j = ϕt j M u m j ω j m j 3 9.9 m = m = m, k = k = k u u =, u = v, f = f sin pt ω k/ m ω i = ξ i ω 9.. () ξ ω ω.68, ξ ω ω.68 ϕ = d, d =.68, ϕ = d u u ω t 3 5 u u 9.35, d =.68 m m i = µ i m (9.) (9.) µ = m m = +.68 = 3.68, µ = m m =.38 q () = u µ, q () = v d µ, q () = u µ, q () = v d µ f = d f sin pt, f = d f sin pt Duhamel d j f q j (t) = A j sin ω j t + B j cos ω j t + ( sin pt m j ω j p)

46 9 d j A j = µ j ξ j A j, B j ( v ω ) d j f µ j ξ j k p/ ω ξ j ( ) p/, B j = u ω µ j v ω =.5 u, f k =.8 u, p ω = 9.35 9-5 5. m = m = m = m 3, k = k = k = k 3 3 u = u, u =, f = f sin pt, f k =.8 u, p ω = (6) 9.. (3) (9.95) (9.7) (9.5) (9.6) N N c i ϕ t i C ϕ j =, j i m i β i ω i, j = i (9.5) β i i f(t) f(t) (9.95) (9.7) N N N q n (t) M ϕ n + q n (t) C ϕ n + q n (t) K ϕ n = n= n= n=

9.. 47 ϕ t j j (9.5) (9.6) (9.5) j n N N N q n (t) ϕ t j M ϕ n + q n (t) ϕ t j C ϕ n + q n (t) ϕ t j K ϕ n = m j q j (t) + m j β j ω j q j (t) + m j ω j q j(t) = n= n= n= q j (t) + β j ω j q j (t) + ω j q j(t) = (9.6) q j (t) = exp ( β j ω j t ) [ A j sin { (ω d ) j t } + B j cos { (ω d ) j t }], (ω d ) j ω j β j (9.7a, b) (9.) A j, B j 9.37 m = m = m, c = c = c, k = k = k ω k m, c = β ω ω β u u =, u = v, v ω =.5 u, β =. β i = β ( ) di + di ξ i µ i (7) Rayleigh (9.5) c i β β m β n β i m + β i k β i k [6] Rayleigh β i m O ωm ω C = ζ m M + ζ k K (9.8) n ω 9.36 (9.5) (9.6) (9.8) ϕ i ϕ t i C ϕ i = ζ m ϕ t i M ϕ i + ζ k ϕ t i K ϕ i = m i ζ m + m i ω i ζk

48 9 (9.5) (9.6) (9.7) m i β i ω i β i β i m + β i k, β m i ζm, β k ω i ζk ω i i (9.9) ϕ t i C ϕ, j i j = m i β i ω i, j = i (9.) ζ m, ζ k m β m n β n (9.9) ζ m ζ k = ω m ω n ω m ω n ω m β n ω n β m β m β n ω n ω m (9.) 9.36 (8) (9.95) (9.7) N N N q n (t) M ϕ n + q n (t) C ϕ n + q n (t) K ϕ n = f(t) n= n= n= ϕ t j j (9.5) (9.6) (9.) j n N N N q n (t) ϕ t j M ϕ n + q n (t) ϕ t j C ϕ n + q n (t) ϕ t j K ϕ n = ϕ t j f(t) n= n= n= m j q j (t) + m j β j ω j q j (t) + m j ω j q j(t) = f j (t) f j (t) (9.) q j (t) + β j ω j q j (t) + ω j q j(t) = m j f j (t) (9.) N M ( N) M

9.. 49 9..3 9..3 (6) u u(t t), u u(t), u + u(t + t) u(t) = ( u + u ), ü(t) = t t { u + u (9.95) u + = D t u } u = ( u u + u +) (9.3a, b) t ( t) M ( u u + u +) + C ( u + u +) + K u = f(t) ( t) t [ ( f(t) K ) ( t M u t M ) ] ( t C u, D t M + ) t C (9.4a, b) u (9.3) u + u = u t u + t ü t = t = ü() = M { f() C u() K u()} 3 ) u( t) = (I t M K u() t ( I + t ) M C u() + t M f() (9.5) t = u I N N 9.4.7 9.55 9.. (6) 9.37 ω k/ m ω t =.4 ω t =.3 t T N π (9.6) [8] T N ω t ω T.4 u u u u ω t 3 9.37 N Newmark Wilson θ [8]

4 9 9.3 9.3. () α T x u(x, t) ρ x 9.38 β T u(x + x, t) ) ) 3) 9 9.38 T sin β T sin α = ρ x A u(x, t) t ρ A sin α α = u(x, t) u(x + x, t) u(x, t), sin β β = = + u(x, t) x x x x x T u(x, t) x = ρa u(x, t) t (9.7) x c c T ρa (9.8) (9.7) (6.9) (3.55) u(x, t) x = c u(x, t) t (9.9) l u(, t) =, u(l, t) = (9.3a, b) u(x, ) = u (x), u(x, ) = v (x) (9.3a, b) 9

9.3. 4 () (9.9) u(x, t) = q(t) ϕ(x) ϕ (x) ϕ(x) = q(t) c q(t), t >, l > x > x x t t x ϕ (x) ϕ(x) = q(t) c q(t) = Λ (const.) q(t) Λ c q(t) =, ϕ (x) Λ ϕ(x) = (9.3a, b) (9.3) ϕ() q(t) =, ϕ(l) q(t) = ϕ() =, ϕ(l) = (9.33a, b, c, d) (9.3b) Λ > Λ = µ ϕ(x) = A sinh (µx) + B cosh (µx) (9.33c) (9.33d) B =, A sinh (µl) = µl > A =, B = ϕ(x) Λ = ϕ(x) = A x + B B =, Al = Λ Λ = µ < ϕ(x) = A sin µx + B cos µx B =, A sin µl = ϕ(x) A sin µl = µl = nπ, Λ = ( n π ), n =,, (9.34a, b, c) l A ( nπ x ) ϕ(x) sin µx = sin l (9.35) (9.3a) (9.34c) Λ q(t) { q(t) = L (sin cµt, cos cµt) = L sin ( nπc t l ) ( nπc t )}, cos l

4 9 ω ω n ω n = cµ = nπc l = nπ l T ρa (9.36) T 3 l µ ϕ(x) u(x, t) = (A n sin cµ n t + B n cos cµ n t) sin µ n x, µ n nπ l n= (9.37a, b) µ (9.3) u(x, ) = B n sin µ n x = u (x), u(x, ) = cµ n A n sin µ n x = v (x) n= n= A n, B n < x < l u (x) = B n sin µ n x n= B n u (x) sine-fourier Fourier Fourier A n, B n sine ϕ(x) (3) ϕ(x) (9.9) x t x (9.3b) u(x, t) = u(x, t) x c t d ϕ(x) dx = Λ ϕ(x), Λ = µ (9.38a, b, c) (9.3) u(, t) =, u(l, t) =, ϕ() =, ϕ(l) =, (9.39a, b, c, d) 3 3 (,, ) C H T l A

9.3. 43 9. N u(x, t) K u(t) = M ü(t) = u(x, t), u(, t) =, u(l, t) = x c t K ϕ n = ω n M ϕ n ϕ n (x) = µ n ϕ n (x), ϕ n () =, ϕ n (l) = (i j) ϕ t i M ϕ j =, ϕ t i K ϕ j = ϕi, ϕ j = N u(t) = q n (t) ϕ n u(x, t) = q n (t) ϕ n (x) n n j q j (t) + ω j q j(t) = q j (t) + ( cµ j ) q j (t) = Λ ϕ(x) Λ ϕ(x) d / dx ϕ n (9.) K ϕ n = ω n M ϕ n ω n ϕ K M ϕ(x) Λ µ ϕ(x) (9.36) c Λ = cµ 9., 3 9. 4 ϕ ϕ(x) (9.37b) µ n ϕ(x) Λ ϕ n (x), Λ n n (9.38b) n ϕ n (x) ϕ n (x) Λ n ϕ n (x) = (9.4) ϕ i (x) l l ϕ i (x) { ϕ n (x) Λ n ϕ n (x) } dx = ϕ i ϕ n dx = ϕ i ϕ l l n ϕ i ϕ n dx (9.39c) (9.39d) l l ϕ i ϕ n dx Λ n ϕ i ϕ n dx = i ϕ n l ϕ n ϕ i dx Λ i l ϕ n ϕ i dx = (Λ i Λ n ) l ϕ n ϕ i dx =, i n

44 9 (9.34c) l, i n ϕ n, ϕ i ϕ n (x) ϕ i (x) dx = l (9.4), i = n (9.4) (5.57) m =, w = (4) (9.38b) (9.39c) (9.39d) ϕ n (x) Λ n µ n (9.9) u(x, t) = ( n π ) q n (t) ϕ n (x), ϕ n sin µ n x, Λ n = µ n = (9.4a, b, c) l n= (9.4a) (9.9) q n (t) ϕ n (x) = q c n (t) ϕ n (x) (9.4) Λ n q n (t) ϕ n (x) = c n= n= n= q n (t) ϕ n (x) i (9.4) i ϕ i, ϕ i ( q i (t) Λ i c q i (t) ) = q i (t) + (cµ i ) q i (t) = q i (t) (9.3a) u(x, t) = n= q i (t) = A i sin cµ i t + B i cos cµ i t (A n sin cµ n t + B n cos cµ n t) sin µ n x (a) n= (a) (9.3) u(x, ) = B n sin µ n x = u (x), u(x, ) = cµ n A n sin µ n x = v (x) n= i n = i B i = ϕ i, u (x) ϕ i, ϕ i n= ϕ i, ϕ i B i = ϕ i, u (x), ϕ i, ϕ i cµ i A i = ϕ i, v (x) = l l u (x) sin µ i x dx, A i = ϕ i, v (x) cµ i ϕ i, ϕ i = cµ i l l v (x) sin µ i x dx Fourier-sine 9. (b)

9.3. 45 [77] u x u l, < x < l u (x) = ( u x ) l, l < x < a, v (x) (b) u(x, t) u O A n =, B n = 8u ( nπ ) (nπ) sin. tc a =.3.5.5. 9.39 9.39 4 5 9.4.7 x a (5) (a) u(x, t) = (A n sin cµ n t + B n cos cµ n t) sin µ n x = n= n= A n [ cos {µn (x ct)} cos {µ n (x + ct)} ] + f (x ct) g(x + ct) n= B n [ sin {µn (x ct)} + sin {µ n (x + ct)} ] u(x, t) = f (x ct) + g(x + ct) (c) f (x ct) t = t x = x f f (x ct ) t t = t + t x = x + c t f (x ct ) = f (x + c t c (t + t)) = f (x ct ) = f t = t, x = x t = t x = x f (x ct) x c g(x + ct) x c c c 9.4 (c) t = c c c g(x + ct) f (x ct) u(x, t) O 9.4 l x (9.9)

46 9 9.3. () σ t x 9.38 σ y x σ σ σ 9.4 T T = σ t y y 9.4 x (9.7) y T u y ( x y t) R o O y r θ x u(x, y, t) + u(x, y, t) = u(x, y, t) σ, c x y c t ρ (9.43a, b) u(x, y, t) 9.4 (r, θ) r r r r ( r ) + r (9.44) r θ C.4 R o ( r u(r, θ, t) r ) + r u(r, θ, t) θ = c u(r, θ, t) t, r < R o, π θ < π, < t (9.45) u(r o, θ, t) = (9.46) (9.45) r = u(, θ, t) < (9.47) R i > (9.47) u(r i, θ, t) = θ = ±π u(r, π, t) = u(r, π, t), u θ u (r, π, t) = (r, π, t) (9.48a, b) θ 3 θ π t = u(r, θ, ) = u (r, θ), u t (r, θ, ) = v (r, θ) (9.49a, b) 3

9.3. 47 () 9., 3 ( r r r ϕ(r, θ) r ) + r ϕ(r, θ) θ = Λ ϕ(r, θ) (9.5) ϕ(r, θ) Λ r (r R ) R ϕ(r, θ) = R(r) Θ(θ) (9.5) Λ r = Θ Θ = ζ (const.) ζ R (r) = dr / dr Θ (θ) = dθ / dθ ϕ (x) = dϕ / dx R (9.46) (9.47) Θ (9.48) r ( r R ) ( ζ + Λ r ) R =, Θ + ζ Θ = (9.5a, b) R(R o ) =, R() < (9.53a, b) Θ( π) = Θ(π), Θ ( π) = Θ (π) (9.54a, b) Θ(θ) (9.54) (9.5b) ζ π (9.54) ζ ζ ζ = n, n =,,, (9.55) Θ(θ) = Θ n (θ) = A n sin nθ + B n cos nθ, n =,,, (9.56) n n = ζ R (9.5a) r ( r R ) ( Λ r + n ) R = r R + r R ( Λ r + n ) R = (9.57a, b) R(r) Λ (Λ ) (9.57a) r R(r) r = R o R o R ( R ( r R ) o Λ r + n ) dr = R dr r R o R ( r R ) dr = R ( r R ) Ro R o r ( R R ) o dr = r ( R ) dr < (a)

48 9.5 J (x) J (x) J (x).5 O Y (x) Y (x) 4 8 x O.5.5 4 8 x 9.4 Bessel Y (x) (a) (Λ ) Λ = µ, µ R (9.57b) n Bessel Bessel r R + r R ( n µ r ) R = (9.58) R(r) = L {J n (µr), Y n (µr)} (9.59) J n n Bessel Y n n Bessel 9.4 (9.53b) r = Bessel r = Bessel R(r) R = R n (r) = J n (µ n r), n =,,, (9.53a) 9. Bessel m 3 n = (J ).4 5.5 8.65 n = (J ) 3.83 7.. n = (J ) 5.4 8.4.6....... J n (µ n R o ) = (b) µ n 9.4 Bessel (b) µ n n m =,, µ nm J n (µ nm R o ) =, n =,,,, m =,, 3, (9.6) 9. µ =.4 R o,, µ = 8.4 R o, µ nm R(r) R(r) = R nm (r) = J n (µ nm r), n =,,, m =,, 3, (9.6) Bessel µ π 3 3 C C

9.3. 49 (3) (9.56) Θ n (θ) Θ n, Θ m θ π Θ n (θ) Θ m (θ) dθ =, n m (9.6) Bessel Bessel R nm (9.5a) (9.58) ) ( r R ) r nm + (µ nm n R nm = R nk r dr dθ = R o { r r ) } ( r R ) nm + (µ nm n R nm R nk r dr R o ( r R ) nm R nk dr = r R Ro R o R nm R nk r R nm R nk dr = o r R nm R nk dr R o R = r R nm R nk dr o R n r R nm R nk dr + µ o nm R nm R nk r dr R nk R nm R o R = r R nk R nm dr n o R r R nk R nm dr + µ o nk R nk R nm r dr, 3 r = ( ) µ nm µ Ro nk R nk R nm r dr m k R nm, R nk r R o R nm (r) R nk (r) r dr =, m k (9.63) Bessel (9.63) (5.57) m =, w = r (4) (9.5) ϕ s nm(r, θ) = J nm (µ nm r) sin nθ, ϕ c nm(r, θ) = J nm (µ nm r) cos nθ, n =,,,, m =,, 3, (9.64a, b) (9.6) (9.63) ϕ a nm, ϕ b jk rθ π R o ϕ a nm(r, θ) ϕ b jk (r, θ) r dr dθ =, n j m k a b (9.65)

43 9 ϕ a nm(r, θ) (9.5) r r ( ) r ϕa nm(r, θ) + ϕ a nm(r, θ) = µ r r θ nm ϕ a nm(r, θ) (9.66) u(r, θ, t) = q a nm(t) ϕ a nm(r, θ) (9.67) a=s,c n= m= (9.45) a=s,c n= m= q a nm(t) { r r ( ) r ϕa nm(r, θ) + ϕ a } nm(r, θ) = c r r θ a=s,c n= m= (9.66) ϕ a nm a=s,c n= m= { q a nm (t) + (c µ nm ) q a nm(t) } = q a nm(t) ϕ a nm(r, θ) ϕ b jk (9.65) a = b, n = j, m = k q b jk (t) q b jk (t) + ( ) c µ jk q b jk (t) = (9.68) q b jk (t) = L { sin ( µ jk ct ), cos ( µ jk ct )} (9.69) u(r, θ, t) = n= m= [ Jnm (µ nm r) sin nθ { A s mn sin (µ mn ct) + B s mn cos (µ mn ct) } +J nm (µ nm r) cos nθ { A c mn sin (µ mn ct) + B c mn cos (µ mn ct) }] (9.7) (9.49) (9.7) u(r, θ, ) = u(r, θ, ) = n= m= n= m= { B s mn ϕ s nm(r, θ) + B c mn ϕ c nm(r, θ) } = u (r, θ), (µ mn ct) { A s mn ϕ s nm(r, θ) + A c mn ϕ c nm(r, θ) } = v (r, θ) ϕ s jk ϕc jk (9.65) v A s jk = (r, θ), ϕ s jk v rθ (µ mn ct) ϕ s jk,, A c ϕs jk = (r, θ), ϕ c jk u rθ jk (µ mn ct) ϕ c jk,, B s ϕc jk = (r, θ), ϕ s jk jk ϕ s jk, ϕs jk rθ rθ rθ rθ u, B c jk = (r, θ), ϕ c jk ϕ c jk, ϕc jk 9.43 θ cos nθ < θ < π / π / < θ < π rθ rθ

9.3. 43 ω = µ c ω = µ c ω = µ 3 c ω = µ c ω = µ c ω = µ 3 c ω = µ c ω = µ c ω = µ 3 c 9.43 r Bessel Fourier sine, cosine Fourier Bessel 3 Legendre [77] 9-6 6. (x = ) (x = l) u(x, t) x u (, t) =, u(l, t) = x 7. R i, R o ( < R i < R o ) u(r o, t) =, u(r i, t) = ( < θ < β < π) (9.55) Bessel?

