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or Mn-added interstitial free steels
Acta Materialia 51 (2003) 4437–4446 www.actamat-journals.com
Solid solution softening at high strain rates in Si- and/or Mn-added interstitial free steels A. Uenishi ∗, C. Teodosiu LPMTM-CNRS, University of Paris 13, 93430 Villetaneuse, France Received 5 December 2002; received in revised form 10 May 2003; accepted 10 May 2003
Abstract In order to understand the high strain rate properties of high strength steel sheets, stress–strain relations of Fe, Fe– Mn and Fe–Si at high strain rates were investigated. The addition of Mn and Si caused both solid solution hardening at lower strain rates and a decrease of strain rate sensitivity of the flow stress. In addition, the ductility at high strain rates was improved by the addition of solute atoms, an effect which can be related to the decrease in the strain rate sensitivity of the flow stress. Thermal activation analysis revealed that the Peierls–Nabarro mechanism controls the strain rate sensitivity of these steels. The effects of solid solution were discussed based on the theories proposed to explain alloy softening at low temperatures. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Steels; Thermally activated processes; High strain rates; Mechanical properties
1. Introduction The development of a new generation of materials for crash energy absorption has been constantly pursued, because material substitution is one of the best solutions both for vehicle safety and light body weight. Steels are known to have a high strain rate sensitivity of flow stress, which explains why they are favourite materials for crash energy absorbing parts. In particular, high strength steels are expected ∗
Corresponding author. Present address: Steel Research Laboratories, Nippon Steel Corp., 20-1 Shintomi, Futtsu City, Chiba, 293-8511, Japan. Tel.: +81-439803113; fax: +81439802743. E-mail address: [email protected] (A. Uenishi).
to have high crash energy absorbing properties. However, several studies have revealed that the higher the strength of steel, the less sensitive the flow stress to the strain rate [1–3]. The high strain rate properties of sheet steels have not been fully investigated, due to the experimental difficulties in impact testing sheet materials. In addition, commercial high strength steels are so complicated in chemical composition, thermal treatment histories, multiple hardening mechanisms, etc., that it is difficult to obtain a comprehensive insight into the rate-controlling mechanisms. The aim of the present study is to understand the relations between the effects of strain rate and material strengthening. Solid solution hardening is a typical and widely used strengthening mechanism for high strength steels. Therefore, the
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00279-9
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room temperature at a strain rate of 0.0033 s⫺1. The specimen was of 12.5 mm width and 50 mm gauge length, with a radius of curvature of 25 mm at the shoulders (JISZ2201–13Btype).
model materials chosen for our study have been elaborated by addition of Si and/or Mn to an interstitial free steel. The high strain rate behaviour of these materials has been investigated both by mechanical tests and by thermal activation analysis.
3. Experimental results Stress–strain curves at several strain rates for the host IF steel, called steel A in the following, are shown in Fig. 1. The upper yield limit increases rapidly and the stress difference between the upper and lower yield point increases with strain rate, in agreement with the results obtained by Klepaczko [6] for similar materials. The flow stress at lower strains increases with strain rate. However, the
2. Materials and experimental procedure The base material (IF steel) was Fe alloyed with about 0.2 mass% Ti, in order to remove interstitial atoms from the ferrite matrix. Binary alloys were made by nominal addition of 1% or 2% by weight of Si and Mn to this base. Table 1 shows the composition of the materials. The materials were melted in a 300 kg vacuum furnace. After casting, they were hot rolled to the final thickness of 2 mm and annealed at 700 °C for 60 min. The average grain size was about 50 µm for all materials, although it decreased slightly with the addition of solute atoms. It is known that slight differences in grain size have no significant effects on mechanical properties [4], and it will be shown later that, indeed, the differences in the flow stress among the tested materials were mainly due to the hardening caused by solute atoms. The mechanical tests at different strain rates were carried out from 0.001 to 1000 s⫺1 at room temperature. The stress–strain curves at strain rates higher than 100 s⫺1 were measured by uniaxial tensile tests using the one-bar technique [1,2,5], whereas an ordinary Instron type screw-driven test machine was employed at strain rates below 1 s⫺1. In addition, low-temperature properties were also examined by uniaxial tensile tests, which were performed over temperatures ranging from 220 K to
Fig. 1. Nominal stress–strain curves of the IF steel (A) at different strain rates, measured at room temperature.
Table 1 Chemical composition of materials (mass%)
A B C D E
IF +1 at.%Mn + 2at.%Mn + 2at.%Si +2 at.%Si +4 at.%Si
C
Si
Mn
P
S
Al
Ti
N
0.0016 0.0013 0.0015 0.0014 0.0012
– 0.014 0.99 0.98 1.97
0.10 0.97 1.92 0.10 0.10
0.005 0.005 0.004 0.005 0.005
0.001 0.001 0.001 0.001 0.001
0.003 0.003 0.003 0.003 0.004
0.015 0.014 0.023 0.019 0.023
0.0014 0.0018 0.0018 0.0013 0.0014
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work hardening rate decreases rapidly at higher strain rate, which leads to early fracture. The adiabatic heating is expected to have a strong influence at high strain rates and hence the necking, which is inevitable in tensile tests, is accelerated by the thermal softening induced by plastic work during deformation. A detailed finite element analysis of thermal softening during our high strain rate tensile tests will be presented elsewhere [7]. The variation of the flow stress with the strain rate at 5% strain is shown in Fig. 2. Addition of solute atoms increases the flow stress at lower strain rates, whereas at higher strain rates, its effects become less pronounced. In particular, the flow stress of 1 at.% Mn-added steel shows a crossover with that of the host steel at around 1 s⫺1, which means that the solid solution hardening by Mn is no longer effective at higher strain rates. This behaviour is similar to the effects of alloy softening that have been reported by Pink [8] by lowering the temperature. The variation of the maximum uniform elongation with strain rate is shown in Fig. 3. An abrupt decrease at higher strain rates is observed in IF steel and 1 at.% Mn steel (steel B). On the other hand, the ductility of the other three steels is less sensitive to strain rate and the flow stress of these steels is not much affected by strain rate, as shown in Fig. 2. As already stated, the effect of the thermal softening increases with the strain rate sensitivity. Therefore, the increased ductility at high strain rates in steels C, D and E, which contain high concentrations of solute atoms, can be attributed to their lower strain rate sensitivity. Stress–strain curves measured at several temperatures below room temperature for the IF steel are shown in Fig. 4. A decrease in deformation temperature causes an increase in the flow stress and the yield phenomena are more pronounced at low temperatures, as in the case of high strain rates. Fig. 5 shows the variation of the flow stress at 5% tensile strain vs. the absolute temperature. As expected from the results of high strain rate tests, the sensitivity of the flow stress on the deformation temperature decreases in solution hardened steels, since an increase in strain rate is equivalent to a decrease in temperature in a thermally activated process. Thus, it can be concluded that solid
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Fig. 2. Variation with the strain rate of (a) the true tensile flow stress and (b) the effective stress at 5% strain and room temperature. Solid curves in (a) show the results of fitting data by quadratic expressions of logarithmic strain.
