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1 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ).

2 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a a a a,. a e x

3 .. 7. a e =(a i e i ) e =(a e + a e + a e ) e (.) = a e e + a e e + a e e (.4) = a +a 0+a 0 (.5) = a (.6),...,,. a 0 0 {, 0, 0} a = a {, 0, 0} 0 + a {, 0, 0} + a {, 0, 0} 0 = a 0 0 a.. a, b θ a b = a b cos θ., a b = a b + a b + a b = a i b i., a b e a b = a b e a b e = a b e + a b e + a b e a b e a b e a b e =(a b a b )e +(a b a b )e +(a b a b )e det[a] =det[a T ],, (, ), a b e a b = a b e a b e = e a b e a b e a b = a a a b b b e e e = e e e a a a b b b a,,.,... a = a i e i = a e + a e + a e (.7) =ā i ē i =ā ē +ā ē +ā ē (.8)

4 8 x a x a x ā ā x a.:.4., a, a ā a = ā a.,.. x x x x 0. x x a {a,a } = {, }, x x,, {ā, ā } = {, }.,,. a x x a,a a = a e,a = a e,, x x ā, ā ā = a ē, ā = a ē.. ā ā i = a ē i (.9) =(a e + a e ) ē i (.0) = a e ē i + a e ē i (.).5... { } { },., { { } = 0} { }

5 .. 9 { 0 = } { } + { } {a,a } T, { } [ { } a = a a { } ] [ + a { } ( a = a a + a ā, ā. ā = ) { { } + } ( + a + a a + a, ā = a + { a ) { { } ] {a,a } = {, }, {ā, ā } = {, }.. } e ē = {, 0} =, e ē = {0, } { } e ē = {, 0} =, e ē = {, 0},. ā = a e ē + a e ē = a + a ā = a e ē + a e ē = a + a } { } = { } = [P ]. P ij i ē i, j e j ē e ē e ē e P ij = ē i e j = e j ē i, [P ]= ē e ē e ē e (.) ē e ē e ē e P ij e, ē. e ē i (i =,, ). e =(e ē )ē +(e ē )ē +(e ē )ē (.) = P ē + P ē + P ē (.4)

6 0 ē e i (i =,, ). ē =(ē e )e +(ē e )e +(ē e )e (.5) = P e + P e + P e (.6) a i, ā i.,.,, P ij, i j. P ij a j, P ji a j, [P ].. δ ij Kronecker i = j δ ij = δ ij =0., a k δ jk = a j, X ij δ jk = X ik.. e i e j = δ ij. a i e i =ā i ē i (.7) a i e i ē k =ā i ē i ē k (.8) P ki a i =ā i δ ik =ā k (.9) ā k = P ki a i (.0) a i e i e k =ā i ē i e k (.) a k = P ik ā i (.). ā P P P a a P P P ā ā = P P P a a P P P a a = P P P P P P ā ā ā (.) {ā} =[P ]{a} {a} =[P ] T {ā} (.4).6. [P ]. 0 [P ]= {ā} =[P ]{a} {a} =[P ] T {ā} (.5)

7 .. ā k = P ki a i a i = P ki ā k = = (.6) (.7).7.,. 0, [P ].,, (, 0), (0, ) θ (cos θ, sin θ),( sin θ, cos θ). ( sin θ, cos θ) (cos θ, sin θ).:, [R], [ ]{ } { } [ ]{ } R R cos θ R R =, 0 R R 0 sin θ R R = { } sin θ cos θ (.8) [ ][ ] R R 0 = R R 0 [ cos θ ] sin θ sin θ cos θ (.9) 0,, [P ]. [ ] [ ] cos θ sin θ = (.0) sin θ cos θ,.,,,,..

