38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1:
5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds
40 5 Cauchy t l n mds df n 5.3: t n n x 1 x 2 x 3 x 1 e 1 t 1 = F /A x 1 x 2 x 3 e 3 t 3 =0 x 3 x 2 F A x 1 n t 1 = F /A n t 3 =0 5.4: n e 1, e 2, e 3 t 1, t 2, t 3 t 1, t 2, t 3 T ij t 1 = T 11 e 1 + T 12 e 2 + T 13 e 3 (5.2) t 2 = T 21 e 1 + T 22 e 2 + T 23 e 3 (5.3) t 3 = T 31 e 1 + T 32 e 2 + T 33 e 3 (5.4)
5.2. Cauchy 4 Cauchy 41 t i = T ij e j (5.5) T ij T 11,T 22,T 33 T 12,T 21,T 23,T 32,T 31,T 13 e i e j T = T ij e i e j (5.6) Cauchy Cauchy Cauchy 2 5.2 Cauchy 4 Cauchy Cauchy t n = T T n (5.7) (5.7) T t i = T ij e j (5.8) (5.8) (5.7) 4 (Cauchy 4 ) ABC n t n t n ABC Δs t n ds = t n Δs (5.9) ABC e 1, e 2, e 3 ABC t 1, t 2, t 3 OCB, OAC, OBA
42 5 Cauchy OCB, OAC, OBA Δs 1, Δs 2, Δs 3 t 1 ds = t 1 Δs 1, t 2 ds = t 2 Δs 2, t 3 ds = t 3 Δs 3 (5.10) OCB OAC OBA x 3 x 2 C B t n t 1 Δs Δs 1 O Δs 3 Δs 2 A x 1 t 2 t 3 ρ, Δv, a, g Cauchy 4 ρ(a g)δv = t n Δs t 1 Δs 1 t 2 Δs 2 t 3 Δs 3 (5.11) 4 4 Δs Δv/Δs 0 4.1 Δs i /Δs = n i (5.12) n = n i e i Δs 1 t n = t 1 Δs + t Δs 2 2 Δs + t Δs 3 3 Δs = t i n i = T ij e j n i = T ij (e j e i )n k e k = T T n (5.13)
5.3. 43 Cauchy T T n t n Cauchy 5.1 A, B, C {a, 0, 0}, {0,b,0},{0, 0,c} CA = a, CB = b n a, b a 0 bc a b = 0 b = ca, n = 1 bc ca (5.14) c c ab b3 c 2 + c 2 a 2 + a 2 b 2 ab Δs 1 = 1 2 bc, Δs 2 = 1 2 ca, Δs 3 = 1 2 ab Δs = 1 2 a b Δs = 1 2 b3 c 2 + c 2 a 2 + a 2 b 2 (5.15) n 1 = ΔS 1 ΔS, n 2 = ΔS 2 ΔS, n 3 = ΔS 3 ΔS (5.16) n a, b 2 ΔS a, b θ a b sin θ 5.1 ΔS = a b sin θ = a b 1 cos 2 θ (5.17) { } 2 a b = a b 1 = a a b 2 b 2 (a b) 2 (5.18) = (a 2 + c 2 )(b 2 + c 2 ) c 4 = a 2 b 2 + c 2 a 2 + b 2 c 2 (5.19) 5.3 T 11 T 12 T 13 T 21 T 22 T 23 (5.20) T 31 T 32 T 33 e i e j
44 5 Cauchy T 1, T 2, T 3 ē i T 11 T 12 T 13 T 1 0 0 T 21 T 22 T 23 = 0 T2 0 (5.21) T 31 T 32 T 33 0 0 T3 T ij e i e j = T i ē i ē i (5.22) = T 1 ē 1 ē 1 + T 2 ē 2 ē 2 + T 3 ē 3 ē 3 (5.23) T 11 ε x ε 11 T 22 ε y = ε 22 T 12 γ xy 2ε 12 (5.24) {ε x,ε y,γ xy } 5.4 9 σ y σ y σ = ( ) 1 3 2 σ ijσ ij 2 (5.25)
5.4. 45 σ ij σ ij = σ ij 1 3 σ kkδ ij (5.26) = σ ij 1 3 (σ 11 + σ 22 + σ 33 )δ ij (5.27)
46 6 Cauchy Cauchy 3 2 6.1 2 3 e 1, ē 1, θ [ [ T ]=[P][T ][P ] T = sin θ ][ sin θ T 11 T 11 T 12 T 21 T 22 ][ sin θ ] sin θ (6.1) T 11 = T 11 cos 2 θ + T 12 sin θ + T 21 sin θ + T 22 sin 2 θ (6.2) 1 = T 11 2 (cos 2θ +1)+T 1 22 2 (1 cos 2θ)+T 12 sin 2θ (6.3) = 1 2 (T 11 + T 22 )+ 1 2 (T 11 T 22 )cos2θ + T 12 sin 2θ (6.4) T 12 = T 21, T12 = T 21 (6.5)
