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Analytical Equations
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ANALYTICAL EQUATIONS INTRODUCTION A N D S Y N O P S I S
T
here is much commonaUty in the principles underlying the different methods of thermal laser material processing. The temperature field generated during vaporization cutting is analogous to that found in keyhole welding - both involve lateral heat flow from a through-thickness energy source. The phase transformations responsible for surface hardening in ferrous alloys are the same as those that determine the heat affected zone hardness of a weld - only the thermal cycle is different. Physical processes are therefore presented here as modular components of heat flow and structural change. The modules can be used interchangeably, and applied across a range of processes. This appendix contains analytical solutions for the temperature fields around a variety of energy sources and the structural transformations that they induce in materials. Analytical equations describe expUcit relationships between the process variables. Thus the effects of changes in process variables on the structure and properties of materials can be estimated quickly, or, if a particular result is desired, combinations of processing parameters can be assessed rapidly. The notation used is defined in Table E.l.
Table E.1
Notation
Symbol
Definition
Units
A A
Absorptivity Constant of precipitate solubility product Beam area Constant of precipitate solubility product Carbon equivalent Concentration of metal species of precipitate MaCb in solution Concentration of non-metal species of precipitate MaCb in solution Diffusion coefficient, D = DQ exp —(Q/RT) Pre-exponential of diffusion coefficient Beam power density Heat flux Vickers hardness number Vickers hardness number of bainite
m^ K wt% wt%
AB
B ^eq
^m
Cc D Do E F H
m
wt% m^s-i m^s-i Js-im-2 Jm-^s"^ VPN VPN (Contd)
526
Laser Processing of Engineering Materials
Table E.1
(Contd)
Symbol
Definition
Units
Hjp Hm I
Vickers hardness number of ferrite-pearlite mixture Vickers hardness number of martensite Kinetic strength of a thermal cycle Volumetric latent heat of melting Normalized latent heat of melting, L^^=Lm/[pc{Tm-To)] Volumetric latent heat of vaporization Normalized latent heat of vaporization, ^=Ly/[pc{Tm-To)] Martensite finish temperature Martensite start temperature Activation energy Gas constant, 8.314 Temperature Initial temperature Temperature at which pearlite transforms to austenite on heating Temperature at which ferrite transforms to austenite on heating Temperature at which martensite starts to form on cooling Temperature at which martensite formation is 50% complete on cooling Temperature at which martensite formation is complete on cooling Melting temperature Dissolution temperature of a precipitate Vaporization temperature Volume fraction Volume fraction of bainite Volume fraction of ferrite-pearlite mixture Volume fraction of martensite Cooling rate at 923 K Thermal diffusivity, A/(pc) Specific heat capacity Plate thickness Base of natural logarithms, 2.718 Matrix volume fraction available for precipitate dissolution Grain size Original grain size Kinetic constant Depth of treatment Precipitate size Original precipitate size
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Jm"^ —
Jm"^ — K K Jmol-i Jmol-^K-i K K K K K K K K K K Kh-i m^s-i Jkg-iK-i m — m m Various m m m
(Contd)
Appendix E: Analytical equations Table E.1
527
(Contd)
Symbol
Definition
q qmax qy r r
Js~^ J s~^ J s~^ m~^ m m
t to tp V w z ZQ Zt At2/i At8/5 At^^
Beam power Peak beam power in a Gaussian distribution Volumetric rate of heat generation Radial distance from centre of a surface heat source Lateral distance from centre of a through-thickness heat source Beam radius defined where q — qmaxl^y or beam half-width Time Time for heat to diffuse over beam radius, r|/(4a) Time taken to attain peak temperature Beam traverse rate Width (spacing of two isotherms) Depth Model displacement of workpiece surface Thermal penetration depth Time to cool from T2 to Ti Time to cool from 800 to 500°C Characteristic Ats/5 for 100% martensite formation
At^
Characteristic Ats/5 for 50% martensite formation
s
TB
Units
ni s s s ms~^ m m m m s s s
At^
Characteristic Ats/5 for 0% martensite formation
s
At?
