Materials Transactions, Vol. 54, No. 4 (2013) pp. 626 to 629
© 2013 The Japan Institute of Metals and Materials
RAPID PUBLICATION
Integration of Temperature, Stress State, and Strain Rate
for the Ductility of Ductile Metals
Hai Qiu
Research Center for Strategic Materials, National Institute for Materials Science, Tsukuba 305-0047, Japan
The ductility of ductile metals, evaluated in terms of fracture strain, is strongly affected by temperature, stress state and strain rate. A
parameter integrating the three parameters was proposed, which can be used as a single variable representing the fracture strain of ductile metals.
[doi:10.2320/matertrans.M2012389]
(Received December 27, 2012; Accepted January 11, 2013; Published March 1, 2013)
Keywords: ductility, triaxiality, strain rate, temperature, metallic materials
1.
Introduction
Temperature, stress state (stress triaxilality), and strain rate
are three most vital parameters which control the fracture of a
metal. Among them, temperature is a very familiar variable
whose effect on fracture has been widely investigated.
Dislocation theory is a powerful tool to explore this effect
in metallurgy. It is well known that decreasing temperature
lowers activation energy, and as a result inhibiting the
movement of dislocation, which usually increases strength
and decreases ductility.
Stress triaxiality has a strong negative impact on fracture.1­5)
An increment in stress triaxiality makes dislocation to move
difficult, playing a similar role as decreasing temperature.
Differing from temperature and stress triaxiality, the effect
of strain rate is complicated, sometimes showing positive
impact for some metals, and sometimes negative for other
metals.1,6,7) Bennett and Sinclair8) described strain rate, ¾_ ,
in the form
¾_ ¼ B expðHð¸Þ=ðRg T ÞÞ
ð1Þ
where B is the frequency factor, H(¸) the stress-modified
activation energy term, Rg the universal gas constant, T
absolute temperature. If temperature remains constant in
eq. (1), large ¾_ corresponds to small activation energy,
indicating an increment in strain rate will reduce the
activation energy and as a result inhibit the movement of
dislocation (usually increased yield strength is the evidence),
like the effect of dropping temperature. On the other hand,
at the higher strain rates, plastic work of deformation
transforms into heat, inducing adiabatic heating, i.e.,
adiabatic temperature rise.9­12) Therefore, the competition
of the two opposite effects determines the role of strain rate
in fracture process.
If we use fracture strain, ¾f, to represent the ductility of
metals, it will be a function of stress state, strain rate and
temperature. Johnson and Cook13) isolated the effects of
stress state, strain rate and temperature, and proposed an
empirical equation based on extensive database
¾f ¼ ½D1 þ D2 expðD3 · Þ½1 þ D4 ln ¾_ ½1 þ D5 T ð2Þ
The stress triaxiality ·* is defined as ·m/·eq where ·m is the
average of the three normal stresses and ·eq is the von Mises
equivalent stress. ¾_ and T* are normalized strain rate and
normalized temperature, respectively. D1­D5 are material
constants. Equation (2) is a function of three parameters.
As aforementioned, regardless of stress state, strain rate
and temperature, their effects on fracture can be ultimately
attributed to the dislocation characteristic from the viewpoint
of dislocation theory. This implies that the variation in
fracture behavior caused by stress state or strain rate can be
regarded to be equivalent to that by temperature. If we use
the “equivalent concept”, i.e., converting the effects of stress
state and strain rate into the equivalent amount induced by
temperature, we can combine the three parameters (stress
state, strain rate and temperature) into one parameter, denoted
as “equivalent temperature (Teq)”, and assume fracture strain
monotonically relates to the “equivalent temperature”. This
approach will simplify a function of three variables, for
example eq. (2), into a function of one variable, and the
“equivalent temperature” becomes the sole source inducing
the variation of fracture strain. In the present study, we make
an attempt to determine the “equivalent temperature”, and
using it to represent the fracture strain of ductile metals.
2.
Proposition of Teq
Prior to the proposition of “equivalent temperature Teq”,
individual effect of temperature, stress triaxiality and strain
rate on fracture strain is firstly clarified.