8 . a,,.. x c X b x.4:.,. [ ]{ } { } X X b c = (.) X X b c b, c,,.,,,... a, b X,. b = Xa (.) X(a + b) =Xa + Xb (.) X(ca) =cxa (.4). (a b) c =(b c)a (.5) b (a c) =(b a)c (.6)

9 a b a a b = a {b b b } = a b a b a b a b a b a b a a b a b a b,. (a b) c,. a c (a b) c = a {b b b } c = a b a b a b c a b a b a b c a c a b a b a b c a b c + a b c + a b c (b c + b c + b c )a a = a b c + a b c + a b c = (b c + b c + b c )a = b c a =(b c)a a b c + a b c + a b c (b c + b c + b c )a a X = X ij e i e j (.7). X X = a b a, b.9.,, X ij e i e j X ij e i e j = X e e + X e e + X e e + X e e + X e e + X e e + X e e + X e e + X e e.,. [X] =X 0 {, 0, 0} + X 0 {0,, 0} + X 0 {0, 0, } X {, 0, 0} + X {0,, 0} + X {0, 0, } X 0 {, 0, 0} + X 0 {0,, 0} + X 0 {0, 0, } = X X X X X X X X X

10 4.0., a b a b = a b e e + a b e e + a b e e + a b e e + a b e e + a b e e + a b e e + a b e e + a b e e. X X ij a i b j. (.8),. b X X X a X a + X a + X a b b = X X X a X X X a = X a + X a + X a (.8) X a + X a + X a b = b i e i = b i e i i= = Xa =(X ij e i e j ) (a k e k )= i= j= k= (X ij e i e j ) (a k e k ) (.9)..,.. Xa =(X ij e i e j )(a k e k ) = X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e + X a (e e ) e. Xa = X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e + X a (e e )e

11 .. 5, e i e j =0(i j), Xa =(X a + X a + X a )e. +(X a + X a + X a )e +(X a + X a + X a )e Xa = X ij a j e i (e i e j ) e k = δ jk e i,(x ij e i e j ) (a k e k )=X ij a k δ jk e i = X ij a j e i. δ ij Kronecker i = j δ ij = δ ij =0.,. X e l = X ij (e i e j ) e l = X ij e i δ jl = X il e i (.40) e k X e l = e k (X il e i ) = X il δ ki,. = X kl (.4) b (X a) =a (X T b) (.4),,. X T =(X ij e i e j ) T = X ij e j e i (.4) X T = X ji e i e j,.. X = X ij e i e j = X kl ē k ē l (.44) P ij = ē i e j. X ij = e i X e j = e i ( X kl ē k ē l ) e j (.45) = X kl (e i ē k )(ē l e j ) (.46) = X kl P ki P lj (.47) = P ki Xkl P lj (.48)

12 6 kl lj ki kl,. X kl = ē k X ē l = ē k (X ij e i e j ) ē l (.49) = X ij (ē k e i )(e j ē l ) (.50) = X ij P ki P lj (.5)... [X] =[P ] T [ X][P ], [ X] =[P ][X][P ] T X X X X X X = P P P X X X P P P X X X P P P P P P X X X P P P X X X P P P X X X X X X = P P P P P P X X X X X X P P P P P P X X X P P P X X X P P P X Y =(X ij e i e j ) (Y kl e k e l ) (.5) = X ij Y kl (e i e j ) (e k e l ) (.5) = X ij Y kl δ jk e i e l (.54) = X ij Y jl e i e l (.55) = X ik Y kj e i e j (.56) X Y T =(X ij e i e j ) (Y kl e l e k ) (.57) = X ij Y kl (e i e j ) (e l e k ) (.58) = X ij Y kl δ jl e i e k (.59) = X ij Y kj e i e k (.60) = X ik Y jk e i e j (.6).,,.,., λ

13 .4. 7., λ, φ. Xφ = λφ (.6), Cauchy, ē i Xē i = λ i (.6) ē i Xē j =0 (i j) (.64). X φ i = λφ i e i,. X X X φ i φ i X X X φ i X X X φ = λ φ i (.65) i φ i. φ φ φ X X X φ φ φ φ φ φ X X X φ φ φ = λ λ φ φ φ X X X φ φ φ λ (.66),. [P ] P ij P ij = ē i e., φ i. P ij = φ i e j., P = φ e = φ P = φ e = φ P = φ e = φ (.67) P = φ e = φ P = φ e = φ P = φ e = φ (.68) P = φ e = φ P = φ e = φ P = φ e = φ (.69), [P ][X][P ] T =[ X]..4,.

14 8,,.,,.,,,,.,,, T T T T 0 0 T T T = 0 T 0 (.70) T T T 0 0 T,, T ij e i e j = T i ē i ē i (.7) = T ē ē + T ē ē + T ē ē (.7),,,,,.

