Materials Science And Engineering, An Introduction: by William D. Callister, Jr., John Wiley & Sons, Inc. Mechanical Metallurgy, G.E.Dieter, McGraw Hill, 1987 Fundamentals of Metal Forming, Robert H. Wagoner, Jean-Loup Chenot, John Wiley & Sons, 1996 Structure of Metals, Charles Barrett and T.B.Massalski, Pergamon Press, 1980 W.D. 1999 1986 1985 1999 1997 1971 1998 2000 1
1../ ' 2 1 #$%& " 3 0,- )*,- Elastic Deformation " = E # 1.1) E : Young's modulus " = G # 1.2) G : Shear modulus G = E 2 1 " ) 1.3) ν : Poisson's ratio Material E GPa) G GPa) Aluminum alloys 72.4 27.5 0.31 Copper 110 41.4 0.33 Steel plain carbon & low-alloy) 200 75.8 0.33 Stainless steel 18Cr-8Ni) 193 65.6 0.28 Titanium 117 44.8 0.31 Tungsten 400 157 0.27 Plastic Deformation 2
x3 "$ "# 0 x u x u x1 x2 Displacement u " u 1,u 2,u 3 ) 1.4 u u Distortion "u i "x j i,j =1,2,3 ) 1.5 StrainijRotation ij " ij # ij $ %u i %x j " ij # 1 % $u i $u j 2' & $x j $x * i ) 1.6) 3
" ij # 1 & $u i % $u ) j 2 ' $x j $x i * " ij # $u i $u j = 2% ij i & j) $x j $x i 1.7) #" 11 " 12 " 13 & % " ij = % " 21 " 22 " 23 $ %" 31 " 32 " 33 ' ) "u 1 1# "u 1 "u & 2 1# "u 1 % "u &, 3 %. "x 1 2$ "x 2 "x 1 ' 2$ "x 3 "x 1 '. 1# "u = 2 "u & 1 "u 2 1 "u 2 % "u )/ # &. 11 / 12 / 13, 3. %. = 2$ "x 1 "x 2 ' "x 2 2$ "x 3 "x 2 ' / 12 / 22 / 23. 1# "u 3 "u & 1 1# "u 3 % "u. & * / 2 "u 13 / 23 / 33 -. 3. % * 2$ "x 1 "x 3 ' 2$ "x 2 "x 3 ' "x. 3 - # 1 )u 0 1 * )u. 2 1 )u 1-0 * )u.& 3 % - 0 #" 11 " 12 " 13 & % 2, )x 2 )x 1 / 2, )x 3 )x 1 / % % 1 )u " ij = % " 21 " 22 " 23 = 2 * )u. 1 1 )u - 0 0 2 * )u. 3 % - 0 2, )x $ %" 31 " 32 " 33 ' % 1 )x 2 / 2, )x 3 )x 2 / 1 )u 3 * )u. 1 1 )u 3-0 * )u %. 2 % - 0 0 $ 2, )x 1 )x 3 / 2, )x 2 )x 3 / ' 1.8) 1.9) traceinvariant " 11 " 22 " 33 = 1.10) V "V = V 1 # 11 ) 1 # 22 ) 1 # 33 ) " V 1 # 11 # 22 # 33 ) 1.11) " 11 " 22 " 33 = #V V 1.12 4
force, load traction body force etc. Stress x3 x3 33 F = F1, F2, F3) 32 F3 31 23 A F2 x2 11 13 12 21 22 x2 F1 x1 x1 X = F A 1.13) Normal Stress " 33 # F 3 A 1.14) Shear stress " 13 # F 1 A, " 23 # F 2 A 1.15) #" 11 " 12 " 13 & % " ij = % " 21 " 22 " 23 $ %" 31 " 32 " 33 ' #" ii & # % 0 0 3 " 11 ) " ii % 3 % " % = % 0 ii 0 % % 3 % % " 0 0 ii % $ 3 ' $ Hydrostatic " ii = " 11 " 22 " 33 ) " 12 " 22 ) " ii 3 " 12 " 13 " 23 " 13 " 23 " 33 ) " ii 3 Deviatoric & ' 1.16) 5
