2
3
4
5 DEFORM-2D, -3D Altan, Ohio State Univ. SFTC FORGE 23 Chenot, CEMEF VirtualForging QForm Quantor Ltd. RIPLS-FORGE FS DD
6
7
! "#%*+,-.0123456789:;<=>?@! EF567! AB! "# x "x + " yx "y + F x = 0 " xy "x + "#! y "y + F y = 0! GHIJKL!! V # x % x +# y % y + xy % xy dv T x %u +T y %v ds = 0 s t! CDB! EF567! "# x "x + " yx "y + F x = 0 " xy "x + "# y "y + F y = 0!! GHIJKL8MNO@!! V # x % x +# y % y + xy % xy dv T x % u +T y % v ds = 0 s t! 8!!
9! "#%*+,-.0123456789:;<=>?@! AB!! ;<=EF! " x = #u #x!!! " y = #v #y!!!! xy = #v #x + #u #y! CDB!! ;<=GHEF!! " x = # u #x!!! xy = # v #x + # u #y! " y = # v #y!
10!"#%*+,-.01*2345 6785 9:785 %;*,-.5 %;*^_,-.5 <==>?+@A45 `?abcdef?f+gha45 " x %" y = D # xy * [ D] = [ ] E 1-. 1+. 1-2. + x % + y 5, xy *. 2 1 1 0 1-. 4 1. 4 1 1 0 4 11-. 4 1 1-2. 0 0 4 0 1-. 3 5 BCD7E5 F55GHIJKLMNOOO>PQRSS L T5 8UVWX,YZD7E5 [5HIJKO\]45 5 " 5 x + x %" 5 y = [ D] % + y 5 # xy *, xy * 2 [ D] = 2" 1 1 0 0 4 10 1 0 3 + 1 0 0 1 4 4 01 234 5 5 BCD7X,YZij.5 " = f +, +, T 5 8Uklmn5 + x + + y = 05 5 = 0. 5 5. Xop45
11!"#%*+,-.0 0 1230 45230 678#9:0 ;<9=0 ;<>?0 @7#ABCD:0 EF0 G0 H7#IJK;<IJA0 00 LMDN0 O7#42PQRAST0 00 UVWX3420 00 PQRAYZ[\0 ]7;<^D_`Ua0 00 bcade0 f7ghbcade0 l742pqra2m0 00 ij,%0 ;<9=ij0 ;<k^0 no0 pqars0 EF0 EF0 EF0 G3wpxbcyzbcKX3A{?ADN}~452,- ;<>?ij0 ;<k^0 tno0 uvrs0 ƒ # 7
12 " 2 = 1 % 2 " x # " y 2 + " y # " z 2 + " z # " x = 1 2 " 1 # " 2 2 + 6 xy { 2 + " 2 # " 3 2 + " 3 # " 1 2 } 2 + 2 yz + 2 zx * = " 2 Y
13 s! 1! 2! 3
14 %# " x = # x # m # " y = # y # m # " z = # z # m " m = " x + " y + " z 3 " 2 = 1 % 2 " x # " y 2 + " y # " z 2 + " z # " x 2 + 6 xy = 1 % + 2 " x + " m # " + y + " m 2 + " + y + " m = 1 % + 2 " x # " + y + " + z + " m 2 + 2 yz + 2 zx * # " + z + " m 2 # " + x + " m 2 +6 xy 2 + " + y # " + z 2 + + 2 + 2 yz + 2 zx } " z # " + x 2 + 6 2 xy + 2 yz + 2 zx Mises *
15 " 2 = 1 % 2 " x # " y 2 + " y # " z 2 + " z # " x 2 + 6 2 xy + 2 yz + 2 zx * + g +" m 2 = " 2 Y
16 s 1 Mises s Y 2 s 2, 2 s 3 - s Y
17 e de
18! 2 "f "f : : "f "# 1 "# 2 "# 3! Y "! Y! Y " x = # f % x " y = # f % y " z = # f % z " xy = # f % xy " yz = # f % yz " zx = # f % zx "! Y! 1
19 f = 1 % 4 " x # " y " 2 + " y # " z 2 + " z # " x 2 + 6 xy 2 + 2 yz + 2 zx *, " x = # x % 1 2 y % 1 2. + z *. " y = # - % 1 2 x + y % 1 2 z.. " z = # % 1 2 x % 1 2 y + z. + * + * % " xy = 3# xy " yz = 3# yz " zx = 3# zx