15 9 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.? l, l (l l) ε ε = l l l (4.) F l l F

16 0 4 Green-Lagrange Δz Δδ γ = Δδ (4.) Δz π/ φ γ = π φ (4.) γ tan γ γ,sin γ γ ( π ) γ tan γ =tan = Δδ (4.4) Δz Δδ X Δz γ X 4.: 4..,,.,.,.

17 4.. 4.:. t 0, X, X t x, X t 0 t u. u = x X (4.5) t 0 X u x t O 4.:,,. X, (, 0) (a, 0) (X,X ) (x,x ) u,u. (b), (c). x = X + ax (4.6) x = X (4.7) u = ax (4.8) u =0 (4.9)

18 4 Green-Lagrange x X u X X X X (a) X +a x X u X X X (b) (c) 4.4: x u X, (0, ) (a, ) (X,X ) (x,x ) u,u. (b), (c). x = X + ax (4.0) x = X (4.) u = ax (4.) u = 0 (4.) x X u X X X X X x X u X a (a) X X (b) (c) 4.5: x u

19 4.. X X, (0, ) (a, b), (X,X ) (x,x ) u,u. (b), (c). x = X + ax (4.4) x = X bx (4.5) u = ax (4.6) u = bx (4.7) x X u X X b X X a X x X u X (a) X X (b) (c) 4.6: x u.,, t 0. {(X,X ) 0 X, 0 X }., X,X..,,,,..

20 4 4 Green-Lagrange 4.,,,,. M 4.7:,,,.,.,. 4.8:,,,,.,.,. dx, X = X (4.8) X = X + dx (4.9)

21 4.. 5,. x = X + u(x )=X + u(x) (4.0) x = X + u(x )=X + dx + u(x + dx) (4.) X + dx + u(x)+ u dx j X j (4.) dx = x x dx = dx + u X j dx j (4.) dx dx = dx i dx i (4.4) ( = dx i + u )( i dx j dx i + u ) i dx k (4.5) X j X k, u i X j = dx i dx i + u i X j dx j dx i + u i X k dx k dx i + u i X j u i X k dx j dx k (4.6) u i X k, dx dx dx i dx i + u i dx j dx i + u i dx k dx i (4.7) X j X k ( ui = dx dx + + u ) j dx i dx j (4.8) X j X i ε ij. ε ij = ( ui + u ) j X j X i (4.9). dx dx dx dx =ε ij dx i dx j (4.0) 4.. u = ax, u =0 ε = ( u + u ) X X = (a + a) =a

22 6 4 Green-Lagrange u = ax, u =0 ε = ( u + u ) X X = (a) =a γ γ =ε (4.) 4.., u X,,., +,., ε = ε. u = ax (4.) u = bx (4.) ε = (0 + 0) = 0 (4.4) ε = (a +0)= a ε = (0 + a) = a (4.5) (4.6) ε = ( b b) =b (4.7),, u = ax (4.8) u = 0 (4.9) ε = (ax +ax )=ax (4.40) ε = (0 + 0) = 0 (4.4) ε = (0 + 0) = 0 (4.4) ε = (0 + 0) = 0 (4.4)., ε ij, t 0 X.,., ε ij., u X, u = a,u = a, u i X j =0 ε ij =0.

23 X,X θ, { } [ ]{ } { } x cos θ sin θ X X cos θ X sin θ = = (4.44) x sin θ cos θ X X sin θ + X cos θ u i = x i X i { } { } u X (cos θ ) X sin θ = X sin θ + X (cos θ ) u [ ] cos θ 0 [ε] = 0 cosθ (4.45) (4.46),,,.,,. θ 0 cos θ,. [ε] = [ ] [ cos θ 0 0 cosθ ] (4.47),,., = Δt (4.48).,, Δt,. n x X u θ u X 4.9:

24 8 4 Green-Lagrange θ cos θ, sin θ θ (4.49), X. X (X,X ). n,.,, x = X + u = X + θn X (4.50) u = θn X (4.5) u = θ(n X n X ) (4.5) u = θ(n X n X ) (4.5) u = θ(n X n X ) (4.54) u u u =0, = n, = n X X X (4.55) u u u = n, =0, = n X X X (4.56) u u u = n, = n, = 0 X X X (4.57) u X = u X, u X = u X, u X = u X (4.58) ε ij = 0 (4.59), ε ij =0.,, ε ij = Green-Lagrange, θ ε ij =0..,

