ASM Metals HandBook Vol. 14 - Forming and Forging
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steels, similar increases have been observed at the low temperature range of 150 to 200 °C (300 to 390 °F) (Ref 16).
Therefore, Eq 19 has been found to be invalid in regions in which the increase in flow stress occurs.
Combined Effect of Temperature and Strain Rate
To correlate flow stress and strain rate at different forming temperatures, it has been proposed that (Ref 20):
= f ( e
[Q/(RT)]
= f(z)
(Eq 21)
where Q is the activation energy and R is the gas constant.
A slightly different approach to this problem has also been taken (Ref 21). It has been proposed that strain rate and
temperature be combined into velocity-modified temperature. The flow stress at a particular strain is then a function of
velocity-modified temperature T
Av
, which is defined as:
(Eq 22)
where k is the Boltzmann constant. Equation 22 has been verified for the case of steel and aluminum over a large
temperature range, but only for a small range of strain rates (Ref 20). Based on Eq 22, values of activation energy, Q,
independent of temperature have been obtained for a number of pure metals and simple alloys deformed at temperatures
greater than 0.5 T
m
(Ref 22).
However, this approach has been found to be invalid at lower temperatures and for more complex alloys, in which
precipitation processes can occur within the temperature range of interest. In addition, in high strain rate tests where
adiabatic conditions are approached, there is some ambiguity as to whether the initial temperature or a corrected
temperature should be used.
The combined effect of temperature and strain rate has been described through the use of the following exponential
relationship (Ref 23):
=
3
e
[-Q/(RT)]
(Eq 23)
Because Eq 23 does not correspond to any theoretical model, the parameters
3
and
3
do not have simple physical
interpretations. This difficulty has been overcome through the use of Eq 24 (Ref 24, 25). Equation 24 is based on the
thermally activated motion of dislocations over local and long-range barriers:
=
5
e
(Eq 24)
where is the internally developed back stress and V is the activation volume.
It has been proposed that, under steady-state conditions, Eq 24 can be written as (Ref 25):
=
5
e e
(Eq 25)
where V' = V( - 1/ ) is the apparent activation volume and = / . However, other researchers have related
experimental steady-state stress, strain rate, and temperature by using the following hyperbolic equation (Ref 26):
=
6
[sin h (
6
)]
1/m
e
[Q/(RT)]
(Eq 26)
In Eq 23, 24, 25, and 26,
3
to
6
are constants that depend on strain and temperature, and
3
to
6
are also constants.
With Eq 26, the average activation energy has been calculated on a computer for copper at somewhat lower strain rates.
One researcher has used Eq 26 to fit his experimental data for various steels, aluminum, and copper and has also
calculated activation energies (Ref 27).
For process modeling applications, continuous low-order polynomial equations are adequate because they generally help
obtain rapid convergence and yield good accuracy in finite-element modeling. For example, this method has been used to
describe the constitutive behavior of Ti-6Al-2Sn-4Zr-2Mo for both the transformed and the + microstructures (Ref
28). The polynomial representations describe flow softening and steady-state behavior with a high degree of accuracy.
References cited in this section
1. E.W. Hart, C.Y. Li, H. Yamada, and C.L. Wire, Phenomenological Theory: A Guide to Constituti
ve
Relations and Fundamental Deformation Properties, in Constitutive Equations in Plasticity,
A.S. Argon,
Ed., The MIT Press, 1975, p 149-197
2. R.J. Green, A Plasticity Theory for Porous Solids, Int. J. Mech. Sci., Vol 14, 1972, p 215
3. M. Oyane, S. Shima, and Y. Kono, Theory of Plasticity for Porous Metals, J. Soc. Mech. Eng.,
Vol 16,
1973, p 1254
4. S. Shima and M. Oyane, Plasticity Theory for Porous Metals, Int. J. Mech. Sci., Vol 18, 1976, p 285
5. H.A. Kuhn and G.L. Downey, J. Eng. Mater. Technol., 1973, p 41
6. S.M. Doraivelu et al., Int. J. Mech. Sci., Vol 26, 1984, p 527
7. J.E. Hockett, Trans. TMS-AIME, 1967, p 239
8. C. Crussard and B. Jaoul, J. Mech. Solids, Vol 5, 1956, p 95-114
9. S.M. Doraivelu, Ph.D. dissertation, 1979
10.