Hooke's law " ij = C ijkl e kl 1.17) Cijkl Elastic Constant Elastic Stiffness e ij = C "1 ijkl # kl 1.18) Cijkl -1 Elastic Compliance C ijkl = C jikl = C ijlk = C klij 1.19) Lame's constant " 11 = #2µ) e 11 # e 22 e 33 ) " 22 = #2µ ) e 22 # e 33 e 11 ) " 33 = #2µ) e 33 # e 11 e 22 ) 1.20) " 12 = 2 µ e 12 " 23 = 2 µ e 23 " 31 = 2 µ e 31 Bulk Modulus 11 E " # 11 e 11 " # $e 22 e 11 # $e 33 e 11 1.21) 112233H K " # 11# 22 # 33 3 e 11 e 22 e 33 ) " # H e ii 1.22) E = µ 3"2µ ) "µ " = # 2 #µ ) E = 2µ 1 ") 1.23) K = 3"2µ 3 = E 3 1#2$ ) 6
"F k F #$"x E S = 1 2 k x2 = 1 2 F x 1.24) 0 E 0 = 1 2 C ijkl e ji e kl = 1 2 " kl e kl 1.25) el E el = 1 2 " C ijkl e ji e kl dv = 1 2 V " # kl e kl dv V 1.26) 7
dii=d11 d22 d33 ) d ij dw = " ij d# ij = " $ ij d# ij P d# ij 1.27) Tresca von Mises '2 Taylor Quinney von Mises J " 2 = # $ " x $ " y # $ " y $ " z # $ " z $ " x $ xy " $ # xx $ # yy " $ # yy $ # zz " $ # zz $ # xx = 1% 6 $ xx "$ yy &' J " 2 = # xx $# yy ) 2 # yy $# zz ) 2 # zz $# xx 2 $ 2 yz $ 2 ) zx 1.28) ) 2 $ yy "$ zz ) 2 $ zz "$ xx ) 2 )* 1.29) ) 2 6 # 2 xy # 2 yz # 2 ) zx J " 2 = 2Y 2 =const. ) 2 " yy #" zz ) 2 " zz #" xx ) 2 6 " xy ) = 2 Y 2 = 6 $ 2 " xx #" 2 yy " 2 " 2 yz zx 1 $ 3 2 #" ' 2 % ij #" & ij = Y ) 8
s " P A 0 e " L #L 0 L 0 " # P A " # ln L = 2ln D 0 L 0 D " = s 1 e) " = ln 1 e) " = d" dt = 1 dl L dt # v L p " # A 0 $A f A 0 9
s B M N"#$%&' x ) yu yl E P e eu el s ef B x 0.2 e = 0.2% 10 e
F F A L A da L dl d " d" ) A da) # " A " A d# # da $ 0 " da A = dl L = d# d" d# $ " " = K # n " = Y K # n ) n " = K B # 11
s #$ x " e 12
slip plane slip direction #" " a b µ τ x b b $ " = " m sin 2# x ' & ) % b 2.1) τ τm 0 x b x/a # " = µ % x& $ a ' 2.2) $ " = " m sin 2# x ' & ) * 2# " m x % b b 2.3) 13
" m = µb 2#a 2.4) a b τm µ / 10 µ / 10 5 Dislocation glide motion a) b) x d) c) b 14
"#$%&')*,-... /012&3456-)7#89:;<= />1?&@45AAB5CD,EF)G7H Burgers vector FCCBCC crystal system slip plane slip direction Burgers vector FCC {111} <110> a/2 <110> {110}, {112}, BCC {123},... <111> a/2 <111> 3.2. FCC {111}{111} <110> F A n d " As F F n d 15
A F " = F A 2.5) F cosas = A/cos resolved shear stress " = Fcos# A S = F A cos$ cos# = %cos$ cos# 2.6) 2.6coscos Schmid factor == 450.5 primary slip systemsecondary slip system tbedge dislocation screw dislocationmixed dislocation b t b // t 16
$% &' "# b &' t &' "# &' $% )*,-'."#/01$%/023456789:;<= a) b) c) τ a)b) a) b b) -b or c) Dislocation Density ρ m/m 3 = m -2 ρ=10 9 10 11 m -2 ρ=10 14 10 16 m -2 17