25 4.4. Green-Lagrange 9 dx, X = X, X = X + dx,. x = X + u(x )=X + u(x) (4.60) x = X + u(x )=X + dx + u(x + dx) (4.6) X + dx + u(x)+ u dx j X j (4.6) dx = x x dx = dx + u X j dx j (4.6), u i X j. u i X k dx dx = dx i dx i (4.64) ( = dx i + u )( i dx j dx i + u ) i dx k (4.65) X j X k = dx i dx i + u i X j dx j dx i + u i X k dx k dx i + u i X j u i X k dx j dx k (4.66).,, (4.66),,4 ( ui dx dx = dx dx + + u j + u ) k u k dx i dx j (4.67) X j X i X i X j ε ij E ij. E ij = ( ui + u j + u ) k u k X j X i X i X j (4.68) dx dx dx dx =E ij dx i dx j (4.69), dx dx dx dx ε ij dx i dx j (4.70). E ij =0, ε ij..

26 40 4 Green-Lagrange { u u } { = X (cos θ ) X sin θ X sin θ + X (cos θ ), (4.68), E ij. E = ( u + u + u ) k u k X X X X = ( u + u + u u + u u + u ) u X X X X X X X X } (4.7) (4.7) (4.7) = { (cos θ ) + (cos θ ) + (cos θ ) +sin θ +0 } (4.74) = 0 (4.75) E = ( u + u + u ) k u k X X X X = ( u + u + u u + u u + u ) u X X X X X X X X (4.76) (4.77) = { (cos θ ) + (cos θ ) + ( sin θ) +(cosθ ) +0 } (4.78) = 0 (4.79) E = ( u + u + u ) k u k X X X X = ( u + u + u u + u u + u ) u X X X X X X X X (4.80) (4.8) = { sin θ +sinθ +(cosθ )( sin θ) sin θ(cos θ ) + 0} (4.8) = 0 (4.8) E = E.,. Green-Lagrange. 4.5 Green- Lagrange.

27 4.5. 4, u = ax, u =0, u = 0 (4.84).,,.,,, 0. u = ax, u = νax,u = νax (4.85). ν Poisson. ε,e ε = (a + a) =a (4.86) E = ( ) a + a + a = ( ) a + a (4.87)., L l. u = l L ε = u L E = ( ) l L (4.88) (4.89).. l =.0L, E =0.0005,ε =0.0,. l =.L, E =0.05,ε =0.. l =L, E =.5,ε =.,,.,. u X = γ 0 (4.90) 0. u = γx (4.9) 0.,.. u / X = γ.

28 4 4 Green-Lagrange X X 4.0: [ ] ε = 0 γ γ 0 (4.9),, Green-Lagrange E 0. [ ] E = (F T F I) = 0 γ, (4.9) γ γ 4.6 Green-Lagrange X a θ Green-Lagrange. X θ a X 4.: (X,X ) (x,x ) { } { } x ( + a)x = X x (4.94)

29 4.6. Green-Lagrange 4 (x,x ) θ (x,x ) { } [ ]{ } { } { } x cos θ sin θ x x = = cos θ x sin θ ( + a)x cos θ X sin θ x sin θ cos θ x x sin θ + x cos θ = ( + a)x sin θ + X cos θ (4.95) u,u { } { } u X (( + a)cosθ ) X sin θ = (4.96) X ( + a)sinθ + X (cos θ ) u. Green-Lagrange E = ( u + u + u ) k u k (4.97) X X X X = ( u + u + u u + u u + u ) u (4.98) X X X X X X X X = { (( + a)cosθ ) + (( + a)cosθ ) + (( + a)cosθ ) +(+a)sin θ +0 } (4.99) = (a + a ) (4.00) E = ( u + u + u ) k u k X X X X = ( u + u + u u + u u + u ) u X X X X X X X X (4.0) (4.0) = { (cos θ ) + (cos θ ) + ( sin θ) +(cosθ ) +0 } (4.0) = 0 (4.04) E = ( u + u + u ) k u k X X X X = ( u + u + u u + u u + u ) u X X X X X X X X (4.05) (4.06) = { sin θ +(+a)sinθ + (( + a)cosθ )( sin θ)+(+a)sinθ(cos θ ) + 0} (4.07) = 0 (4.08)

30 44 4 Green-Lagrange, Green-Lagrange.,., Green-Lagrange,. Green-Lagrange.