C.H. Sellars and W.J. McG. Tegart, Review 158, Int. Metall. Rev., Vol 17, 1972, p 1-24
11.
G. Dickenson, J. Iron Steel Inst., 1962, p 2-7
12.
J.M. Bechthold, Trans. ASM, Vol 65, 1954, p 1449-1469
13.
J.D. Hollomon and C. Zener, Trans. AIME, Vol 96, p 237-296
14.
J.H. Westbrook, Trans. ASM, Vol 45, 1953, p 221-248
15.
M. Grothe, Ph.D. dissertation, University of Berlin, 1969
16.
S.K. Samanta, Int. J. Mech. Sci., Vol 10, 1968, p 613-626
17.
H.W. Wagoner, Ph.D. dissertation, University of Lerbeck, 1965
18.
H.G. Hoptner, Ph.D. dissertation, University of Ramschield, 1967
19.
M.E. Backman and S.A. Finnegan, Metallurgical Effects at High Strain Rate, Plenum Press, 1976, p 531-
541
20.
C. Zener and J.H. Holloman, J. Appl. Phys., Vol 15, 1944, p 22-32
21.
C.W. MacGregor and J.C. Fisher, J. Appl. Phys., Vol 15, 1946, p 11-17
22.
J.J. Jonas and J.P. Immarigeon, Z. Metallkd., Vol 60, 1969, p 227-231
23.
T.A. Trozera, O.D. Sherby, and J.E. Dorn, Trans. ASM, Vol 49, 1957, p 173-188
24.
A. Seeger, Philos. Mag., Vol 46, 1955, p 1194
25.
J.J. Jonas, Acta Metall., Vol 17, 1969, p 3
26.
C.M. Sellars and W.J. McG. Tegart, Met. Sci. Rev., 1966, p 731-746
27.
S.K. Samanta, in Proceedings of the Eleventh International MTDR Conference
(Birmingham, AL), Vol B,
1970, p 827-883
28.
P. Dadras and J.F. Thomas, Jr., Characterization and Modeling for Forging Deformation of Ti-6242,
Metall.
Trans. A, Vol 12A, 1967, 1981
Modeling Techniques Used in Forging Process Design
H. L. Gegel and J.C. Malas, Air Force Wright Aeronautical Laboratories/Materials Laboratory; S.M. Doraivelu and V.A. Shende,
Universal Energy Systems, Knowledge Integration Center
Deformation Mechanisms and Deformation Maps
Deformation Mechanisms. The activation analysis discussed in Ref 29 has been used to obtain quantitative data on
the deformation mechanisms that take place in metals and alloys. According to Ref 29, the activation enthalpy H is
given by:
(Eq 27)
and the activation volume V is given by:
(Eq 28)
Substituting Eq 28 in Eq 27 gives:
(Eq 29)
Because Eq 27, 28, and 29 are used for creep and interrupted tests, they are modified for continuous compression tests as
follows:
(Eq 30)
(Eq 31)
The experimental activation energy values have been calculated for various metals and alloys using several independent
flow stress-true strain curves rather than the more often used single-interruption test, which accounts for sudden changes
in strain rate and temperature. One researcher, while calculating the activation energy of Fe-30Cr alloy steel, has shown
that correct values of ( / ) or ( / T) can also be obtained from flow stress measurements made on continuous tests
at varying temperatures and strain rates (Ref 30). Therefore, there is justification for the use of flow stress-true strain
curves obtained from continuous tests for the calculation of experimental activation energies. These experimental values
are compared with the available creep and self-diffusion data to predict the deformation mechanism based on available
dislocation models. The activation energy values for various materials under self-diffusion, creep, and hot-working
conditions are available in the literature.