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Fig. 3. Variation of the maximum uniform elongation with strain rate. Curves fitted to experimental data: 1: IF steel, 2: IF + 1at.%Mn, 3: IF + 2at.%Si, 4: IF + 2at.%Mn + 2at.%Si and 5: IF + 4at.%Si.
Fig. 5. Variation of the true tensile stress at 5% strain with respect to the deformation temperature at a strain rate of 0.0033 s⫺1.
solution causes the decrease of the sensitivity of the flow stress on both strain rate and temperature. The changes of the maximum uniform elongation with temperature are shown in Fig. 6. For the
Fig. 4. Nominal stress–strain curves at several deformation temperatures measured at a strain rate of 0.0033 s⫺1. Fig. 6. Variation of the maximum uniform elongation with temperature. Curves fitted to experimental data: 1: IF steel, 2: IF + 1at.%Mn, 3: IF + 2at.%Si, 4: IF + 2at.%Mn + 2at.%Si and 5: IF + 4at.%Si.
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IF steel, it is almost constant until 240 K and it decreases rapidly at lower deformation temperatures, whereas it has a tendency to increase with decreasing temperature for the other investigated steels. It has been reported that, with decreasing temperature, the dislocation distribution becomes more uniform at any given strain, and thus the work hardening rate tends to increase in iron [9]. Then, it is expected that the maximum uniform elongation increases with decreasing temperature, which explains the temperature sensitivity of the ductility of the tested steels, except for the IF steel. An explanation of the latter exception could be again provided by considering the effects of the adiabatic heating. Because IF steel is a strain rate and temperature sensitive material, the plastic work dissipated during deformation increases rapidly with a decrease in temperature and it might cause early necking even at lower strain rates.
4. Discussion 4.1. Origin of the increase in flow stress at low strain rates Before discussing the results, the origin of the increase in flow stress at low strain rates for the tested steels will be examined. For iron-based alloys, Leslie [10] made an extensive study on the effects of alloying elements. The effects of Si and Mn on the increase in flow stress can be evaluated based on his results. The differences in flow stress at 5% tensile strain between solution hardened steels and the host steel are plotted as a function of the atomic concentration of solute atoms in Fig. 7. Although it was pointed out that the dilation of the lattice in Fe–Si alloys does not vary linearly with solute concentration, the strengthening by Si and Mn seems to be proportional to concentration, as shown in this figure. The slopes ⌬s/⌬c of the two straight lines are 54.3 (MPa/at.%) and 36.5 (MPa/at.%) for Fe–Si and Fe–Mn alloys, respectively. Because one of the tested steels contains both Si and Mn, the increase in the flow stress due to solid solution hardening was calculated by cumulating the weighted contributions of the alloying elements and again compared with the
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Fig. 7. Increase in the flow stress vs. atomic percent of solute atoms. The slopes of the two straight lines show the solution hardening ability of the corresponding elements.
experimental data. In Fig. 8, the abscissa axis shows the increase in the flow stress due to solid solution estimated from the solute concentrations and the slopes in Fig. 7, while the ordinate axis shows the corresponding increase obtained from
Fig. 8. Comparison of the increase in flow stress due to solid solution between the calculations estimated from the solute concentration and the slopes in Fig. 7 and the experimental data.
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experiments. The data are located on the diagonal line, which shows that almost all the increase in flow stress of the tested steels is due to the alloying by Si and/or Mn. 4.2. Decrease in the strain rate sensitivity of the flow stress caused by solute atoms In this section, we will use thermal activation analysis in order to get some supplementary insight into the deformation mechanisms and thus to explain the temperature and strain rate effects on the flow stress in the investigated materials. A gliding dislocation, subjected to a resolved shear stress t, encounters during its motion two kinds of obstacles: extended obstacles generating a long-range stress field, whose resolved shear stress is denoted by tm, and local obstacles, which act only over a few atomic distances and, unlike the former, can be overcome with the help of thermal fluctuations, under the action of the local effective stress t∗ ⫽ t⫺tm.
(1)
The long-range stress has been denoted by tm because it is proportional to some effective shear modulus m, and depends on temperature only via the slight temperature dependence of the elastic constants. On the other hand, it depends strongly on the microstructure and generally evolves with progressing strain. By using rate theory and some statistical assumptions concerning the distribution and strength of the local obstacles, the slip rate on any active slip system may be written as g˙ ⫽ g˙ 0exp(⫺⌬Gt / kT),
(2)
where k is Boltzmann’s constant, T is the absolute temperature, g˙ 0 is a material dependent preexponential factor, and ⌬Gt denotes the difference in free enthalpy between the configuration of the dislocation segment in the saddle point and in the ground state for forward jumps, which depends in turn on the interaction between the dislocation segments and the local obstacles (see, e.g., [11]). For the present study, the experiments were conducted in uniaxial tension and the specimens were polycrystalline materials. Thus, the tensile stress
should be converted to the resolved shear stress acting in each grain and on each slip system, by eventually taking into account the long-range stresses produced by the grain interaction, whereas the tensile strain rate results by volume averaging the contributions of all slip systems and grains. While this conversion can be eventually carried out for a given texture, the quantitative result obtained depends critically on the simplifying hypotheses employed in the micro–macro transition (see, e.g., [12]). We shall, therefore, adopt here an alternative, more phenomenological approach, namely the equations governing the thermally activated plastic slip will be transcribed at the macroscopic scale, by simply replacing the shear rate g˙ and the shear stress t by the tensile flow stress s and the tensile strain rate e˙ , respectively: s∗ ⫽ s⫺si,
(3)
e˙ ⫽ e˙ 0exp(⫺⌬G / kT).