43 9 9.4 9.4. 4 9.44 p(x, t) M V dx z Newton V(x, t) x + p(x, t) = m w(x, t) t (9.7) w(x, t) z m ρ A m = ρ A A M(x, t) O z V(x, t) A p(x, t) dx M(x + dx, t) x V(x + dx, t) 9.44 M(x, t) x V(x, t) = (9.7) 33 (9.7) (9.7) M(x, t) x + p(x, t) = m w(x, t) t (9.73) x (4.3b) M(x, t) = EI w(x, t) x (9.74) (9.73) x ( ) EI w(x, t) + p(x) = m w(x, t) x t m w(x, t) t + EI 4 w(x, t) x 4 = p(x, t) (9.75) x 4 (9.9) (4.8) x i, (i =, ), x =, x = l ( ) w(x i, t) = w i (t) n i EI 3 w x (x i, t) 3 θ(x i, t) w x (x i, t) = θ i (t) n i = S i (t), (9.76a) ( ) EI w x (x i, t) = C i (t) (9.76b) θ n i (4.6) w(x, ) = w (x), w t (x, ) = v (x) (9.77a, b) 33 9.4.8 ()

9.4. 433 9.4. 9. d4 ϕ(x) = Λ ϕ(x) dx 4 w(x, t) = ϕ(x) q(t) p(x, t) (9.75) q(t) q(t) = EI m ϕ (x) ϕ(x) = Λ (const.) (9.78) (9.76) w w x ϕ() =, ϕ () =, ϕ(l) =, ϕ (l) = ϕ(x) ϕ (x) = Λ m EI ϕ(x) q(t) Λ Λ m EI = 4µ4 < ϕ(x) = exp (µx) (a sin µx + b cos µx) + exp ( µx) (c sin µx + d cos µx) sin µl cosh µl cos µl sinh µl cos µl sinh µl sin µl cosh µl a c = a, c Λ = Λ m EI = µ4 > ϕ(x) = a sin µx + b cos µx + c sinh µx + d cosh µx (9.79) sin µl sinh µl sin µl sinh µl a c = a c sin µl sinh µl = n =,, sin µl = µl = nπ µ n = nπ l

434 9 a ϕ(x) ( nπ x ) ϕ(x) = ϕ n (x) = sin µ n x = sin l µ n ϕ n (x) Λ n = EI m µ4 n = EI m n q(t) (9.78) q n (t) + EI ( nπ ) 4 qn (t) = m l ( nπ l ) 4 (9.8) ω n ω n EI ( nπ ) 4 EI ( nπ ) ω n = (9.8) m l m l w(x, t) = q n (t) = A n sin ω n t + B n cos ω n t (A n sin ω n t + B n cos ω n t) ϕ n (x) (9.8) n= (9.8) k/ m EI/ m A n, B n 9.4.4 9.4.3 () ω w(x, t) = ϕ(x) exp (iωt) (9.83) (9.78) q(t) (9.83) p(x, t) (9.75) { EI ϕ (x) m ω ϕ(x) } exp (iωt) = t ϕ(x) (9.84) ϕ (x) m ω EI µ 4 m ω EI ϕ(x) = (9.84) (9.85) ϕ(x) = c sin µx + c cos µx + c 3 sinh µx + c 4 cosh µx (9.86) (9.79)

9.4. 435 ω = ω = ω 3 = EI ( π ), ( πx ) sin m l l ( ) ( ) EI π πx, sin m l l ( ) ( ) EI 3π 3πx, sin m l l 3 9.45 9.46 () ϕ() =, ϕ () =, ϕ(l) =, ϕ (l) = (9.86) c + c 4 =, µ ( c + c 4 ) =, c sin µl + c cos µl + c 3 sinh µl + c 4 cosh µl =, µ ( c sin µl c cos µl + c 3 sinh µl + c 4 cosh µl) = c 4 cosh µl sinh µl cos µl sin µl cosh µl sinh µl cos µl sin µl c 3 c c = (a) det ( ) = det sinh µl cos µl cosh µl sin µl = 4 sin µl sinh µl = sinh µl cos µl cosh µl sin µl sin µl = µl = nπ µ n = nπ l (b) (a) c (9.8) (b) (9.85) (9.8) 9.45 3 (3) (9.76) x = w w x = l x w x 3 w x 3 (9.83) ϕ(x) ϕ() =, ϕ () =, ϕ (l) =, ϕ (l) =

436 9 (9.86) c + c 4 =, µ (c + c 3 ) =, µ ( c sin µl c cos µl + c 3 sinh µl + c 4 cosh µl) =, µ 3 ( c cos µl + c sin µl + c 3 cosh µl + c 4 sinh µl) = c 4 cosh µl sinh µl cos µl sin µl sinh µl cosh µl sin µl cos µl c 3 c c = (c) det ( ) = det sinh µl cos µl cosh µl sin µl = cos µl cosh µl + = cosh µl sin µl sinh µl cos µl cos µ n l cosh µ n l = (9.87) µ n (n =,, ) (c) c = Ξ n c, c 3 = c, c 4 = Ξ n c, Ξ n cos µ nl + cosh µ n l sin µ n l sinh µ n l (9.88a, b, c, d) ϕ n (x) = (sin µ n x sinh µ n x) + Ξ n (cos µ n x cosh µ n x) (9.89) (9.87) 34 µ l =.875, ω = 3.56 l µ 3 l = 7.855, ω 3 = 6.7 l 9.46 EI m, µ l = 4.694, ω =.3 l EI m EI, (9.9a, b, c, d) m (9.9e, f) 9.4.4 () (9.8) (9.77) (9.84) n EI ϕ n (x) m ω n ϕ n (x) = (9.9) i l ϕ i (x) { EI ϕ n (x) m ω n ϕ n (x) } dx = 34 N.3. Scilab p.9 6

9.4. 437 l EI ϕ i (x) ϕ n (x) dx = ( EIϕ i ϕ n EIϕ ) l l i ϕ n + EI ϕ i (x) ϕ n (x) dx = l EI ϕ i (x) ϕ n (x) dx l EI ϕ i (x) ϕ n (x) dx ω n l m ϕ i (x) ϕ n (x) dx = i n l EI ϕ n (x) ϕ i (x) dx ω i l m ϕ n (x) ϕ i (x) dx = ( ω i ωn) l m ϕ n (x) ϕ i (x) dx = ϕ n, ϕ i m l m ϕ n (x) ϕ i (x) dx =, i n (9.9) m ϕ n m ϕ n, ϕ n m = l m ϕ n(x) dx = m n (9.93) m n (9.9) ϕ n, ϕ i m = l m ϕ n (x) ϕ i (x) dx =, i n m n, i = n (9.94) (9.9) (5.57) m =, w = m (9.9) (9.94) ϕ n ϕ n l m ϕ n (x) ϕ i (x) dx = l l ϕ n, ϕ i EI EI ϕ n (x) ϕ i (x) dx = EI ω n ϕ n (x) ϕ i (x) dx, i n m n ω n, i = n (9.95) ϕ n, ϕ i (symm) EI l EI ϕ n (x) ϕ i (x) dx =, i n m n ω n, i = n (9.96) 35 (9.96) (5.57) m =, w = EI 35 (9.95) (9.96)

438 9 () (9.9) ω n ϕ n (x) w(x, t) = q n (t) ϕ n (x) (9.97) n= q n (t) p(x, t) (9.) n= { m qn ϕ n + EI q n ϕ } n = i l ϕ i dx (9.94) (9.95) i q i (t) + ω i q i(t) = w(x, t) = q n (t) = A n sin ω n t + B n cos ω n t (A n sin ω n t + B n cos ω n t) ϕ n (x) (9.98) n= (9.8) (9.77) (9.98) w w(x, ) = B n ϕ n (x) = w (x), t (x, ) = A n ω n ϕ n (x) = v (x) n= ϕ i (x) (9.94) B n ϕ n, ϕ i m = w, ϕ i m, A n ω n ϕ n, ϕ i m = v, ϕ i m n= i n= n= B i ϕ i, ϕ i m = w, ϕ i m, A i = m i ω i l m v (x) ϕ i (x) dx, A i ω i ϕ i, ϕ i m = v, ϕ i m B i = l m w (x) ϕ i (x) dx (9.99a, b) m i 9-7 8. (9.8) (9.9) (9.89) 9.. k

9.4. 439 9.4.5 (4.7) κ σ xx (x, z, t) = E z κ(x, t) + η z κ(x, t), κ(x, t) w(x, t) (9.a, b) t x η (9.74) (9.75) M(x, t) = EI w(x, t) x η I 3 w(x, t) t x (9.) m w(x, t) t + η I 5 w(x, t) t x 4 + EI 4 w(x, t) x 4 = p(x, t) (9.) (9.76) ( w(x i, t) = n i EI 3 w x (x i, t) η I 3 w x (x i, t) = n= n i ( EI w x (x i, t) η I n= 4 ) w t x (x i, t) = S 3 i (t), (9.3a) 3 ) w t x (x i, t) = C i (t) (9.3b) (p(x, t) ) (9.97) { m qn ϕ n + η I q n ϕ n + EI q n ϕ } { ( η ) } n = m q n ϕ n + E q n + q n EI ϕ n = i l ϕ i dx (9.94) (9.95) i ( η ) m i q i + m i ω i E q i + q i = q i (t) + η E ω i q i (t) + ω i q i(t) = (9.7) β be i ω i Bernoulli-Euler β i be = η ω i E (9.4) (9.9) Rayleigh β i k q i (t) + β i be ω i q i (t) + ω i q i(t) = (9.5) q i (t) (9.) 9.4.6 () (9.97) (9.75) n= { m qn ϕ n + EI q n ϕ } n = p(x, t)

44 9 i l ϕ i dx (9.94) (9.95) i m i q i (t) + m i ω i q i(t) = p i (t) (9.6) p i (t) p i (t) l p(x, t) ϕ i (x) dx (9.7) (9.6) Duhamel (9.49) (9.6) p q i (t) = A i sin ω i t + B i cos ω i t + i (τ) H(t τ) sin {ω i (t τ)} dτ m i ω i w(x, t) = (A n sin ω n t + B n cos ω n t) ϕ n (x) + n= n= (9.7) w(x, t) = p n (τ) m n ω n ϕ n (x) H(t τ) sin {ω n (t τ)} dτ (A n sin ω n t + B n cos ω n t) ϕ n (x) (9.8) n= + n= l m n ω n ϕ n (x) ϕ n (ξ) H(t τ) sin {ω n (t τ)} p(ξ, τ) dτ dξ Duhamel A n, B n (9.99) 9.47 /4 t = P(t) = P p(x, t) = P δ (x l / 4 ) (9.8) Duhamel ξ l ϕ n (ξ) p(ξ, τ) dξ = P ϕ n ( l / 4 ) τ w τ EI l m t, w P l 3 768EI (9.9a, b) w( l /, t) w O O P(t) l/ 4 3l/ 4 τ.3 τ =..55.7 x l.5 9.47 w P.65 τ (n = ) π / π.637

9.4. 44 () (9.8) p(ξ, τ) w i (x; ξ, t τ) ϕ n (x) ϕ n (ξ) H(t τ) sin {ω n (t τ)} (9.) m n ω n w(x, t) = (A n sin ω n t + B n cos ω n t) ϕ n (x) + n= n= l w i (x; ξ, t τ) p(ξ, τ) dτ dξ (9.) ξ τ p(ξ, τ) w i (9.) w i ξ τ w i (x; ξ, t τ) (9.) Duhamel (9.) w i (x; ξ, t τ) w i (x, t) x = ξ t = τ 9.48 Dirac 4.4. Dirac p(x, t) = δ (x ξ) δ (t τ) ξ δ(t τ) 9.48 w i (x, t) w i (, t) =, m w i (x, t) t + EI 4 w i (x, t) x 4 = δ (x ξ) δ (t τ) (a) w i x (, t) =, w i(l, t) =, w i (l, t) = x (b) Fourier t = w i (x, t) Fourier W i (x; α) F(w i ) = w i (x, t) exp ( iαt) dt (a) Fourier Fourier ( ) w i wi exp ( iαt) dt = + iα w t i exp ( iαt) t α w i exp ( iαt) dt (c) ( ) wi = + iα w i exp ( iαt) α W i t t Fourier α α = a + ib, b exp ( iαt) = exp (bt) exp ( iat) t b < I(α) < (d) x (c)

44 9 α I(α) Fourier w i t exp ( iαt) dt = α W i (a) Fourier Fourier (c) 4 w i x 4 exp ( iαt) dt = d4 W i dx 4 (a) Fourier Dirac (4.57) (a) Fourier (b) Fourier W i (; α) =, δ(x ξ) δ(t τ) exp ( iαt) dt = δ(x ξ) exp ( iατ) d 4 W i dx m 4 EI α W i = δ(x ξ) exp ( iατ) (e) EI d W i dx (; α) =, W i(l; α) =, d W i (l; α) = dx W i (9.8) (e) n= W i (x; α) = c n ϕ n (x) ( f ) n= { c n ϕ m } n c n EI α ϕ n = δ(x ξ) exp ( iατ) EI (9.9) c n m ( ω n α ) ϕ n (x) = δ(x ξ) exp ( iατ) n= ϕ i (x) l ϕ i dx (9.94) i c i m i ( ω i α ) = l ϕ i (x) δ(x ξ) exp ( iατ) dx = ϕ i (ξ) exp ( iατ) Dirac (4.57) c i = m i ( ω i α ) ϕ i(ξ) exp ( iατ) ( f ) w i (x, t) Fourier W i (x; α) = n= ϕ n (x) ϕ n (ξ) ( m n ω n α) exp ( iατ) (9.55) (9.6) W i

9.4. 443 ω I(α) I(α) n ω n ω n R(α) Γ Γ + Γ ω n ω n Γ R(α) (a) (b) t < τ (c) t > τ ω n I(α) R(α) 9.49 Fourier Γ + w i (x, t) w i = π exp (iαt) n= w i (x, t) = F (W i ) = π ϕ n (x) ϕ n (ξ) ( m n ω n α ) exp ( iατ) dα = W i (x; α) exp (iαt) dα n= exp {iα (t τ)} ϕ n (x) ϕ n (ξ) dα πm n α ω n (g) I f (α) dα, f (α) exp {iα (t τ)} α ω n α I α = ±ω n (d) α 9.49 (a) Γ f (α) dα I α α = a + ib exp {iα (t τ)} = Γ exp {i (a + ib) (t τ)} = exp {ia (t τ)} exp { b (t τ)} b ± f (α) t < τ f (α) b 9.49 (b) Γ f (α) dα (h) ( Γ Γ ) f (α) Cauchy f (α) dα = Γ+Γ Γ f (α) dα + f (α) dα = I + f (α) dα = Γ Γ (h) I = f (α) dα w i = (9.) Γ t > τ f (α) b + 9.49 (c) f (α) dα (i) Γ + (Γ Γ + ) f (α) dα = I + f (α) dα = πi Res [ ] [ ] f ; ω n + πi Res f ; ωn Γ+Γ + Γ +

444 9 Res [ f ; β ] α = β f (α) (i) I = πi Res [ f ; ω n ] + πi Res [ f ; ωn ] Res [ f ; ±ω n ] = lim {(α ω n ) f (α)} = ± exp {±iω n (t τ)} α ±ω n ω n ζ n ω n (t τ) Euler (9.6) I = πi exp {i ζ n } exp { i ζ n } = π exp {i ζ n } exp { i ζ n } = π ω n ω n i ω n sin ζ n = π ω n (g) t > τ w i (x; ξ, t τ) = ϕ n (x) ϕ n (ξ) sin {ω n (t τ)} m n ω n (9.) n= sin {ω n (t τ)} (3) Duhamel Duhamel (9.) (9.3) w i (x; x, t t ) m w i (x; x, t t ) t + η I 5 w i (x; x, t t ) t x 4 + EI 4 w i (x; x, t t ) x 4 = δ(t t ) δ(x x ) (a) 9..3 (5) w i (x; x, t t ) m w i (x; x, t t ) t η I 5 w i (x; x, t t ) t x 4 + EI 4 w i (x; x, t t ) x 4 = δ(t t ) δ(x x ) (b) w i (x, ) =, w i t (x, ) = (9.) w i l { w i m w t + η I 5 } w t x + EI 4 w l dx dt = w 4 x 4 i p(x, t) dx dt w i m w t dt = ( mw w i m w i w t t ) + m w i t w dt = m w i t w (c) x l ( ) w i ηi 5 w t x + EI 4 w dx 4 x 4 { ( ) = w i η 4 w t x + EI 3 w w i 3 x 3 x = l w i x ( ηi 3 w t x + EI w x ( )} η 3 w t x + EI w l l w i + x x ) dx w dt ( ) ηi 3 w t x + EI w dx x (c) (d)

9.4. 445 l w i x ( ) ηi 3 w t x + EI w dx dt x ( ) = w i x ηi w x ηi 3 w i w t x x dt = ηi 3 w i t x w x dt w w x (x, ) = w i w i x l ( w ηi 3 w i x t x + EI w ) i dx dt x x l ( w ηi 3 w i x t x + EI w ) i dx x = w ( ηi 3 w i x t x + EI w ) ( i w ηi 4 w i x t x + EI 3 w i 3 x 3 l ( = w ηi 5 w i t x + EI 4 w ) i dx 4 x 4 (d) l w(x, t) { m w i t (c) l ) l + η I 5 w i t x + EI 4 w } i dx dt = 4 x 4 w(x, t) δ(x x ) δ(t t ) dx dt = w(x, t ) = l l l l = ( w ηi 5 w i t x + EI 4 w ) i dx 4 x 4 w i p(x, t) dx dt w i (x; x, t t ) p(x, t) dx dt w i (x; x, t t ) p(x, t) dx dt (a) w i l { w i m w i η I 5 w i t t x + EI 4 w } l i dx dt = w 4 x 4 i δ(x x ) δ(t t ) dx dt (b) l w i (x; x, t t ) δ(x x ) δ(t t ) dx dt = l w i (x ; x, t t ) = w i (x ; x, t t ) w i (x; x, t t ) δ(x x δ(t t ) dx dt ( f ) (e) w(x, t ) = l w i (x ; x, t t) p(x, t) dx dt w(x, t) = l (e) ( f ) w i (x; ξ, t τ) p(ξ, τ) dξ dτ (9.) Duhamel (9.) (9.47) (9.4) w i (x; ξ, t τ) ϕ m n ω d n (x) ϕ n (ξ) exp { β be n ω n (t τ) } H(t τ) sin { ω d n (t τ)} (9.3) n n= ω d n ω n ( β n be ) (9.4)

446 9 (4) V x P t = 9.5 t s x = V t Dirac p(x, t) = P δ(x V t) P V t 9.5 x P t > l / V p(x, t) Duhamel (9.) w(x, t) = l w i (x; ξ, t τ) P δ(ξ V τ) dξ dτ ξ τ (9.) (9.8) m n = ml w(x, t) = P ( nπx ) l ( nπξ ) sin sin δ(ξ V τ) dξ mlω n l l n= H(t τ) sin ω n (t τ) dτ ξ Dirac (4.57) V τ < V τ < l < τ < l / V H ( l/ V τ ) l ( nπξ ) ( ) l l ( nπξ ) ( ) l ( nπ V τ ) sin δ(ξ V τ) dξ = H τ sin δ(ξ V τ) dξ = H τ sin l V l V l l / V w(x, t) = n= k n nπ V, t l (9.5a, b) l V P ( nπx ) sin H(t τ) H (t τ) sin k n τ sin ω n (t τ) dτ mlω n l Heaviside t > τ t > τ t t min (t, t ) > τ w(x, t) = P ( nπx ) min(t, t ) sin sin k n τ sin ω n (t τ) dτ mlω n l n= s t t s sin k nτ sin ω n (t τ) dτ P k ( n nπx ) { ( ) } ωn w(x, t) = ( m l ω n= n k n ω ) sin sin ω n t sin k n t, t l, (9.6a) n l k n V P k ( n nπx ) w(x, t) = ( m l ω n k n ω ) sin {sin ω n t ( ) n sin ω n (t l )}, t > l (9.6b) n l V V n=