Equation (1) physically interprets the strain rate. Because
activation energy is usually independent of time, if integrating eq. (1) against time, the left term of eq. (1) becomes to
strain, and the right term remains the form of exponential
function. This deduction from eq. (1) indicates that fracture
strain should be related to temperature (T) in the form of
e¹1/T. The experimental fracture strain14­16) against temperature is shown Fig. 1(a). The correlation coefficient (R)
shows excellent coincidence between experimental data and
fitting curves, which strongly supports the above conclusion.
Ductile fracture in an engineering material usually occurs
in fibrous fracture mode, in which void is its characteristic.
Rice and Tracey17) investigated the growth of an initially
spherical void in an infinite, rigid and perfectly plastic
material, and found that stress triaxiality strongly influences
the void growth. Based on their research, Hancock and
Mackenzie18) proposed an equation descripting fracture strain
¾f in the form
Integration of Temperature, Stress State, and Strain Rate for the Ductility of Ductile Metals
(a)
627
(c)
(b)
Fig. 1 Dependence of fracture strain on (a) temperature, (b) stress triaxiality and (c) strain rate.
¾f ¼ ¾n þ ¡0 expð1:5 · m =· eq Þ
ð3Þ
where ¾n is the void nucleation strain, and ¡A is a constant.
Equation (3) demonstrates that ¾f is an exponential function
of ·m/·eq. Figure 1(b) verifies eq. (3). Many researches’
work1,3,15,16) also support it.
Figure 1(c) shows the experimental fracture strain6,7,19,20)
over a wide range of strain rate (10¹4­10+4 s¹1) for various
metals. Strain rate does not show the same tendency for all the
metals in Fig. 1(c) ® positive impact on brass and aluminum
alloy; negative effect on stainless steel and the heat-affected
zone of SN490 steel; almost no impact on SN490 steel.
Higher strain rate usually produces two phenomena: increased strength and adiabatic temperature rise. Increased strength
lowers ductility while adiabatic temperature rise plays an
opposite role. The competition of the two opposite effects
determines the effect of strain rate. When adiabatic heating
is dominant, fracture strain will be enhanced, otherwise
strain rate shows negative impact or no effect on ductility.
Therefore, various strain rate-dependences exist in metals.
According to the exponential relationship between fracture
strain and temperature described above, we can assume
¾f ¼ C0 þ C1 expðC2 =Teq Þ
ð4Þ
When T and ¾_ remain constant, eq. (4) returns to eq. (3),
and thus constant C0 is similar to the void nucleation strain,
and C0 = 0. According to eq. (1), C2 is a term representing
activation energy, and C1 is a correction factor. C0 and C1 are
dimensionless, but C1 is in K. The next problem we have to
handle is how to establish Teq. It should obey the following
basic principles.
① An increase in stress triaxiality (·m/·eq) decreases Teq.
② Positive impact of strain rate increases Teq; negative
impact decreases Teq; no effect remains Teq constant.
③ Teq should have a certain form making eq. (4) return to the
initial equation for single parameter when any two parameters keeping unchanged among temperature, stress triaxiality
and strain rate.
Bennett and Sinclair8) investigated the yield strength (·ys)
under various temperatures and strain rates. They found
that the effect of temperature (T) and strain rate (_¾) can be
represented by one parameter RTS ð¼ T lnðA=_¾ÞÞ, where A is
a constant, and that ·ys £ exp(1/RTS). It is well known that
strength and ductility usually have inverse relationship,
i.e., higher strength corresponding to lower ductility while
decreased strength increasing ductility. Therefore, it is
rational to assume that ¾f £ exp(¹1/RTS), which is similar
to eq. (4) when stress triaxiality is also involved into RTS.
Modifying RTS by incorporating ·m/·eq and simultaneously
obeying the aforementioned basic principles (①­③) yields
T0 þ ¡ðT T0 Þ=T0
¾_
Teq ¼
1 þ ¢ ln
ð5Þ
· m =· eq
¾_ 0
where T0 and ¾_ 0 are reference temperature and reference
strain rate, respectively, ¡ and ¢ are dimensionless conversion
coefficients showing the contribution of temperature and
strain rate to Teq, respectively. The more sensitive to the two
parameters, the larger the values of ¡ and ¢ are. Because
quasi-static tensile test at room temperature is usually a
routine test, we take it as a benchmark, i.e., setting
T0 = 293 K and reference strain rate ¾_ 0 ¼ 103 s¹1. T is in
K, and ¾_ is in s¹1. The terms of ·m/·eq and lnð_¾=_¾0 Þ are
dimensionless, the unit of Teq is K. In the definition of Teq, the
terms of [T0 + ¡(T ¹ T0)/T], (·m/·eq) and ½1 þ ¢ lnð_¾=_¾0 Þ
represent temperature, stress state and strain rate, respectively. At higher strain rate, adiabatic temperature rise
becomes significant, and T in eq. (5) should be replaced
with test temperature plus adiabatic temperature increment.