When the structure changes because of the occurrence of other mechanisms along with the thermally activated diffusion
or recovery or recrystallization mechanism, the activation energy given by Eq 27 is modified as follows:
(Eq 32)
In such cases (for example, copper), it would be difficult to predict the mechanism from the existing dislocation models
alone. Other mechanisms have lower activation energies, so their possible frequency of occurrence is higher than the
frequency demanded by the thermally activated process (Ref 31). It can therefore be concluded that compression, like
creep, is also a thermally activated process, despite the difference of order in strain rate values (Ref 31).
Rate-controlling dislocation mechanisms (see below) are often identified from the magnitude of the measured activation
parameters and their variation with temperature. Details on the effects of stress and temperature on the activation
parameters are available in the literature. Some of the important mechanisms include:
• Overcoming P-N stress
• Intersection of dislocation
• Nonconservative motion of jogs
• Cross slip of screw dislocation
• Climb of edge dislocations
• Recovery and recrystallization
The deformation mechanisms that control the plastic behavior of various materials at different temperatures and strain
rates have been identified by calculating the values of activation energy Q and activation volume V and comparing them
with those of the above mechanisms.
Deformation Maps. During the past two decades, various investigators have extensively researched and reviewed the
effects of strain, strain rate, temperature, and microstructure on the flow behavior of metals during deformation
processing (Ref 1, 10, 32, 33). More recently, attempts have been made by H.J. Frost and M.F. Ashby (Ref 34) and R. Raj
(Ref 35) to describe the deformation and fracture processes that occur during deformation processing. Both approaches
are deterministic in the sense that shear strain rate equations (valid for steady state) are written, assuming that the
equations are dependent on a number of basic atomic processes, such as dislocation motion, diffusion, grain-boundary
sliding, twinning, and phase transformation. These approaches, as well as an extension of the work of Raj, are briefly
described below.
Ashby-Frost Deformation Maps. Deformation maps of normalized stress versus absolute temperature for any
polycrystalline material can theoretically be constructed, showing the area of dominance of each flow mechanism. Each
flow mechanism must have an equation relating to shear strain rate , shear stress , absolute temperature T (°K), and
structure. The term structure includes all parameters describing the atomic structure, such as bonding, crystal class, defect
structure, grain size, dislocation density and arrangement, solute concentration, and volume fraction of second-phase
particles. Maps of this type are limited to pure polycrystalline materials, simple alloys, and steady-state conditions. A
typical example is shown in Fig. 3.
Fig. 3 Example of an Ashby-Frost deformation map
Damage Nucleation Maps. R. Raj extended the deformation map concept to a processing map that represents the
nucleation of damage as a function of temperature T and effective strain rate (Ref 35). A Raj processing map, such as
that shown in Fig. 4, is a composite processing map based on various damage mechanisms. The processing map is very
useful in that it defines regions in which it is safe to process the workpiece material and avoid defect nucleation. The
boundaries are the loci of bifurcation points, where the differential equation takes on a new solution or where the flow
process changes from a stable to an unstable process.
Fig. 4 Example of a Raj damage nucleation deformation map
Like the Ashby-Frost deformation maps, the Raj processing map is highly idealized. Neither approach to describing the
workability of materials is suitable for complex engineering alloys, because these models consider only the deterministic
aspect of predicting the behavior of a workpiece material under processing conditions. Experience in designing bulk
forming processes clearly indicates that predicting the future behavior of a material involves not only deterministic
aspects but also statistical and indeterministic features.