9.4. 447 w(x, t) w.5 O x l =. x l =.5 4 ζ = 3 x l =.7 τ w(x, t) w.5 O 9.5 x l =. x l =.5 x l =.7 ζ = 5 5 τ ω n k n 9.5 3 (9.9a) τ ζ lv EI m, w P l 3 48EI w P τ = ζ T ζ = 3, l / V = πζ / T 4.7 T ζ =, l / V 5.7 T τ = ζ (9.6) k n = ω n n = (9.5a) k = ω πv l = ω l V = π ω = T, ( ζ = ) π Zimmermann w(x, t) w O τ ζ.5.5 x l =. x l =.5 x l =.7 ζ =. π 9.5 ζ =. / π 9.5 τ = ζ [5] k = ω (9.6) n = < t < t t cos ω t.5 9-8. 9.47

448 9 w(x, t) w.5 O x l =. Q P d x l =.5 4 ζ = 3 x l =.7 d l =.3 τ w(x, t) w.5 O x l =.5 9.53 x l =. ζ = d l =.3 x l =.7 5 5 τ. 9.47 P(t) = P sin pt 3. 9.53 V 3 m T d l =.3, Q P = 4, w (P + Q) l3 48EI Heaviside Dirac d 9.4.7 () (9.) (9.3) l (9.) l δw { m w t + x ( ) ηi 3 w + t x x, 3 l { δw m w t δw + δ w x ( ηi 3 w t x ( ηi 4 w t x EI 3 w 3 x 3 ) + δ w x ) l δ ( w x ( ) } EI w p dx = x ( EI w x ) } δw p dx ) ( ηi 3 w t x + EI w x (9.3) l ) l = { ( ) ( ) } δw m w t + δ w ηi 3 w + δ w EI w δw p dx δw x t x x x S δθ C δw S δθ C = (9.7) w, θ x = w, θ x = l S i, C i

9.4. 449 (5.) w(x, t) = w (t) ψ (x) + θ (t) ψ (x) + w (t) ψ 3 (x) + θ (t) ψ 4 (x) (9.8) (9.8) (9.7) δ w b (t) t { m b ẅ b (t) + c b ẇ b (t) + k b w b (t) f b (t) g b (t) } = δ w b (t) m b ẅ b (t) + c b ẇ b (t) + k b w b (t) = f b (t) + g b (t) (9.9) w b (t) w (t) θ (t) w (t) θ (t) t, f b (t) S (t) C (t) S (t) C (t) t, g b (t) q (t) q (t) q 3 (t) q 4 (t) t (9.a, b, c) q i (t) (5.3b) q(x) p(x, t) q i q i (t) l p(x, t) ψ i(x) dx m b ( l m b ) m ψ i (x) ψ j (x) dx = m 3l/ 35 l / 9l/ 7 3l / 4 l 3 / 5 3l / 4 l 3 / 4 3l/ 35 l / Symm. l 3 / 5 (9.) ( l ) k b EI ψ i (x) ψ j (x) dx, c b ( l ) ηi ψ i (x) ψ j (x) dx = η E k b (9.a, b) (5.5) (9.8) Rayleigh ζ m =, ζ k = η / E (9.9) 5.3. () (9.95) M ü(t) + C u(t) + K u(t) = f(t) (9.3) f(t) f b (t) g b (t) (9.3) n n ü n (t) = N i =,,, N M ni =, M in =, C ni =, C in =, K ni =, K in =, f i = M nn = (9.4) n M ni M n i

45 9 () u(t) = U exp(iωt) U (9.3) ( K ω M ) U = U det ( K ω M ) = (9.5) ω (9.5) (9.9), 4 = EI 4/ l / l / l 4/ l m ω l 3 / 5 l 3 / 4 l 3 / 4 l 3 / 5 U U 4 (9.5) det 4 4 mω l 4 EI / 5 / 4 / 4 / 5 = U, U 4 θ (t), θ (t) ω l m EI = 3, 6 7, (9.8) ω l m EI = 3.99 (π), ω l m EI = 6 7.757 (π) % 7% 9.3 ζ k k ζ k = ω k (kπ/l) EI/m 9.3 ζ ζ ζ 3 ζ 4 ζ 5.99.757.395.99.3994.757 4.6.395.87.99.99 8..6.9.395.97 6...8.6.63 4 %

9.4. 45 u(t) U (9.8) 9.54 4 W n 3 9.54 9.45 (3) (9.3) 9..3 9.55 9.5 4 9.56 t V t n x x n a a = V t x n (9.c) l Dirac q i = l P δ(x a) ψ i (x) dx = P ψ i (a) 4 9.57 τ = ζ τ =.5ζ 9.58 (a) 9.47 4 (9.8) 9.58 (b) β be η E l EI m η E = βbe l m EI (9.6) β be =. (9.4) β be i w t = P x = l / 4

45 9 start t N n = f(t) u := u, u := u M, C, K, f Eq.(9.4) M, D Eq.(9.4), Eq.(9.5) u( t) = Eq.(9.5) t = n t u + := Eq.(9.4) u, ü := Eq.(9.3) t u, u, ü u := u, u := u + n = n + no n = N yes end 9.55 w( l /, t) w w( l /, t) w.5 O.5.5 O O ζ = 3 ζ = 4 4 5 5 9.56 w( l /, t) w ζ = 3 P v 4 l l/ 6 9.57 τ τ τ w( l /, t) w O O.3 τ τ =..55.7 x l.5 (a) w( l /, t) w 9.58 O 4 (b) β be =. τ 3

9.4. 453 9-9 4. 9.47 P(t) = P 5. 9.47 P(t) = P sin pt 6. 9.3 (4) : l (5.4) l EA u(x, t) x δ u x dx + l m u(x, t) t l δu dx p a (x, t) δu dx F (t) δu F (t) δu = (9.7) p a (x, t) F i (t), (i =, ) (5.6) u(x, t) u (t) ϕ (x) + u (t) ϕ (x) (9.8) (9.7) m a ü a (t) + k a u a (t) = f a (t) + g a (t) (9.9) u a (t) u (t) u (t) t, f a (t) F (t) F (t) t, g a (t) p (t) p (t) t (9.3a, b, c) p i (t) (5.7a) p(x) p a (x, t) p i p i (t) l p a(x, t) ϕ i (x) dx m a ( l m a ) m ϕ i (x) ϕ j (x) dx / 3 / 6 = ml / 6 / 3 (9.3) ( ) l k a EA ϕ i (x) ϕ j (x) dx (5.9b) (9.9). EA T 9.3. (9.a) N(x, t) = EA u(x, t) x + η A u(x, t) t x (9.3) η ( l ) c a ηa ϕ i (x) ϕ j (x) dx = ηa l = η E k a (9.33)

454 9 : (9.9) (9.9) 5.5 ξ-ζ z-x α (5.34) (5.39) (9.) (9.3) ml/ 3 ml/ 6 3ml/ 35 ml / 9ml/ 7 3ml / 4 ml 3 / 5 3ml / 4 ml 3 / 4 m a f m b f m f = ml/ 3 (m 3ml/ b 35 ml f / )t m c f Symm. ml 3 / 5 (9.34) u f (t) u (t) w (t) θ (t) u (t) w (t) θ (t) t, f f (t) F (t) S (t) C (t) F (t) S (t) C (t) t (9.35a, b) (5.35) m f k f (9.34) (5.39) T t m a f m f = T Tt m b f T Symm. T t m c f T, k T t k a f f = T Tt k b f T Symm. T t k c f T z-x (9.36a, b) m f ü f (t) + k f u f (t) = f f (t) (9.37) : n n 9.59 L L 4 6 7 9.59 (a) 9.4 3 3 4 3 9.59 (b) 9.4 3 3 4 4

9.4. 455 5 4 L 4 3 3 5 L L L L (a) 9.59 (b) 9.4 ωl m/ EI θ = 3 θ =.5 θ = 45 θ = 36 3.4 6.8 38.79 36.9 6.6 55.47.6 4.78 47.5 39.9 66.4 6.63 3.6.58 9.8 89.49 4.6.9 4.8 45.4 57. 57.5 97.4 98.3 5 44.79 58. 46.4 46.6 98. 98. 3 θ = 3 5 4 3 5 θ =.5 4 (a) (b) 9.6 5 4 4 5 : 4 L L / r = 9.6 θ = 3 9.6 (a)

456 9.5 9.6 (b) 9.4 5 θ 4.385 ωl m/ EI 39.4 36.5 9.6 ω L m EI 4.9 θ 45 36 9.6 9.4 6, 7 45, 3, 5, 4 36, 4 9.4 6 4 4 θ ( ) 9.6 4 5 4 5 3 θ = 45 3 θ = 36 (a) (b) 9.6 9.4.8 () (9.7) (.3) (9.7) M(x, t) x V(x, t) = J θ(x, t), θ(x, t) = dw t dx J (.4) dx J = A ρ z da = ρ I 36? 6.4. (3)

9.4. 457 M(x, t) x V(x, t) = ρ I 3 θ(x, t) t x (9.38) (9.7) (9.3) ( n i EI 3 w x (x i, t) η I 3 m w(x, t) t + EI 4 w(x, t) x 4 ρ I 4 w(x, t) t x = p(x, t) (9.39) 4 w t x 3 (x i, t) + ρ I 3 ) w t x (x i, t) = S i (t) (9.4) l (9.39) [ v(ξ, t) t ξ x w(x, t), v(ξ, t) l l + EI m l 4 4 v(ξ, t) ξ 4 (9.4a, b) ] p(ξ, t) 4 v(ξ, t) m l λ t ξ = (9.4) λ (6.4) 3 4 λ (9.4) () Timoshenko A Timoshenko A. x, z u x = u + z ϑ, u z = w (9.43a, b) ϑ(x) γ = ϑ ( w ) (9.44) Bernoulli-Euler A. V x + p = m w t, M x V = ρ I ϑ (9.45a, b) t V = Gk t A γ, M = EI ( ) γ x w = EI ϑ x x (9.46a, b)

458 9 G k t (A.7) Poisson ν [7] { w = wi n i V = S i, } { ϑ = ϑi n i M = C i } (9.47a, b) u i γ w ϑ (9.45) (9.46) ϑ, γ w (9.4b) v [ EI 4 v ml 4 ξ v 4 t + p ] + 4 [ v ml λ t ξ + α 4 v t t ξ ml4 4 v EI λ t { p 4 ml ξ ml4 }] p = (9.48) EI λ t (9.4a) ξ Bernoulli-Euler Timoshenko α t (A.5) α t (9.48) ξ v(ξ, t) = V(ξ) exp(iωt) V + ω µ 4 ( + Ξ) λ V + Ξ µ8 ω 4 µ 4 λ 4 ω λ 4 V = Ξ E k t G, µ l m EI V = exp(pξ) p 4 + ω µ 4 ( + Ξ) λ p + Ξ µ8 ω 4 µ 4 λ 4 ω λ 4 V = α t = Ξ λ (9.49a, b, c) p = [ ω µ 4 ( + Ξ) λ ± µ λ ω 4 µ 4 ( Ξ) + 4ω λ 4 ] ω < λ4 Ξµ 4 ±p ±ip p = [ ω µ 4 ( + Ξ) λ + µ λ ω 4 µ 4 ( Ξ) + 4ω λ 4 ] (a) V(ξ) = c sin p ξ + c cos p ξ + c 3 sinh p ξ + c 4 cosh p ξ (b) (9.47) w =, ϑ x = = w x + m w Gk t A t

9.4. 459 Ω n Ω n.8 st.8.6.4. 3rd.6.4. 4 λ 4 9.63 nd 4th λ (b) c 4 p p cosh p sinh p cos p sin p p cosh p p sinh p p cos p p sin p c 3 c c = ( p + p ) sin p sinh p = p = nπ c sin p ξ (a) Ξ Ω 4 n β n ( βn + + Ξ ) Ω n + β n = Ω n µ ω (nπ), ( λ ) β n (9.5a, b) nπ Ξ Ξ = Bernoulli-Euler Ω n = (9.8) ν =.3 4 9.63 Ω n = Bernoulli-Euler Bernoulli- Euler 3 %

46 9 9.5 9.5. () 6 Northwestern Davis (98 Asymptotic and Perturbation Methods in Applied Mathematics 37 (9.4) θ + g l sin θ = (9.5) θ() = a, θ() = b (9.5a, b) a b sine a = b (9.4) () (9.) ω θ θ θ + ω θ sin θ = {( θ ) } + ω ( cos θ) = 9.. (3) ( θ ) ω cos θ = C (const.) (9.53) 3 O 3 θ ω 9.64 θ π C = b ω cos a (9.53) m l m v (t), (v(t) l θ(t) = ) mg l cos θ 9.64 π (9.8) θ π 9.65 π 9.65 37 sit-in

9.5. 46 (3) : ϵ x ϵ x + = x = ϵ ± i ϵ ϵ x = x + ϵ x + ϵ x + O(ϵ 3 ) O p.9 9 Landau ϵ ( x + ) + ϵ (x x x ) + ϵ ( x + x ) x x + O(ϵ 3 ) = ϵ x = ±i, x =, x = i O() x = ±i + ϵ ϵ + O(ϵ 3 ) Taylor ϵ : (x ) + ϵ = O() : (x ) = x = O(ϵ) : (x ) x + = = ±i ϵ ϵ x = x + ϵ x + ϵ x + O(ϵ 3 / ) O() x = O( ϵ) : (x ) x = x, O(ϵ) : x + (x ) x + = x = ±i, x x k =, (k )

46 9 : ϵ ϵ x x + = ϵ Taylor x = ( ) [ { ± ϵ = ± ϵ ϵ ϵ }] 8 ϵ + x + = ϵ 8 ϵ +, x = + 8 ϵ + x = x + ϵ x + ϵ x + O(ϵ 3 ) x + ( x + ) + ϵ ( x x ) + = x =, x = 8 x x + ϵ x ϵ x 3 x ϵ ϵ x x ϵ x ϵ x x ϵ x x x ϵ 3 ϵ x x x / ϵ 38 X ϵ x (9.54) X X + ϵ = X = X + ϵ X + O() X X = X =, O(ϵ) X X X + = X = +/ X = / X = 38 [5]

9.5. 463 X = ϵ + x = ϵ + x +, X = + ϵ + x = + x x +, x (4) a τ ωt, Θ(τ) θ a (9.55a, b) Θ + a sin (aθ) =, Θ() =, Θ() = B b aω (9.56a, b, c) τ sine Taylor B = Θ + Θ = a 6 Θ3 + O(a 4 ), Θ() =, Θ() = (9.57a, b, c) : a a Θ(τ) = Θ + a Θ + a 4 Θ + O(a 6 ) a ( ) ( Θ + Θ + a Θ + Θ ) 6 Θ3 + O(a 4 ) = Θ + a Θ + O(a 4 ) =, Θ + a Θ + O(a 4 ) = O() Θ + Θ =, Θ () =, Θ () = Θ = A sin τ + B cos τ A =, B = Θ = cos τ O(a ) Θ + Θ = 6 Θ3, Θ () =, Θ () = Θ 6 Θ3 = 6 cos3 τ = (3 cos τ + cos 3τ) 4

464 9 Θ = A sin τ + B cos τ 9 cos 3τ + 6 τ sin τ A =, B = 9 Θ = 9 (cos τ cos 3τ) + 6 τ sin τ [ Θ = cos τ + a 9 (cos τ cos 3τ) + ] 6 τ sin τ + O(a 4 ) a τ a : ω a ω a Ω(a ) ω Ω(a ) s Ω(a ) ω t = Ω(a ) τ (9.58) 39 τ s π d dτ Ω d ds, d dτ Ω d (9.57) ds Ω (a ) Θ + Θ = 6 a Θ 3 + O(a 4 ), Θ() =, Θ () = s Θ = Θ (s) + a Θ (s) + O(a 4 ), Ω(a ) = Ω + a Ω + O(a 4 ) (9.59a, b) Ω Ω = Ω + a Ω Ω + O(a 4 ) O() Ω Θ + Θ =, Θ () =, Θ () = ( ) s Θ = cos 39 t s = ω t, t Ω(a ) t (9.54) Ω

9.5. 465 s π / Ω Ω = Θ = cos s, Ω = (9.6a, b) O(a ) Ω Θ + Θ = 6 Θ3 Ω Ω Θ O() (9.6) Θ + Θ = ( ) 8 + Ω cos s + 4 cos 3s cos s cos s Ω = 6 Θ = A sin s + B cos s 9 cos 3s [{ θ(t) = a cos } ] 6 a + O(a 4 ) ω t + O(a 3 ) a θ(t) = a cos ω t + O(a 3 ) { ω } 6 a + O(a 4 ) ω (9.6) Poincaré θ(t) = a cos ω t + O(a 3 ) (9.6) t Ω(a ) t = { } 6 a + O(a 4 ) t method of strained coordinates [5] method of strained coordinates Poincaré

466 9 (5) θ = ±π 9.65 θ = ±π θ = ±π (9.5) ω sin θ = θ = n π, (n =, ±, ) 6.. (3) θ = θ = θ = θ = π θ θ = θ + θ, sin θ = sin (θ + θ) θ cos θ (9.5) θ θ + ω cos θ θ = O θ ω 9.65 θ π : θ = θ + ω θ = θ = L {sin(ωt), cos(ωt)}, : θ = π θ ω θ = θ = L {sinh(ωt), cosh(ωt)} θ = 9.5. (9.7) van der Pol ϵ > ü(t) + u(t) = ϵ { u (t) } u(t), u() = A, u() = (9.63a, b, c) c u c ϵ { u (t) } u

9.5. 467 u u(t) = u (t) + ϵ u (t) + ϵ u (t) + O(ϵ 3 ) (9.63a) ϵ =. u [ü + u ] + ϵ [ ü + u ( ] u) u + O(ϵ ) = O 3 4 (9.63b) (9.63c) u () + ϵ u () + O(ϵ ) = A, u () + ϵ u () + O(ϵ ) = O() ü + u =, u () = A, u () = 3 9.66 u (t) = A cos t O(ϵ) ü + u = ( u ) u, u () =, u () = u u (u ) p (u ) p (t) = A ( 4 A ) t cos t 3 A3 sin 3t A 4 A A = ± A = ± u = 4 sin 3t + B sin t + C cos t C =, B = ± 3 / 4 u = ± cos t ± ϵ 4 (3 sin t sin 3t) + O(ϵ ), u = ± (3 sin t sin 3t) 4 u = sin t ± 3ϵ 4 (cos t cos 3t) + O(ϵ ) t 9.66 O(ϵ) u u β := ϵ { u (t) } A = A = 4 ϵ

468 9 9.5.3 () 9.67 l v f (t) v f v f v f l v f v f (t) l v f (t) l θ v l θ v [38] u u 9.67 u = l sin θ, v = l ( cos θ) + v f ü = l θ cos θ l ( θ ) sin θ, v = l θ sin θ + l ( θ ) cos θ + v f (a) T T sin θ = m ü, T cos θ mg = m v T ü cos θ + v sin θ + g sin θ = (b) (b) (a) l θ + ( g + v f ) sin θ = cosine v f = v cos ωt, ω g l (9.64a, b) θ + } v {ω + l ω cos ωt sin θ = sine } v θ + {ω + l ω cos ωt θ = (9.65) Mathieu