20 w w = {" } T { #} { } {# x, # y, # z, % xy, % yz, % zx } {" } = " x," y," z, xy, yz, zx { #} = w = ", 1 % 2 # x # -. y = " 2# 2 2 + # y # z 2 + # z # x 2 * + 6+ xy 2 + + 2 yz + + 2 zx 01
21 w = ", 1 % -. 2 # x # y = " 2# 2 2 + # y # z 2 + # z # x 2 * + 6+ xy 2 + + 2 yz + + 2 zx 01! w = " # " = # + " x = " % # # x 1 2 # y 1 2 # - z * - " y = " % # # y 1 2 # z 1 2 #, x * - - " z = " % # # z 1 2 # x 1 2 # y *.- " xy = 3 # % xy " yz = 3 # % yz " zx = 3 # % zx
- " x 1, 1 2, 1 0 0 0 0 2 2 + x " y, 1 0, 1 2 0 0 02 + y " z % = 2 2 " #, 1 xy + 2, 1 2 + z 1 0 0 02 % 2 2 3 xy # yz 0 0 0 3 0 02 3 yz # zx * 0 0 0 0 3 02 2 3 zx *. 0 0 0 0 0 31 22 " x - 1 +, +, 0 0 0 0 3 x 2 " y +, 0 +, 0 0 0 2 3 y " z % = 1 +, +, 1 0 0 0 2 3 z # xy E 0 0 0 2 1+, 0 0 2 % 2 4 xy # yz 0 0 0 0 2 1+, 0 2 4 yz # zx *. 0 0 0 0 0 2 1+, 2 1 4 zx *
23 "
24 - " x 1, 1 2, 1 2 0 0 0 0 2 + x " y, 1 0, 1 " z % = 2 2 0 0 0 2 2 + y " #, 1 xy + 2, 1 2 + z 1 0 0 02 % 2 2 3 xy # yz 0 0 0 3 0 02 3 yz # zx * 0 0 0 0 3 02 2 3 zx *. 0 0 0 0 0 31 " " = 2 9 % " x # " y 2 + " y # " z 2 + 2 + 3 2 " z # " x xy 2 + 2 yz + 2 zx *
25 " m = " x + " y + " z 3 %# " x = # x # m # " y = # y # m # " z = # z # m # " xy = # xy %# " yz = # yz # " zx = # zx 3.2.16 " x = " % # # x 1 2 # y 1 2 # z * = " % 3 # 2 # x 1 2 # x + # y + # z * = " % 3 # 2 # x 3 %# x + # y + # z ** 2 3 = " % 3 # 2 # x 3 2 # m * = 3 " # + x 2 # 3.2.17
% " x = 3 " 2 # " y = 3 " 2 # " z = 3 " 2 # # x # y # z " xy = 3 # % xy " yz = 3 # % yz " zx = 3 # % * zx 3.2.18 26 " x, 1 0 0 0 0 0 + 2 x. 1 " x 0 1 0 0 0 0 2. 1 + y " x % = 3 ". 0 0 1 0 0 01 + 2 z #. 1 % xy 2 + 0 0 0 2 0 0. 1 3 2 xy # yz. 0 0 0 0 2 01 3 2 yz. * - 0 0 0 0 0 2 1 0 2 * # zx 3 zx 3.2.19
27-1 0 0 0 0 00 %# " x 2 % 0 1 0 0 0 0, x 2 # " y 0 0 1 0 0 0 # " z * = 2 # 0 0 0 1 2, y 2 0 0, z " xy 3, 2 2 * 1 2 3 xy " yz 0 0 0 0 02 3 2 yz " zx + 1 2 3 0 0 0 0 0 2 zx +. 21 3.2.20 [ ] {# "} = D " { }
28 f = " 2 2 = 1 % 4 " x # " y 2 + " y # " z 2 + " z # " x 2 + 6 xy 2 + 2 yz + 2 zx * + g 2 " m 2 " x = # f % x = # % x 1 2 % y 1 2 % z + g 3 % m = # - 1+ g * 9 +,% x + 1 2 + g * 9 +,% y + 1 2 + g * 0. 9,% + z 1 2 *, +
29, " x = # x % 1 2 y % 1 2 z + g 3. m + *. " y = # - % 1 2 x + y % 1 2 z + g 3 m.. " z = # % 1 2 x % 1 2 y + z + g 3 m. + * + * % " xy = 3# xy " yz = 3# yz " zx = 3# zx