Characteristic Ats/5 ior 0% ferrite formation
s
Atp
Characteristic Ats/s for 0% pearlite formation
s
Aty^
Characteristic Ats/5 for 50% bainite formation
s
At^ A, p r
Characteristic Ats/5 for 0% bainite formation Thermal conductivity Density Beam interaction time
s Js~^ m""^ K~^ kgm~^ s
El
EQUATIONS OF HEAT FLOW
Heat flow in laser processing can be complex. However, for many processes it may be approximated to three ftindamental conditions: steady state, transient, or quasi-steady state. Fourier's first law describes steady state conditions: F = -XVT
(ELI)
where F is the heat flux (Jm~^s~^), VT is the thermal gradient (Km~^), and k is the thermal conductivity (Js~^ m~^ K~^). In this state, the temperature field does not change with time at a location in the material. Fourier's second law describes transient conditions: 3L = ^_aW^T pc dt
(E1.2)
528
Laser Processing of Engineering Materials
-^r Figure El.l Schematic illustration of limiting geometries of heatflow:(a) radial from a point source, (b) lateral from a line source; note that a single temperature profile is used for clarity - in practice the peak temperature varies as 1/r^ and 1/r for radial and lateral heatflow,respectively where q^ is the energy generation per unit time and volume (J s ^ m ^), p is the density (kg m ^),cis the specific heat capacity (Jkg~^ K~^), T is the temperature (K), a is the thermal diffiisivity (m^ s~^), and t is the time (s). In this state, a thermal cycle is experienced at a location in the material. Quasi-steady state heatflovyrdescribes a condition in which the temperature field observed from a moving energy source remains constant. The variable ^ =x — vt is defined, where ^ is the distance of the point of interest from the source, measured along the x axis (the direction of source motion) and V is velocity. If no energy is generated, equation (El.2) becomes d^T d^T d^T _j_ _l_ a|2 ^ 3^2 ^ 9^2
—
vdT ad^
(E1.3)
In a state of quasi-steady state heat flow, transient effects can be ignored, which enables analytical expressions for temperature fields to be derived. In addition, heat flow in most laser-based processes may be considered to be limited to one of two types of source: a surface source, from which heat is conducted radially into the material (three-dimensional or *thick plate' heat flow); or a through-thickness source, from which heat is conducted laterally into the material (two-dimensional or *thin plate' heat flow). Radial and lateral heat flow are illustrated schematically for a structural steel in Figs El. 1(a) and (b), respectively. Analytical solutions to equations (E1.1)-(E1.3) can be derived by making assumptions, such as: the material is homogeneous and isotropic; heat flow occurs exclusively by conduction; and no energy is generated by material transformations. Under conditions typical of thermally induced change during laser processing, these are justifiable for the level of accuracy sought in this treatment (±5%).
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Laser Processing of Engineering Materials
E3
TEMPERATURE FIELD A R O U N D A THROUGH-THICKNESS ENERGY SOURCE
Table E3.1
Analytical equations for the temperature field around a through-thickness energy source
Type
Shape
Equation
Eqn no.
Source
Stationary
Line
Aq 27tk (Ti - T2) ~d ''
E3.1
(Klemens, 1976)
E3.2
(Rosenthal, 1941; Ashby and Easterling, 1982)
•"(^) Moving
Line
Aq T(r,t) — To =
1/2
1
dT _ dt
(vd^ \Aq)
Tp-To
^Aq( \7Te) vd \2neJ
r^
1 .r A{7tkpcy/^^
T^3/2 ^^
E3.3
1/2
E3.4
pa per 2
E3.5
E3.6
47rXpce {Tp - ro)2 Atl9| e(rp - ToV
T-To
^2
la
E3.7
exp
1 1 where — = • ^""^^^el {T,-To)'
1 {T2-T0)'
and T2 > Ti 1/2
E3.8
Ine) vd \27t 1 pc {Tp,-To)
(Tp,-To)
Appendix E: Analytical equations
E4
533
KINETIC EFFECT OF A THERMAL CYCLE A N D DIFFUSION-CONTROLLED STRUCTURAL CHANGES
Table E4.1 Analytical equations for the kinetic effect of a thermal cycle and diffusion-controlled structural changes Equation Kinetic effect
I = atptxp-
—
Eqn no.