(4)
Clearly, the newly introduced phenomenological quantities and parameters cannot be uniquely defined by using a rigorous micro–macro transition. Nevertheless, they can be supposed to inherit some of the physical significance of their microscopic counterparts. Thus, si is related to the intensity of the long-range obstacles to dislocation glide; e˙ 0 depends essentially on the density of the mobile dislocations, on the average spacing of the pinning points along the dislocation lines and on the average distance between the local obstacles; ⌬G is an apparent activation free enthalpy, which depends on the effective tensile stress s∗ and on the apparent free energy of interaction between the gliding dislocations and the local obstacles. By definition, the apparent activation volume V∗ is expressed as ∂(⌬G) V∗ ⫽ ⫺ ∂s∗
|
.
(5)
T=const
Solving Eq. (4) for ⌬G and introducing the result obtained into Eq. (5) yields ∂(lne˙ ) V∗ ⫽ kT ∂s∗
|
.
(6)
T=const
This alternative expression has been used to
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determine the dependence of the activation free enthalpy on the effective tensile stress from the results of our high strain rates experiments. Previous researchers investigating the strain rate sensitivity of steels [4,14,15] have in general arbitrarily taken t∗ = s∗ / 2, and represented the variation of the activation volume ∂(lne˙ ) Vt∗ ⫽ kT ∗ ∂t
|
⫽ 2V∗
(7)
T=const
with respect to t∗. For the thermal activation analysis, it is preferable to use the flow stress at the beginning of deformation, because the effects of the thermal softening are not negligible, especially at high strain rate. On the other hand, all internal variables change rapidly at yield point and the deformation is not steady-state. Furthermore, the accurate measurement of the upper yield limits for high strain rate tests is difficult, because it is influenced by experimental configurations, such as the stiffness of the machine, specimen size, loading conditions, etc. The present analysis has been done, therefore, by using the strain rate sensitivity of the flow stress at 5% tensile strain, which appeared to be the lowest strain value at which the transitory yield point phenomena can be considered negligible. Then, the activation volume could have been calculated directly from the data shown in Fig. 2a. However, this procedure proved to be insufficiently accurate, because of the fluctuation of the experimental errors. Therefore, the experimental variation of the flow stress has been first fitted by a quadratic polynomial in terms of the logarithm of the strain rate and then si was considered as a minimum value of the interpolating function. Subtracting this latter value from the experimental flow stress resulted in the variation of the effective stress s∗ in terms of the logarithm of the strain rate, as represented in Fig. 2b. Finally, the activation volume has been evaluated from the slopes of the curves in Fig. 2b, by using Eq. (6). As shown by Uenishi and Teodosiu [7], this procedure is equivalent to directly using Eqs. (3)–(5) in conjunction with the phenomenological formula for
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the activation free enthalpy proposed by Kocks et al. [13],
冋 冉 冊册
⌬G ⫽ (⌬F) 1⫺
s∗ s0
p q
,
(8)
where s0 characterizes the average resistance to slip of the local obstacles and ⌬F is the apparent free energy of interaction between the gliding dislocations and the local obstacles, provided that we set p = 1 / 2 and q = 1. Fig. 9 shows our results for IF steel, converted for the sake of comparison to t∗⫺V∗t data, along with results obtained by other authors for similar materials. The results of Harding [14] were obtained from experiments at a fixed temperature over a very wide range of high strain rate tests, as in the present work. On the other hand, the results of Spitzig and Leslie [4] were derived from a change in stress due to strain rate change, at low temperatures and relatively low strain rates. The good agreement of these different sets of experimental data is not trivial. We have already seen similar behaviour of the flow stress at high strain rates and at low temperatures (cf. Figs. 2 and 5). The conformity of the activation volume at high strain rates and at low temperatures for the IF steel
Fig. 9. Comparison of the activation volume of IF steel with previous results for similar materials. The activation volume was evaluated from experiments carried out either at high strain rates ([14] and the present study) or at low temperatures [4].
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gives supplementary support to the conjecture that the rate-controlling mechanism at high strain rates in this material is the same as that at low temperatures. Furthermore, since the activation volume of the IF steel ranges from 10 to 100 b3, and thus corresponds to the activation volume that is generally ascribed to the Peierls–Nabarro mechanism, it can be concluded that this mechanism controls the deformation rate of the IF steel even at high strain rates, as already assumed by other authors [4,15]. Fig. 10 shows the variation of the activation volume of solution hardened steels as a function of the effective stress. At a lower stress level, the activation volume increases with increasing strength of the materials, i.e. with increasing concentration of solute atoms. The activation volume of 1 at.% Mn steel decreases more rapidly than that of IF steel and shows a crossover with that of IF steel. Similar changes of the activation volume due to solution hardening have been observed for Fe–Ni and Fe–Si by several authors [4,15], although the corresponding data have been obtained from low temperature tests. Their results show that the plots
Fig. 10. Activation volume of solution hardened steels as a function of the effective stress. At a lower stress level, the activation volume increases by solid solution. At higher effective stress, the activation volume of solution hardened steel decreases more rapidly than the host steel, while the variation of the activation volume of 1 at.% Mn steel shows a crossover with that of the host metal.