However, because it is difficult to know the exact value of
adiabatic temperature increment, in actual application, it is
convenient to reflect the adiabatic heating by adjusting ¢
value while keeping T = test temperature. When ¢ > 0, Teq
increases with an increase in strain rate, corresponding to
the positive impact of strain rate; when ¢ < 0, Teq decreases
with an increase in strain rate, corresponding to the negative
impact; ¢ = 0 corresponding to no impact.
3.
Representation of Fracture Strain by Teq
Tensile tests were performed on round bar specimens
(diameter 8 mm, gage length 40 mm) and axisymmetric
notched tension specimens (notch radius r = 0.5, 1.0, 1.5,
2.0, 3.0, 4.0, 6.0 mm; radius of the minimum cross-section
a = 5 mm) to measure the fracture strain of HT590 steel
(0.10C­0.44Si­1.53Mn­0.057V­0.045Nb) at various temperatures, strain rates and stress states. The longitudinal
specimen direction is parallel to the rolling direction. The
initial stress state at the minimum cross-section is evaluated
in terms of the triaxiality ratio, ·m/·eq, given by21) ·m/
·eq = 1/3 + ln(1 + 0.5a/r), and the fracture strain, ¾f, was
628
H. Qiu
(a)
(b)
Fig. 2 Experimental fracture strains of (a) HT590 steel at various
temperatures, stress triaxialities and strain rates, and (b) their representation by Teq.
(a)
(b)
Fig. 3 Experimental fracture strains of (a) mild steel at various stress
triaxialities and strain rates, and (b) their representation by Teq.
(a)
(b)
(c)
(d)
Fig. 4 Experimental fracture strains of (a) OFHC copper,13) (b) Armco iron13) and (c) 4340 steel13) at various temperatures, stress
triaxialities and strain rates, and (d) their representation by Teq.
measured from the post-tested specimens, and is given by
¾f = ln(A0/Af ) where A0 and Af are the area of the initial and
fractured cross-section, respectively. The strain rate was
determined by the fracture strain divided by the consumed
time. The fracture strain at various temperatures, stress
sates and strain rates is given in Fig. 2(a). The ranges of
temperature, stress sate (·m/·eq) and strain rate are,
respectively, 193­293 K, 0.33­1.61, 10¹4­101 s¹1.
In the application of eqs. (4) and (5), the determination of
¡ and ¢ is crucial. Using the experimental data of (T, ¾f )
(_¾ and ·m/·eq remain constant; usually quasi-static tensile test
results on round bar specimen) to determine the ¡, and the
process can be briefly stated as follows: ① setting a certain
value to ¡; ② calculating Teq by eq. (5); ③ plotting ¾f
against Teq, fitting the data with eq. (4), and obtaining the
corresponding correlation coefficient; ④ repeating ①­③
for a series of ¡, and obtaining the corresponding R values.
The ¡ corresponding to the minimum value of R is the
optimum value. The ¢ is determined by the similar approach
by using the data of (_¾, ¾f ) (T and ·m/·eq remain constant).
The representation of Fig. 2(a) by eqs. (4) and (5) is shown
in Fig. 2(b).
To apply the concept of Teq to as wide alloys as possible,
experimental data in the literature were cited in Figs. 3 and 4.