9.5. 469 9.67 u = ( l v f ) sin θ, v = l ( l v f ) cos θ ü = { θ ( ) } { l v f θ v f cos θ v f + ( θ ) ( )} l v f sin θ, v = { θ ( ) } { l v f θ v f sin θ + v f + ( θ ) ( )} l v f cos θ (b) θ ( l v f ) θ v f + g sin θ = w ( l v f ) θ ẅ + v ( f w w + g sin l v f l v f θ, v f l ) = (9.64a) ) v ẅ + (ω + l ω cos ωt w = (9.65) Mathieu τ ωt d dt = ω d dτ, d d dt = 4 ω dτ w(t) u(τ) ü + 4 ω ω + v l cos τ u = τ δ 4 ω ω, ϵ v l (9.66a, b) Mathieu ü(τ) + {δ + ϵ cos τ} u(τ) = (9.67) () u = u + ϵ u + O(ϵ ) (9.68) (9.67) [ü + δ u ] + ϵ [ü + δ u + u cos τ] + O(ϵ ) =

47 9 O() u = sin δτ, cos δτ π δ = n δ = n, (n =,, ) O(ϵ) ü + n u = u cos τ n = u =, ü = cos τ u 4 cos τ n = u ü + u = (sin τ sin 3τ) = (cos τ + cos 3τ) sin τ, cos τ τ sin τ, τ cos τ (3) (9.68) δ ϵ δ = δ + ϵ δ + O(ϵ ) (9.69) (9.67) [ü + δ u ] + ϵ [ü + δ u + (δ + cos τ) u ] + ϵ [ü + δ u + δ u + (δ + cos τ) u ] + O(ϵ 3 ) = O() δ = n, u = sin nτ, cos nτ n =, u = O(ϵ) ü = δ cos τ ( ) u = δ τ + c τ + c + 4 cos τ + c δ = O(ϵ ) u = 4 cos τ + c ü = δ c cos τ 4 cos τ = δ 8 ( + cos 4τ) c cos τ

9.5. 47 δ = / 8 δ δ = 8 ϵ + O(ϵ 3 ) (c) n = u = sin τ, δ = O(ϵ) ( ) ü + u = (δ + cos τ) sin τ = δ sin τ sin 3τ δ =, u = 6 sin 3τ O(ϵ ) ü + u = δ sin τ ( ) ( + cos τ sin 3τ = δ + ) sin τ (sin 3τ sin 5τ) 6 3 3 δ = / 3 δ δ = + ϵ 3 ϵ + O(ϵ 3 ) (d) n = u = cos τ, δ = O(ϵ) ( ) ü + u = (δ + cos τ) cos τ = + δ cos τ cos 3τ δ =, u = 6 cos 3τ O(ϵ ) ü + u = δ cos τ ( cos τ ) 6 ( cos 3τ = δ + ) cos τ + (4 cos 3τ cos 5τ) 3 3 δ = / 3 δ δ = ϵ 3 ϵ + O(ϵ 3 ) (e) n =, u = sin τ, u = cos τ u = sin τ u = 4 sin 4τ, δ =, δ = 48 δ = 4 48 ϵ + O(ϵ 3 ) ( f ) u = sin τ u = 8 + 4 cos 4τ, δ =, δ = 5 48 δ = 4 + 5 48 ϵ + O(ϵ 3 ) (g) (c) (g) 9.68 ϵ =. (d) (e) δ = δ =.8 Mathieu 9.68

47 9 ϵ.5 (c) (e).5 O u(t) (d) (g).4 δ =. ( f ). O 5. δ =.8.4 ϵ =. 3 4 δ 9.68 t 9.68 ϵ δ Floquet γ ϕ = ϕ + ϵ ϕ + O(ϵ ), δ = δ + ϵ δ + O(ϵ ), γ = + ϵ γ + O(ϵ ) Mathieu δ = γ γ = γ Floquet Theorem: T u(t) = exp (γt) ϕ(t) ϕ T γ R(γ) 9.67 Mathieu 98 Northwestern Technological Institute 9.67 4 9. 4 99 TAINSbbms (p.iv)

9.5. 473 9.5.4 Duffing () 9.69 f u f = α u + ϵ β u 3, < ϵ (9.7) f β > β < O u 9.69 α > β ϵ q cos pt ü(t) + α u(t) + ϵ β u 3 (t) = q cos pt (9.7) Duffing u = u + ϵ u + O(ϵ ) (9.7) [ü + α u q cos pt ] + ϵ [ ] ü + α u β u 3 + O(ϵ ) = O() u = q cos pt α p O(ϵ) ( α p ) () q = ϵ q (a) τ pt d dt = p d dτ τ (9.7) p ü + α u + ϵ β u 3 = ϵ q cos τ

474 9 u u 6 4 α = ϵβ =.3 q = 3 6 4 α = ϵβ =.3 q = 3 O 3 p O 3 9.7 Duffing p p α p p p = α + ϵ p + O(ϵ ), u = u + ϵ u + O(ϵ ) p = α + α p ϵ + O(ϵ ) α p p α O() ϵ (b) α ü + α u = O(ϵ) u = c sin τ + c cos τ α (ü + u ) = α p ü β u 3 + q cos τ u sin τ, cos τ sin 3τ, cos 3τ [ α p c 3 4 β c ( ) ] [ c + c sin τ + α p c 3 4 β c ( ) ] c + c + q cos τ c {α p 3 4 β ( c + ) } { c =, c α p 3 4 β ( c + ) } c + q = q = c = α p c 3 4 β c3 + q = (a) q (b) 3, p α c 3 ϵ 4 β c3 + q ϵ =

9.5. 475 4 O 4 u(t) p =.5 4 8 ϵβ =.3 t 4 O 4 u(t) p =.95 t 4 6 ϵβ =.3 u u 4 4 4 u 4 p =.5 ϵβ =.3 p =.95 ϵβ =.3 4 4 u 4 4 u 9.7 Duffing u ϵβ =.4 ϵβ = 5 5 u 5 5 9.7 Duffing u (ϵβ) c 3 4 (ϵβ) c3 + ( α p ) c q = 4 9.7 c u α =, q = 3 ϵβ =.3 ϵβ =.3 p α = 9.7 α =, q = 3 4 3 u := A cos pt [6]

476 9 p =.5, ϵβ =.3 p =.95, ϵβ =.3 9.7 ϵβ α =, p =, q = 3 (3) Duffing ϵ ϵ ü + u = ϵ u (9.7) u() =, u() = method of strained coordinates t = s + ϵ s (s) +, u = u + ϵ u + O(ϵ) ü + u = sin s ( s ) + cos s ( s ) s s u = exp ( ϵ ) ( ) t sin t ϵ / 4 sine t method of multiple scales Northwestern

9.5. 477 Gauss 4 : x = x = l {g(x) f (x)} Gauss n x = V x = l n x l g(x) d f (x) dx dx = V g d f dx dv = n x (g f ) ds S = n x g f x= + n x g f x=l l V dg dx f dx dg dx f dx S 9.73 n x n x x= =, n x x=l = + l g(x) d f (x) dx dx = g(x) f (x) l l dg(x) dx f (x) dx g(x) l d f (x) dx dx = f (x) n Cauchy l lazybones: goof off pissed off 4 Green

478 9 9. 95 E talent: N?!

... E Northwestern Achenbach Wave Propagation 98 [].. (4.9a) u(x, t) u(x, t) x = c b u(x, t) t (.) c b (4.9b) t = u(x, ) u (x, ) t u 3 σ (4.7) σ(x, t) = E u(x, t) x σ ϵ u x (.)..3 d Alembert (.) y = x + c b t, z = x c b t 3 σ 479

48 u(x, t) v(y, z) u x = v y + v z, u x = v y (.) + v y z + v z, u t ( v = c b y v ), z u t = c b ( v y ) v y z + v z v y z = (.3) (.3) v(y, z) = w(z) dz + V(y) = W(z) + V(y) u(x, t) = f (x c b t) + ḡ(x + c b t) (.4) d Alembert 9.3. (5) f (x c b t) x c b ḡ(x + c b t) x c b (.) Hooke σ(x, t) σ(x, t) = E ( f (x c b t) + ḡ (x + c b t) ) = F(x c b t) + Ḡ(x + c b t) (.5) f (x c b t) x = f (x c b t) = f (x c b t) s b c b (.6) u(x, t) = f (t s b x) + g(t + s b x) (.7). u = f (t s b x) (.) Hooke σ = E u x = E s b f f = E s b t = E s b u t = ρ c b u t u(t s b x) x = f (t s b x) (t s b x) = s b f, (t s b x) x u(t s b x) t = f (t s b x) (t s b x) = f = f t σ u t σ u = f (t s b x) F(t s b x) t

.. 48. u = g(t + s b x) σ = ρ c b u t, σ u t = g (t + s b x) G(t + s b x) σ(x, t) σ(x, t) u(x, t) =, u(x, t) = (.8a, b) ρ c b ρ c b..4. l p(t) p(t) l x u(x, ) =, u(x, ) =. σ(, t) = p(t), u(l, t) = (.8) σ(, t) = p(t), u(l, t) =, u(l, t) = (.9a, b, c) x = l < t < l / cb = l s b p(t) σ(x, t) σ(x, t) = f (t s b x) 4 x = (.9a) σ(, t) = f (t s b ) = f (t) = p(t) ( ) O O t c b. l x σ(x, t) = f (t s b x) = p(t s b x) ( ) f p (.8a). u(x, t) = ρ c b p(t s b x) 4 u f σ F F f

48 x = l l / cb < t < l / cb O σi c b l x O c b σ r x σ r c b σ r u i (x, t) = ρ c b p(t s b x) σ c b σ σ r (x, t) = g(t + s b x).3 u r (x, t) = ρ c b g(t + s b x) i r x = l (.9c) u i (l, t) + u r (l, t) = ρ c b p(t s b l) + ρ c b g(t + s b l) = ξ t + s b l g g(ξ) = p(ξ s b l) g(t + s b x) = p(t + s b x s b l) σ r (x, t) = p(t + s b x s b l) σ(x, t) = p(t s b x) p(t + s b x s b l) = p(t s b x) p((t s b l) + s b (x l)) t > s b l x = l x.3 x = l / cb < t < 3l / cb σ r σ i = p(t s b x), σ r = p(t s b l + s b (x l)), σ r = h(t s b x) (.9a) x = p(t) = σ i (, t) + σ r (, t) + σ r (, t) p(t) = p(t) p(t s b l) + h(t) h(t) = p(t s b l)

.. 483 h(t) σ r = p(t s b l s b x) σ(x, t) = p(t s b x) p(t s b l + s b x) + p(t s b l s b x) p(t).3 -..4 p(t) = p, < t < a c b < t < 3l / cb O a = l p / 3 p p p(t) [] a/ cb l x M p(t) = F t H(t).4..4 A x = l M t < x = p(t) = F H(t) x H(t) (9.46) Heaviside x = l [] (9.46) dh(t t ) dt d = δ(t t ) (.) δ(t t ) (4.57) Dirac H(t t ) t = t (.) ϕ(t) dh(t t ) dt ϕ(t) dt = H(t t ) ϕ(t) = ϕ(t ) = δ(t t ) ϕ(t) dt H(t t ) dϕ(t) dt dt = (.) d = Heaviside t dϕ(t) dt dt... 3 u(x, t) 3 (3.5) X u j u i (λ + µ) + µ = ρ ü x j x i x i (.) j j j Σ

484 5 3 δ kk = δ + δ + δ 33 = 3, δ i j σ jk = σ ik, δ i j σ i j = σ ii = σ j j = σ kk = σ + σ + σ 33 δ i j Kronecker Hooke (3.46) (3.49a) σ i j = µ ϵ i j + λ δ i j ϵ kk, σ i j = C i jmn ϵ mn (.) 3 (3.6) (3.) ϵ i j = ( ui, j + u j,i ), σ ji, j + X i = (.3a, b) 6 (.) (λ + µ) u j, ji + µ u i, j j = ρ ü i (.4) Fourier.. (.4) f (x c b t) x = const. x x 3 O x p.5 x c b.5 p 3 p x + p x + p 3 x 3 = const. x p = x k p k = const. 3 u = d f (x p c t) u i = d i f (x k p k c t) (.5) d c (x p c t) 5 (.9) 6 C

.. 485 (.5) (.4) (.5) u i, j = d i p j f (x k p k ct), u i, j j = d i p j p j f (x k p k ct) = d i f, u j, ji = d j p j p i f, ü i = c d i f p (p j p j = ) { (λ + µ) ( d j p j ) pi + µ d i ρ c d i } f (x k p k ct) = ( µ ρ c ) d i + (λ + µ) ( d j p j ) pi = (.6) (.6). p = ±d p i = ±d i (.6) {( ) µ ρc + (λ + µ) } d i = (λ + µ) ρ c = (3.56a) λ + µ c = c l ρ (.7) P u = d p f (x p c l t) = ±d d f = [ d d = ]. p d = p i d i = (.6) µ ρ c = c = c t µ ρ < c l (.8) (3.56b) S u = d p f (x p c t t) = [ d p = ] x -x 3 p x -x.6 3 O x P p SH p SV p d d d x.6 P SH SV

486 P : d x -x SH : d x 3 d = t x -x x 3 H[orizontal] SV : d x -x x -x V[ertical]..3 () f ( ) A f (x p c t) = A exp (i η), η k (x p c t) = k (x p) ω t (.9a, b) k ω / c ω.7 x = () SH SH.7 d = t SH SH x 3 u 3 p () sin θ = cos θ, sin θ p() = cos θ (.a, b) d O x p () p () θ θ.7 SH x u () 3 = A exp {i k (x sin θ + x cos θ c t t)} (.) u () 3 = A exp {i k (x sin θ x cos θ c t t)} (.) A, A : x > x = u 3 = u 3 (x = ) = u () 3 (x = ) + u () 3 (x = ) (.3) = A exp {i k (x sin θ c t t)} + A exp {i k (x sin θ c t t)} = x, t (exp) x t k sin θ = k sin θ, k = k

.. 487 k = k, θ = θ (.4a, b) (.3) A = A (.5) x 3 (.) (.) u 3 (x, x, t) = A exp {i k (x sin θ + x cos θ c t t)} A exp {i k (x sin θ x cos θ c t t)} = i A sin (k x cos θ ) exp {i k (x sin θ c t t)} (.6) sin (k x cos θ ) x sine (x sin θ c t t) x : x = x = σ i = (i =,, 3) SH u =, u =, u 3 = u 3 (x, x ) ϵ =, ϵ =, ϵ 33 =, ϵ = Hooke σ = σ = σ 3 = µ ϵ 3 = µ u 3, = u 3, = (.7) (.7) u 3, (x = ) = u () 3, (x = ) + u () 3, (x = ) (.8) = i k A cos θ exp {i k (x sin θ c t t)} i k A cos θ exp {i k (x sin θ c t t)} = x, t k = k, θ = θ A = A (.9a, b, c) u 3 (x, x, t) = A cos (k x cos θ ) exp {i k (x sin θ c t t)} (.3) (3) P SH SH P P, SV :. : u =, u =

488. : σ =, σ = P :. u =, σ =. : u =, σ =.8 n =,, O x d (), p () θ θ p () d (), p () x u (n) i P : d () = p () sin θ = cos θ = A n d (n) i exp (i η n ) (i =, ), (.3a) η n k n ( x p (n) + x p (n) c n t ) (.3b) ϵ (n) = u(n), = i k n p (n), c = c l, P : d () = p () = SV : d () = d(n) ϵ (n) = u(n), + u(n), = i k n cos θ sin θ, p() = Hooke P sin θ cos θ sin θ cos θ θ d ().8 P P SV, c = c l, (.3a, b, c, d), c = c t (.3e, f, g) A n exp (i η n ), ϵ (n) = u(n), = i k n p (n) d(n) A n exp (i η n ), (.33a, b) ( ) d (n) p (n) + d (n) p (n) An exp (i η n ) (.33c) σ (n) = (λ + µ) ϵ(n) + λ ϵ(n) = i k { n (λ + µ) d (n) σ (n) = µ ϵ(n) = i k n µ { d (n) p (n) + d (n) p (n) } p (n) + λ d (n) p (n) An exp (i η n ), (.34a) } An exp (i η n ) (.34b) x = σ =, σ = (.34) σ (x = ) = σ (x = ) = n= n= σ (n) (x = ) = σ (n) (x = ) = n= n= { } i k n (λ + µ) d (n) p (n) + λ d (n) p (n) An exp ( ) i η n =, (.35a) i k n µ { } d (n) p (n) + d (n) p (n) An exp ( ) i η n = (.35b) η n η n (x = ) = k n ( x p (n) c n t ) (.3) i k ( λ + µ cos θ ) A exp ( i η ) + i k ( λ + µ cos θ ) A exp ( i η ) i k µ sin θ cos θ A exp ( ) i η =, i k µ sin θ cos θ A exp ( ) i η i k µ sin θ cos θ A exp ( ) i η (.36a) + i k µ ( sin θ cos θ ) A exp ( i η ) = (.36b)

.. 489 x, t k c l = k c l = k c t = ω, k sin θ = k sin θ = k sin θ = (.37a, b) (.37b) Snell k = k, k = κ k, κ c l c t = λ + µ ( + ν) = > (.38a, b, c) µ ν θ = θ, sin θ = k k sin θ = κ sin θ < θ < θ = θ (.39a, b, c) (.38c) κ θ θ (.36) ( λ + µ cos θ ) A A κ µ sin θ A A = ( λ + µ cos θ ), sin θ A A + κ cos θ A A = sin θ ( ) µ κ = λ / µ + ( κ sin θ ) A A κ sin θ A A = ( κ sin θ ) sin θ = κ sin θ κ sin θ = κ cos θ κ cos θ A A sin θ A A = κ cos θ ( ) κ cos θ sin θ sin θ κ cos θ A / A A / A = κ cos θ sin θ A / A A / A = κ cos θ sin θ sin θ κ cos θ κ cos θ sin θ, sin θ sin θ + κ cos θ A A = sin θ sin θ κ cos θ sin θ sin θ + κ cos θ, A κ sin θ cos θ = (.4a, b) A sin θ sin θ + κ cos θ. θ = θ = θ =, A = P A = A A d () = A d () u. θ = π / A = P A d () = A d () u u =? 3. sin θ sin θ = κ cos θ θ A = SV