of the activation volumes vs. effective stress of all solution hardened steels intersect with that of the host metal and that the intersection point moves to higher stresses with increasing concentration of the solute atoms. In our results, not all solution hardened steels show a crossover. This is probably due to the fact that for higher concentrations of the solute atoms, the effective stress does not reach the level that is needed for the crossover for strain rates below 1000 s⫺1. It is known that in bcc metals a small addition of alloying elements can decrease the temperature sensitivity of the flow stress at low temperatures [8]. This phenomenon has been called “alloy softening”. At room temperature, alloying almost always increases the strength of the material. However, alloy softening indicates that the properties at room temperature cannot be extrapolated monotonically to lower temperatures. It should be pointed out that Fe–Ni and Fe–Si alloys are typical materials that show “alloy softening” at low temperatures [4,15]. Thus, the behaviour of the activation volume of these alloys should be interpreted in connection with the decrease of the temperature sensitivity due to alloying. The decrease in the strain rate sensitivity of the flow stress of the tested steels can be ascribed to the solid solution effect, because all other conditions, such as hot rolling and annealing conditions, are almost identical. According to the theory of thermally activated processes, an increase in strain rate is equivalent to a decrease in temperature. Furthermore, as shown in Fig. 5, the temperature sensitivity of the flow stress of solution hardened steels is lower than that of host metal, a tendency already emphasized by previous studies carried out at low temperatures [8]. Thus, the decrease in strain rate sensitivity due to alloying at high strain rates can be discussed in the same manner as the “alloy softening” at low temperatures. The models for alloy softening can be roughly divided into two groups, extrinsic and intrinsic theories. One of the most widely used extrinsic theories is the scavenging model. It assumes the presence of residual interstitial atoms, which increase the yield stress of base metals. The attraction or repulsion between newly added alloying atoms and residual atoms promotes a clustering of
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dispersed interstitial atoms and then the yield stress decreases due to alloying. However, there are experimental results showing that the alloy softening is insensitive to variations in purity level (see, e.g., [15]), thus directly contradicting the scavenging theories. In addition, the steels tested in this study, which contain Ti in order to obtain excess interstitial atoms such as carbon and nitrogen, also exhibit alloy softening at low temperatures, which can hardly be explained by the scavenging model. Thus, the intrinsic changes in lattice resistance due to alloying seem to be more probable. As seen before, it can be concluded that the strain rate sensitivity of IF steel is controlled by the Peierls–Nabarro mechanism, in which the elementary processes are the nucleation and propagation of kinks. Thus, in order to understand the effects of solid solution, it is important to take into account the effects of solute atoms on these elementary processes. Ono and Sommer [16] were the first to calculate the flow stress of a crystal with a high Peierls–Nabarro stress, by considering that point obstacles, such as solute atoms, affect the formation of kinks by limiting the length of the dislocation segments. They found that point obstacles have little or no effect on the nucleation process of double kinks when the applied stress is nearly equal to the Peierls–Nabarro stress, e.g. at low temperatures. At higher temperatures, the flow stress becomes independent of the temperature and proportional to the concentration of point obstacles. The predictions of Ono and Sommer [16] compare favourably with the experimental data obtained by Nakada and Keh [17] for Fe–N alloys. However, their calculation cannot reproduce the crossover of the activation volume observed in low temperature experiments of Fe–Ni and Fe–Si alloys and in the present high strain rate experiments. The theory proposed by Sato and Meshii [18] can explain both solid solution softening and solid solution hardening by a unified model. These authors extended the Ono–Sommer model, by considering explicitly the interaction between a kink on a screw dislocation and solute atoms, described as misfit strain centres. The couple forces due to this interaction promote the nucleation of double kinks in the vicinity of misfit strain centres. Solid
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solution softening is attributed to this promotion. On the other hand, strong misfit centres will provide greater resistance to the overcoming of the solute atoms by the dislocations. Therefore, these combined effects on the overall dislocation motion generate two antagonistic softening and hardening effects of the solute atoms. Sato and Meshii [18] calculated the dependence of the activation volume on the strength of the misfit strain centre. The activation volume increases with increasing strength of the misfit strain centre and shows a crossover with the activation volume of the host metal for some particular conditions. A quantitative comparison of our results with their calculation is difficult, because (i) their calculation is based on several simplifications concerning the interaction between dislocations and point obstacles and (ii) the evaluation of the strength and the spacing of misfit strain centres is difficult for the present materials due to the fluctuation of the local concentration and the clustering of solute atoms. However, the effects of solute atoms at high strain rates can be qualitatively explained by both the promotion of the nucleation of kinks and the inhibition of their movements, which may yield a correlation of the strain rate sensitivity of the flow stress and solid solution hardening. In Fig. 11, the difference in the flow stress at 5% tensile strain between 0.001 and 1000 s⫺1 is plotted against the flow stress at 5% tensile strain and 0.001 s⫺1. Clearly, the strain rate sensitivity decreases sharply with increasing quasi-static strength, i.e. solid solution hardening at lower strain rates. 5. Conclusions The high strain rate properties of solution hardened steels have been investigated. The strain rate sensitivity of the flow stress decreases with quasistatic strength, i.e. by addition of solute atoms. The elongation at high strain rates decreases rapidly for a strain rate sensitive material, due to the adiabatic heating produced by the plastic work during deformation. For solution hardened steels, these effects are less pronounced because of the low strain rate sensitivity of their flow stress. As a result, the elongation at high strain rates is improved by addition of solute atoms.
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Acknowledgements This work has been supported by Nippon Steel Corporation.
References
Fig. 11. Strain rate sensitivity shown as the difference in the flow stress at 5% strain between 0.001 and 1000 s⫺1 vs. the flow stress at 5% strain at a strain rate of 0.001 s⫺1. The strain rate sensitivity of the flow stress decreases with increasing static strength.
From the results of the thermal activation analysis, it can be concluded that the Peierls–Nabarro mechanism controls the strain rate sensitivity of the flow stress in the IF steel and the effects of solute atoms on high strain rate properties are similar to those at low temperatures. The model proposed by Sato and Meshii [18], in which the interaction between solute atoms and a kink on a screw dislocation is explicitly considered, can explain the peculiar behaviour of the activation volume of solution hardened steel. In particular, the decrease of strain rate sensitivity of the flow stress in solution hardened steel may be ascribed to the promotion of the nucleation of kink pairs by the misfit strain produced by solid atoms. The correlation between solid solution hardening at low strain rates and the decrease in the strain rate sensitivity at high strain rates can be at least qualitatively explained by this model.
