応力とひずみ.ppt

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1 in

2 2

3 3

4 4

5 5 x 2

6 6 Continuum)

7 7

8 8

9 9 F F

10 10 F L L F L 1 L F L F L F

11 11 F L F F L F L L L 1 L 2

12 12 F L F! A A! S! = F S

13 13 F L L F F n = F " cos# F t = F " sin# S $ = S cos# S S " = F n S # = F S $ cos2 % & = F t # S = F S $ sin% $ cos%

14 14 F L L F " = F S # = 0 F L L F " = F n S # = F S $ cos2 % & = F t S # = F S $ sin% $ cos%

15 15 P

16 16 P ds df t (n) = lim ds"0 df ds P

17 17 x t (e x ) = lim ds x "0 df x ds x e x : x t (e y ) = t (e z ) = lim ds y "0 lim ds z "0 df y ds y df z ds z

18 18 " x " xy, " xz y z " y, # yx, # yz " z, # zx, # zy

19 19

20 20 t (e x ) (e, t y ) (e, t z ) t (n) P n S x S y S t x (n) S x f x = t x (n) " S S z Sx,Sy,Sz-x " f x = # x $ S x + % yx $ S y + % zx $ S z

21 21 ( ) ( n =1) n = n x, n y, n z t x (n) " S = # x " S x + $ yx " S y + $ zx " S z = # x " S " n x + $ yx " S " n y + $ zx " S " n z S S x t x (n) t x (n) = " x # n x + $ yx # n y + $ zx # n z S y S z t y (n) = " xy # n x +$ y # n y + " zy # n z t z (n) = " xz # n x + " yz # n y +$ z # n z

22 22 x y z 1,2,3 " t (n)& x + ) $ $ x * xy * xz. T t (n) - 0 # y ' = -* yx ) y * yz 0 $ t (n) $ - % z * (, zx * zy ) 0 z / + ) 11 ) 12 ) 13. T - 0 = ) 21 ) 22 ) 23-0,) 31 ) 32 ) 33 / t (n) =" T #n Cauchy " P t (n) " nx & $ $ # n y ' $ % n $ z ( " n1 & $ $ # n 2 ' $ % n $ 3 (

23 23 " (n) t & 1 $ * (n) $, # t 2 ' = $ (n) $, % $ t 3 ( $ + ) 11 ) 12 ) 13 - / ) 21 ) 22 ) 23 / ) 31 ) 32 ) 33. T " n1 & $ $ # n 2 ' $ % n $ 3 ( i, j t j (n) = 3 $ " ij # n i i=1 " t j (n) = " ij # n i Σ

24 24

25 25

26 26 F M a M " a = F F 2 F 1 a M " a = # F i a = 0 M F 3 " F i = 0

27 27 dx dy dz

28 28 X " + P x dx " x + = " x + #" x #x dx +L " + yx, " + zx P " + yx = " yx + #" yx #y " + zx = " zx + #" zx #z dy +L dz +L

29 29 X P & "# x $ dy $ dz + (# x + %# x ' %x dx ) + dy $ dz * & ", yx $ dx $ dz +, yx + %, yx ( ' %y dy ) + dx $ dz * & ", zx $ dx $ dy +, zx + %, zx ( ' %z dz ) + dx $ dy = 0 *

30 30 X "# x "x dx $ dy $ dz + "% yx "y dy $ dx $ dz + "% zx "z dz $ dx $ dy = 0 dx dy dz "# x "x + "$ yx "y + "$ zx "z = 0 "# xy "x + "$ y "y + "# zy "z = 0 "# xz "x + "# yz "y + "$ z "z = 0 "# ij "x i = 0 j =1L3 ( )

31 31 L = # m i r i " r i F 1 F 2 dl dt = ( ) # m i r i " r i + r i " r i #( ) #( ) = m i $ r i " r i = F i " r i = N F 3 N: N = 0

32 32 z " xy # dy # dz # dx 2 + % " xy + $" xy ' & $x dx ( * dy # dz # dx ) 2 +" yx # dx # dz # dy 2 + % " yx + $" yx ' & $y dy ( * dx # dz # dy ) 2 = 0

33 33 " xy # dy # dz # dx + $" xy $x dy # dz # dx2 2 %" yx # dx # dz # dy % $" yx $y dy # dz # dy2 2 = 0 dx dy dz " xy + #" xy #x $ dx 2 %" yx % #" yx #y dy 2 = 0 dx, dy " xy = " yx " yz = " zy " zx = " xz Cauchy! x,! y,! z, " xy, " yz, " zx