Rao-Raj Deformation Maps. K.P. Rao superimposed room-temperature ductility data for specimens deformed under
various elevated-temperature and strain rate combinations onto a Raj damage nucleation map (Ref 36). Rao observed that
low ductility values are located in the region where the Raj map predicts wedge cracking, which is a form of plastic
instability. Peak values of ductility were found in the middle of the safe region. This behavior is shown in Fig. 5.
Fig. 5 Raj damage nucleation map with superimposed room-
temperature ductility data for specimens
previously deformed under various temperature-strain ratio conditions
References cited in this section
1. E.W. Hart, C.Y. Li, H. Yamada, and C.L. Wire, Phenomenological Theory: A Guide to Const
itutive
Relations and Fundamental Deformation Properties, in Constitutive Equations in Plasticity,
A.S. Argon,
Ed., The MIT Press, 1975, p 149-197
10.
C.H. Sellars and W.J. McG. Tegart, Review 158, Int. Metall. Rev., Vol 17, 1972, p 1-24
29.
H. Conrad and H. Wiedersich, Acta Metall., Vol 8, 1960, p 128-130
30.
R. Langeburg, Acta Metall., Vol 15, 1967, p 1737-1745
31.
H.J. McQueen, J. Met., Vol 20, 1968, p 31-38
32.
J.J. Jonas, C.M. Sellars, and W.J. McG. Tegart, Strength and Structure Under Hot Working
Conditions,
Met. Rev., 1969, p 1
33.
W.J. McG. Tegart, The Role of Ductility in Hot Working, in Ductility,
American Society for Metals, 1968,
p 133
34.
H.J. Frost and M.F. Ashby, Deformation Mechanism Maps, Pergamon Press, 1982
35.
R. Raj, Development of a Processing Map for Use in Warm-Forming and Hot-Forming Process,
Metall.
Trans. A, Vol A12, 1981, p 1089
36.
K.P. Rao, S.M. Doraivelu, H.M. Roshan, and Y.V.R.K. Prasad, Deformation Processing of an Aluminum
Alloy Containing Particles: Studies on Al-5Si Alloy (4043), Metall. Trans. A, Vol 14A, 1983, p 1671
Modeling Techniques Used in Forging Process Design
H. L. Gegel and J.C. Malas, Air Force Wright Aeronautical Laboratories/Materials Laboratory; S.M. Doraivelu and V.A. Shende,
Universal Energy Systems, Knowledge Integration Center
Dynamic Material Modeling
Control of microstructure during any material-processing operation requires answers to two fundamental questions:
• What conditions does the process require the workpiece material to withstand?
• How does the workpiece material respond to the demands imposed by the process?
The first question can be answered by modeling the process with finite-element methods. The second question, which
deals with how the workpiece material responds to the demands of the process, can be answered only if the flow behavior
of the workpiece material has been adequately characterized under processing conditions. This necessitates an
understanding of how the workpiece will dissipate the instantaneous power applied to it by the process.
For example, during the forging process, the press provides instantaneous power to the workpiece in the amount described
by:
=
1 1
+
2 2
+
3 3
(Eq 33)
where is the effective stress, is the effective strain rate, and the terms on the right-hand side are the products of
principal stresses and principal strain rates.
The workpiece will dissipate the instantaneous power applied by a metallurgical process commensurate with the level of
power supplied. For example, when energy is supplied at a very high strain rate, the material dissipates it by fracture
processes. On the other hand, when energy is supplied at a lower and more controlled rate, the material dissipates it by
superplastic flow if the material has the correct microstructure and is deformed under superplastic conditions. The
efficiency of energy dissipation of these metallurgical processes in both cases may be the same, but the variation of
efficiency with respect to strain rate may not be favorable for achieving a steady-state condition.