a) w b) c) d h x b h x d x w a) x b) " 32 = b h # x d 2.7) " = w hdw = 1 hd 2.8) " 32 = #b x 2.9) 2.9 stress upper yield point lower yield point A B 0.2) twinning a) b) c) d) 2.9) 0.2% strain strain " 32 = d" dt = b x d# dt # b dx dt 2.10) 18
" 32 = d" dt = #bv 2.11) v ρ JohnstonGilman τ v " # m 2.12) LiF 25Fe-Si 40 1030 510 ρ 2.11 2.12 τ i) ii) b) yield drop; discontinuous yielding bcc fcc hcp bcc fcchcp bcc fcc, hcp ρ v ρ v bcc fcc bcc fcc 19
x3 core Volterra dislocationr, θ, x3x1 x3 b r, θ, x3 x3 u3 u 3 = "b 2# 2.13) tanθ = x2 / x1 x1, x2, x3 u 3 = b $ x 2" tan#1 2 ' & ) % x 1 2.14) 1.6 e S ij = $ & & & & & &" b % 2# 0 0 " b x ' 2 2# x 2 2 ) 1 x 2 ) b x 0 0 1 ) 2# x 2 2 1 x ) 2 ) x 2 x 1 2 x 2 2 b 2# x 1 x 1 2 x 2 2 1.20 % 0 0 # µb x ' 2 2$ x 2 2 * ' 1 x 2 * " S ' µb x ij = 0 0 1 * ' 2$ x 2 2 1 x * ' 2 # µb * x 2 µb x 1 ' 2$ x 2 2 1 x 2 2$ x 2 0 2 * & 1 x 2 ) 20 0 ) 2.15) 2.16)
Volterra dislocation model u1 θ = π b & " b 2# e E ij = ' & µb # 2$ 1#% " E ij = ' { } ) 2 ) x 2 2 1 x 2 ) 2 x 2 2$ 3µ ) x 2 2 1 µ x 2 $2µ ) x 2 2 1 x 2 b x 1 x 2 2 1 " x 2 2# 1"% ) ) ) x 2 2 1 x 2 ) 2 0 { ) x 2 2 1 "µ x 2 } $2µ ) 2 0 b x 1 x 2 2 1 " x ) 2 2# 1"% ) b x 2 2 $µ 2# ) x 2 2 1 x 2 0 0 0 * ) x 2 2 1 x 2 ) 2 ) x 2 2 1 x 2 ) 2 x 2 3x 1 2 x 2 2 µb x 1 x 2 2 1 #x 2 2$ 1#% ) ) x 2 2 1 x 2 ) 2 0 ) x 2 2 1 x 2 ) 2 0 µb x 1 x 2 2 1 #x 2 2$ 1#% ) µb x 2 x 2 2 1 #x 2 2$ 1#% ) 0 0 # µ %b $ 1#% ) ) x 2 x 2 2 1 x 2 * 2.17) 2.18) 21
1.26 S S e ij E S = 1 % V#$ " ij dx 2 ' &' $ 2% R S S S S $ r0 ) = 1 $ 0 " # x 2 3 e # x 3 " x 3 # e x 3 # r dr d# dx 3 2.19) r r0 R r0 r0 5 b R # E S µ b 2 & 2" R = )* ) 0 ) r0 % $ 8 " 2 r 2 r dr d, dx 3 ' 2.20) dx3 r θ E S 0 = µb2 4" ln # R & % $ r 0 ' 2.21 ln R / r0 ) 2.21) 4 π E 0 S = "µb 2 a1/21 2.22) E 0 E = µb 2 4" 1#$ % ' ) ln R & r 0 * ) 2.23) 1/3 2.22 22
TL dislocation TL TL 2.22 Δl ΔE "E = # µ b 2 "l 2.24) T L = de dl = lim $ "E ' & ) = * µ b 2 "l#0% "l 2.25) 23
Peach-Koehler τ L x S = L x ) b S F F = τ S W W = F b = " S b = " L x b 3.1) τ b L ) x W L τ b L x f = " b 3.2) b=b1, b2, b3) t=t1, t2. t3) f=f1, f2, f3) f = G " t 3.3) G " # $b Peach-Koehler 3.3 x3 t=0,0,1) b=b,0,0) b=0,0,b) G G = " 11 b, " 12 b, " 13 b ) G = " 13 b, " 23 b, " 33 b ) 3.3 f = " 21 b, #" 11 b, 0) 3.4) f = " 23 b, #" 13 b, 0) 3.5) 24