49 (4) P.9 n = 3, 4 (.3) P : d (3) = p (3) sin θ 3 = cos θ 3, c 3 = c b l SV : d (4) cos θ 4 = sin θ 4, sin θ 4 p(4) = cos θ 4 λ b + µ b, ρ b, (.4a, b) c 4 = c b t µb ρ b (.4c, d, e) µ b, λ b Lamé ρ b σ (n) = i k { } n (λb + µ b ) d (n) p (n) + λ b d (n) p (n) An exp (i η n ), σ (n) = i k { } n µ b d (n) p (n) + d (n) p (n) An exp (i η n ) (.4a, b) x = u, u σ, σ (.34) (.4) A n d (n) i exp ( ) 4 i η n = A n d (n) i exp ( ) i η n, (.43a) n= n= n= n=3 { } i k n (λ + µ) d (n) p (n) + λ d (n) p (n) An exp ( ) i η n = 4 n=3 { } i k n (λb + µ b ) d (n) p (n) + λ b d (n) p (n) An exp ( ) i η n, (.43b) i k n µ { } d (n) p (n) + d (n) p (n) An exp ( ) i η n = 4 n=3 { } i k n µ b d (n) p (n) + d (n) p (n) An exp ( ) i η n (.43c) P µ b, λ b, ρ b x θ 3 p (4) θ 4 O d (), p () θ θ p () d (4) SV θ d () P d (3), p (3) SV P x d (), p ().9 P η n η n (x = ) = k n ( x p (n) c n t ) (.3) (.4) k = k, k = κ k, k 3 = α b k, k 4 = β b k (.44a, b, c, d) α b c l c b, β b c l l c b > α b (.45a, b) t κ (.38c) θ = θ, sin θ = κ sin θ <, sin θ 3 = α b sin θ, sin θ 4 = β b sin θ < sin θ 3 (.46a, b, c, d) (.43)

.. 49 (5) SV.8 SV SV : d () cos θ = sin θ, sin θ p() = cos θ, c = c t (.47a, b, c) (.3) i k µ sin θ cos θ A exp ( i η ) + i k ( λ + µ cos θ ) A exp ( i η ) i k µ sin θ cos θ A exp ( ) i η =, i k µ ( ) sin θ cos θ A exp ( ) i η i k µ sin θ cos θ A exp ( ) i η (.48a) + i k µ ( sin θ cos θ ) A exp ( i η ) = (.48b) k c t = k c l = k c t, k sin θ = k sin θ = k sin θ (.49a, b) k = k, k = κ k, θ = θ, sin θ = κ sin θ (.5a, b, c, d) (.48) ( λ + µ cos θ ) A A κ µ sin θ A A = κ µ sin θ, µ sin θ A A κ µ cos θ A A = κ µ cos θ A κ sin 4θ =, A sin θ sin θ + κ cos θ A A = sin θ sin θ κ cos θ sin θ sin θ + κ cos θ (.5a, b). θ =, π / 4, π / A = SV. sin θ sin θ = κ cos θ θ A = P (.38c) κ > (.5d) ( ) θ < θ cr sin κ (.5) θ θ > θ cr cos θ = sin θ = κ sin θ < cos θ = i κ sin θ = i κ β, p () = sin θ = κ sin θ, p () = cos θ = i κ β β sin θ κ (.53a, b, c)

49 P u () i = A d () i exp { ( i k x p () + x p () c l t )} ( )} = A d () c l i exp (k β x ) exp {i k sin θ x t κ sin θ (.54) x x < exp (k β x ) - 3. (.) (.3) (.4) 4. x = µ b µ.9 SH θ (n = ) SH (n = ) SH (n = 4) A 4 A θ 4 [].. ρ (kg/m 3 ) µ (MN/m ) 5 67 8 85 56 95 49.8 SH θ = 6.3.3. Rayleigh. x x 7 x x = O x u = A exp ( b x ) exp {i k (x c t)}, (.55a) u = B exp ( b x ) exp {i k (x c t)}, u 3 = (.55b, c) x. Raileigh R( ) R(b) > (.56) x = σ =, σ = Hooke (λ + µ) u, + λ u, =, µ ( u, + u, ) = (.57a, b) 7 (.55)

.3. 493 (.4) (λ + µ) ( u, + u, ), + µ ( u, + u, ) = ρ ü, (.58a) (λ + µ) ( u, + u, ), + µ ( u, + u, ) = ρ ü (.58b) (.55a) (.55b) (.58) exp {i k (x c t)} (λ + µ) i k {i k A b B} + µ { (i k) + ( b) } A = ρ ( i k c) A, (λ + µ) b {i k A b B} + µ { (i k) + ( b) } B = ρ ( i k c) B (λ + µ) { k A i k b B } + µ ( b k ) A + ρ k c A =, (λ + µ) { i k b A + b B } + µ ( b k ) B + ρ k c B = k (λ + µ) + µ ( b k ) + ρ k c (λ + µ) i k b (λ + µ) i k b (λ + µ) b + µ ( b k ) + ρ k c A B = (.59) k (λ + µ) + µ b + ρ k c (λ + µ) i k b det (λ + µ) i k b b (λ + µ) µ k + ρ k c = c l c t k c l + b c t + k c det i k b ( ) c l c t i k b ( ) c l c t b c l k c t + k c = { c t b k ( c t c )} { c l b k ( c l c )} = (.6) c b, b b = k c, b c = k l c c t (.6a, b) c l > c t (.56) R(b) > (.6) c < c t < c l (.59) c l, c t ( ) B A = k c c l + b c t i k b ( ), B c l c A = i k b ( ) c l c t t k ( c ct) + b c l (.6) ( B A) = i b k = b i k, ( B A ) = k i b = i k b

494 u = { A exp ( b x ) + A exp ( b x ) } exp {i k (x c t)}, { u = b i k A exp ( b x ) + i k } A exp ( b x ) exp {i k (x c t)} b (.6a) (.6b) c (.6) (.57) b (λ + µ) i k A i k A + λ i k (A + A ) =, { b A b A } + i k { b i k A + i k } A = b c l, c t { ) } c l k ( c A c + k A k ( ) c l ct) (A + A ) =, b A (b + k A = l b ( c t c ) A + c t A =, b A k b ) ( c A c = t c b c t ( ) b k c c t A A / b = A, A ( ) ) k c 4 b c b = k ( c 4 k c c = t c t c l c t R(c) 4 c c l c c t ) ( c = (.63) c t c Rayleigh c r < ε R(c t ) = <, R(ε c t ) = 4 ε c t c l ε ( ε ) ( 4 ε c ) ( ) t ( ) 4+4ε ε = ε c t > c l c l < c < c t c r 8 R(c) = R(c) c Rayleigh < c r < c t < c l (.64) 8 Achenbach

.3. 495.3. Love Rayleigh SV SH (.55a) (.55b) u 3 = A exp ( b x ) exp {i k (x c t)} x 3 µ ( u 3, + u 3, ) = ρ ü3 (.65) c t = µ / ρ exp ( b x ) exp {i k (x c t)} k A + b A = k c A c t { k + b + k c } A = b = k c c t c t x = σ 3 = u 3, = u 3, (x = ) = b A exp {i k (x c t)} = (.66a, b, c) A b =. Love f (x ) exp {i k (x c t)}, > x > H u 3 = A exp ( b x ) exp {i k (x c t)}, x > (.67a, b) b c (.66c) ρ b, µ b H ρ, µ O x. Love x x = u 3 σ 3 lim ϵ (.65) u 3, + u 3, = c t c b t u 3, (x = H) = (.68) u 3 ( + ϵ) = u 3 ( ϵ), µ u 3, ( + ϵ) = µ b u 3, ( ϵ) (.69a, b) ü 3, x >, u 3, + u 3, = ( c b t ) ü 3, > x > H (.7a, b) c b µb t (.7) ρ b x > (.67b) (.66c) (.67a) (.7b) { k f (x ) + f (x ) } exp {i k (x c t)} = k c ( c b t ) f (x ) exp {i k (x c t)}

496 f (x ) ( c f + q b f =, q b k c b t ) (.7a, b) f (x ) = B sin (q b x ) + B cos (q b x ) u 3 = {B sin (q b x ) + B cos (q b x )} exp {i k (x c t)} (.73) x = H u 3, (x = H) = q b {B cos (q b H) + B sin (q b H)} exp {i k (x c t)} = B cos (q b H) + B sin (q b H) = (.74) x = (.69) (.67b) (.73) exp {i k (x c t)} B = A, µ b q b B = µ b A (.75a, b) (.74) (.75) A µ b q b µ b B cos (q b H) sin (q b H) = B, B µ b cos (q b H) µ b q b sin (q b H) = tan (q b H) µ b µ b q b = (.66c) (.7b) b q b ( ) ( ) c µ c L(c) tan k H c b t c t = (.76) ( ) c µ b c b t B Love c c t > c b t (.76) ( ) L(c = c t ) = tan k H ct c b t >, L(c cb t) = c b t < c < c t Love Rayleigh c k Love (.76) c k Love Fourier

.3. 497 x ρ ρ T T T T x x P θ x. [35] RC -3 5.. [] 6.. [] syllabus, office hours, portfolio, effort, faculty development, good practice, manifesto initiative, solution, action plan and front runner: hours s syllabus 5 6 syllabus

498. E : 9 7 4 4 6 6? 4 9 T T [43] (Don t)!

III 499

Machinac 橋, Michigan 州 957 年竣工 985, 987 年当時の児島坂出ルート

... (). σ ϵ ϵ ϵ ϵ A B A B A B A B C C A B C C C..4.,.4.3. A E E S E E S. A 5

5 A B A C D E (a) (b).3 3 (c) S E S S E : : : : :.3 AB

.. 53 : C.3.3 (a) (b) (c) :.3 D Northwestern Nemat-Nasser (98 Plasticity % ()..4 U

54. U 3. 4. 3 5. 4 Bauschinger.4..4 σ.3 (a) (b) U.3 (b) σ, ϵ O ϵ.4.3 (c).4.. () 4 3 4 ( [5, 6])

.. 55 (3.43).5 I A B C D II A E I C ϵ F III G H K σ A, B, E, D, F, H, K 7 H C K F ϵ D G σ O A B E.5 ϵ = F(σ, ϵ) σ dϵ = F(σ, ϵ) dσ G(σ, ϵ) = G(σ, ϵ) = G(σ, ϵ) =.3.3 3.4.5 ().6 F µ s f S µ s N = F (cos θ µ s sin θ) = (.) N γ p ϵ σ F θ S F(+α).6 +α F F ± F, N N ± N, S S ± S, θ θ ± θ f > (.)

56 S ( ) γ p = λ S S, γp S, λ = λ(f, S, θ, F, S, θ) (.3a, b, c) γ p.6 γ p λ > ().7 45.7 τ y >.4 τ, σ τ y σ y O γ τ σ ϵ γ, ϵ.7 τ y γ τ y τ = τ y (.4) τ < τ y γ = τ = τ y τ < τ τ < γ = τ < τ y γ = (.5) τ = τ y τ τ < τ τ = τ τ τ y τ γ = τ τ = τ y τ τ = (.6)

.. 57 (.b) γ..3 γ (3).8 γ = τ µ { } τ < τy { τ = τy τ τ < } (.7a) τ τ y µ µ γ = τ { τ = τ y τ τ = } (.7b) O γ.8 µ 5 (H ) (4) γ = γ e + γ p (.8) e p Hooke γ γ e = τ µ τ τ H µ µ O γ.9 (.9) { γ p = τ } (.3a) γ p = λ τ τ, γp = λ τ, λ > (.a, b, c) λ 6 τ λ = { τ < τ y (γ p ) } { τ = τ y (γ p ) τ τ < } (.a) λ > { τ = τ y (γ p ) τ τ } (.b) τ y τ y (γ p ) γ p γ p τ y τ τ = τ τ 5 G µ 6 λ (.b) λ Lamé µ Lamé λ

58 : (.8) γ e = τ (.) µ Hooke 7 τ y τ O H H = H µ γ p. γ = γ e + γ p (.3).9 ( τ = τ + H γ τ ) µ (.4) (.) (.3) τ = τ + H µ τ + H γ p H µ τ ( τ = τ + H ) γ p µ τ τ y (γ p ) γ p γ p = γ p τ y (γ p ) τ y (γ p ) = τ + H γ p, H τ ( y γ p = H ) (.5a, b) µ H. f (τ) τ ( τ + H γ p) = (.6) f (τ) H H H µ H H λ τ > (.6) (.6) f = τ H γ p = (.b) λ = τ H τ = τ τ γ p = τ τ H τ H τ τ = τ τ τ (.7a, b) H τ y 7 γ p

.. 59 τ τ > H µ H > λ > (.7b) τ τ 3 τ (.6) τ y (.b) (.7b)..3 () [] P O + cos 3 α + cos α N / cos α α δ N α P, δ. 3 Young σ ϵ σ = E ϵ τ y σ y ϵ y σ y E. : N + N cos α = P : δ = δ cos α, δ : N = EA δ l, N = EA δ l/ cos α N = σ y A, N = σ y A A 3 3 N = δ = P l EA + cos 3 α P + cos 3 α > N = P cos α + cos 3 α δ = l σ y E δ = l EA cos 3 α (P σ y A) N l

5 P = P p P y ( + cos α), δ = l σ y E cos α δ. P P A σ y = P P y, δ δ E l σ y = P y A σ y P p.5 δ l ϵ y (). (a) (b) z y (c) h b σ y z y σ y σ y σ y σ y σ y (a) M y (b) (c) M p. M ϕ M = EI ϕ, σ e = M W e, W e I h/ σ e W e (a) σ e M y M y W e σ y = EI ϕ y = bh 6 σ y, ϕ y σ y Eh ϕ y (b) 3 M = M y ( ) ϕy ϕ, z y = σ y E ϕ = h ϕ y ϕ ϕ (c) (.8a, b) M M p = bh 4 σ y = 3 M y (.9) M p (3).3 3 a 8 x P > P y l a < x < l 3 z y = h 4 P x P y l, P y 4 M y a, l l = P y (.a, b, c) P 8.3

.. 5 P.5 P x z y z l/ a M M y.5 cl O / 9 w.3 3.4 w( l / ) w ( l / ) = l / w = { 5 ( P + 3 ) } 3 P P ϕ(η) η dη 9 P y w w(l / ) δ y, δ y M y l EI, P P P y δ y P P p 3 P y = 4 M p l (.) P = P p z z y ( l / ) M p.4 3.5.3 -..,.3,.4 9?

5. 3.. Prandtl-Reuss (.8) ϵ p ϵ e ϵ i j = ϵ i e j + ϵp i j = ) ( ui, j + u j,i dϵ i j = dϵi e j + dϵp i j = ( dui x j + du ) j x i (.a, b) d u u u u du, u u t, t = (.) (3.6) (.) [7]/ [3]..,..3 (3.43) 3 (3.43) J (3.36) ( J σ kk = ), 3 σ J ( ) σ i j σ ji, J 3 det σ (.3a, b) J (3.36b) I s 3 J s i = 3 cos θ J s, s ii = 3 cos s 3 J s J 3 = ( θ s π 3 ) ( J, s iii = 3 cos θ s + π ) 3 (.4a, b, c) θ u u θ t 3.43 (.7) 3

.. 3 53 θ s (.58a) Lode cos 3θ s = 3 3 J 3 J 3, θ s π 3 (.5a, b) s i + s ii + s iii = (.4) (.5) J J 3 J J 3 Tresca J J 3 Mises Mises (.3a) Euclid 6 σ 6 x -x σ (.3a) σ = σ σ σ σ f (σ, ϵ p ) σ τ y (ϵ p ) f = (.6a, b) (.53) (.54) f < f > Mises f τ y ϵ p (.3a) ϵ p ϵ p i j ϵp ji dt (.7) t 4 5 ϵ p γ τ y x σ (.3a) σ = / 3 (σ ) σ = 3 τ y σ y 6 (.3a) (.6) σ y = 3 τ y (Mises) (.8) f σ σ y ( ϵ p ), σ 3 σ i j σ ji = 3 σ (.9a, b) 4 5 6 [7]

54 σ 7 (.35) (.7) ϵ p 3 ϵp i j ϵp ji dt (.3) 3 σ ϵ p.5 8 9 3.4.5 () σ A B σ a < τ y σ b = τ y C σ c < τ y σ.5 B A.6 ABC B C Miller ().6 3 AB, BC, CA A C Miller [ ] 4 3 4 (.6b) 7 Mises 8 9 {} < > : =

.. 3 55 3.4. () (3.54). 3 σ = σ a, σ = σ 33 = σ c σ = 3 (σ a σ c ), σ = σ 33 = 3 (σ a σ c ) σ = 3 σ a σ c (σ a σ c ) 3 ϵ p = ϵ a, ϵ p = ϵp 33 = ϵ l. 3 (.7) ϵ p = 3 ϵ a ϵ l = 3 3 ϵa p p p ϵ p ii = ϵ a + ϵ l 3 σ ϵ p ϵ l ϵ a p (σ a σ c ) ϵ a.. Hooke Hill [39] Saint-Venant 87 ( ϵ ) (σ ) [4]

56 Lèvy (87) Mises (93) ϵ xx σ xx = ϵ yy σ = = ϵ xy yy σ xy Prandtl (94) Reuss (93) ϵ xx p σ xx = ϵp yy σ = = ϵp xy yy σ = = λ pr ϵ p i j = λ pr σ i j, λ pr (.3a, b, c) xy Prandtl-Reuss ϵ p σ (.3b) (.3) (.3b) λ pr (.3b) (.7) (.3a) ϵ p = ϵ p i j ϵp ji = λ pr σ i j σ ji = λ pr σ λ pr = ϵ p σ λ pr λ pr (.3b) Saint-Venant p, ϵ p σ (.33a, b) (3.44) (3.45) e σ ave, ϵ e σ ; e σ ave, ϵ e σ σ B σ ξ η σ B σ ϵ p ξη = ϵp ξη > ξ η (.3).7 ξ-η σ e p (3.4) σ ave (3.4).5 3.7 3

.. 3 57 ẇ p σ i j ϵ p i j (.34) (.3b) (.34) (.3a) σ (.9b) σ ẇ p = σ i j ϵ p i j = λ pr σ = 3 λ pr σ (.7) ϵ p (.3) ϵ p (.3b) ϵ p = ϵ p i j ϵp ji = λ pr σ i j σ ji = λ pr σ, ϵ p = 3 ϵp i j ϵp ji = λ pr 3 σ i j σ ji = 3 λ pr σ λ pr 3, 4 ẇ p = σ i j ϵ p i j = σ ϵ p = σ ϵ p (.35) [ ] ( ) = ( ) cos( ) = ( ) ( ) = (.3a) σ (.9b) σ (.7) ϵ p (.3) ϵ p ϵ p = ϵp = γ > (.7) ϵ p ϵ p = ϵ p i j ϵp ji = 4 ( ϵ p ) = γ σ = σ (.3b) ϵ p = 3 λ pr σ, ϵ p = ϵp 33 = 3 λ pr σ ϵ p = ϵ >, ϵ p = ϵp 33 = ϵ (.3) ϵ p ϵ p = ( ) 3 ϵp i j ϵp ji = 3 3 ϵ = ϵ σ ϵ p..3 f = f f = (.36)

58 (.6a) f = σ σ i j τ y(ϵ p ) σ σ i j ϵ p ϵp = σ i j τ y(ϵ p ) σ i j ϵ p ϵ p i j ϵp i j = (.3b) (.3a) σ σ i j = σ i j σ (.37) λ pr f = σ σ i j τ y(ϵ p ) σ i j ϵ p λ pr σ i j σ i j = σ kl σ σ kl τ y(ϵ p ) ϵ p λ pr σ = λ pr = H σ kl 4 σ σ kl, H τ y(ϵ p ) ϵ p (.38a, b) H (.3b) ϵ p i j = H σ i j σ kl 4 σ σ kl (.39) (.7b) ( ϵ p kk ) 4 5 τ τ 3 (.38a) λ pr > H > ( σ i j σ i j > ) : λ pr = f < (.4a) : λ pr = f = σ i j σ i j < (.4b) : λ pr = f = σ i j σ i j = (.4c) : λ pr > f = σ i j σ i j > (.4d) : f > (.4e) 6 ( σ i j σ i j > ) 9 Hooke (3.5b) ϵ ϵ p ϵ p 4 5 (3.57a) 6