The applied power also sets up an entropy production rate by a material system. This rate is controlled by the second law
of thermodynamics and is directly related to grain size. The rate of entropy production by the material system reaches a
maximum level when the material system has the potential to develop very fine grain structure. The rate of entropy
production begins decreasing when grain growth takes place. To study this phenomenon, a parameter known as entropy
rate ratio s has been defined by the dynamic material modeling approach (Ref 37, 38). Therefore, to control
microstructures and to avoid defect formation, a new type of material behavior model termed a dynamic material model is
required in order to understand how the workpiece dissipates power while producing entropy to satisfy the demands of the
process. This model is capable of producing information consistent with the unifying aspects of the finite-element model
in such a way that it can become a nonholonomic constraint in the finite-element model for obtaining optimal solutions
during the design of various unit processes.
Basic Concepts
According to the dynamic material models described in Ref 37 and 38, power P (rate of work done) is dissipated
coherently by partitioning it into J, whose energy primitive is potential energy, and G, which is heat. Kinetic energy K is
the primitive for G. The partitioning is given by:
(Eq 34)
Power dissipated by plastic work is denoted by the area G under the curve, which is illustrated schematically in Fig. 6.
This is designated as the dissipator content. Area J, above the curve, is designated as the dissipator co-content. The
fundamental assumption in Ref 37 and 38 is that J is related to structural changes and that G is related to continuum
effects. Plastic instability and fracture processes are associated with G, and microstructure evolution is associated with J.
Both J and G are complementary functions, and they are related by a Legendre transformation (Ref 39).
Fig. 6 Schematic showing the constitutive relation of the material system a
s an energy dissipator during
forming. See text for details.
It follows from Eq 34 that the partitioning of power between J and G is given by:
(Eq 35)
This ratio is generally known as the strain rate sensitivity parameter, m, of the workpiece material. Using this parameter,
the dynamic constitutive equation given below can be developed:
= k
m
(Eq 36)
Equation 36 is termed a dynamic constitutive equation because it is a result of integrating along the dynamic (actual) path
that an element of material takes according to the principle of least action, when it is instantaneously deformed at a
particular strain rate. The strain rate sensitivity is assumed to be constant along this trajectory up to that strain rate and at
the temperature and effective strain under consideration. The strain rate sensitivity parameter, calculated from continuous
flow stress values measured at different constant strain rate tests, has generally been observed to be independent of
temperature and strain rate for pure metals, but in engineering alloys it has been shown to vary with temperature and
strain rate:
(Eq 37)
(Eq 38)
Equations 37 and 38 have been obtained by integrating dJ = d and dG = d (from Eq 35). The upper limit of m = 1
fixes the maximum value of J to be 0.5P. By normalizing instantaneous J with this maximum value, an efficiency factor
can be defined as = J/J
max
= 2m/(m + 1). This dimensionless parameter becomes important for controlling the
dissipated power J and for formulating a Liapunov function, because the system reaches a maximum value of at the
lowest energy state during stable conditions. The Liapunov function is a system quantity associated with Liapunov
stability criteria and is regarded as a general and accepted method in engineering design (Ref 40). This function is an
arbitrary term that relates changes in the total energy of a given system. The Liapunov criteria for stability require the
system to lower its total energy continuously.
Therefore, the Liapunov function V
1
is formulated as shown below:
V
1
= (log )
(Eq 39)
Because itself forms the condition for stability, / (log ) should be less than 0 in the stable region. This condition
definitely ensures that the system is approaching the steady-state condition in which it experiences a minimum energy
state and maximum without fracture.
The next step is to identify a dimensionless control parameter in order to study the influence of temperature on material
behavior and workability. For this purpose, the total power dissipated is represented in terms of the rate of entropy applied
app
and temperature T as shown below, and a methodology has been established to determine the relationship between
the rate of entropy produced by the system
sys
, and the rate of entropy applied to the system
app
, using the following
steps:
P = =
app
T
(Eq 40)
and by definition:
(Eq 41)
Furthermore:
(Eq 42)
where:
(Eq 43)
define a new coefficient s such that:
(Eq 44)
The minus sign in Eq 42, 43, and 44 can be neglected and the value can be treated as a positive because flow stress
decreases with respect to T.