Figure 3(a) shows the experimental fracture strain of mild
steel1) at 293 K and within the strain rate range 4.2 © 10¹3­
Integration of Temperature, Stress State, and Strain Rate for the Ductility of Ductile Metals
Table 1 Values of the constants in eqs. (4) and (5).
eq. (4)
eq. (5)
C0
C1
C2 (K)
¡
¢
HT590 steel
0.169
1.302
227.273
240
0.014
Mild steel
OFHC copper
0.449
0.638
1.184
6.131
529.101
1203.901
®
95
0.002
0.015
Armco iron
0
4.299
448.936
38
0.019
4340 steel
0.072
3.782
689.655
28
0.0016
1.96 © 103 s¹1 and the range of ·m/·eq 0.33­1.87. The
experimental fracture strain of OFHC copper,13) Armco
iron13) and 4340 steel13) is, respectively, depicted in
Figs. 4(a)­4(c) over a wide range of temperature 293­
673 K, strain rate 10¹3­105 s¹1 and ·m/·eq 0.4­1.4. These
experimental data in Fig. 3(a) and Figs. 4(a)­4(c) were
converted into the fracture versus Teq in Figs. 3(b) and
4(d), respectively.
The correlation coefficient of the fitting curves in
Figs. 2(b), 3(b) and 4(d) is given in those figures. These
higher R values indicate that eqs. (4) and (5) are suitable
to those metals. The coefficients of the fitting curves are
summarized in Table 1. As shown in Figs. 3(a) and 4(c),
mild steel and 4340 steel are insensitive to strain rate,
and thus the value of ¢ is extremely small. For a given
temperature change, resultant variations in fracture strain
for the metals shown in Figs. 2­4 are different; these
sensitivities to temperature agree with the ¡ values.
combined into one single parameter, Teq, expressed by
eq. (5). It is confirmed that the fracture strain of ductile
metals can be represented by Teq in the form of eq. (4).
REFERENCES
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
4.
Summary
In the present study, by using “equivalent concept”, the
effects of strain rate, stress state and temperature are
629
20)
21)
M. Alves and N. Jones: J. Mech. Phys. Solid 47 (1999) 643­667.
K. Enami and K. Nagai: Tetsu-to-Hagané 91 (2005) 285­291.
Y. B. Bao: Eng. Fract. Mech. 72 (2005) 505­522.
M. S. Mirza, D. C. Barton and P. Church: J. Mater. Sci. 31 (1996)
453­461.
J. Sun: Eng. Fract. Mech. 39 (1991) 799­805.
H. G. Baron: J. Iron Steel Inst. 182 (1956) 354­365.
H. Qiu, M. Enoki, H. Mori, N. Takeda and T. Kishi: ISIJ Int. 39 (1999)
955­960.
P. E. Bennett and G. M. Sinclair: Trans. ASME 88 (1966) 518­524.
J. Castellanos, I. Rieiro, M. Carsí, J. Muñoz, M. El Mehtedi and O. A.
Ruano: Mater. Sci. Eng. A 517 (2009) 191­196.
R. Krishna Kumar, R. Narasimhan and O. Prabhakar: Int. J. Fract. 48
(1991) 23­40.
S. Mandal, V. Rakesh, P. V. Sivaprasad, S. Venugopal and K. V.
Kasiviswanathan: Mater. Sci. Eng. A 500 (2009) 114­121.
J. H. Park, Y. Tomota, S. Takagi, S. Ishikawa and T. Shimizu: Tetsu-toHagané 87 (2001) 657­664.
G. R. Johnson and W. H. Cook: Eng. Fract. Mech. 21 (1985) 31­48.
B. Erice, F. Gálvez, D. A. Cendón and V. Sánchez-Gálvez: Eng. Fract.
Mech. 79 (2012) 1­17.
T. Børvik, O. S. Hopperstad, S. Dey, E. V. Pizzinato, M. Langseth and
C. Albertini: Eng. Fract. Mech. 72 (2005) 1071­1087.
A. H. Clausen, T. Børvik, O. S. Hopperstad and A. Benallal: Mater. Sci.
Eng. A 364 (2004) 260­272.
J. R. Rice and D. M. Tracey: J. Mech. Phys. Solids 17 (1969) 201­
217.
J. W. Hancock and A. C. Mackenzie: J. Mech. Phys. Solids 24 (1976)
147­160.
H. Qiu, Y. Kawaguchi, M. Enoki and T. Kishi: Mater. Sci. Technol. 19
(2003) 1045­1049.
H. Qiu, H. Mori, M. Enoki and T. Kishi: Mater. Sci. Eng. A 316 (2001)
217­223.
P. W. Bridgman: Studies in Large Plastic Flow and Fracture,
(McGraw-Hill, New York, 1952) Chp. 1.