[1] Nakanishi E, Itabashi M, Kawata K. The fast and slow deformation behaviour in commercial aluminium alloys and steel sheets for automobile bodies. In: Rossmanith HP, editor. Structural failure, product liability and technical insurance. Elsevier; 1993. p. 423. [2] Uenishi A, Suehiro M, Kuriyama Y, Usuda M. Crashworthiness of high strength steel for automobile bodies. In: Kelkar S, editor. Proceedings of International Body Engineering Conference, vol. 23. 1996. p. 89. [3] Uenishi A, Suehiro M, Kuriyama Y, Usuda M. Improvement of crashworthiness by application of high strength steel for light weight autobodies. In: Uddin MN, Roessler DM, editors. Proceedings of International Body Engineering Conference, vol. 29. 1997. p. 59. [4] Spitzig WA, Leslie WC. Acta Met 1971;19:1143. [5] Kawata K, Hashimoto S, Takeda N, Sekino S. On highvelocity brittleness and ductility of dual-phase steel and some hybrid fiber reinforced plastics. In: Vinson JR, Taya M, editors. Recent advances in composites in the United States and Japan, ASTM STP 864. Philadelphia: American Society for Testing and Materials; 1985. p. 700. [6] Klepaczko JR. Int J Solids Struct 1969;5:533. [7] Uenishi A, Teodosiu C. Int J Plasticity 2003 [in press]. [8] Pink E, Arsenault RJ. Prog Mater Sci 1979;24:1. [9] Keh AS, Weissmann S. Deformation substructure in bodycentered cubic metals. In: Thomas G, Washburn J, editors. Electron microscopy and strength of crystals. New York: Interscience; 1961. p. 231. [10] Leslie WC. Metall Trans 1972;3:5. [11] Gibbs GB. Mater Sci Eng 1969;4:313. [12] Clausen B. Characterization of polycrystal deformation by numerical modelling and neutron diffraction measurements. PhD thesis, Risø National Laboratories, Roskilde, Denmark, 1997. [13] Kocks UF, Argon AS, Ashby MF. Prog Mater Sci 1975;19:230. [14] Harding J. Acta Met 1969;17:949. [15] Tanaka T, Watanabe S. Acta Met 1971;19:991. [16] Ono K, Sommer AW. Metall Trans 1970;1:877. [17] Nakada Y, Keh AS. Acta Met 1968;16:903. [18] Sato A, Meshii M. Acta Met 1973;21:753.
Solid solution softening at high strain rates in Si- and/or Mn-added interstitial free steels A. Uenishi ∗, C. Teodosiu LPMTM-CNRS, University of Paris 13, 93430 Villetaneuse, France Received 5 December 2002; received in revised form 10 May 2003; accepted 10 May 2003
Abstract In order to understand the high strain rate properties of high strength steel sheets, stress–strain relations of Fe, Fe– Mn and Fe–Si at high strain rates were investigated. The addition of Mn and Si caused both solid solution hardening at lower strain rates and a decrease of strain rate sensitivity of the flow stress. In addition, the ductility at high strain rates was improved by the addition of solute atoms, an effect which can be related to the decrease in the strain rate sensitivity of the flow stress. Thermal activation analysis revealed that the Peierls–Nabarro mechanism controls the strain rate sensitivity of these steels. The effects of solid solution were discussed based on the theories proposed to explain alloy softening at low temperatures. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Steels; Thermally activated processes; High strain rates; Mechanical properties
1. Introduction The development of a new generation of materials for crash energy absorption has been constantly pursued, because material substitution is one of the best solutions both for vehicle safety and light body weight. Steels are known to have a high strain rate sensitivity of flow stress, which explains why they are favourite materials for crash energy absorbing parts. In particular, high strength steels are expected ∗
Corresponding author. Present address: Steel Research Laboratories, Nippon Steel Corp., 20-1 Shintomi, Futtsu City, Chiba, 293-8511, Japan. Tel.: +81-439803113; fax: +81439802743. E-mail address: [email protected] (A. Uenishi).
to have high crash energy absorbing properties. However, several studies have revealed that the higher the strength of steel, the less sensitive the flow stress to the strain rate [1–3]. The high strain rate properties of sheet steels have not been fully investigated, due to the experimental difficulties in impact testing sheet materials. In addition, commercial high strength steels are so complicated in chemical composition, thermal treatment histories, multiple hardening mechanisms, etc., that it is difficult to obtain a comprehensive insight into the rate-controlling mechanisms. The aim of the present study is to understand the relations between the effects of strain rate and material strengthening. Solid solution hardening is a typical and widely used strengthening mechanism for high strength steels. Therefore, the
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00279-9
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room temperature at a strain rate of 0.0033 s⫺1. The specimen was of 12.5 mm width and 50 mm gauge length, with a radius of curvature of 25 mm at the shoulders (JISZ2201–13Btype).
model materials chosen for our study have been elaborated by addition of Si and/or Mn to an interstitial free steel. The high strain rate behaviour of these materials has been investigated both by mechanical tests and by thermal activation analysis.
3. Experimental results Stress–strain curves at several strain rates for the host IF steel, called steel A in the following, are shown in Fig. 1. The upper yield limit increases rapidly and the stress difference between the upper and lower yield point increases with strain rate, in agreement with the results obtained by Klepaczko [6] for similar materials. The flow stress at lower strains increases with strain rate. However, the
2. Materials and experimental procedure The base material (IF steel) was Fe alloyed with about 0.2 mass% Ti, in order to remove interstitial atoms from the ferrite matrix. Binary alloys were made by nominal addition of 1% or 2% by weight of Si and Mn to this base. Table 1 shows the composition of the materials. The materials were melted in a 300 kg vacuum furnace. After casting, they were hot rolled to the final thickness of 2 mm and annealed at 700 °C for 60 min. The average grain size was about 50 µm for all materials, although it decreased slightly with the addition of solute atoms. It is known that slight differences in grain size have no significant effects on mechanical properties [4], and it will be shown later that, indeed, the differences in the flow stress among the tested materials were mainly due to the hardening caused by solute atoms. The mechanical tests at different strain rates were carried out from 0.001 to 1000 s⫺1 at room temperature. The stress–strain curves at strain rates higher than 100 s⫺1 were measured by uniaxial tensile tests using the one-bar technique [1,2,5], whereas an ordinary Instron type screw-driven test machine was employed at strain rates below 1 s⫺1. In addition, low-temperature properties were also examined by uniaxial tensile tests, which were performed over temperatures ranging from 220 K to
Fig. 1. Nominal stress–strain curves of the IF steel (A) at different strain rates, measured at room temperature.