b b 11 21 21 f2 f1 x3 11 x2 x1 b b 23 23 f2 13 13 f1 τ a)c) b/2 b)a), c) a) x = 0 b) x = b/2 c) x = b b) a,c)b) b/2 Peierls potential 25
Peierls stress Peierls potential 0 b/2 b a b b/a b/a BCC, FCC, HCP b/a b/a b b) dislocation obstacle slip plane 26
L L F x F m b L b L 0 x τ L τ b L F F τ b L τ m τ m b L τ m b L long-range obstacleshort-lange obstacle 27
a) long-range obstacle m b L a b L x b) short-range obstacle m b L * b L x "#$%&'%)*,-./0 a) τa τm b) τ* τ* τm t " = " a " * 3.6) τa athermal stressτ* effective stressτ* 28
work-hardening strain-hardening,- ' 0 / #$%& ". 1 )* 29
stress I II III strain FCC Ieasy glide region IIlinear hardening region IIIparabolic hardening region dynamic recovery " = # µ b $ 0.5 3.7) Bailey-Hirsch II grain boundary d " y = " 0 k d # 1 2 3.8) Hall-Petch 30
relationship d -1/2 <> Frank-Read n grain boundary 0 x1 x2 x3 xn-1 L dislocation source slip plane f0 f 0 = " n # b 3.9) back stress n n " = " c 3.10) L L = n µ b 1 " #) $ % 3.11) 1-) d L L " d 2 3.12) 3.103.12 n L 31
& " = 2 " c µ b ) ' * 1# $ ) % 1 2 # 1 d 2 3.13) -1/2 primary slip system "# $% single crystal polycrystal constraint effect 1.2.1. " 11 " 22 " 33 = 0 3.14) von Mises' condition FCC BCC HCP HCP FCC BCC 32
Tresca Taylor multiple slip y y " y = # y M 3.15) Taylor factor texture FCC M = 3.06 BCC M = 2.0 33
L0 L "#$%&')*,-. c /#$0&')*,-. c L c c c = 0 Fm a " a b L = F m 3.16) Fm TL " a b L = 2 T L cos # c 2 3.17) a c Fm L L c c L L0 LL0 34
"#$% L0 L &"#$% " c = 0, F m 2 T L =1 3.18) 3.17 L = L0 " m # " OR = 2 T L b L 0 = µ b L 0 3.19) 2.250.5 Orowan stress dislocation Orowan loop L0 particle 2r Orowan loop Orowan mechanism 35
3.19 L0 " L 0 = $ 2% ' # f & 1 2 r " OR = µ b # 0.7 µ b f L 0 r 3.20) Fm/2TL) 1 r f 1/2 solution hardening F m s = µ b r " 3.21) " = r s # r m r m 3.22) rs rm F m m " µ b2 # 20 3.23) " = d lnµ ) dc 3.24) c 36
"#$%&' )#$*,-. r " b 2 3.25) 3.21 " 3.23 Fm " /20 3.21, 3.23 F m 2 T L <<1 a) Friedel limit b) Labusch limit ~L 2 "c L L0 ab a) # % " m = $ % & 2 F m ) 3 2 b 3 ' %* c, - / % µ. ) 1 2 0 2 µ 1 3 2 c 3.26) 1/2 a) Friedel limit3.26 b) 37
1 1 # " m = F m 4 c 2 w & 3 # F 4 % $ 8 b 7 = m c 2 w & 3 % T L ' $ 4 µ b 9 ' 3.27) w 5b Labusch limit 2/3 precipitation hardening) r f L 1 " L = T % L 2 " $ ' L 0 = µ b2 % $ ' # F m & # 2 F m & 1 2 L 0 3.28) Fm # L = r b & % $ " f ' 1 2 3.29) 2 a = µ " 3 # f r& % $ b ' 1 2 3.30) r f 1/2 3.193.20 38