34 34 $ " x # xy # zx ' & ) " = &# xy " y # yz ) & %# zx # yz " ) z ( #" & % ( 0 " 2 0 % ( $ 0 0 " 3 '

35 35 n " n # S = (" x cos$ + % xy sin$ ) # S x + (" y sin$ + % xy cos$ ) # S y = " x cos 2 $ # S + % xy sin$ cos$ # S +" y sin 2 $ # S " n = " x cos 2 # +" y sin 2 # + 2$ xy sin# cos# Sx S Sy = " x +" y 2 + " x %" y 2 = " x +" y 2 ( cos 2 # + sin 2 #) ( cos 2 # % sin 2 #) + 2$ xysin# cos# + " x %" y 2 cos2# + $ xy sin2# 2sin" cos" = sin2" cos 2 " # sin 2 " = cos2"

36 36 n ( ) " n = # x sin$ cos$ %# y sin$ cos$ %" xy cos 2 $ % sin 2 $ = # x %# y sin2$ %" xy cos2$ 2 S Sx Sy

37 37 n " n = " x +" y 2 " n = # x $# y 2 + " x #" y cos2$ 2 + % xy sin2$ sin2% $ " xy cos2%

38 38 3 $ " x # xy # zx ' " & ) 1, " 2, " 3 &# xy " y # yz ) 3 & %# zx # yz " ) z ( (" x #" ) $ xy $ zx $ xy " y #" ( ) $ yz ( ) $ zx $ yz " z #" = 0

39 39 " 3 #" 2 (" x +" y +" z ) #" #( 2 " x " y +" y " z +" z " x )+ $ xy + $ 2 yz + $ 2 zx { } ( ) = 0 2 # " x " y " z #" x $ yz #" y $ 2 zx #" z $ 2 xy + 2$ xy $ yz $ zx " 3 # J 1 " 2 # J 2 " # J 3 = 0 J 1 = " x +" y +" z 2 J 2 = #(" x " y +" y " z +" z " x ) + $ xy + $ 2 yz + $ 2 zx 2 J 3 = " x " y " z #" x $ yz #" 2 y $ zx #" 2 z $ xy + 2$ xy $ yz $ zx J 1, J 2, J 3

40 40 J 1 = " 1 +" 2 +" 3 ( ) J 2 = # " 1 " 2 +" 2 " 3 +" 3 " 1 J 3 = " 1 " 2 " 3

41 41

42 42

43 43

44 44 d d d" 0 = dl l 0 d" = dl l " 0 = l # l 0 l 0 " = l dl $ # l = ln l 0 & l % l 0 ' ) (

45 45 " 0 = l # l 0 l 0 # " = ln% l $ l 0 & ( ' " 0 # "

46 46 " &% & #% $% $%% # $ ' ' $ %! (! (!%! dx, dy P P X X x P : x X : x + dx P! : x + u X!!: x + dx + u! x = x + dx + u + "u "x dx

47 47 "! & &% #% # $ ' ' $ % (! (!% $% $%% PX = dx # P' X" = % x + du + u + "u $ "x dx & () x + u ' = dx + "u "x dx x P' X" # PX " x = PX %! ' dx + $u & = $x dx ( *# dx ) dx y " y = #v #y ( ) = $u $x

48 48 " ' " % &% &%% ) " %! " ("% &!! #% $% $%% (" ) " # $ ' ' $ %! (! (!% )! %! " xy =# x +# y X' X" = P' X" + Y'Y" P'Y" % ' v + $v & = $x dx ( % * + v u + $u ) $y dy ( ' * + u & ) % ' 1+ $u + ( % * dx 1+ $v ( & $x) ' * dy & $y), $v $x + $u $y

49 49 $ &" x = #u #x & %" y = dv & #y & " z = #w '& #z $ &" xy = #v #x + #u #y & %" yz = #w #y + #v & #z " zx = #u #z + #w '& #x

50 50 " xy = " yx = # xy 2 " yz = " zy = # yz 2 " zx = " xz = # zx 2 # " x " xy " xz & % ( " = %" yx " y " yx ( % $ " zx " zy " ( z ' #" 11 " 12 " 13 & % ( = " 21 " 22 " 23 % ( $ " 31 " 32 " 33 ' " ij = 1 $ #u i + #u ' j 2& %#x j #x ) i (

51 51 # " x " xy " zx & % ( " = %" xy " y " yx ( % $ " zx " yz " ( z ' #" & % ( 0 " 2 0 % ( $ 0 0 " 3 '

52 52

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, 変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +

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