Source
E4.1
(Ion eta/., 1984)
E4.2
(Avrami, 1939; Johnson and Mehl, 1939)
E4.3
(Lifshitz and Slyozov, 1961; Wagner, 1961)
E4.4
(Ion eta/., 1984)
where Aq 1 1 Q V lirXe Tp — TQ for radial heat flow, and iTtRTty
enidtt)
a =
a=2
[WRJ^
/AqV
1 AnXpce {Tp — TQY
for lateral heat flow 9
9
Q
Grain growth
g'-g^
Precipitate coarsening
p' - PQ = -y- ^^P " ^
Precipitate dissolution
Ts =
= katpexp-
—
^ P
B A-log
/'
where I 3/2
/ = 1 — exp
oitp
_ Q
1 Tp
1 T;
and starred variables denote calibration conditions
534
E5
Laser Processing of Engineering Materials
P H A S E T R A N S F O R M A T I O N S A N D H A R D N E S S IN CARBON-MANGANESE STEELS
Because of their importance in engineering, analytical equations describing phase transformations and hardness in plain carbon-manganese steels are provided in Table E5.1. Figure E5.1 illustrates the critical cooling times for phase formation. Note that the carbon equivalent Ceq used here to calculate critical cooling times is that defined in equation (E5.7). Table E5.1 in wt%
Empirical and theoretical equations for phase transformations in steels: alloying additions
Phase transformation temperatures
Carbon equivalent Critical cooling times
Phase volume fractions Phase hardness
Average hardness
Equation
Eqn no.
Source
= 996 - 30Ni - 25Mn - 5Co + 25Si + 30A1 + 25MO + 50V TAC3(K) = 1183 --416C + 228C2 -40Cr - 50Mn - 40Ni + 800P + 60V + 130Mo + 50W + 50S T^(K) = 1810-90C TMSCK) = 812 - 423C - 30.4Mn -17.7Ni-12.lCr-7.5Mo
E5.1
(Andrews, 1965)
E5.2
(Kumar, 1968)
E5.3 E5.4
(Hansen, 1958) (Andrews, 1965)
TM50 (K) = TMS — 47
E5.5
TMf (K) = T M S - 2 1 5
E5.6
Q^ = C + Mn/12 +Si/24
E5.7
At^ = exp (17.724Ceg - 2.926) At^ = exp {l9.954Ceq - 3.944) At^ = exp(16.929Ce^ + 1.453) At^o = exp(ln(At^o • Atp/2) y^=exp{ln(0.5).(At/A40)2} n = exp{ln(0.5) • (At/At'^^f) - Vm
E5.8 E5.9
(Steven and Haynes, 1956) (Steven and Haynes, 1956) (Inagaki and Sekiguchi, 1960) (Ion eta/., 1984) (Ion eta/., 1984)
TACI (K)
Vfp = l-{Vm + Vb) Hm = 295-{-515 Ceq Hb = 223-\-U7 Ceq Hjp = UO-\-139 Ceq Hmax = VmHm + VyHy + VfpHjp
E5.10 E5.11 E5.12 E5.13 E5.14 E5.15 E5.16 E5.17 E5.18
(Ion eta/., (Ion eta/., (Ion eta/., (Ion eta/.,
1984) 1984) 1984) 1984)
(Ion eta/., (Ion eta/., (Ion eta/., (Ion eta/.,
1984) 1984) 1984) 1984)
Appendix E: Analytical equations
Cooling time 800-500°C, At (s)
At? 1
•w—•*-~....^—w
—
*^ ^ \ ^N
Bainite V ^
\
\\ \ \ \ \ ^ ^ cu
535
Martensite
\
\
\ \
^'"^''
\\\ \ \ \ \
\ \
\
^
\ v
\
\.
\ Pearlite
\
\ Hypoeutectoid steel
\
-^^=#
AfO
^^^^
^^
Hardness
-^^«
Afg
Applied energy (J mm~2)
Figure E5.1 Critical cooling times for the formation of various microstructures in hypoeutectoid carbonmanganese steels, with a schematic hardness variation superimposed (broken line)
BIBLIOGRAPHY Andrews, K.W. (1965). Empirical formulae for the calculation of some transformation temperatures Journal of the Iron and Steel Institute, 203, 721-727. Ashby, M.F. and Easterling, K.E. (1982). A first report on diagrams for grain growth in welds. Acta Metallurgical 30, (11), 1969-1978. Ashby M.R and Easterling, K.E. (1984). The transformation hardening of steel surfaces by laser beams - 1. Hypoeutectoid steels. Acta Metallurgical 32,(11), 1935-1948. Avrami, M. (1939). Kinetics of phase change I. Journal of Chemical Physics, 7,1103-1112. Bass, M. (1983a). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 83. Bass, M. (1983b). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 86. Bass, M. (1983c). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 87. Carslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids. 2nd ed. Oxford: Clarendon Press.