" " = # 30 " x " y " z % # xy # yz # zx - 1+ g, 1 9 2 + g 9, 1 2 + g 0 0 0 0 9 2 " #, 1 =! = 2 + g 1+ g, 1 9 9 2 + g 2 0 0 02 9 " +, 1 2 + g 9, 1 2 + g 1+ g 2 0 0 02 % 9 9 2 0 0 0 3 0 02 * 0 0 0 0 3 02 2. 0 0 0 0 0 31 + x + y + z 3 xy 3 yz 3 zx * g
31 % " x " y " z # xy # yz # zx - 4 9 + 1, 2 g 9 + 1 g, 2 9 + 1 0 0 0 0 g 2, 2 9 + 1 4 g 9 + 1, 2 g 9 + 1 2 0 0 02 g 2 = ", 2 9 + 1 g, 2 9 + 1 4 g 9 + 1 2 0 0 0 g 2 + 1 2 % 0 0 0 3 0 0 2 2 1 * 0 0 0 0 3 0 2 2 12 0 0 0 0 0. 312 + x + y + z 3 xy 3 yz 3 zx * [ ] {" } = D {#}
32 " 2 = 2 % 9 " x + " y 2 + " y + " z " v = " x + " y + " z 2 + 2 + 3 2 # xy 2 + # yz 2 + #zx 2 " z + " x " v + 1 g " v 2 " v = " # g # m
33 %" z = # yz = # zx = 0 yz = zx = 0 % " x " y " z # xy # yz # zx - 4 9 + 1, 2 g 9 + 1 g, 2 9 + 1 0 0 0 0 g 2, 2 9 + 1 4 g 9 + 1, 2 g 9 + 1 2 0 0 02 g 2 = ", 2 9 + 1 g, 2 9 + 1 4 g 9 + 1 2 0 0 0 g 2 + 1 2 % 0 0 0 3 0 0 2 2 1 * 0 0 0 0 3 0 2 2 12 0 0 0 0 0. 312 + x + y + z 3 xy 3 yz 3 zx *
34 % " x " x # xy * = " + - 4 9 + 1, 2 g 9 + 1 g, 2 9 + 1 4 g 9 + 1 g 0 0. 0 02 2 + x 02 % + y 2 12 312 3 xy * " z = " % # 2 9 + 1 * # x + g = " % # 2 9 + 1 * + g # y # v
35 % " x " x # xy * = " + = D D 3, 2, 0 0 2 2. 3 1 9 + 1 g 2 2 9 + 1 6 5. 018 5. 2 1. g. 0 01 + 2 2. 3 1 9 + 1 g 2 2 9 + 1 18 5. 018 % 5. g 18 5. 1 0 0 1. 18 5-30. 0 0 018 4-07 [ ] + [ D V ]{ + } + x + y 9 xy *
g=0 %" z = # yz = # zx = 0 yz = zx = 0 - " x 1+ g, 1 9 2 + g 9, 1 2 + g 0 0 0 0 9 2 " y, 1 " z % = 2 + g 1+ g, 1 9 9 2 + g 2 0 0 02 " 9 #, 1 xy + 2 + g 9, 1 2 + g 1+ g 2 0 0 02 % 9 9 2 # yz 0 0 0 3 0 02 # zx * 0 0 0 0 3 02 2. 0 0 0 0 0 31 + x + y + z 3 xy 3 yz 3 zx * 36
- 1+ g, 1 " x % " y = 9 2 + g 0 0 9 2 ", 1 # + 2 + g 1+ g + 2 x 02 % + y 9 9 2 xy * 3 0 0 32 xy *. 1 37 " x %" y # xy * = " + 3 1+ g 1 5 9 2 2 g 9 1 51, 3. 4 + g 5 2 2 g 1+ g 9 9 15-305 0 0 4 6 0 8 + 8 x 0 8 % + y 1 4 + g 8 9 8 xy * 97 =0
38 Mises " x %" y # xy * = " +, 4 2. 3 3. 2 4.. 3 3. 0 0-01 + 1 x 01 % + y 1 1 2 1 xy * 30 " z = # 1 2 " % % x + % y = # 3 2 " % % m " y = # " x +
41 u, v, w " x = # u #x % " y = # v #y " z = # w #z " xy = # v #x + # u #y % " yz = # w #y + # v #z " zx = # u #z + # w #x