Table 1 Chemical composition of materials (mass%)
A B C D E
IF +1 at.%Mn + 2at.%Mn + 2at.%Si +2 at.%Si +4 at.%Si
C
Si
Mn
P
S
Al
Ti
N
0.0016 0.0013 0.0015 0.0014 0.0012
– 0.014 0.99 0.98 1.97
0.10 0.97 1.92 0.10 0.10
0.005 0.005 0.004 0.005 0.005
0.001 0.001 0.001 0.001 0.001
0.003 0.003 0.003 0.003 0.004
0.015 0.014 0.023 0.019 0.023
0.0014 0.0018 0.0018 0.0013 0.0014
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work hardening rate decreases rapidly at higher strain rate, which leads to early fracture. The adiabatic heating is expected to have a strong influence at high strain rates and hence the necking, which is inevitable in tensile tests, is accelerated by the thermal softening induced by plastic work during deformation. A detailed finite element analysis of thermal softening during our high strain rate tensile tests will be presented elsewhere [7]. The variation of the flow stress with the strain rate at 5% strain is shown in Fig. 2. Addition of solute atoms increases the flow stress at lower strain rates, whereas at higher strain rates, its effects become less pronounced. In particular, the flow stress of 1 at.% Mn-added steel shows a crossover with that of the host steel at around 1 s⫺1, which means that the solid solution hardening by Mn is no longer effective at higher strain rates. This behaviour is similar to the effects of alloy softening that have been reported by Pink [8] by lowering the temperature. The variation of the maximum uniform elongation with strain rate is shown in Fig. 3. An abrupt decrease at higher strain rates is observed in IF steel and 1 at.% Mn steel (steel B). On the other hand, the ductility of the other three steels is less sensitive to strain rate and the flow stress of these steels is not much affected by strain rate, as shown in Fig. 2. As already stated, the effect of the thermal softening increases with the strain rate sensitivity. Therefore, the increased ductility at high strain rates in steels C, D and E, which contain high concentrations of solute atoms, can be attributed to their lower strain rate sensitivity. Stress–strain curves measured at several temperatures below room temperature for the IF steel are shown in Fig. 4. A decrease in deformation temperature causes an increase in the flow stress and the yield phenomena are more pronounced at low temperatures, as in the case of high strain rates. Fig. 5 shows the variation of the flow stress at 5% tensile strain vs. the absolute temperature. As expected from the results of high strain rate tests, the sensitivity of the flow stress on the deformation temperature decreases in solution hardened steels, since an increase in strain rate is equivalent to a decrease in temperature in a thermally activated process. Thus, it can be concluded that solid
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Fig. 2. Variation with the strain rate of (a) the true tensile flow stress and (b) the effective stress at 5% strain and room temperature. Solid curves in (a) show the results of fitting data by quadratic expressions of logarithmic strain.
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Fig. 3. Variation of the maximum uniform elongation with strain rate. Curves fitted to experimental data: 1: IF steel, 2: IF + 1at.%Mn, 3: IF + 2at.%Si, 4: IF + 2at.%Mn + 2at.%Si and 5: IF + 4at.%Si.
Fig. 5. Variation of the true tensile stress at 5% strain with respect to the deformation temperature at a strain rate of 0.0033 s⫺1.
solution causes the decrease of the sensitivity of the flow stress on both strain rate and temperature. The changes of the maximum uniform elongation with temperature are shown in Fig. 6. For the
Fig. 4. Nominal stress–strain curves at several deformation temperatures measured at a strain rate of 0.0033 s⫺1. Fig. 6. Variation of the maximum uniform elongation with temperature. Curves fitted to experimental data: 1: IF steel, 2: IF + 1at.%Mn, 3: IF + 2at.%Si, 4: IF + 2at.%Mn + 2at.%Si and 5: IF + 4at.%Si.
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IF steel, it is almost constant until 240 K and it decreases rapidly at lower deformation temperatures, whereas it has a tendency to increase with decreasing temperature for the other investigated steels. It has been reported that, with decreasing temperature, the dislocation distribution becomes more uniform at any given strain, and thus the work hardening rate tends to increase in iron [9]. Then, it is expected that the maximum uniform elongation increases with decreasing temperature, which explains the temperature sensitivity of the ductility of the tested steels, except for the IF steel. An explanation of the latter exception could be again provided by considering the effects of the adiabatic heating. Because IF steel is a strain rate and temperature sensitive material, the plastic work dissipated during deformation increases rapidly with a decrease in temperature and it might cause early necking even at lower strain rates.
4. Discussion 4.1. Origin of the increase in flow stress at low strain rates Before discussing the results, the origin of the increase in flow stress at low strain rates for the tested steels will be examined. For iron-based alloys, Leslie [10] made an extensive study on the effects of alloying elements. The effects of Si and Mn on the increase in flow stress can be evaluated based on his results. The differences in flow stress at 5% tensile strain between solution hardened steels and the host steel are plotted as a function of the atomic concentration of solute atoms in Fig. 7. Although it was pointed out that the dilation of the lattice in Fe–Si alloys does not vary linearly with solute concentration, the strengthening by Si and Mn seems to be proportional to concentration, as shown in this figure. The slopes ⌬s/⌬c of the two straight lines are 54.3 (MPa/at.%) and 36.5 (MPa/at.%) for Fe–Si and Fe–Mn alloys, respectively. Because one of the tested steels contains both Si and Mn, the increase in the flow stress due to solid solution hardening was calculated by cumulating the weighted contributions of the alloying elements and again compared with the
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Fig. 7. Increase in the flow stress vs. atomic percent of solute atoms. The slopes of the two straight lines show the solution hardening ability of the corresponding elements.
experimental data. In Fig. 8, the abscissa axis shows the increase in the flow stress due to solid solution estimated from the solute concentrations and the slopes in Fig. 7, while the ordinate axis shows the corresponding increase obtained from
Fig. 8. Comparison of the increase in flow stress due to solid solution between the calculations estimated from the solute concentration and the slopes in Fig. 7 and the experimental data.
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experiments. The data are located on the diagonal line, which shows that almost all the increase in flow stress of the tested steels is due to the alloying by Si and/or Mn. 4.2. Decrease in the strain rate sensitivity of the flow stress caused by solute atoms In this section, we will use thermal activation analysis in order to get some supplementary insight into the deformation mechanisms and thus to explain the temperature and strain rate effects on the flow stress in the investigated materials. A gliding dislocation, subjected to a resolved shear stress t, encounters during its motion two kinds of obstacles: extended obstacles generating a long-range stress field, whose resolved shear stress is denoted by tm, and local obstacles, which act only over a few atomic distances and, unlike the former, can be overcome with the help of thermal fluctuations, under the action of the local effective stress t∗ ⫽ t⫺tm.