Hansen, M. (1958). Constitution of Binary Alloys. New York: McGraw-HiU. Inagaki, M. and Sekiguchi, H. (1960). Continuous cooling transformation diagrams of steels for welding and their applications. Transactions of the National Institute for Metals, Japan, 2, 102-125. Ion, J.C, Easterling, K.E. and Ashby M.R (1984). A second report on diagrams of microstructure and hardness for heat-affected zones in welds. Acta Metallurgical 32, (11), 1949-1962. Johnson, W.A. and Mehl, R.R (1939). Reaction kinetics in processes of nucleation and growth. Transactions of the American Institute of Mining and Metallurgical Engineering, Iron Steel Division, 135, 416-458. Klemens, RG. (1976). Heat balance and flow conditions for electron beam and laser welding. Journal ofApplied Physics, 47, (5), 2165-2X7 A. Kumar, R. (1968). Physical Metallurgy of Iron and Steel. London: Asia. Lifshitz, I.M. and Slyozov, V.V. (1961). The kinetics of precipitation from supersaturated solid solutions. Journal of the Physics and Chemistry of Solids, 19, 35-50. Ready J.R (1971). Effects of High-Power Laser Radiation. New York: Academic Press.
536
Laser Processing of Engineering Materials
Rosenthal, D. (1941). Mathematical theory of heat Steven, W. and Haynes, A.G. (1956). The temperature distribution during welding and cutting. Welding of formation of martensite and bainite in low-alloy steels. Journal of the Iron and Steel Institute, 183, Journal 20,220s-234s. 349-359. Rosenthal, D. (1946). The theory of moving sources of heat and its application to metal treatments. Wagner, C. (1961). Theory of precipitate ageing Transactions of the American Society of Mechanical during re-solution (Ostwald ripening). Zeitschrift Engineers, 68,849-866. der ElektrochemiCy 65, 581-591. (In German.) Rykalin, N., Uglov, A. and Kokora, A. (1978). Laser Machining and Welding. Oxford: Pergamon Press.
ANALYTICAL EQUATIONS INTRODUCTION A N D S Y N O P S I S
T
here is much commonaUty in the principles underlying the different methods of thermal laser material processing. The temperature field generated during vaporization cutting is analogous to that found in keyhole welding - both involve lateral heat flow from a through-thickness energy source. The phase transformations responsible for surface hardening in ferrous alloys are the same as those that determine the heat affected zone hardness of a weld - only the thermal cycle is different. Physical processes are therefore presented here as modular components of heat flow and structural change. The modules can be used interchangeably, and applied across a range of processes. This appendix contains analytical solutions for the temperature fields around a variety of energy sources and the structural transformations that they induce in materials. Analytical equations describe expUcit relationships between the process variables. Thus the effects of changes in process variables on the structure and properties of materials can be estimated quickly, or, if a particular result is desired, combinations of processing parameters can be assessed rapidly. The notation used is defined in Table E.l.
Table E.1
Notation
Symbol
Definition
Units
A A
Absorptivity Constant of precipitate solubility product Beam area Constant of precipitate solubility product Carbon equivalent Concentration of metal species of precipitate MaCb in solution Concentration of non-metal species of precipitate MaCb in solution Diffusion coefficient, D = DQ exp —(Q/RT) Pre-exponential of diffusion coefficient Beam power density Heat flux Vickers hardness number Vickers hardness number of bainite
m^ K wt% wt%
AB
B ^eq
^m
Cc D Do E F H
m
wt% m^s-i m^s-i Js-im-2 Jm-^s"^ VPN VPN (Contd)
526
Laser Processing of Engineering Materials
Table E.1
(Contd)
Symbol
Definition
Units
Hjp Hm I
Vickers hardness number of ferrite-pearlite mixture Vickers hardness number of martensite Kinetic strength of a thermal cycle Volumetric latent heat of melting Normalized latent heat of melting, L^^=Lm/[pc{Tm-To)] Volumetric latent heat of vaporization Normalized latent heat of vaporization, ^=Ly/[pc{Tm-To)] Martensite finish temperature Martensite start temperature Activation energy Gas constant, 8.314 Temperature Initial temperature Temperature at which pearlite transforms to austenite on heating Temperature at which ferrite transforms to austenite on heating Temperature at which martensite starts to form on cooling Temperature at which martensite formation is 50% complete on cooling Temperature at which martensite formation is complete on cooling Melting temperature Dissolution temperature of a precipitate Vaporization temperature Volume fraction Volume fraction of bainite Volume fraction of ferrite-pearlite mixture Volume fraction of martensite Cooling rate at 923 K Thermal diffusivity, A/(pc) Specific heat capacity Plate thickness Base of natural logarithms, 2.718 Matrix volume fraction available for precipitate dissolution Grain size Original grain size Kinetic constant Depth of treatment Precipitate size Original precipitate size