.. 3 59 (3.5b) σ i j = C i jkl ( ϵkl ϵ p kl) = Ci jkl ϵ e kl (.4) Hooke (3.59) ϵ i e j = D i jkl σ kl = µ σ i j + ( 3 3K ) δ i j σ kk (.4) µ ϵ e i j σ i j (i j) (.39) ϵ p i j σ kl σ i j (i j) (.39) (.4) (.) ϵ i j = µ σ i j + 3 ( 3K ) δ i j σ kk + σ i j σ kl σ µ H 4 σ kl (.43) (.39) dϵ p i j = H σ i j σ kl 4 σ dσ kl dϵ p dϵ p = F dσ + F dσ + F 3 dσ 33 + F 4 dσ 3 + F 5 dσ 3 + F 6 dσ (a) F σ F σ, F σ 3 F 4 σ, (.44a, b) (a).3.3 σ σ (σ, τ ) N d d N σ σ σ τ d d N ϵ p dϵ p = ϵ p dt ϵ p ϵ p = ϵ p (.45a, b)

5 7 (.) ϵ = ϵ e + ϵ p ϵ p ( ) : (.43) (.43) i j 3 8 ϵ ii = µ σ ii + ( 3 3K ) δ ii σ kk = µ 3K σ kk σ kk = 3K ϵ kk (b) (.43) σ i j i j 3 σ i j ϵ i j = µ σ i j σ i j + σ i j σ i j σ H 4 σ kl σ kl (.3a) σ σ i j ϵ i j = µ σ i j σ i j + H σ kl σ kl = µ + H µ H σ i j σ i j σ i j σ i j = µ H µ + H σ i j ϵ i j (c) (b) (c) (.43) µ ( σ i j = µ ϵ i j + K µ ) δ i j ϵ kk µ σ i j σ kl ϵ 3 µ + H σ kl (.46) (3.56) C σ i j = C i jkl ϵ kl µ σ i j σ kl ϵ µ + H σ kl (.47) µ H µ + H.9 H (.43) (3.58) ϵ i j = D i jkl σ kl + H σ i j σ kl 4 σ σ kl (.48) C ep i jkl C i jkl χ µ σ i j σ kl, D ep µ + H σ i jkl D i jkl + χ σ i j σ kl (.49a, b) H 4 σ 7 [7] (algebraic sum) 8

.3. 5 σ i j = C ep i jkl ϵ kl, ϵ i j = D ep i jkl σ kl (.5a, b) χ χ = (.5) : H (.43) (σ = σ ) ϵ = E σ + 3H σ = E t σ E Young E t 3H = E t 3H = E t E E t E 3H 3 σ σ ϵ p ϵ p 3 σ = τ γ = ϵ = µ τ + H τ = H τ H H = H µ H = H H µ (.5b).4.4 -. (.49) Poisson (3.84).3.3. () f = f (J, J 3, ) Prandtl-Reuss f = 6

5 Mises : Mises f = f (J, ) = J τ y ( ) = (.5) π e = σ iii / 3 / 3 / 3 J { (σ σ 33 ) + (σ 33 σ ) + (σ σ ) } σ ii J = 6 + σ 3 + σ 3 + σ (.53) O σ i f = { (σii σ iii ) + (σ iii σ i ) + (σ i σ ii ) } τ y = 6 (.54).8 π Mises.8 3 σ i = σ ii = σ iii e π e 9 Tresca :.6 4.5 3.6.3 (3) Mohr, / σi σ j, (i, j = I, II, III) Tresca f (σ, ) { (σ i σ ii ) 4 τ y( ) } { (σ ii σ iii ) 4 τ y( ) } { (σ iii σ i ) 4 τ y( ) } = (.55).9 f (J, J 3, ) = 4J 3 7J 3 36 (J ) τ y + 96 J (τ y ) 4 64 (τ y ) 6 = σ i (.55) f f f = ( ) max σi σ j 4 τ y ( ) (.56) i, j=i, II, III s x (x=i, ii, iii) σ i σ ii 3 σ iii σ i, 3 σ ii σ iii 3 σ i σ ii, 3 σ iii σ i 3 σ ii σ iii 3 (.4) s x Tresca f = J cos θ l τ y = (.57) 9 Mises [9] H.

.3. 53 [7] θ l Lode θ l 3 3 3 J 3 sin, π J 3 6 θ l π 6, θ l = θ s π 6 (.58a, b, c) (.5) θ s τ y σ iii σ ii σ iii σ ii.9 B 3τy σ i σ ii σ i σ ii A, B σ i = σ iii = O τ y O 3τy ±τ y, σ ii = A Tresca.9 Mises Tresca σ y σ y = τ y ( Tresca ) (.59) Mises (.8) 3.9 Tresca Mises (.8) Mises Mises Tresca Mises [] Mises 3 () Drucker [] (σi ) j σ () p i j ϵ i j dt (.6) Drucker σ () i j (.6) ( ) σi j σ () p i j ϵ i j (.6) τ C A τ y B O γ p. (.6). B C σ σ () ϵ p [39] 3.5 (.7) 3 Mises

54 (3) (.6b) f h(j, J 3 ) τ y (ϵ p ) =, (.6) τ y ϵ p h J J 3.4 σ () (.6) σ ( σ σ ()) ϵ p. (.6). (.3b) σ : : ϵ p ϵ p f = B B A A.. ( σ σ ()) 9 (B) (.6). (.6) (B) Mises Tresca ϵ p i j 3 f ϵ p i j σ = λ f, λ (.63a, b, c) i j σ i j f σ i j ( f = ) 33 λ pr λ (.6) f (.63b) (.3b) 3 Tresca.9 (.63b) [5] [].4.3 [8] 33 f (x, y) = x + y r = x y ν f f t x y = x y t

.3. 55 (4) (.6) f = f σ i j τ y(ϵ p ) f σ i j ϵ p ϵp = σ i j τ y(ϵ p ) σ i j ϵ p (.63b) f = f σ i j λ τ y(ϵ p ) σ i j ϵ p f σ i j f σ i j = ϵ p i j ϵp i j = λ = H f σ i j, H τ y(ϵ p ) σ i j ϵ p f σ i j f σ i j (.64a, b) H (.63b) ϵ p i j = H f σ i j f σ kl σ kl (.65) Prandtl-Reuss Mises (.37) (.65) (.39) (.64a) : λ = f < (.66a) : λ = f = f σ i j σ i j < (.66b) : λ = f = f σ i j σ i j = (.66c) : λ > f = f σ i j σ i j > (.66d) : f > (.66e) Mises (.37) (.66) (.4).3. () Hill [39] (.6) h τ y f : f =, f =, ḣ >, τ y > ϵ p >, f = (.67)

56 (.6) τ y h : f =, f =, ḣ =, τ y = ϵ p =, f = (.68) τ y : f =, f <, ḣ <, τ y = ϵ p =, f < (.69) ϵ p ḣ ϵ p i j = G i j ḣ (.7) G (G ) (σ ), G kk = (.7) G f G i j = H g(j, J 3 ) σ i j, g(j, J 3 ) σ kk = (.7a, b) g(j, J 3 ) H g (.7a) (.7) ϵ p i j = H g(j, J 3 ) σ i j ḣ (.73) f = (.6) f = = ḣ τ y ϵ p i j (.73) ḣ ḣ = τ y ϵ p i j H ϵ p i j g σ i j ḣ H = τ y ϵ p i j (.73) (.3b) ḣ H λ ϵ p i j = λ g(j, J 3 ) σ i j, λ (.74a, b) (.63b) g f Drucker (.6) g σ i j

.3. 57 () f (J, J 3, ϵ p ) h(j, J 3 ) τ y (ϵ p ) (.75) f g(j, J 3 ) (.74a) Hill (.67) ḣ = h σ i j = f σ i j > σ i j σ i j (.68) (.69) (.66) (.75) f = h σ i j σ i j τ y ϵ p ϵ p i j ϵp i j = (.74a) f σ i j f = f σ i j σ i j τ y ϵ p λ h σ i j = g(j, J 3 ) σ i j g(j, J 3 ) σ i j = (a) f = g = const. H (.64b) g H τ y ϵ p f σ ϵ p σ. g(j, J 3 ) σ mn g(j, J 3 ) σ mn (.76) λ = H f σ kl σ kl (b) (.74a) ϵ p i j = H g(j, J 3 ) σ i j f (J, J 3 ) σ kl σ kl (.77) g H. Hooke (.4) (.77) (.) ϵ i j = { D i jkl + χ H g σ i j f σ kl } σ kl (.78) D ep i jkl = D i jkl + χ H g σ i j f σ kl (.79) χ (.5)

58 : λ Simo Hughes [83] (a) (.76) f σ i j σ i j = H λ (c) (.4) ( σ i j = C i jkl ϵ kl λ g ) σ kl (d) (d) (c) f C i jkl ϵ kl λ f C i jkl σ i j σ i j g σ kl = H λ λ λ = H f C abcd ϵ cd, H f g C mnpq + H (.8a, b) σ ab σ mn σ pq (b) (d) { σ i j = C i jkl ϵ kl χ g } f C abcd ϵ cd σ kl H σ ab σ i j = { C i jkl χ H C i jab g σ ab } f C cdkl ϵ kl (.8) σ cd C ep i jkl = C i jkl χ H C i jab g σ ab f σ cd C cdkl (.8) χ (.5) J : Prandtl-Reuss (.65) (.77) f (J, J 3 ) = h(j, J 3 ) = g(j, J 3 ) = σ i j h = g (= σ) (.83a, b) σ i j σ i j σ i j σ Prandtl-Reuss (.39) f g (.83) (.77) ϵ p i j = P i j Q kl σ kl, ϵ p = (P Q) : σ P Q Prandtl-Reuss (.39) J J ϵ p = λ J σ i j σ ji = λ (.84) λ

.3. 59? (.49a) C ep H >, H < µ H µ > + ν 6 3 f = ḣ < τ y < (λ > ) ḣ = h σ i j < σ H < h = f f σ i j < σ i j σ i j σ i j (.66) σ i j : λ = f < (.85a) : λ = f = f < f σ i j < (.85b) σ i j : λ = f = f = f σ i j σ i j = (.85c) : λ > f = f = H : f > f > (.8a) λ : λ > f σ ab C abcd ϵ cd > f σ i j σ i j > (.85d) (.85e) λ > (.8b) H > H ϵ, σ? -3 3. (.43) (.46) (.8) Prandtl-Reuss (.43) (.46).3.3 ()

53 34 Prandtl-Reuss f σ τ y, σ J, J σ i j σ i j, τ y τ y + H ϵ p, ϵp ϵ p i j ϵp i j (.86a, b, c, d, e) H f = σ i j σ σ i j f σ i j > σ i j σ σ i j σ i j > (.87) ϵ p i j = H 4 σ σ i j σ kl σ kl (.88) σ = σ, σ = τ, (.89a, b, c) (.4) Hooke ϵ e = E σ, ϵe = ν E σ, ϵe 33 = ϵe, ϵe = τ (.9a, b, c, d) µ σ = 3 σ, σ = 3 σ, σ 33 = σ (.9a, b, c) (.86b) (.86c) σ = 3 σ + τ (.9) (.87) σ i j σ i j = 3 σ σ + τ τ σ σ + τ τ > (.93) 3 (.88) (.93) ϵ p = σ 3H ( σ + 3τ ) (σ σ + 3 τ τ), ϵp = ϵp, ϵp 33 = ϵp, ϵp = τ H ( σ + 3τ) (σ σ + 3 τ τ) (.94a, b, c, d) ϵ p i j = λ pr σ i j 3 ϵp σ = ϵp τ (.95) 34 Nemat-Nasser! p.76 4

.3. 53 (.94) (.86e) ϵ p = 3 ( ϵ ) p ) { } p + 4 ( ϵ = H σ σ + τ τ / 3 σ + τ 3 (.9) ϵ p = H σ H (.96) ϵ p = H 3 σ + τ + const. σ = H ( ϵ p const. ) σ = τ y + H ϵ p (.97) (.86a) (.86d) σ τ τ 3H ϵ p = σ, σ H ϵp = τ (.98a, b) (.94) 3H ϵ p ( H ϵ p = A ) σ τ, ( A ) σ + 3τ σ 3σ τ σ τ 3τ (.99a, b) A τ = A σ, A τ = A σ (.a, b) A τ = 6σ τ (σ + 3τ ), A σ = 3τ ( 3τ σ ) (σ + 3τ ), A τ = σ ( σ 3τ ) (σ + 3τ ), A σ = 6στ (σ + 3τ ) (.) (.44) σ / τ = const. [6] (.) τ σ? () (.99) p.43 Alternative Theorem ( A ) t { } { } w = { } w = 3H ϵ p t { } w = 3 ϵ p H ϵ p τ ϵp σ = 3 ϵp σ = ϵp τ τ σ

53 (.95) ( A ) { } { } v = { } v = (.95) (.98) { } = 3H ϵ p H ϵ p 3τ σ σ τ = 3H ϵ p H ϵ p + c 3τ σ (.) (.) c (.93) c ( f/ σi j σ i j = ) (.) ( ) ( ) ( ) = + c < (3) σ (.9) (.98b) ϵ p = H τ, σ = τ Hooke (.9) ϵ e = µ τ τ τ y O µ τ y µ ( H + ) µ µ µ + H τ y ϵ.3 ϵ = µ τ + H τ τ > ( ϵ = H + ) τ µ H τ y τ = µ H µ + H (ϵ µ ) + µ + H τ y = µ (ϵ ) µ µ + H { (ϵ ) τ y µ.3. (.4) }

.3. 533 (4) (.6) σ = τ y + h ( ϵ p ) m H = m h ( ϵ p ) m (.a, b) (.96) (.) (.94) ϵ = τ ( τ ) µ + τy / m h (.3).4.46 h τ τ y ( µ τ y ) τ τ / y m h O ϵ µ.4 (5) 3 σ + τ = ( ) τ y + H ϵ p current, 3 (σ + σ) + (τ + τ) = { τ y + H ( ϵ p current + ϵ p)} (.4a, b) (.96) ( τ y + H ϵ p current) 3 (σ + σ) + (τ + τ) = 3 σ + τ + (σ τ τ σ) = / 3 σ + τ ( ) /3 σ + τ 3 σ σ + τ τ σ τ τ σ = σ τ = σ τ ln σ = ln τ + const. σ τ = const. (.5)

534 ( f = ) ( ) { ( )} 3 (σ + σ) + (τ + τ) 3 σ + τ + / 3 σ + τ 3 σ σ + τ τ ( ) 3 σ + τ + /3 σ + τ 3 σ σ + τ τ (.96) (.4) (.6) σ ϵ p = σ ϵ p + τ ϵp (.7) = τ y + H ( ϵ p p) current + ϵ (.6) (.) c 35 c C ep D ep (6) E = GN/m, ν =.3, H = E /, τ y = 3 MN/m.5 3 ts prop τ = α σ (prop) (σ final, τ final ) (σ start, τ start ) σ τ σ final (st: shear-to-tension) τ final (ts: tension-to-shear) O τ st α f σ τ y = σ.5 35 c

.3. 535 α α = / 3 α. N N = ts st N =.. (MN/m ) (.97) α σ start τ start σ final τ final / 3 367 433 5 / ( 3 ) 57 9.9 589 34. / 3 5.7 99 64. 37 % α = / 3.4.7% α = / ( 3 ) α = / 3. (%) α prop-ortional ts: torsion-to-stretch st: stretch-to-torsion ϵ.48.688 ( 4.%:.7%).584 (3.9%:.6%) / 3 ϵ 9.695.7 (3.9%:.4%) 9.3 ( 4.%:.5%) ϵ p 6.777 6.777 6.777 ϵ.7.63 (.65%:.38%).78 (.57%:.34%) / ( 3 ) ϵ.593.75 (6.4%:.33%).9939 ( 6.%:.3%) ϵ p.848.848.848 ϵ.957.884 ( %:.59%).36 (%:.63%) / 3 ϵ 8.3 8.3 (.94%:.47%) 8.9 (.%:.58%) ϵ p 35.93 35.93 35.93 (.4).3.3 (5) (.5) f 5 α = / 3 f.6 (σ start, τ start ) (σ final, τ final ) N = 3 µ N =

536 4 ( ) f log τ y 6 N = N = N = 4 ( ) f log τ y 6 N = N = N = 8 ts α = 3 8 st α = 3...3 ϵ p...3.6 ϵ p.3 (%) α prop ts st ϵ.698.389 (.6%:.5%).99 (.5%:.5%) / 3 ϵ.978.9975 (.6%:.3%).94536 (.7%:.3%) ϵ p.9957.9957.9957 α = / ( 3 ) f = f = (.97) ϵ p τ y % H = E / 5, τ y = 5 MN/m α = / 3 (σ start, τ start ) = (6, 354) % (σ final, τ final ) = (7, 4).3 %.5% (7)

.3. 537 return-mapping [83] (.95) (.9) (.96) ϵ p = 3 λ pr σ, ϵ p = λ pr τ (.8a, b) ϵ p = λ pr σ 3 + τ (.9) Hooke (.9) (.8) ( σ = E ϵ ) 3 λ pr σ, τ = µ ( ϵ λ pr τ) (.a, b) (.4) 3 (σ + σ) + (τ + τ) = 3 σ + τ + H ϵ p (.9) (.) λ pr {( ) f E (λ pr ) a λ pr + b λ pr + c =, a 4 σ + ( µ H ) } τ, (.a, b) {( E b 4 9 + H 3 c E 3 7 H 3 ) } σ + (µ + H) τ + E 9 σ ϵ + µ τ ϵ, (.c) { σ ϵ + E ( ϵ ) } + 4µ { τ ϵ + µ ( ϵ ) } (.d) (.4a) f = /3 σ + τ ( 3 σ σ + τ τ ) H ϵ p = (.9) (.) λ pr {( E 9 + H ) } ( ) σ + (µ + H) τ λ pr + 3 3 σ E ϵ + µ τ ϵ = (.) λ pr f (.) Newton-Raphson ( d f/ dλ pr = a λ pr + b ) (.3) Newton-Raphson (.94) Hooke (.9) ( ϵ = ϵ D ep ) σ τ (.) D ep = E + σ 3H ( σ + 3 τ ), Dep = σ τ H ( σ + 3 τ ), Dep = D, D ep = µ + 3τ H ( σ + 3 τ )