(1)
The long-range stress has been denoted by tm because it is proportional to some effective shear modulus m, and depends on temperature only via the slight temperature dependence of the elastic constants. On the other hand, it depends strongly on the microstructure and generally evolves with progressing strain. By using rate theory and some statistical assumptions concerning the distribution and strength of the local obstacles, the slip rate on any active slip system may be written as g˙ ⫽ g˙ 0exp(⫺⌬Gt / kT),
(2)
where k is Boltzmann’s constant, T is the absolute temperature, g˙ 0 is a material dependent preexponential factor, and ⌬Gt denotes the difference in free enthalpy between the configuration of the dislocation segment in the saddle point and in the ground state for forward jumps, which depends in turn on the interaction between the dislocation segments and the local obstacles (see, e.g., [11]). For the present study, the experiments were conducted in uniaxial tension and the specimens were polycrystalline materials. Thus, the tensile stress
should be converted to the resolved shear stress acting in each grain and on each slip system, by eventually taking into account the long-range stresses produced by the grain interaction, whereas the tensile strain rate results by volume averaging the contributions of all slip systems and grains. While this conversion can be eventually carried out for a given texture, the quantitative result obtained depends critically on the simplifying hypotheses employed in the micro–macro transition (see, e.g., [12]). We shall, therefore, adopt here an alternative, more phenomenological approach, namely the equations governing the thermally activated plastic slip will be transcribed at the macroscopic scale, by simply replacing the shear rate g˙ and the shear stress t by the tensile flow stress s and the tensile strain rate e˙ , respectively: s∗ ⫽ s⫺si,
(3)
e˙ ⫽ e˙ 0exp(⫺⌬G / kT).
(4)
Clearly, the newly introduced phenomenological quantities and parameters cannot be uniquely defined by using a rigorous micro–macro transition. Nevertheless, they can be supposed to inherit some of the physical significance of their microscopic counterparts. Thus, si is related to the intensity of the long-range obstacles to dislocation glide; e˙ 0 depends essentially on the density of the mobile dislocations, on the average spacing of the pinning points along the dislocation lines and on the average distance between the local obstacles; ⌬G is an apparent activation free enthalpy, which depends on the effective tensile stress s∗ and on the apparent free energy of interaction between the gliding dislocations and the local obstacles. By definition, the apparent activation volume V∗ is expressed as ∂(⌬G) V∗ ⫽ ⫺ ∂s∗
|
.
(5)
T=const
Solving Eq. (4) for ⌬G and introducing the result obtained into Eq. (5) yields ∂(lne˙ ) V∗ ⫽ kT ∂s∗
|
.
(6)
T=const
This alternative expression has been used to
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determine the dependence of the activation free enthalpy on the effective tensile stress from the results of our high strain rates experiments. Previous researchers investigating the strain rate sensitivity of steels [4,14,15] have in general arbitrarily taken t∗ = s∗ / 2, and represented the variation of the activation volume ∂(lne˙ ) Vt∗ ⫽ kT ∗ ∂t
|
⫽ 2V∗
(7)
T=const
with respect to t∗. For the thermal activation analysis, it is preferable to use the flow stress at the beginning of deformation, because the effects of the thermal softening are not negligible, especially at high strain rate. On the other hand, all internal variables change rapidly at yield point and the deformation is not steady-state. Furthermore, the accurate measurement of the upper yield limits for high strain rate tests is difficult, because it is influenced by experimental configurations, such as the stiffness of the machine, specimen size, loading conditions, etc. The present analysis has been done, therefore, by using the strain rate sensitivity of the flow stress at 5% tensile strain, which appeared to be the lowest strain value at which the transitory yield point phenomena can be considered negligible. Then, the activation volume could have been calculated directly from the data shown in Fig. 2a. However, this procedure proved to be insufficiently accurate, because of the fluctuation of the experimental errors. Therefore, the experimental variation of the flow stress has been first fitted by a quadratic polynomial in terms of the logarithm of the strain rate and then si was considered as a minimum value of the interpolating function. Subtracting this latter value from the experimental flow stress resulted in the variation of the effective stress s∗ in terms of the logarithm of the strain rate, as represented in Fig. 2b. Finally, the activation volume has been evaluated from the slopes of the curves in Fig. 2b, by using Eq. (6). As shown by Uenishi and Teodosiu [7], this procedure is equivalent to directly using Eqs. (3)–(5) in conjunction with the phenomenological formula for
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the activation free enthalpy proposed by Kocks et al. [13],
冋 冉 冊册
⌬G ⫽ (⌬F) 1⫺
s∗ s0
p q
,
(8)
where s0 characterizes the average resistance to slip of the local obstacles and ⌬F is the apparent free energy of interaction between the gliding dislocations and the local obstacles, provided that we set p = 1 / 2 and q = 1. Fig. 9 shows our results for IF steel, converted for the sake of comparison to t∗⫺V∗t data, along with results obtained by other authors for similar materials. The results of Harding [14] were obtained from experiments at a fixed temperature over a very wide range of high strain rate tests, as in the present work. On the other hand, the results of Spitzig and Leslie [4] were derived from a change in stress due to strain rate change, at low temperatures and relatively low strain rates. The good agreement of these different sets of experimental data is not trivial. We have already seen similar behaviour of the flow stress at high strain rates and at low temperatures (cf. Figs. 2 and 5). The conformity of the activation volume at high strain rates and at low temperatures for the IF steel
Fig. 9. Comparison of the activation volume of IF steel with previous results for similar materials. The activation volume was evaluated from experiments carried out either at high strain rates ([14] and the present study) or at low temperatures [4].
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gives supplementary support to the conjecture that the rate-controlling mechanism at high strain rates in this material is the same as that at low temperatures. Furthermore, since the activation volume of the IF steel ranges from 10 to 100 b3, and thus corresponds to the activation volume that is generally ascribed to the Peierls–Nabarro mechanism, it can be concluded that this mechanism controls the deformation rate of the IF steel even at high strain rates, as already assumed by other authors [4,15]. Fig. 10 shows the variation of the activation volume of solution hardened steels as a function of the effective stress. At a lower stress level, the activation volume increases with increasing strength of the materials, i.e. with increasing concentration of solute atoms. The activation volume of 1 at.% Mn steel decreases more rapidly than that of IF steel and shows a crossover with that of IF steel. Similar changes of the activation volume due to solution hardening have been observed for Fe–Ni and Fe–Si by several authors [4,15], although the corresponding data have been obtained from low temperature tests. Their results show that the plots
Fig. 10. Activation volume of solution hardened steels as a function of the effective stress. At a lower stress level, the activation volume increases by solid solution. At higher effective stress, the activation volume of solution hardened steel decreases more rapidly than the host steel, while the variation of the activation volume of 1 at.% Mn steel shows a crossover with that of the host metal.