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Jm"^ —
Jm"^ — K K Jmol-i Jmol-^K-i K K K K K K K K K K Kh-i m^s-i Jkg-iK-i m — m m Various m m m
(Contd)
Appendix E: Analytical equations Table E.1
527
(Contd)
Symbol
Definition
q qmax qy r r
Js~^ J s~^ J s~^ m~^ m m
t to tp V w z ZQ Zt At2/i At8/5 At^^
Beam power Peak beam power in a Gaussian distribution Volumetric rate of heat generation Radial distance from centre of a surface heat source Lateral distance from centre of a through-thickness heat source Beam radius defined where q — qmaxl^y or beam half-width Time Time for heat to diffuse over beam radius, r|/(4a) Time taken to attain peak temperature Beam traverse rate Width (spacing of two isotherms) Depth Model displacement of workpiece surface Thermal penetration depth Time to cool from T2 to Ti Time to cool from 800 to 500°C Characteristic Ats/5 for 100% martensite formation
At^
Characteristic Ats/5 for 50% martensite formation
s
TB
Units
ni s s s ms~^ m m m m s s s
At^
Characteristic Ats/5 for 0% martensite formation
s
At?
Characteristic Ats/5 ior 0% ferrite formation
s
Atp
Characteristic Ats/s for 0% pearlite formation
s
Aty^
Characteristic Ats/5 for 50% bainite formation
s
At^ A, p r
Characteristic Ats/5 for 0% bainite formation Thermal conductivity Density Beam interaction time
s Js~^ m""^ K~^ kgm~^ s
El
EQUATIONS OF HEAT FLOW
Heat flow in laser processing can be complex. However, for many processes it may be approximated to three ftindamental conditions: steady state, transient, or quasi-steady state. Fourier's first law describes steady state conditions: F = -XVT
(ELI)
where F is the heat flux (Jm~^s~^), VT is the thermal gradient (Km~^), and k is the thermal conductivity (Js~^ m~^ K~^). In this state, the temperature field does not change with time at a location in the material. Fourier's second law describes transient conditions: 3L = ^_aW^T pc dt
(E1.2)
528
Laser Processing of Engineering Materials
-^r Figure El.l Schematic illustration of limiting geometries of heatflow:(a) radial from a point source, (b) lateral from a line source; note that a single temperature profile is used for clarity - in practice the peak temperature varies as 1/r^ and 1/r for radial and lateral heatflow,respectively where q^ is the energy generation per unit time and volume (J s ^ m ^), p is the density (kg m ^),cis the specific heat capacity (Jkg~^ K~^), T is the temperature (K), a is the thermal diffiisivity (m^ s~^), and t is the time (s). In this state, a thermal cycle is experienced at a location in the material. Quasi-steady state heatflovyrdescribes a condition in which the temperature field observed from a moving energy source remains constant. The variable ^ =x — vt is defined, where ^ is the distance of the point of interest from the source, measured along the x axis (the direction of source motion) and V is velocity. If no energy is generated, equation (El.2) becomes d^T d^T d^T _j_ _l_ a|2 ^ 3^2 ^ 9^2
—
vdT ad^
(E1.3)
In a state of quasi-steady state heat flow, transient effects can be ignored, which enables analytical expressions for temperature fields to be derived. In addition, heat flow in most laser-based processes may be considered to be limited to one of two types of source: a surface source, from which heat is conducted radially into the material (three-dimensional or *thick plate' heat flow); or a through-thickness source, from which heat is conducted laterally into the material (two-dimensional or *thin plate' heat flow). Radial and lateral heat flow are illustrated schematically for a structural steel in Figs El. 1(a) and (b), respectively. Analytical solutions to equations (E1.1)-(E1.3) can be derived by making assumptions, such as: the material is homogeneous and isotropic; heat flow occurs exclusively by conduction; and no energy is generated by material transformations. Under conditions typical of thermally induced change during laser processing, these are justifiable for the level of accuracy sought in this treatment (±5%).