u = n 1# i=1 n 1# % v = i=1 n 1# w = N i " u e i e N i " v i N i " w e i i=1 42 n 1% " x = i=1 n 1% " y = i=1 n 1% " z = #N i #x u i e #N i #y v i e #N i #z w i e i=1,. " xy =.. - " yz =... " zx = n 1+ % i=1 n 1+ % n 1+ % i=1 i=1 #N i #y u i e + #N i #x v i e #N i #z v i e + #N i #y w i e #N i #x w i e + #N i #z u i e * * *
43 e u 1 v e 1 e w 1 " x + B 1 0 0 B 2 0 0 B n1 0 0. u e 2-0 e " y 0 C 1 0 0 C 2 0 0 C n1 0 v - 0 2 e " z - 0 0 D 1 0 0 D 2 0 0 D n1 0 w 2 % = - # xy C 1 B 1 0 C 2 B 2 0 C n1 B n1 0 0 % - 0 # yz - 0 D 1 C 1 0 D 2 C 2 0 D n1 C n1 0 - # zx *, D 1 0 B 1 D 2 0 B 2 D n1 0 B 0 n1 e u n v e n e w n * B i = "N i "x C i = "N i "y D i = "N i "z
44 u = a + bx + cy + dxy v = e + fx + gy + hxy " u 1 = a + bx 1 + cy 1 + dx 1 y 1 u 2 = a + bx 2 + cy 2 + dx 2 y 2 # u 3 = a + bx 3 + cy 3 + dx 3 y3 % u 4 = a + bx 4 + cy 4 + dx 1 y 4
45
4 x = # N i " x i % i=1 4 y = # N i " y i i=1 u = % v = 4 # N i " u i i=1 4 # N i " v i i=1 46 N i = 1 4 1+ " i #" 1+ i # % N 4 = 1 4 1" # N 1 = 1 4 1" # 1" N 2 = 1 4 1+ # 1" N 3 = 1 4 1+ # 1+ 1+ * * * * * * *
" x = # u #x, " y = # v #y, xy = # v #x + # u #y 47, +. " x. +x % " y =. 0 #. xy *. +. - +y 0 1 1 + 1 u % +y1 v * + 1 +x 1 0 4 u = # N i " u i % i=1 4 v = # N i " v i i=1, +N. 1 " x. +x % " y =. 0 #. xy *. +N 1. - +y { u e } { "} = [ B] 0 +N 1 +y +N 1 +x +N 2 +x 0 +N 2 +y 0 +N 2 +y +N 2 +x +N 3 +x 0 +N 3 +y 0 +N 3 +y +N 3 +x +N 4 +x 0 +N 4 +y u 0 1 1 1 v +N 4 1 1 % M +y 1 +N 4 1 u 4 +x 1 0 v 4 *
- 48 %"N i "# = "N i "x "N i " = "N i "x "x "# + "N i "y "x " + "N i "y "y "# "y " %"N i "# * "N i " +,"x. "# =.. "x -" "y %"N i "# 1 1 "x "y "N 1 i " 0 "y * + [ ] = J %"N i "x "N i "y * +
- 49 %"N i "# * "N i " +,"x. "# =.. "x -" "y %"N i "# 1 1 "x "y "N 1 i " 0 "y * + [ ] = J %"N i "x "N i "y * + #"N i % "x % "N = J % i % "y "N i [ ] *1 % "+ # % = 1 % "N i % J ", - "y ", * "x. ", * "y 0 #"N i "+ 2 % "+ % 2 "x 2 % "N i % "+ 1 ", J = "x "y "# " % "x "y " "#
" B 11 0 B 12 0 B 13 0 B 14 0 % [ B] = 0 B 21 0 B 22 0 B 23 0 B24 # B 21 B 11 B 22 B 12 B 23 B 13 B 24 B 14 50 + B 1i = 1 %"y "N i J "# " + "y "N i - * " "#, - B 2i = 1 %"x "N i J "# " + "x "N i *.- " "# "N i "# = 1 4 1+ i % %#, "N i " = 1 4 1+# i %# % "x "# = 4 "N % i "# x i, i=1 "y "# = 4 "N % i "# y i, i=1 "x " = "y " = 4 % i=1 4 % i=1 "N i " x i "N i " y i
51 y x