6.1. 47 cos 2θ =cos 2 θ sin 2 θ =2cos 2 θ 1= 2sinθ +1 (6.6) sin 2θ =2cosθsin θ (6.7) T 12, T 22 T 12 = T 11 sin θ T 21 sin 2 θ + T 12 cos 2 θ + T 22 sin θ (6.8) 1 = T 11 2 sin 2θ + T 12(cos 2 θ sin 2 1 θ)+t 22 sin 2θ 2 (6.9) = 1 2 (T 11 T 22 )sin2θ + T 12 cos 2θ (6.10) T 21 = T 12 (6.11) T 22 = T 11 sin 2 θ T 21 sin θ T 21 sin θ + T 22 cos 2 θ (6.12) = 1 2 (T 11 + T 22 ) 1 2 (T 11 T 22 )cos2θ T 12 sin 2θ (6.13) Cauchy 4 θ A C B Cauchy 4 ABC AB t n n = ē 1 t n = T 11 ē 1 (6.14) BC e 1 t 1 = t 1 t 1 = (T 11 e 1 + T 12 e 2 ) (6.15)
48 6 Cauchy AC e 2 t 2 = t 2 t 2 = (T 21 e 1 + T 22 e 2 ) (6.16) AB 1 BC sin θ, AC e 1, e 2 T 11 = T 11 + T 21 sin θ (6.17) T 11 sin θ = T 12 + T 22 sin θ (6.18) (6.17), (6.18) sin θ T 11 = T 11 cos 2 θ + T 21 sin θt 12 sin θ + T 22 sin 2 θ (6.19) 6.2 2 T 22 = T 12 = T 21 =0 T 11 = 1 2 T 11(1 + cos 2θ) (6.20) T 12 = 1 2 T 11 sin 2θ (6.21) T 12 θ = ±45 T 11 = 1 2 T 11, 1 2 T 11 (6.22) T 12 = 1 2 T 11, 1 2 T 11 (6.23)
6.2. 2 49 T 11 =1 e 2 ē 2 e 1 ē 1 T 11 =0.5 T 12 =0.5 T 21 =0.5 T 22 =0.5 T 11 = T 22 =0 θ = ±45, 0 T 11 = T 12 sin 2θ (6.24) T 12 = T 12 cos 2θ (6.25) (6.26) T 11 = T 12, T 12 (6.27) T 12 =0, 0 (6.28) T 21 =1 T 12 =1 e 2 e 1
50 6 Cauchy 2 T 11 = T 22 = T,T 12 =0 T<0 T 11 = T (6.29) T 12 = 0 (6.30) T 22 = T T 11 = T e 2 e 1 6.3 Cauchy 2 T φ = λφ (6.31) (6.4), (6.10), (6.13) =0 θ T 12 =0 tan 2θ = 2T 12 T 11 T 22 (T 11 T 22 ) (6.32) (6.4), (6.13) θ T 11 θ =2( 1 2 (T 11 T 22 )sin2θ + T 12 cos 2θ) (6.33) = T 12 = 0 (6.34)
6.3. 51 T 22 θ =2(1 2 (T 11 T 22 )sin2θ T 12 cos 2θ) (6.35) = T 12 = 0 (6.36) T 11, T 22 ẽ i T = T ij e i e j = T ij ē i ē j = T ij ẽ i ẽ j (6.37) T 12 =0 T 12 = 1 2 ( T 11 T 22 )sin2φ + T 12 (6.38) φ =45 T 12 T 12 T 12 = 1 2 ( T 11 T 22 ) (6.39) T 11, T 22 cos 2φ =0, T 12 =0 T 11 = 1 2 ( T 11 + T 22 )+ 1 2 ( T 11 T 22 )cos2φ + T 12 sin 2φ (6.40) = 1 2 ( T 11 + T 22 ) (6.41) T 22 = 1 2 ( T 11 + T 22 ) 1 2 ( T 11 T 22 )cos2φ T 12 sin 2φ (6.42) = 1 2 ( T 11 + T 22 ) (6.43) e 2 ē 2 ẽ 1 φ ẽ 2 ē 1 θ e 1
52 6 Cauchy 6.4 Cauchy θ θ 6.1 T 11 = T 11 cos 2 θ + T 21 sin θt 12 sin θ + T 22 sin 2 θ (6.44) [ T11 T12 T 21 T22 ] [ = sin θ ][ ][ sin θ T 11 T 12 T 21 T 22 sin θ [ T ]=[P][T ][P ] T ] (6.45) sin θ (6.46) T ij [ ] [ T 11 T 12 = T 21 T 22 sin θ sin θ ][ T11 T 21 T12 T22 ][ sin θ [T ]=[P] T [ T ][P ] ] (6.47) sin θ (6.48) e i T ij θ T ij θ T ij θ