31 45 5 Green-Lagrange

32 46 6 Cauchy.,,,,., σ.,,,,. 6.?,,,,,,,.,,,. 6.:,,.,,,

33 6.. Cauchy 47.,,,,,.,,,,,.,,,,,,.,.,,,,.,.,, (a) (b),,,.,,,,,. (a) (b) (c) 6.: 6. Cauchy.,.

34 48 6 Cauchy.,,,.,, F., Q,.,,.,,., Newton,. F F F Q F Q 6.:, n. n., ds df n ( 6.). df n n df(n), df n., t n. t n = df n (6.) ds

35 6.. Cauchy 49 t l n mds df n 6.4:,,,,.. t n n.,,,. x. x x x e, t = F /A., x x, x e, t =0. : n = e t = e +0 e +0 e : n = e t =0 e +0 e +0 e : n = e t =0 e +0 e +0 e (6.) x x F A x n t = F /A n t =0 6.5:

36 50 6 Cauchy, n e, e, e, t, t, t. t, t, t T ij. t = T e + T e + T e (6.) t = T e + T e + T e (6.4) t = T e + T e + T e (6.5) t i = T ij e j (6.6) T ij,, T,T,T, T,T,T,T,T,T ,,, (6.)., 0,,,. : n = e t =0 e + e +0 e : n = e t =0 e +0 e +0 e? : n = e t =0 e +0 e +0 e? (6.7) T ij e i e j T = T ij e i e j (6.8) Cauchy. Cauchy. Cauchy,,,.,. 6. Cauchy 4 Cauchy,,, Cauchy t n = T T n (6.9)

37 6.. Cauchy 4 Cauchy 5. (6.5), T t i = T ij e j (6.0). (6.0), (6.5). 4 (Cauchy 4 ),. ABC n, t n. t n ABC,. Δs, t n ds = t n Δs (6.) ABC., e, e, e ABC t, t, t., OCB, OAC, OBA, OCB, OAC, OBA Δs, Δs, Δs t ds = t Δs, t ds = t Δs, t ds = t Δs (6.) OCB OAC OBA x x C B t n t Δs Δs O Δs Δs A x t t ρ, Δv, a, g, Cauchy 4. ρ(a g)δv = t n Δs t Δs t Δs t Δs (6.)

38 5 6 Cauchy,., 4, 4. Δs, Δv/Δs Δs i /Δs = n i (6.4), n = n i e i Δs t n = t Δs + t Δs Δs + t Δs Δs = t i n i = T ij e j n i = T ij (e j e i )n k e k = T T n (6.5) Cauchy., T T n t n. Cauchy. 6.. (6.5), (e i e j ) e k = δ jk e i t n = t i n i = T ij e j n i = T ij e j δ ik n k = T ij (e j e i )n k e k = T T n 6.. 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 = bc ca (6.6) c c ab b c + c a + a b ab Δs = bc, Δs = ca, Δs = ab, Δs = a b, Δs = b c + c a + a b (6.7) n = Δs Δs, n = Δs Δs, n = Δs Δs (6.8)

39 6.4. 5,, n a, b ΔS(= Δs. a, b θ, a b sin θ ΔS = a b sin θ = a b cos θ (6.9) { } a b = a b = a a b b (a b) (6.0) = (a + c )(b + c ) c 4 = a b + c a + b c (6.) 6.4,.. T T T T T T (6.) T T T,, e i e j,,,,.. T, T, T, ē i, T T T T 0 0 T T T = 0 T 0 T T T 0 0 T (6.),, T ij e i e j = T i ē i ē i (6.4) = T ē ē + T ē ē + T ē ē (6.5),. T ε x ε T ε y = ε T γ xy ε (6.6)

40 54 6 Cauchy,. {ε x,ε y,γ xy },...,,,,. 6.5,, 9.,.., σ y,, σ y.., σ ij σ = ( ) σ ij σ ij (6.7) σ ij = σ ij σ kkδ ij (6.8) = σ ij (σ + σ + σ )δ ij (6.9),,,.,.

all.dvi

all.dvi 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