FEM %"# x "x + " yx "y + " zx + b x = 0 "z " xy "x + "# y "y + " zy + b y = 0 "z " xz "x + " yz "y + "# z "z + b z = 0 " x = # u #x % " y = # v #y " z = # w #z " xy = # v #x + # u #y % " yz = # w #y + # v #z " zx = # u #z + # w #x - 4 9 + 1, 2 g 9 + 1 g, 2 9 + 1 0 0 0 0 g 2 " x, 2 " 9 + 1 4 g 9 + 1, 2 g 9 + 1 2 0 0 02 + x g 2 y " z % = ", 2 9 + 1 g, 2 9 + 1 4 g 9 + 1 2 + y 0 0 0 g 2 + z # xy + 1 2 % 0 0 0 # yz 3 0 0 2 3 xy 2 3 yz 1 # zx * 0 0 0 0 3 0 2 3 2 zx * 12 0 0 0 0 0. 312 52
53 { T} = { T 0 } { u } = { u 0 }
54 {! u } # V {" } T { %}dv = # { T0 } T { u }ds + # b { u }dv S f V { } T { u e } { u } = [ N] { "} = [ B] { u e } " { #} = B { u e } [ ] " {" } = [ D] { #} {" u e }# V [ B] T [ D] [ B] dv { u e } = " u e # S [ ] T { T f 0 } ds + # V [ N ] T { b } dv % { } N
55 {" u e } # V [ B] T [ D] [ B] dv u e { } = {" u e } N [ ] T { T 0 } [ ] T { b } # S ds + # N f V dv % { } " u e " V [ B] T [ D] [ B] dv u e [ K e ] =! v e [ B ]T [ D] [ B]dV { } = " S [ N] T { T f 0 } ds + V [ N ] T { b } = { F e } " dv [ K e ] u e { } = { F e } 3.4.8
56 [ K] U { } = F { } D] 1
57 Ko
Newton-Raphson 58 x n f x = 0 f xn " 0 f x = f xn + df dx + O dx x=xn dx 2 = 0 f xn dx = " # df % x n+1 = x n + dx dx x=x n
59 " U { { }} = [ K { U }] U { }# { F} = 0 { U n } {" { U } } = {" { } }+ K T U U n [ { }]{ du n } = 0 [ K T ] K Tij = "# i %" u = "K K ij + + ik j U = U k " u n % % j u k * "F i " u j U = U n
60 Newton-Raphson
61 " V [ B] T [ D ] B [ ]dv { u e } = F e { } [ K e ] = [ B ] T [ D] [ B]" A e
62 Gauss m 1 1 #1 f " d" = % w i f "i i=1 m 2 m 1 % 1 1 1 % 1 f ", # d# d" = w j w i f "i j=1i=1, # j
63 m 2 m 1 % 1 1 1 % 1 f ", # d# d" = w j w i f "i j=1i=1, # j! i, " j, # k w i,w j, w k ±1 3 ± 3 5
64
65! v = A + Bx + Cy A! 0, B! 0, C! 0
66,
67! v = A
68
69,
70
71 = % V % 0 # d dv % { F} T { u }ds " u S f
72 = % % 0 # d dv % { F} T { u }ds + % V v dv " u V S f = " D + " F + " V = % % 0 # d dv % { F} T { u }ds + 1 2 % V v 2 dv " u V S f = " D + " F + " P = % % 0 # d dv % { F} T { u }ds " u V S f = " D + " F
73 " 2 = 1 % 2 " x # " y 2 + " y # " z 2 + " z # " x 2 + 6 xy 2 + 2 yz + 2 zx * + g +" m 2 2 = " Y " 2 = 2 9 % " x + " y 2 + " y + " z 2 + " u 2 + 3 2 # xy 2 + # yz 2 + #zx 2 " z + " x V = % % 0 # d dv % { F} T { u }ds = " D + " F "# u % = 0 k=1nn "u k S f + 1 g " v 2