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532
Laser Processing of Engineering Materials
E3
TEMPERATURE FIELD A R O U N D A THROUGH-THICKNESS ENERGY SOURCE
Table E3.1
Analytical equations for the temperature field around a through-thickness energy source
Type
Shape
Equation
Eqn no.
Source
Stationary
Line
Aq 27tk (Ti - T2) ~d ''
E3.1
(Klemens, 1976)
E3.2
(Rosenthal, 1941; Ashby and Easterling, 1982)
•"(^) Moving
Line
Aq T(r,t) — To =
1/2
1
dT _ dt
(vd^ \Aq)
Tp-To
^Aq( \7Te) vd \2neJ
r^
1 .r A{7tkpcy/^^
T^3/2 ^^
E3.3
1/2
E3.4
pa per 2
E3.5
E3.6
47rXpce {Tp - ro)2 Atl9| e(rp - ToV
T-To
^2
la
E3.7
exp
1 1 where — = • ^""^^^el {T,-To)'
1 {T2-T0)'
and T2 > Ti 1/2
E3.8
Ine) vd \27t 1 pc {Tp,-To)
(Tp,-To)
Appendix E: Analytical equations
E4
533
KINETIC EFFECT OF A THERMAL CYCLE A N D DIFFUSION-CONTROLLED STRUCTURAL CHANGES
Table E4.1 Analytical equations for the kinetic effect of a thermal cycle and diffusion-controlled structural changes Equation Kinetic effect
I = atptxp-
—
Eqn no.
Source
E4.1
(Ion eta/., 1984)
E4.2
(Avrami, 1939; Johnson and Mehl, 1939)
E4.3
(Lifshitz and Slyozov, 1961; Wagner, 1961)
E4.4
(Ion eta/., 1984)
where Aq 1 1 Q V lirXe Tp — TQ for radial heat flow, and iTtRTty
enidtt)
a =
a=2
[WRJ^
/AqV
1 AnXpce {Tp — TQY
for lateral heat flow 9
9
Q
Grain growth
g'-g^
Precipitate coarsening
p' - PQ = -y- ^^P " ^
Precipitate dissolution
Ts =
= katpexp-
—
^ P
B A-log
/'
where I 3/2
/ = 1 — exp
oitp
_ Q
1 Tp
1 T;
and starred variables denote calibration conditions
534
E5
Laser Processing of Engineering Materials
P H A S E T R A N S F O R M A T I O N S A N D H A R D N E S S IN CARBON-MANGANESE STEELS
Because of their importance in engineering, analytical equations describing phase transformations and hardness in plain carbon-manganese steels are provided in Table E5.1. Figure E5.1 illustrates the critical cooling times for phase formation. Note that the carbon equivalent Ceq used here to calculate critical cooling times is that defined in equation (E5.7). Table E5.1 in wt%
Empirical and theoretical equations for phase transformations in steels: alloying additions
Phase transformation temperatures
Carbon equivalent Critical cooling times
Phase volume fractions Phase hardness
Average hardness
Equation
Eqn no.
Source
= 996 - 30Ni - 25Mn - 5Co + 25Si + 30A1 + 25MO + 50V TAC3(K) = 1183 --416C + 228C2 -40Cr - 50Mn - 40Ni + 800P + 60V + 130Mo + 50W + 50S T^(K) = 1810-90C TMSCK) = 812 - 423C - 30.4Mn -17.7Ni-12.lCr-7.5Mo
E5.1
(Andrews, 1965)
E5.2
(Kumar, 1968)
E5.3 E5.4
(Hansen, 1958) (Andrews, 1965)
TM50 (K) = TMS — 47
E5.5
TMf (K) = T M S - 2 1 5
E5.6
Q^ = C + Mn/12 +Si/24
E5.7
At^ = exp (17.724Ceg - 2.926) At^ = exp {l9.954Ceq - 3.944) At^ = exp(16.929Ce^ + 1.453) At^o = exp(ln(At^o • Atp/2) y^=exp{ln(0.5).(At/A40)2} n = exp{ln(0.5) • (At/At'^^f) - Vm
E5.8 E5.9
(Steven and Haynes, 1956) (Steven and Haynes, 1956) (Inagaki and Sekiguchi, 1960) (Ion eta/., 1984) (Ion eta/., 1984)
TACI (K)
Vfp = l-{Vm + Vb) Hm = 295-{-515 Ceq Hb = 223-\-U7 Ceq Hjp = UO-\-139 Ceq Hmax = VmHm + VyHy + VfpHjp
E5.10 E5.11 E5.12 E5.13 E5.14 E5.15 E5.16 E5.17 E5.18
(Ion eta/., (Ion eta/., (Ion eta/., (Ion eta/.,
1984) 1984) 1984) 1984)
(Ion eta/., (Ion eta/., (Ion eta/., (Ion eta/.,
1984) 1984) 1984) 1984)
Appendix E: Analytical equations
Cooling time 800-500°C, At (s)
At? 1
•w—•*-~....^—w
—
*^ ^ \ ^N
Bainite V ^
\
\\ \ \ \ \ ^ ^ cu
535
Martensite
\
\
\ \
^'"^''
\\\ \ \ \ \
\ \
\
^
\ v
\
\.
