5.18.The same stress–strain curves as in Figure5.17plotted logarithmically. Note that the slope decreases at higher strain rates.
From W. F. Hosford,op. cit.
5.19.The decrease ofninσ=Kεn calcu- lated from Figure5.18. From W. F. Hosford, op. cit.
with increasing strain rate indicating thatndecreases. The average slope,n, is plotted against strain rate in Figure5.19.
5.6 TEMPERATURE DEPENDENCE OF FLOW STRESS
At elevated temperatures, the rate of strain hardening falls rapidly in most metals with an increase in temperature, as shown in Figure5.20. The flow stress and tensile strength, measured at constant strain and strain rate, also drop with increasing temperature as
Strain-hardening exponent, n
0.6
0.4
0.2
00 200 400 600
Temperature, K 5.20.Decrease of the strain-hardening expo-
nent, n, of pure aluminum with temperature.
Adapted from R. P. Carreker and W. R. Hibbard, Jr.,Trans. TMS-AIME, 209 (1957), pp. 1157– 63.
1000
100
10
1
Tensile strength, MPa
0 0.2 0.4 0.6 0.8 1
T/TM copper
aluminum 5.21.Decrease of tensile strength of pure cop- per and aluminum with homologous tempera- ture. Adapted from R. P. Carreker and W. R.
Hibbard, Jr.,op. cit.
5.22.Schematic plot showing the temper- ature dependence of flow stress for some alloys. In the temperature region where flow stress increases with temperature, the strain-rate sensitivity is negative.
illustrated in Figure5.21. However, the drop is not always continuous; often there is a temperature range over which the flow stress is only slightly temperature dependent or in some cases even increases slightly with temperature. The temperature dependence of flow stress is closely related to its strain-rate dependence. Decreasing the strain rate has the same effect on flow stress as raising the temperature, as indicated schematically in Figure5.22. Here it is clear that at a given temperature the strain-rate dependence is related to the slope of theσ-versus-Tcurve; whereσ increases withT,mmust be negative.
The simplest quantitative treatment of temperature dependence is that of Zener and Hollomon∗who argued that plastic straining could be treated as a rate process using Arrhenius rate law,†rate∝exp(–Q/RT), which has been successfully applied to many rate processes. They proposed that
ε˙ = Ae−Q/RT (5.15)
∗ C. Zener and J. H. Hollomon,J. Appl. Phys., 15 (1994), pp. 22–32.
† S. Arrhenius,Z. Phys. Chem., 4 (1889), p. 226.
5.6. TEMPERATURE DEPENDENCE OF FLOW STRESS 67
5.23.Strain rate and temperature combinations for various levels of stress. The data are for aluminum alloy 2024 and stresses are taken at an effective strain of 1.0. From D. S. Fields and W. A. Backofen, op. cit.
whereQis an activation energy, Tthe absolute temperature, andRthe gas constant.
Here the constant of proportionality,A, is both stress and strain dependent. At constant strain,Ais a function of stress alone,A=A(σ), so equation5.15can be written as
A(σ)=ε˙eQ/RT, (5.16)
or more simply as
σ = f(z), (5.17)
where the Zener–Hollomon parameterz =ε˙eQ/RT. This development predicts that if the strain rate to produce a given stress at a given temperature is plotted on a logarithmic scale against 1/T, a straight line should result with a slope of –Q/R. Figure5.23shows such a plot for 2024-O aluminum.
Correlations of this type are very useful in relating temperature and strain-rate effects, particularly in the high-temperature range. However, such correlations may break down if applied over too large a range of temperatures, strains, or strain rates. One reason is that the rate-controlling process, and henceQ, may change with temperature or strain. Another is connected with the original formulation of the Arrhenius rate law in which it was supposed that thermal fluctuations alone overcome an activation barrier, whereas in plastic deformation, the applied stress acts together with thermal fluctuations in overcoming the barriers as indicated in the following development.
5.24.Schematic illustration of an activation barrier for slip and the effect of applied stress on skewing the barrier.
Consider an activation barrier for the rate-controlling process, as in Figure5.24.
The process may be cross slip, dislocation climb, et cetera. Ignoring the details, assume that the dislocation moves from left to right. In the absence of applied stress, the activation barrier has a height Q and the rate of overcoming this barrier would be proportional to exp (–Q/RT). However, unless the position at the right is more stable—
has a lower energy than the position on the left—the rate of overcoming the barrier from right to left would be exactly equal to that in overcoming it from left to right, so there would be no net dislocation movement. With an applied stress,σ, the energy on the left is raised byσV, whereVis a constant with dimensions of volume, and on the right the energy is lowered byσV. Thus the rate from left to right is proportional to exp [–(Q–σV)/RT] and from right to left the rate is proportional to exp [–(Q+σV)/RT].
The net strain rate then is
ε˙ =C{exp[−(Q−σV)/RT]−exp[−(Q+σV)/RT]}
=Cexp (−Q/RT)[exp (σV/RT)−exp (−σV/RT)]
=2Cexp (−Q/RT) sinh (σV/RT). (5.18) To accommodate data better, and for some theoretical reasons, a modification of equation5.18has been suggested.∗†It is:
ε˙ = A[ sinh (ασ)]1/mexp (−Q/RT). (5.19) Steady-state creep data over many orders of magnitude of strain rate correlate very well with equation5.19, as shown in Figure5.25.
It should be noted that ifασ 1, sinh(ασ) ≈ασ, so equation5.19 reduces to ε˙ = Aexp (−Q/RT)(ασ)1/mor
σ = Aε˙mexp(m Q/RT), (5.20)
or σ = AZm, which is consistent with both the Zener–Hollomon development, equation 5.17, and the power-law expression, equation 5.1. Since sinh (x) → ex/2 forx1, at low temperatures and high stresses equation5.19reduces to
ε˙ =Cexp(ασ −Q/RT). (5.21) But now strain hardening becomes important soCandαare both strain and temperature dependent. Equation5.21reduces to
σ =C +mln ˙ε (5.22)
∗ F. Garofalo,TMS-AIME, 227 (1963), p. 251.
† J. J. Jonas, C. M. Sellers, and W. J. McG. Tegart,Met. Rev., 14 (1969), p. 1.
5.8. HOT WORKING 69
extrusion 320 to 616°C compression 250 to 550°C torsion 195 to 550 °C creep 204 to 593 °C
slope = 4.7 α = 2.1 kPa Q = 157 kJ/mole
0.01 0.1 1 10 100
10-8 10-6 10-4 10-2 1.0 102 104 106
sinh(ασ)
Z/A = (
ε
/A)exp[Q/(RT)].
5.25.Plot of the Hollomon–Zener parameter versus flow stress data showing the validity of the hyper- bolic sine relation (equation5.19). Adapted from J. J. Jonas,Trans. Q. ASM, 62 (1969), pp. 300–3.
which is consistent with equation5.12and explains the often observed breakdown in the power-law strain-rate dependence at low temperatures and high strain rates.