" 2 = 2 % 9 " x + " y 2 + " y + " z 2 + 2 + 3 2 # xy 2 + # yz 2 + #zx 2 " z + " x + 1 g " v 2 74 " 2 = { "} T D # [ ]{ "} = { u e } T B { u e } [ ] T [ D #][ B] 4 9 + 1 # 2 g 9 + 1 g # 2 9 + 1 0 0 0 g # 2 9 + 1 4 g 9 + 1 # 2 g 9 + 1 0 0 0 g # 2 { D "} = 9 + 1 g # 2 9 + 1 4 g 9 + 1 0 0 0 g 1 0 0 0 3 0 0 1 0 0 0 0 3 0 1 0 0 0 0 0 % 3 3.7.7 3.7.8
75 " = " e # % "# D e " u e k. * = "-, 0 + d, 1 -V 0 " u e 3 dv k 2 = =."-, 0 + d, ", 1 -V 0 4 ", " u e 3 dv k 2 - V. + 0, B { u e } [ ] T [ D 5 ][ B] [ ] T [ D] [ B] = - B = K e V dv [ ] { u e } { u e } 1 3 dv 2 3.7.9 " # " u e k = " # " # 2 " # 2 " u e k = 1 2 # " # 2 " u e k = 1 2 # 2[ B]T D % = 1 # [ B]T D % { u e } [ ][ B] { u e } [ ][ B]
76 " e F = # u S f e = # u e { } T F { }ds { } T N [ ] T [ N]dS F % S f e { } 3.7.10 { F e } = N " [ ] T [ N]dS F # S f e { } % "# F e " u k * = + Fe [ K e ] u e { } { } = { F e }
77 {" } = [ D] { #} = " # [ D ] { u e }+ " # % 2 9 + 1 + C g* [ ] { u e } 3.7.14 [ D " "] = 1 3 # 2 % % % % % % % 2 0 2 1 0 1 1 3.2.20 " 1 1 1 % 1 1 1 0 3.7.16 1 1 1 C = 3.7.15 0 0 0 # 0 3.2.24
78 Lagrange = % % 0 # d dv % { T 0 } T { u }ds + % V v dv " u V S f = " D + " F + " V 3.7.17 " v % "# = 0 " u k "# = 0 " l 3.7.18
79 Lagrange " = { u } T D # # [ ]{ u } = { u e } T B { u e } [ ] T [ D # #][ B] 3.7.19 "# e % D " u k * =, V = K e { } [ B]T [ D + ][ B]dV u e [ ]{ u e } 3.7.20 [ D " ] = # [ D " "] 3.7.21 % e "# D " l * + = 0 3.7.23
80 Lagrange % "# F e " u k * = + Fe { } 3.7.24 % "# F e " l * + = 0 3.7.25
81 Lagrange [ N " ],[ B " ] " V = Ve # % v dv = # e V e { } T N # [ ] T { P} T [ B # ] { u e }dv 3.7.26 { P} = { 1 1 1 0 0 0} T % "# V e " u k * =, [ B + ] T { P} [ N + ]dv{ + e } V = K e e + [ ]{ +e } 3.7.28 % "# e V " l * + =, [ N ] T { P} T [ B ]dv{ e } V = K e e [ ] T { e } 3.7.29
82 Lagrange # K e % u % K e T " 0 K " e * + u e " e, -. * + = Fe 0, -. 3.7.30
83 Lagrange " m = # {" } = {" }+ " m { P} = " % [ D ] {%}+ " m { P} 3.7.31 3.7.32
84 = % V % 0 # d dv % { T 0 } T { u }ds + " u S % f V v 2 dv 3.7.33 = " D + " F + " P { u e } " v = { P} [ B] 3.7.34 % "# P " u k = * + V B = * K P e [ ] T { P} T { P} B [ ] { u e } { } [ ]dv u e 3.7.35
85 [ K e ] + " [ K P e ] u e { } = { F e } 3.7.36
86 " { } = {" #}+ P = " % [ D # #] { } % v {%}+ C [ ]{ %} 3.7.37 [ D " "] 3.7.16 [ C] 3.7.15 {" } = [ D] { #} = " [ D ]{ #}+ " # # % 2 9 + 1 + C g* [ ]{ #} 3.7.14
87