CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Hone
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
ON THE DEPENDENCE OF THE YIELD STRENGTH OF METALS
ON TEMPERATURE AND STRAIN RATE.
THE MECHANICAL EQUATION OF THE SOLID STATE
Pietro Paolo Mil el I a
ANPA, Agenzia Nazionale per la Protezione Ambientale, via K Brancati, 48, 00144 Rome, Italy
Abstract The purpose of this paper is to present a new constitutive equation (CE) based on the
experimental results obtained by the author on different carbon steels and those available in the open
literature on carbon steels, niobium ant titanium at different temperatures (-200 to 300 °C) and strain
rates (10~4 to 5000 s"1), as well. The equation stems from the experimental evidence that for BCC
alloys, tested at low strain rates, the trend of the yield strength ay is linear, when plotted in a lnay
versus l/T scale, independently of the strain rate applied, that determines the relative slope. Moreover,
all trends at different strain rates point towards a lower common value that represents the lower
athermal component of the yield strength. Increasing the strain rate, the trend remains linear, but the
lines merge in a new common upper point, at very low temperature, that represents the upper athermal
component of the yield strength. This initial formulation, valid for BCC metals, has been revisited to
consider metals other than BCC, i.e. HCP. The result is a unified equation of the solid state.
axis. This leads to a formulation of a constitutive
equation (CE):
LOW STRAIN RATE BEHAVIOR
To assess the effect of the strain rate £ on the
response of metals and how this effect depends on
temperature we will start to analyse BCC lattice
materials in the low strain rate range. To this
purpose, we will consider, as first, two particular
steels: A 533 B and A 508 C13, used in the nuclear
industry for pressure vessels. The present analysis is
based on data obtained by Kanninen et al [1] and by
the author [2]. As to A 533B steel, figure 1 presents
the results obtained by Kanninen (solid points)
plotted in a lnay versus l/T diagram, together with
the corresponding best-fit lines, where ay is the lower
yield strength. It shall be seen the characteristic
linear trend that is kept at all strain rates considered,
from the quasi-static application of the load, £ =10"3
s1, to the dynamic ones: namely 0.1,1.0 and 530 s"1.
Even more interesting is the finding that all lines in
fig. 1 point back to the same intersection^ with they
D
lnoyv = A + —
(D
6,4-
6,2-
6,0
0,001
0,002
0,003
0,004
0,005
1/T (1/K)
FIGURE 1. Trend of lnay vs l/T in A 533B steel at four different
strain rates and five temperatures.
642
were particular to steels, a completely different BCC
metal was chosen, namely annealed niobium. The
experimental data used were obtained by Campbell
and Briggs [4J. The metal was tested at -73 °C,
-23 °C and room temperature at four different, very
low to low strain rates: 0.00016 s"1, 0.0016 s"l and
0.008 s"1, respectively. The results are shown in
figure 3. Again, the linear trend is maintained at all
in which A is an invariant, i.e., independent of the
strain rate e, and B a characteristic of the material
that depends on s. For the steel considered, A 533B,
these intersections yield ~ 5.9 and, therefore, eq. 1 in
its explicit form becomes:
- 365
(2)
6,0-
This finding, that all lines merge to the same point,
means that the effect of strain rate is less and less
pronounced as the temperature increases, as if
temperature would wash out the strain rate effect. A
testing program was run by the author on a second
steel: A508BC13. This steel is quite similar to
A533B, besides it is used for forging applications.
Three strain rate were selected, namely 0.0001 s"1,
representative of a quasi-static condition, 500 s"1 and
1500 s"1. The results relative to the two lower strain
rates are shown in figure 2. Again, it can be seen the
5,5-
5,0-
0,0030
0,0035
0,0040
0,0045
0,0050
0,0055
1/T (1/ K)
7,0
FIGURE 3. Trend of lnav vs IIT in niobium at different strain
6,8
strain rates and the lines seem to converge towards a
common point, whose value is -3.6.
6,664
' "
BCC METAL CONSTITUTIVE EQUATION
6,2
The linear nature of the lnay versus IIT at all the
strain rate s considered, opens a new issue about the
relationship between the slope of the lines and the
value of the applied s. Let's plot the slopes B of the
four straight lines of fig. 1 versus the natural
logarithm of the strain rate, In s, which they refer to.
This can be seen in figure 4. The surprising result is
another straight line whose equation is:
6,05,8
0,000 0,002 0,004 0,006 0,008 0,010 0,012 0,014
1/T (1/K)
FIGURE 2. Trend of lnay vs 1/T in A 508 C13 steel at different
strain rates and temperatures.
linear trend at both strain rates and the merging
toward a common point at ln(a$ - 5.9. A second
experimental program was run by the author on a
different steel: A 537 Cll, aluminium killed. Twelve
tests have been run at three different temperatures
and four strain rates. The temperatures selected were
300 °C, RT and -100 °C, respectively. The four
strain rates ranged from 10"3 s4 to 103 s"1. Once
again, the trend of lnay vs IIT was linear at all the
strain rates considered and the best fit lines merged
towards a common value equal to ~ 5.5 [3]. At this
point, in order to check whether or not the findings
B = 120 + 7.3-//I*
(3)
This means that the parameter B in eq. 2 depends on
the strain rate s in a linear fashion with In s. We
can, then, introduce in eq. 2 the analytical expression
(3) of the parameter B and get:
= 365-(e uo/T
643
(4)
lower and upper merging point. To verify this, new
strain rate s were investigated for two of the steels
previously considered in this analysis, namely
A508BC13 and A533B. Twelve specimens of A508B
C13 were prepared and tested at ~ 1500 s"1 at ten
different temperatures, namely -70, -50, -30, -10,
0, 20, 40, 50, 80 and 130 °C. The results are shown
in figure 4, together with the previously found ones
obtained at 10"4 and 500 s"1. Effectively, the trend
remains linear, but the slope decreases. A new point
of convergence is reached at about -200° C,
According to the considerations previously done, this
point shall represent the upper limit of the yield
strength that is thermally independent. Its value is
about 2300 MPa. Same results were found for the A
533 B steel, that was tested at 5000 and 900 s"1. The
results are shown in figure 5.