\ Pearlite
\
\ Hypoeutectoid steel
\
-^^=#
AfO
^^^^
^^
Hardness
-^^«
Afg
Applied energy (J mm~2)
Figure E5.1 Critical cooling times for the formation of various microstructures in hypoeutectoid carbonmanganese steels, with a schematic hardness variation superimposed (broken line)
BIBLIOGRAPHY Andrews, K.W. (1965). Empirical formulae for the calculation of some transformation temperatures Journal of the Iron and Steel Institute, 203, 721-727. Ashby, M.F. and Easterling, K.E. (1982). A first report on diagrams for grain growth in welds. Acta Metallurgical 30, (11), 1969-1978. Ashby M.R and Easterling, K.E. (1984). The transformation hardening of steel surfaces by laser beams - 1. Hypoeutectoid steels. Acta Metallurgical 32,(11), 1935-1948. Avrami, M. (1939). Kinetics of phase change I. Journal of Chemical Physics, 7,1103-1112. Bass, M. (1983a). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 83. Bass, M. (1983b). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 86. Bass, M. (1983c). Laser heating of solids. In: Bertolotti, M. ed. Physical Processes in Laser-material Interactions. New York: Plenum Press, p. 87. Carslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids. 2nd ed. Oxford: Clarendon Press.
Hansen, M. (1958). Constitution of Binary Alloys. New York: McGraw-HiU. Inagaki, M. and Sekiguchi, H. (1960). Continuous cooling transformation diagrams of steels for welding and their applications. Transactions of the National Institute for Metals, Japan, 2, 102-125. Ion, J.C, Easterling, K.E. and Ashby M.R (1984). A second report on diagrams of microstructure and hardness for heat-affected zones in welds. Acta Metallurgical 32, (11), 1949-1962. Johnson, W.A. and Mehl, R.R (1939). Reaction kinetics in processes of nucleation and growth. Transactions of the American Institute of Mining and Metallurgical Engineering, Iron Steel Division, 135, 416-458. Klemens, RG. (1976). Heat balance and flow conditions for electron beam and laser welding. Journal ofApplied Physics, 47, (5), 2165-2X7 A. Kumar, R. (1968). Physical Metallurgy of Iron and Steel. London: Asia. Lifshitz, I.M. and Slyozov, V.V. (1961). The kinetics of precipitation from supersaturated solid solutions. Journal of the Physics and Chemistry of Solids, 19, 35-50. Ready J.R (1971). Effects of High-Power Laser Radiation. New York: Academic Press.
536
Laser Processing of Engineering Materials
Rosenthal, D. (1941). Mathematical theory of heat Steven, W. and Haynes, A.G. (1956). The temperature distribution during welding and cutting. Welding of formation of martensite and bainite in low-alloy steels. Journal of the Iron and Steel Institute, 183, Journal 20,220s-234s. 349-359. Rosenthal, D. (1946). The theory of moving sources of heat and its application to metal treatments. Wagner, C. (1961). Theory of precipitate ageing Transactions of the American Society of Mechanical during re-solution (Ostwald ripening). Zeitschrift Engineers, 68,849-866. der ElektrochemiCy 65, 581-591. (In German.) Rykalin, N., Uglov, A. and Kokora, A. (1978). Laser Machining and Welding. Oxford: Pergamon Press.