538.4 (MN/m ) prop ts st (ϵ, ϵ ) start (%) (.837,.3789) (ϵ, ϵ ) intermediate (%) (5.635, 5.93) (6.47, 4.8489) (ϵ, ϵ ) final (%) (.48, 9.695) (σ, τ) (433., 5.) (453.7, 37.53) (4.4, 6.96) prop (%) (4.8, 5.) ( 5., 4.8) ϵ p final (%) 6.78 6.8 6.8 N = (433., 5.) (454.4, 37.75) (4.76, 6.8) N = (433., 5.) (453.74, 37.55) (4.44, 6.98) N = (433., 5.) (453.7, 37.53) (4.4, 6.96) (.) σ ( τ = C ep ) ϵ ϵ = ( D ep ) ϵ ϵ (.3) (.89) (.49) Newton-Raphson α = / 3.4 α = / 3. 3 ts st ϵ intermediate.4 N = 5%..7 σ.8 3 ts st N =.6 µ N = µ

.3. 539 45 σ (MN/m ) ts τ (MN/m ) ϵ p =.39 st ϵ p =.68 prop 5 prop 4 st τ ts α = 3 35.5. ϵ.7 ϵ p = σ 35 4 45 σ (MN/m ).8 36 (.) (8) (.a) h = 5 MN/m, m =.7, τ y = 4 MN/m m < ϵ p ζ ζ = 3 m h ( ϵ p ) m, H = m h (ζ) m if ϵ p > ζ if ϵ p ζ ( ).5 (MN/m ) α σ start τ start σ final τ final / 3 489.9 8.8 779.4 45. / ( 3 ) 689.4 39.8 9 63. / 3 68.94 398. 9. 63. %.5 α = / 3.9 N = 3 ( ) 36

54 ( ) f log τ y N = 3 4 σ 8 (MN/m ) st (N = 8 ) N = 3 4 4 5 6 7 prop (N = 8 ) 7 6 ts (N = 8 ) 6 prop 8 5 α = 3..9 ϵ p.4.6.8 ϵ.3.6 (%) (N = 8 ) [prop] N = 3 4 5 6 7 8 ϵ.78875 ( %).87537.8869.88754.88768.8877 prop ϵ.6388 ( %).739.755.7364.7376.7377 ϵ p.97744 (.%).896.64.94.98.98 ϵ.845 ( 6.%).8495.85487.85548.85555.85556 [ 3.6%] ts ϵ.69787 ( 6.8%).7464.748.7486.74867.74868 [3.4%] ϵ p.849 ( %).38.8.96.98.98 ϵ.8578 ( 6.4%).9949.9567.9637.9645.9646 [3.%] st ϵ.657 ( 6.4%).6969.69534.69587.69593.69593 [ 3.8%] ϵ p.849 ( %).38.8.96.98.98.6 3 N = 3 N = 8 5%.9 (prop).3 3 N = 4 N = 8 3 kn/m.6 3% N = 8

.3. 54 σ (MN/m ) 8 6 σ 5.5. ts α = st ( 3 ) ϵ prop σ (MN/m ) 8 6 6 σ 5 st ts.5. prop α = 3 ϵ 4.5. ϵ 4.5. ϵ.3 ( N = 8 ).3 ( N = 8 ).3 ts st 3% % α = / ( 3 ), / 3 α = / 3 N = 8.3,.3 (.8) (.9) (.) (.a) λ pr f (λ pr ) ( σ + E ϵ pr) 3 3 E σ λ + (τ + µ ϵ µ τ λ pr ) 3 σ + τ + h ( ϵ p ) m h ϵp + λ pr 3 σ + τ Newton-Raphson m { d λ (m+) pr = λ (m) f ( ) } pr λ (m) pr f ( ) λ pr (m) λ (m+), pr λ (m) pr dλ pr λ (m+) < ϵ pr ϵ = 6 λ (m) pr N ϵ.6.6.7 3.6.6.7 N Newton-Raphson 4 m =

54.7 (MN/m ) prop ts st (ϵ, ϵ ) start (%) (.4495,.8385) (ϵ, ϵ ) intermediate (%) (.573,.3864) (.4834,.9455) (ϵ, ϵ ) final (%) (.8877,.7377) (σ, τ) N = (783.7, 447.87) (84.39, 4.58) (734.86, 474.75) (σ, τ) N = 3 (783., 447.9) (83.85, 4.93) (735.43, 474.45) (σ, τ) N = 4 (783., 447.9) (83.8, 4.96) (735.48, 474.4) (σ, τ) N = 5 (783., 447.9) (83.79, 4.97) (735.49, 474.4) prop (%) (5., 5.57) ( 6.7, 5.9) ϵ p final (%).96.5.89 N = (789.5, 45.94) (86.8, 44.79) (738.9, 476.43) N = 3 (783.6, 448.3) (84.9, 43.5) (735.77, 474.6) N = 4 (783.8, 447.95) (83.8, 4.98) (735.5, 474.43) N = 5 (783., 447.9) (83.79, 4.97) (735.49, 474.4).33 N N N.7 4.7 9.6.34 3 N = 3 5 7.5..33 Newton-Raphson.35 N =.34 N = 3 N = 8 µ.7 ϵ p

.3. 543 8 st (N = 8 ) (MN/m ) σ τ 5 (MN/m ) ts ϵ p =. 6 prop (N = 8 ) prop ( : N = ) 4 prop st ts (N = 8 ) α = 3 3 ϵ p =.387 τ 4.4.6.8 ϵ.34 ϵ p = σ 4 6 σ (MN/m ) 8.35 N = 5.3.4 Prandtl-Reuss [69] 3.6.4 Burgers b ν V b ν V x β p x x x.36 β p i j = ρ ϵ jnh V n ν h b i β p = ρ (V ν) b (a) 37 ϵ jnh (3.7) α hi ρ ν h b i α.36 b b t x 3 ν t V V t β p = ρ V b ϵ p i j = ( β p i j + β ji) p (b) 37 x < b β p = n b H( x ) δ(x )

544 (a) (b) ẇ p = σ i j ϵ p i j = σ i j ϵ jnh V n α hi >, ϵ p ii = ϵ inh V n α hi = V n α hi = µ g nhi, g nhi ϵ nh j σ ji, µ > (c) (c) ẇ p = µ ϵ nh j ϵ nhk σ i j σ ki = µ σ i j σ ji >, ϵp ii = µ ϵ nhi ϵ nh j σ ji = g nhi? 3??? ϵ nab ϵ ni j = δ ai δ b j δ a j δ bi (c) (a) (b) ϵ p i j = µ ϵ nh j ϵ nhk σ ki = µ σ ji, µ > µ λ pr (.3b) g g nhi g nhi = 4 ϵ nhk σ ki ϵ nhl σ li = σ ji σ ji (.3a) σ.4.4. ().37 A A.. () 5. Bauschinger O σ y B C A σ y σ ϵ.37.37 B C α.37 α B C A

.4. 545 B C α J f = f (σ i j α i j ) h(σ i j α i j ) τ y, h(σ i j α i j ) = (σ α) ( ) ( σ i j α i j σ i j α i j) (.4a, b) α α τ y (.63b) ϵ p i j = λ f σ i j, λ (.5a, b) (.4) f = h = σ i j σ i j (σ α) ( σ i j α ( ) i j) = σ τ i j α i j y (.6) (.4) f = f σ i j σ i j + f α i j α i j = f σ i j σ i j f σ i j α i j = f σ i j ( σi j α i j ) = (.7) α f σ.38 f =.38 σ = α ( σ α) α () Prager Prager α i j = c ϵ p i j (.8) (.5a) (.6) α i j = c λ f σ i j = c λ σ i j α i j τ y (.9) α (.7) f σ i j = f c λ σ i j α i j = λ H p k λ = σ i j σ i j τ y H p k f σ i j σ i j H p k H p k f σ i j c σ i j α i j τ y = c σ i j α i j τ y σ i j α i j τ y = c (.)

546 c λ ( ) ( ϵ p i j = f f H p σ kl = σ i j α i j σ kl α kl) k σ kl σ i j H p k 4 τ y σ kl (.) α (.8) (.) ( ) ( σ i j α i j σ kl α kl) α i j = σ τ kl (.) y (3) Ziegler (.7) α i j = α i j (ϵ p ) (.3) (.7) f = f σ i j f σ i j σ i j α i j ϵ p ϵ p = (.7) (.5a) (.6) ϵ p = λ f σ kl f σ kl = λ H k f σ i j α i j ϵ p (.4) H k λ = H k f σ i j σ i j ( ) ( ϵ p i j = f f σ kl = σ i j α i j σ kl α kl) σ H k σ kl σ i j H k 4 τ kl (.5) y Prager (.) Prandtl-Reuss (.46) α α Ziegler α i j = ξ ( σ i j α i j ), ξ (.6a, b)

.4. 547 α ξ λ (.7) (.6) ( ) ( ) f σ i j = f ξ ( ) σ i j α i j σ i j σ i j ( ) = ξ ( ) ( ) σ τ i j α ( ) ( ) i j σi j α i j = ξ σ y τ i j α i j σ i j α i j = τy ξ y τ y ξ = f σ i j ξ = ( ) σ σ i j τ y τ i j α i j σi j y ( ) Prager ( ) ( ( ) ( σi j α i j σ kl α kl) σ i j α i j σ kl α kl) Ziegler: α i j = σ τ kl, Prager: α i j = y τ y σ kl (.7a, b) (.9) Prager (.6a) Ziegler Ziegler (.6a): α i j = ξ ( ) σ i j α i j, Prager (.9): αi j = c λ ( ) σ τ i j α i j y (4) Hooke (.4) (.5) (.) ( ) ( σ i j α i j σ kl α kl) ϵ i j = D i jkl σ kl + χ σ 4 H k τ kl (.8) y..3 ( σ i j = C i jkl ϵ kl χ µ σ i j ) ( α i j σ kl α kl) ϵ (µ + H k ) τ kl (.9) y χ (.5) (.7) (5) 38 Ziegler (.4) f (σ α) τ y (ϵ p ) (.3) f = f σ i j f σ i j σ i j α i j ϵ p ϵ p τ y ϵ p ϵ p = 38

548 (.5a) (.38b) (.4) λ = H k + H f σ i j ) ( σ kl α kl) σ i j ( σ ϵ p i j = i j α i j σ 4 (H k + H) (σ α) kl (.3) ( ) ( H σi k j α i j σ kl α kl) α i j = σ H k + H (σ α) kl (.3) H = Ziegler (6) Ziegler.3.3 E = GN/m, ν =.3, H = E /, τ y = 3 MN/m H k = 5H 6 σ (MN/m ) st prop σ σ τ σ.5 3 ζ α, η α 5 ts.3.3 σ τ 4 α = 3 σ := σ ζ, τ := τ η ζ = H k σ ζ H k + H 3 τ y {(σ ζ) σ + 3 (τ η) τ}, η =.5. ϵ.39 H k τ η H k + H 3 τ y {(σ ζ) σ + 3 (τ η) τ} α = / 3 3 (σ, τ) : (367.4,.) (66., 35.) MN/m % ϵ p =.9.39 N = 7.4.4 α st ts

.4. 549 ( ) f log τ y 4 3 4 N = 4 τ (MN/m ) ts 6 8 5 6 7 st α st...4 ϵ p O (MN/m ) 3 6 σ 9.4-4 4. (.8) (.9) 5. (.3) (.3) 6..39.4. () Drucker-Prager [3] [74] f g f σ F(I, p, ϵ p ), g σ + G(I ) (.33a, b) σ (.3a) I (3.36) p p 3 I = 3 σ kk (.34) (.7) ϵ p ϵ p i j ϵp i j dt, p ϵ p kk dt (.35a, b) p 39 39

55 () (.33b) (.74a) ϵ p i j = λ σ i j σ + β δ i j, β = β(i ) G(I ) (.36a, b) I ϵ p i j = λ σ i j σ, ϵp kk = 3λ β (.37a, b) β f = (.37) σ σ i j σ i j σ σ i j F I σ i j F I İ F p σ kk = F p ϵp kk + F ϵ p (.37) (.38) σ i j σ σ i j F I σ kk = λ p F ϵ p ϵ p = { 3 F p β + F } ϵ p ϵ p i j ϵp i j (.38) λ λ = ( σ kl H σ F ) δ kl I σ kl, H 3 F p β + F ϵ p (.39a, b) H ϵ p p F (.36a) ϵ p i j = H σ { i j σ + β δ σ } kl i j σ + α δ kl σ kl, α = α(i, p, ϵ p ) F(I, p, ϵ p ) I (.4a, b) α α = β (3) Hooke (.4) (.4a) (.) ϵ i j = D i jkl σ kl + χ H σ i j σ + β δ i j ( σ ) kl σ + α δ kl σ kl (.4) χ (.5)..3 σ i j = C i jkl ϵ kl χ µ σ i j σ + 3K β δ i j ( µ σ kl σ H + µ + 9K α β + 3K α δ kl ) ϵ kl (.4)

.4. 55 (4) α (.33a) (.4b) f = σ + α I τ y ( p, ϵ p ) = (.43) I σ.4 3 P (.4a) σ i j = 3α δ i j τ y, τ y τ y (, ) ϵ p i j = αβ H δ i j σ kk.3.3 σ σ τ σ α < / 3.577 σ 3 + 3 α τy + 3α α = / 3 τ ( 3α = τ ) y 3α τ y σ: O σ iii P I : P σ 3 τ y + τ = τ y α > / 3 σ 3 3 α τy 3α τ ( 3α = τ ) y 3α σ ii O σ i = σ ii = σ iii σ i.4.4 σ i -σ ii.43 α.4 P (.36a) ϵ p 3 =, ϵp 3 = σ 3 =, σ 3 = ( σ = ϵ p ) 33 = λ 33 σ + β σ 33 = β σ (.44a, b) β (.44b) σ 33 = (σ + σ ) ξ, ξ 6β 3β (σ σ ) + σ

55 α = 3.5.5 O α =.3..5.5.5 τ τ y α =.9.7 σ 3τy α =..43 β =.5.. α =.3.5 τ y.5.5 O.5 α =.4, β =.37.5 τ τ y α =, β = σ α =. β = β =.9.44 β = (3.77) ν = / ν = / σ = (σ σ ) + ξ 6, σ = (σ σ ) + ξ 6, σ 33 = ξ 3 ( σ σ σ ) = + σ + ξ = {( σ σ ) } + σ 3β, I = (σ + σ ) ξ 6 σ σ, τ σ (x + ηα) η ( αβ) + y η =, η 3β ( αβ) ( 3β ) α, x σ τ y, y τ τ y.44 β < 3 β < α α α β (5) (.33a) α (.43) Drucker-Prager [, 3] σ + α ϕ I k = (.45) α ϕ k (.43) α τ y Mohr-Coulomb σ max σ min + σ max + σ min sin ϕ = c cos ϕ τ ϕ + σ ϕ tan ϕ = c, τ ϕ = σ max σ min cos ϕ, σ ϕ = σ max + σ min + σ max σ min sin ϕ (.46a, b, c)

.4. 553 (.58) Lode θ l ( σ cos θ l ) sin θ l sin ϕ + 3 3 I sin ϕ c cos ϕ = (.47) [7] c ϕ σ max σ min Drucker-Prager α ϕ k Mohr-Coulomb c ϕ 3 : α ϕ = sin ϕ 3 (3 sin ϕ), k = 6 c cos ϕ 3 (3 sin ϕ), (.48a) 3 : α ϕ = sin ϕ 3 (3 + sin ϕ), k = 6 c cos ϕ 3 (3 + sin ϕ), (.48b) : α ϕ = sin ϕ 3 ( 3 + sin ϕ ), k = 3 c cos ϕ 3 ( 3 + sin ϕ ) (.48c) [, 34] ϕ 3 α ϕ.6. Mohr- Coulomb Drucker- Prager [] (.4a) β α β (.48) α ϕ [] β (.48) β < α ϕ ψ (ψ < ϕ) -5 7. (.4) (.4).4.3 () [8] Prandtl-Reuss ϵ p pr σ ( ϵ p pr σ ) σ ϵ p nc ) { p ( ϵ nc σ i j i j } σ σ i j σ kl σ kl σ i j (.3a) ) p σ i j ( ϵ nc σ i j σ i j i j σ σ i j σ i jσ kl σ kl = σ i j σ i j σ kl σ kl = ( ) (.6) Drucker (.74a) ϵ p i j = λ g σ i j + h ( σ i j σ σ i j σ kl σ kl ) (.49)

554 h (.49) (i j) (.33b) ( ) σ i j (.49) ϵ p nc J (.3b) ϵ p nc f = ϵ p pr.45 J (.49) Prandtl-Reuss ϵ p pr σ ϵ p nc Prandtl-Reuss ϵ p pr.45 ϵ p nc ϵ p nc (.49) (.33b) ϵ p i j = λ σ i j σ + ( σ i j h ) σ σ i j σ kl σ kl, ϵ p kk = 3 λ β (.38) λ ϵ p i j ϵp i j = λ + { σ h i j σ i j ( ) } σ σ i j σ i j λ H h λ 4 (.39a) λ λ ( σ ) kl H σ + α δ kl σ kl H (.39b) (.49) ϵ p i j = σ { i j H σ + β δ σ } kl i j σ + α δ kl σ kl + ( σ i j h ) σ σ i j σ kl σ kl (.5) (.4) (.) ϵ i j = µ σ i j + ( 3 3K ) δ i j σ kk (.5) µ + σ { i j H σ + β δ σ } kl i j σ + α δ kl σ kl + ( σ i j h ) σ σ i j σ kl σ kl.5. (5) () (.5) [8] ( µ µ + χ ), K h K χ 9 α β ( H ) h h, (.5a, b) 4

.4. 555 ( H H ), α α h ( H ) (, β β H ) (.5c, d, e) h h χ (.5) (.5) ϵ i j = µ σ i j + 3 ( 3 K ) δ i j σ kk + χ µ H σ i j σ + β δ i j { σ } kl σ + α δ kl σ kl (.53) µ σ ( i j µ σ ) ( σ i j = µ ϵ i j + K µ ) σ + 3K β δ kl i j σ + 3K α δ kl δ i j ϵ kk χ ϵ kl (.54) 3 H + µ + 9K α β 4 (.5) h h H h H H h [87].4.4 H E ϵ (σ σ y ) ( ) σ = m ϵ σ y (σ > σ y ) ϵ y (.55) E Young m (< ) m =.65 ϵ y ϵ y σ y E σ, ϵ (.9b) (.3) σ, ϵ p H h (.3a) (.7) σ σ y = 3 σ 3 τy = σ τ y (a) 4 [8]

556 ϵ y = σ y E = 3 τy = 3 µ E E τ y µ = 3 ϵ p = ϵp µ ( + ν) ( + ν) = ϵ y 3 3 τy 3 ϵ p γ y, τ y ( + ν) µ γ y τ y µ µ ν Poisson γ y µ µ (a) σ τ (b) ϵ p γ (.55) σ σ τ ϵ ϵp σ y σ y τ y ϵ y ϵ y ( + ν) γ 3 γ y ( ) m τ ( + ν) γ = τ y 3 γ y Poisson τ = µ ( + ν) γ (τ τ y ; γ 3 γ y ( 3 ) ( + ν) ) m ( + ν) γ τ y (τ > τ y ) 3 γ y (b) (.56) H dτ dγ = m τ y ( + ν) γ y 3 { ( + ν) 3 } m γ = m E γ y 3 { ( + ν) 3 γ γ y } m h h τ γ = τ y ( + ν) γ y 3 { ( + ν) 3 } m γ = E γ y 3 { ( + ν) 3 } m γ, H = m h < h (.57a, b) γ y H h Poisson ν = / 3 8 / 9 Poisson ν = / µ γ (τ τ y ; γ γ y ) ( ) τ = m γ τ y (τ > τ y ) γ y (.58) ( ) m H γ µ = m, γ y ( ) m h γ µ = (.59a, b) γ y m γ ϵ p ϵ y = γ y 3 (.55) σ σ y = 3σ = ϵp 3τy 3 3 γ y m σ ( ϵ p ) m = (.6a, b) τ y γ y