in which the constant C was given the previously
found value of 365 MPa. Eq. (4) represents the
mechanical equation of the solid state for the steel
considered: A 533 B. The same linear trend can be
observed on A 537 Cll steel and niobium whose CE
are:
(5)
a
y
=
respectively. For a generic BCC metal, for which the
parameter B will take the most general form:
B = D + E-lns
(6)
eq. 2 will be written as:
a
-
(7)
with C, D and E to be determined experimentally
with the procedure just mentioned. It is clear that to
determine the three parameters C, D and E, a
minimum of three data points is necessary.
HIGH STRAIN RATE BEHAVIOR
The previous considerations, though interesting, yet
leave elements of uncertainty. At both very low and
very high dislocation velocities, the drag-stress
exerted by its atmosphere has to be temperature
independent. In fact, at extremely high velocities an
atmosphere of solute atoms should not exist any
more, eliminating the temperature dependence of the
yield strength. These facts, lead to the conclusion
that at very high strain rates it is not the temperature
to wash out the strain rate effect, but the opposite
and at sufficiently high strain rate, resulting in
extremely high dislocation velocity, only a thermal
independent component of the yield strength must
exist that appears as a flat, horizontal line in a ln(a$
versus l/T space. This actually means that, by
increasing the strain rate beyond a certain value, the
trend shown in figs 1, 2 and 3 must reverse. Now the
lines, if they still remain straight, must converge
toward a common upper value that represents
precisely the upper athermal component of the yield
strength. Therefore, there will be a transition strain
rate value £ fr between the two behaviors that will
result in a line pointing at the same time toward the
6.0
0,000 0,002 0,004 0,006 0,008 0,010 0,012 0,014
FIGURE 4. High and low strain rate trend of the yield strength in
A 508 Cll steel.
7,6
A533B
7,47,2
^ 7,0-
5*6-8'
1~
6,6
5000S-1
6,4
6,2
6,0
5,8
0,002
0,004
0,006
0,008
0,010
0,012
1/T (1/K)
FIGURE 5. High and low strain rate trend of the yield strength in
A 533 B steel
644
and drop abruptly afterwards, tmin - II Tmax and tmax =
I/ Tmin. The a-titanium data have been reanalysed
GENERALIZED CONSTITUTIVE EQUATION
A question arises about figures 4 and 5 where both
high and low strain rate lines appear to start from a
common point and continue to run without any end
in the opposite direction. This is not acceptable from
a physical point of view. In other words, the diagram
of figures 4 and 5 cannot be left open, but must be a
closed one. Moreover, eqs. 2 and 7 refer to BCC
metals. To check the behavior of HPC metals, that
are very sensitive to strain rate, an a-titanium alloy
has been considered [5], The results are shown in
figure 6. It is clear that the linear trend observed so
far between InOy and l/T is lost, yet the curves
obtained at various strain rates seem still to
using eq. 8, as well as niobium, A 533B, A508C13
and A 537. The results are shown in figure 7. As to
niobium, fig. 7 is showing the behaviour at five more
strain rates not present in figure 3, namely 0.48,1.48,
4.7, 23 and 63 s"1. This is because niobium is such
sensitive a metal that 0.48 and 1.48 s"1 can be already
considered high strain rates.
7.6- a-Titanium alloy
7.4
7.2-
7
-°
6.8
6.6
FIGURE 7. Experimental results and predictions of eq. 8 for all
the metals considered in the present study.
6.4
0.000
0.005
0.010
0.015
ACKNOWLEDGEMENTS
Figure 6. Trend of \nay vs I IT in a-titanium alloy at different
strain rates and temperatures.
This work was funded by Contract F61775-99WE066 from the U.S. Air Force.
converge towards a common point both at very high
and very low temperature (~ -250 °C). These latter
results are of great importance because they suggest,
recalling the previous observation about the closed
form of the diagrams, a more convenient and general
expression for the CE. This general form of the solid
state equation is:
= 1-1-
1.
REFERENCES
Kanninen, M. F. et al, US NRC NUREG CR/4219,
Vol. 4, No. 2, 1987.
2. Milella, P.P., "Temperature and Strain Rate
Dependence of Mechanical Behavior of BodyCentered Cubic Structure Materials", TMS Fall
Meeting '98, Chicago, Illinois, 11-15 October 1998.
(3)
where m is a strain rate exponent, amax the maximum
value of the yield strength (upper athermal
component) achieved at all strain rates at a minimum
temperature Tmin, a0 a reference value (lower
athermal component of the yield strength),
corresponding to Tmax, the temperature at which the
mechanical properties of the material start to flatten
645
3.
Milella, P.P. and Bonora, N., "Strain Rate and
Temperature Effect in Ductile Failure process
Characterization and Modeling Using Continuum
Damage Mechanics55, USAF-EOARD, Research
Contract, July 2000.
4.
Campbell, J.D., and Briggs, T.L.J., "Less Common
Metals", 40, p. 235, 1974.
5.
Harding, J., 7th HERF Conference Proceedings, T. Z.
Blazynski Editor, University of Leeds, p. 1, 1981.