of the activation volumes vs. effective stress of all solution hardened steels intersect with that of the host metal and that the intersection point moves to higher stresses with increasing concentration of the solute atoms. In our results, not all solution hardened steels show a crossover. This is probably due to the fact that for higher concentrations of the solute atoms, the effective stress does not reach the level that is needed for the crossover for strain rates below 1000 s⫺1. It is known that in bcc metals a small addition of alloying elements can decrease the temperature sensitivity of the flow stress at low temperatures [8]. This phenomenon has been called “alloy softening”. At room temperature, alloying almost always increases the strength of the material. However, alloy softening indicates that the properties at room temperature cannot be extrapolated monotonically to lower temperatures. It should be pointed out that Fe–Ni and Fe–Si alloys are typical materials that show “alloy softening” at low temperatures [4,15]. Thus, the behaviour of the activation volume of these alloys should be interpreted in connection with the decrease of the temperature sensitivity due to alloying. The decrease in the strain rate sensitivity of the flow stress of the tested steels can be ascribed to the solid solution effect, because all other conditions, such as hot rolling and annealing conditions, are almost identical. According to the theory of thermally activated processes, an increase in strain rate is equivalent to a decrease in temperature. Furthermore, as shown in Fig. 5, the temperature sensitivity of the flow stress of solution hardened steels is lower than that of host metal, a tendency already emphasized by previous studies carried out at low temperatures [8]. Thus, the decrease in strain rate sensitivity due to alloying at high strain rates can be discussed in the same manner as the “alloy softening” at low temperatures. The models for alloy softening can be roughly divided into two groups, extrinsic and intrinsic theories. One of the most widely used extrinsic theories is the scavenging model. It assumes the presence of residual interstitial atoms, which increase the yield stress of base metals. The attraction or repulsion between newly added alloying atoms and residual atoms promotes a clustering of
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dispersed interstitial atoms and then the yield stress decreases due to alloying. However, there are experimental results showing that the alloy softening is insensitive to variations in purity level (see, e.g., [15]), thus directly contradicting the scavenging theories. In addition, the steels tested in this study, which contain Ti in order to obtain excess interstitial atoms such as carbon and nitrogen, also exhibit alloy softening at low temperatures, which can hardly be explained by the scavenging model. Thus, the intrinsic changes in lattice resistance due to alloying seem to be more probable. As seen before, it can be concluded that the strain rate sensitivity of IF steel is controlled by the Peierls–Nabarro mechanism, in which the elementary processes are the nucleation and propagation of kinks. Thus, in order to understand the effects of solid solution, it is important to take into account the effects of solute atoms on these elementary processes. Ono and Sommer [16] were the first to calculate the flow stress of a crystal with a high Peierls–Nabarro stress, by considering that point obstacles, such as solute atoms, affect the formation of kinks by limiting the length of the dislocation segments. They found that point obstacles have little or no effect on the nucleation process of double kinks when the applied stress is nearly equal to the Peierls–Nabarro stress, e.g. at low temperatures. At higher temperatures, the flow stress becomes independent of the temperature and proportional to the concentration of point obstacles. The predictions of Ono and Sommer [16] compare favourably with the experimental data obtained by Nakada and Keh [17] for Fe–N alloys. However, their calculation cannot reproduce the crossover of the activation volume observed in low temperature experiments of Fe–Ni and Fe–Si alloys and in the present high strain rate experiments. The theory proposed by Sato and Meshii [18] can explain both solid solution softening and solid solution hardening by a unified model. These authors extended the Ono–Sommer model, by considering explicitly the interaction between a kink on a screw dislocation and solute atoms, described as misfit strain centres. The couple forces due to this interaction promote the nucleation of double kinks in the vicinity of misfit strain centres. Solid
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solution softening is attributed to this promotion. On the other hand, strong misfit centres will provide greater resistance to the overcoming of the solute atoms by the dislocations. Therefore, these combined effects on the overall dislocation motion generate two antagonistic softening and hardening effects of the solute atoms. Sato and Meshii [18] calculated the dependence of the activation volume on the strength of the misfit strain centre. The activation volume increases with increasing strength of the misfit strain centre and shows a crossover with the activation volume of the host metal for some particular conditions. A quantitative comparison of our results with their calculation is difficult, because (i) their calculation is based on several simplifications concerning the interaction between dislocations and point obstacles and (ii) the evaluation of the strength and the spacing of misfit strain centres is difficult for the present materials due to the fluctuation of the local concentration and the clustering of solute atoms. However, the effects of solute atoms at high strain rates can be qualitatively explained by both the promotion of the nucleation of kinks and the inhibition of their movements, which may yield a correlation of the strain rate sensitivity of the flow stress and solid solution hardening. In Fig. 11, the difference in the flow stress at 5% tensile strain between 0.001 and 1000 s⫺1 is plotted against the flow stress at 5% tensile strain and 0.001 s⫺1. Clearly, the strain rate sensitivity decreases sharply with increasing quasi-static strength, i.e. solid solution hardening at lower strain rates. 5. Conclusions The high strain rate properties of solution hardened steels have been investigated. The strain rate sensitivity of the flow stress decreases with quasistatic strength, i.e. by addition of solute atoms. The elongation at high strain rates decreases rapidly for a strain rate sensitive material, due to the adiabatic heating produced by the plastic work during deformation. For solution hardened steels, these effects are less pronounced because of the low strain rate sensitivity of their flow stress. As a result, the elongation at high strain rates is improved by addition of solute atoms.
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Acknowledgements This work has been supported by Nippon Steel Corporation.
References
Fig. 11. Strain rate sensitivity shown as the difference in the flow stress at 5% strain between 0.001 and 1000 s⫺1 vs. the flow stress at 5% strain at a strain rate of 0.001 s⫺1. The strain rate sensitivity of the flow stress decreases with increasing static strength.
From the results of the thermal activation analysis, it can be concluded that the Peierls–Nabarro mechanism controls the strain rate sensitivity of the flow stress in the IF steel and the effects of solute atoms on high strain rate properties are similar to those at low temperatures. The model proposed by Sato and Meshii [18], in which the interaction between solute atoms and a kink on a screw dislocation is explicitly considered, can explain the peculiar behaviour of the activation volume of solution hardened steel. In particular, the decrease of strain rate sensitivity of the flow stress in solution hardened steel may be ascribed to the promotion of the nucleation of kink pairs by the misfit strain produced by solid atoms. The correlation between solid solution hardening at low strain rates and the decrease in the strain rate sensitivity at high strain rates can be at least qualitatively explained by this model.
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