.4. 557 σ σ y σ σ y, σ τ y.5 ferrite E = 7 GN/m, σy = 76 MN/m h = 98 MN/m, n =.7 E = 93 GN/m, σy = MN/m h = 688 MN/m, n =.55 austenite.5 m =.65 epoxy E = 3.6 GN/m, σy = 75.86 MN/m h = 3.8 MN/m, n =.6 O 4 ϵ 4 ϵ p, ϵp ϵ y ϵ y γ y.46 Ludwik [89, 98] f σ { σ y + h ( ϵ p) n} (.6) (.9) (.3) { f σ τ y + h p ( ϵ ) } n 3 = σ τ h ( y + ( ) n+ ϵ p) n 3 (.6) (.64b) H = τ y ϵ p = τ y ϵ p ϵ p ϵ p = τ y 3 ϵ p = h n 3 ( ϵ p ) n = h n ( 3 ) n+ ( ϵ p) n (.63).46.4.5 ().5 Mises [39] [] Tresca J (.53) J F (σ σ 33 ) + G (σ 33 σ ) + H (σ σ ) + Lσ 3 + Mσ 3 + Nσ +σ 3 {a 4 (σ σ ) + a 5 (σ σ 33 )} + σ 3 {a 6 (σ σ 33 ) + a 7 (σ σ )} (.64) +σ {a 8 (σ 33 σ ) + a 9 (σ 33 σ )} + a σ 3 σ 3 + a σ 3 σ + a σ σ 3

558 Hill [39] Mises a 4 a =, 3F = 3G = 3H = L = M = N = () Mises Tresca 4 43.5.4 (4).5.5 (3).5.5..5.5 (3)..3 P y M p Northwestern 98 Plasticity 4 phenomenological 43

.5. 559.5. ()..3 Tresca.6.5. (4).5.4 (4) σ, + σ, + X =, σ, + σ, + X = (.65a, b) X 44 Mises ϵ j3 = ϵ p j3 =, j =,, 3 (.66) (.3b) = ϵ 33 = ϵ p 33 = λ pr σ 33 σ 33 = σ 33 = (σ + σ ) (.67a, b, c) (3.77) ν = / ν = / (.67c) σ p 3 σ kk = (σ + σ ), σ J = 4 (σ σ ) + (σ ) (.68a, b) Mises 4 (σ σ ) + (σ ) = τ y (.69) σ = τ y σ = τ y σ y = τ y ( Mises) (.7) Tresca (.59) Mises Tresca 44

56 τ y τ σ φ α τ y σ p y τ y β, s β I p τ y τ y τ y p p σ iii σ O p σ σ i α, s α φ β O III x.47 Mohr () (.65) (.69).5.4 3.5. σ = p τ y sin φ, σ = p + τ y sin φ, σ = τ y cos φ (.7a, b, c) (.69) p (.68a).47 Mohr (.7) τ y α φ α x φ β τ y.5. (4) Mohr I, III σ i σ iii 45 s α -s β σ αα = σ ββ = p, σ αβ = τ y (α, β ) (.7a, b) s α, s β α, β 45 3 (.67) σ ii

.5. 56 (.7) (.65) p x τ y cos φ φ x τ y sin φ φ x + X =, τ y sin φ φ x α β s α, s β = cos φ sin φ, x s α s β p x = sin φ + cos φ x s α s β + τ y cos φ φ x + X = (.73a, b) p s α + τ y φ s α = X cos φ + X sin φ, p s β τ y φ s β = X sin φ + X cos φ (.74a, b) X α p + τ y φ =, β p + τ y φ = (.75a, b) Hencky (3).48 h a T Mises f > x a cl A B C h T x α B A I III x cl C β D O.48.48 x DA : σ =, σ = (.69) σ = ±τ y DA DA σ = τ y DA α p = σ + σ = τ y, φ = π + π 4 = 3 π (.76a, b) 4 DO : DO σ T ADB β 45 DBC D β DCO β 45 DCO φ π 4

56 (.75) β ( p + τ y φ) DA β DO β ( p + τ y φ) da = ( p + τ y φ) do (.76) φ π DO 4 (.7) p do = πτ y τ y 3πτ y = ( + π) τ y σ do = ( p + τ y sin φ) do = ( + π) τ y + τ y = ( + π) τ y = + π σ y.57 σ y (.7) σ y = τ y.57 σ y h σ y.57 T a T max a = ( + π)τ y h T max = ( + π) τ y h a = ( + π) σ y h a [, ] (4) α β x i v i α β v α, v β v = v α cos φ v β sin φ, v = v α sin φ + v β cos φ (.77a, b) (.7) ϵ αα = ϵ p αα = λ pr σ αα =, ϵ ββ = ϵ p ββ = λ pr σ ββ =, ϵ αβ = ϵ p αβ = λ pr σ αβ = λ pr τ y (.78a, b, c) x -x (.)

.5. 563 ϵ αα = cos φ ϵ + sin φ cos φ ϵ + sin φ ϵ = cos φ v s α + sin φ v s α =, ϵ ββ = sin φ v s β + cos φ v s β = (.77) v α φ v β = s α s α α dv α v β dφ = v β φ + v α = β dv β + v α dφ = s β s β (.79) Geiringer (.73) wavefront dx dx = tan φ, dx dx = cot φ α β : Geiringer CO v DO α β Geiringer DA v β 3 v < [, ] (5).4. Mohr-Coulomb c ϕ τ n p n tan ϕ = c (.8) τ n n = cos χ sin χ t p n p n = σ n ϕ α φ Mises.47 Mohr q = (σ σ ) + (σ ) (.8) σ = p + q cos Ψ, σ = p q cos Ψ, σ = q sin Ψ (.8a, b, c) p (.68a) Ψ.47 φ + π (.8) (n) 4 ( π χ = Ψ ± 4 ϕ ) (.83)

564.49 (.8) (.68b) (.8) (.8) q p sin ϕ = c cos ϕ (.84) (.8) (.65) x = tan (Ψ 4 x π ) ϕ x, = tan (Ψ + 4 x π + ) ϕ (.85a, b) α β { cot ϕ dq + q dψ = X sin (Ψ + 4 π + ) ϕ X cos (Ψ + 4 π + )} ϕ ds α, { cot ϕ dq q dψ = X sin (Ψ 4 π ) ϕ + X cos (Ψ 4 π )} ϕ ds β Kötter ϕ (.74) Hencky (.86a) (.86b) (.79) Geiringer cos ϕ v α s α ( v β v α sin ϕ ) Ψ s α Ψ sin ϕ =, cos ϕ v β s β + ( v α v β sin ϕ ) Ψ s β + Ψ sin ϕ = (.87a, b) ν tan ν 3.4.3 (.5) H, h, α, β H =, h = σ (αβ ) α β = σ cos ν sin (ϕ ν), α = sin ϕ, β = sin ν cos (ϕ ν) H = h [64] [6, 63, 64, 84] 46.49 Prandtl [39] [49] [, 49] 46 [75]

.5. 565.5.3 () (.78) [39] φ (.78) ϵ αα = ϵ ββ =.5.5 x σ x φ α x ϵ αα = cos φ ϵ + sin φ cos φ ϵ + sin φ ϵ =, ϵ ββ = sin φ ϵ sin φ cos φ ϵ + cos φ ϵ = (.88a) (.88b) σ = σ ϵ = λ pr 3 σ, ϵ = λ pr 3 σ, ϵ = (.88) tan φ =, φ = 54.7, 35.3 (.89) 45 Hill [39] (.88) (.88) x -x 3 Hill (.67) σ = σ, σ =, σ 33 = σ 35 or 55 σ ϵ = λ pr σ, ϵ = λ pr σ, ϵ = (.88) tan φ = ± φ = ±45 (.9) x -x 3 σ 33 = 45 9.5 σ σ ϵ 33 =, ϵ 3 = sin φ = cos φ = φ =, 9 (.9a, b, c).5

566 ().4. (.36a) (.67) σ > σ [5] σ σ = β + ( σ = ϵ p ) 33 = λ 33 σ + β σ t σ σ σ 33 = β σ (.9a, b) (.93) 3β + t, σ σ = β 3β + t, σ σ = t 3β (.94a, b, c) + t (.9b) σ β (.88) t (.93) (.94) (.88) tan φ = 3β ± 3β 3β 3β (.95).8 45 9 45 [4] 3 45 [93].8 β φ β φ, 9 φ 45. 45.9.33 45.8. 45.86.33 47.85.5 49.33.6 5..8 5.. 53.87.5.4 () ϵ i j = ϵ i e j + ϵp i j = ( ) vi, j + v j,i (.96) u Hooke (.4) Poisson ϵ e i j = µ ( σ i j ν ) + ν δ i j σ kk (.97)

.5. 567 Prandtl-Reuss ϵ p i j = λ pr σ i j, ϵp kk = (.98a, b) Mises f (σ i j ) σ i j σ i j τ y, f = (.99a, b) : ϵ e kk = (.) σ ji, j + X i =, σ ji, j + Ẋ i = (.a, b) X S S ν j σ ji = F i, ν j σ ji = Ḟ i (.a, b) ν F : Ḟ i =, Ẋ i =, ϵp > (.3a, b, c) ϵ p (.7) : σ i j = (.4) : 47 (.96) (.) ) ) ( ) e σ i j ϵ i j dv = σ i j ( ϵ i j + ϵ p i j dv = σ e i j ( ϵ i j + ϵ p i j dv = σ i j µ σ i j + λ pr σ i j dv V V V σ Mises f = σ i j σ i j = µ > σ i j ϵ i j dv = µ σ i j σ i j dv (.5) V V (.96) Gauss ( ) σ i j ϵ i j dv = σ i j vi, j + v j,i dv = σ i j v j,i dv V V V ( ) ( ) = ν j σ i j v i ds σ ji, j vi dv = Ḟ i ds σ ji, j vi dv V 47 (.6) Drucker V V V V

568 (.) (.3) (.5) µ σ i j σ i j dv = (.6) V µ σ i j = (.7) (.) Hooke (.4)) (.97) (.) (.7) σ kk = (.8) : ϵ e i j = (.9) 48 : σ i j ) σ ji, j + m s X i = in V ) f (σ i j ) in V (.) 3) ν j σ ji = m s F i on S X, F m s : v i ) ϵ i j = ( v i, j + v j,i) in V ) v k,k = in V 3) v i = on S (.) 4) X i v i S dv >, F i v i ds > V 48 ϵ e i j < m k = V τ p y ϵ dv + V L τ y v s ds X i v i dv + S F i v i ds (.)

.5. 569 ϵ p ϵ p ϵ p i j ϵ p i j L v s : m m X, m F m s m m k (.3) (m s m): (.6) Drucker ϵ p i j V ( ) σ i j σ i j dv (.4) (.96) ( ) ( ) ϵ p i j σ i j σ i j dv = σi j σ p i j ϵ i j dv = ( ) ) σi j σ i j ( ϵi j ϵ i e j dv V V (.9) (.96) ( ) ( ) ( ) ( ) = σi j σ i j ϵi j dv = σi j σ i j vi, j + v j,i dv = σi j σ i j vi, j dv V V Gauss (.4) ( ) ϵ p i j σ i j σ i j dv = (m F i v i m s F i v i ) ds + (m X i v i m s X i v i ) dv V S V ( ) = (m m s ) X i v i dv + F i v i ds V S V X i v i dv >, S F i v i ds > m m s (m m k ): (.4) 3 σ i j ϵ p i j dv = = = = σ i j v i, j dv V σ L Gauss = m X i v i dv + ν j σ ji v V S i ds + ν + j σ ji v + i ds + ν j σ ji v i ds L + L L + L L ν = ν + = ν = m X i v i S dv + m F i v i ds ( ) ν j σ ji v i v + i ds V L V V V

57 L σ i j ϵ p i j dv = m X i v i S dv + m F i v i ds τ y v s ds (.5) V V V σ i j ϵ p i j dv = V σ i j ϵ p i j dv (.35) σ i j ϵ p i j σ ϵ p σ ϵ p dv σ i j ϵ p i j dv σ ϵ p dv τ p y ϵ dv V V (.5) m X i v i S dv + m F i v i ds τ y v s ds τ p y ϵ dv V ( m V V L ) X i v i S dv + F i v i ds τ p y ϵ dv + τ y v s ds V L (.) (.) m m k V L V ().5 m s A, B, B, C α, β 4a a T a ψ m s T A B B C q P v π/ 4 Q q = τ y ( + sin ψ), m s T = τ y ( sin ψ).5 ψ.5 ( 4a m s T = a a cos ψ ) q m s.6 τ y T m k PQ α v (.) q τ y v ds p m k = T ( v / ) ds S = 3 aτ y v at v =.5 τ y T

.5. 57.6 τ y T m.5 τ y T.63 σ y mt.75 σ y q = σ y mt =.5 σ y [] (3) : Bernoulli-Euler 4-3.53 (a) b σ y σ y (z c = h/ 3 ) z c z y z p h M y σ y σ y σ y σ y (a) M y (b) (c) (d) M p.53 (b) σ y (c) z y = ( 3 3 ) h/ 4 (d) z p = ( ) h/ M p ( ) b h M p = Z p σ y, Z p = 3 (.6a, b) Z p b Z p [5] = bh / 4 :.54 h a a 4 q m s (8.) M x x + M xy x y + M y y + m s q = a y O a a x a σ xx σ xx x z.54 ξ

57 (8.) σ xx = M xx h, σ yy = M yy h, σ xy = M xy h Mises ( ) σ 3 xx + σ yy σ xx σ yy + σ xy = τ y Mxx + Myy M xx M yy + 3Mxy = 3τ yh 4 M xx = c ( a x ), M yy = c ( a y ), M xy =, c = 4 m s q 3τ yh 4 = ( ca ) + ( cb ) ( ca ) ( ca ) = a 4 c c = ± c c m s m s = 4 3 τ y h q a 3τy h ẇ ẇ = c (a x) (x >, x < y < x), ẇ = c (a y) (y >, y < x < y) (.) m k m k = ( Mξξ ) p L a a a a ẇ ds ξ qẇ dx dy L s ẇ = c ξ ( ) ( M ξξ )p = h σ y ( ) = 4 {( h σ y ) c a } = 6 τy c a h (.7) σ y = τ y ( ) = 8q a c (a x) x m k = τ y h q a dy dx = 4 3 τ y h m τ y h q a q a 4q c a3 3 a ( )

.5. 573 Q M M p m s Q M y M y M p M y 6Ql 3 5Ql 3 ẇ R A M p θ Q M p θ R B.55.56 : S C q ) M (x) + m sq =, ) M M p 3) n i M = m s S n i M = m s C (.7) M (x) x M p (.6a) n i (4.6) ) κ = ẇ (x), ) ẇ =, θ ẇ =, 3) q ẇ dx >, θ C >, ẇ S > (.8) ẇ θ (.) m k M p θ m k = q ẇ dx + ( θ C + ẇ S ) = U V (.9) (.) U V.55 l M p M p M p.56 M R A + R B = m s Q, M p = l m s Q l R B, M p l/ = R B R A = 4M p l, R B = M p l, M p = l 6 m s Q

574 P P θ θ.57 m s = 6M p l Q (.) M.56 M p θ = θ, θ = θ U = ( M p ) ( θ ) + Mp θ = 3M p θ V = Q θ = l/ m k = U V = 3 M p θ Q = 6M p l Q (.) m = m s = m k = 6M p l Q? (.) -6 8..53 z y, z p Z p (4).57 l 7 3

.5. 575 4 m k m k 49 m k = M p [] P l.5.5? ().3 3.58 M p l/ = m s Q m s = 4M p l Q Q θ M p M.58 U = M p θ = 4M p, l V = Q, = l θ m k = U V = 4M p l Q m ss m s = m k = 4M p l Q, Q ult = m ss Q = 4M p l (.3) Q ult (.) P p.59 R = m s q l = 4M p l m s = 8M p q l U = M p θ = 4M p, V = q l l, = l θ 49 q M p θ.59 m k = U V = 8M p m ss q l q m s = m k = 8M p (.4) q l

576 [4].6 R = m s q l = 8M p l m s = 6M p q l U = 4M p θ = 8M p, V = q l l, = l θ m k = U V = 6M p q l m = m s = m k = 6M p q l (.5).4 m.55.6 α, β m M p ql ql 4 q M p θ.6 R () M p?. 3 4 46 9 7 R. 3 4 3.55.

.5. 577 (.) m m p = 6M p l Q (.6) M p = 3 M y (.7) 6 m y Q, l 3 = M y m y = 3 M p 9 l Q M y = l m Q 4 M p m = 4 M p 3 l Q (.6) m y < m < m p.6. m (.3) m ss.6 A M p m p m ss m ss 5 m p, m, m y, m ss mql M p 6 m y 4 M = M y, m m ss A mp M = M y, m y O w ( l / ).6 m p [7].6 A m y.7 m p.9 3 m p 5 m ss.6

578 m ss m y m ss m y m ss 3 5.6 m (.5) m p = 6M p q l M y m y ql = M y m y = 8M p q l (.7) (.4) m ss q m p m y m y.7 m p 3.4 mql M p 6 8 M = M y, m B m ss q mp A 4 M = M y, m y O w ( l / ).6 m = 4M p 3 q l.6 A m p B M p m y -7 9..6,.6 5

.5. 579 (3)? %.6 A B. curve-fitting.63.63 5 f f Young E m Young E Voigt H. Young E = f [ ( Young ) ] + ( f ) E m = ( f ) E m (.8) D Young E = ( D) E m (.9) D driving force Voigt 5

58 H f Young E i f Young Voigt E = { f + f } (.3) E i E m Reuss Voigt Voigt σ = f σ i + ( f ) σ m, ϵ = f ϵ i + ( f ) ϵ m (.3a, b) Reuss σ y (.8) Mises ϕ ( ) σ ϕ σ ( f ) σ y ϕ ( f ) (.3a, b) σ y σ (.9) f (.3) [5] (driving force) Gurson [3, 95] ( ) ( ) σ 3 q σ ave ϕ + q f cosh ( + q 3 f ) (.33) σ y σ y σ ave (3.4) q q 3 Tvergaard Gurson Gurson driving force (.33) q i = (i =,, 3) σ ave = (.3) (.3) Gurson curve-fitting q

.5. 58 D 53 ±3% [96].4.5.4 53 cm m

58.3 94(?) m Theodore: first name Ted 54 Ben 54