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Constitutive modelling of the high strain rate behaviour of interstitial-free steel
International Journal of Plasticity 20 (2004) 915–936 www.elsevier.com/locate/ijplas
Constitutive modelling of the high strain rate behaviour of interstitial-free steel A. Uenishi*, C. Teodosiu LPMTM-CNRS, University Paris 13, 93430 Villetaneuse, France Received in final revised form 27 June 2003
Abstract A physically based modelling and experimental investigation of the work hardening behaviour of IF steel covering a wide range of strain rates including complex strain path and/or strain rate changes are presented. In order to obtain isothermal stress–strain curves at high strain rates, a procedure has been proposed with the aid of finite element analysis. The result reveals that the apparent excess of the flow stress after a jump in strain rate, which is frequently observed in bcc metals, is in fact due to the thermal softening at large strains, and that the flow stress after a jump in strain rate tends asymptotically to the values corresponding to the curve at the new strain rate. The strain rate affects not only the short-range stress but also the long-range stress via the strain-rate dependant evolution of dislocation structures. The proposed model is based on the dislocation model of intragranular hardening proposed by Teodosiu and Hu [Teodosiu, C., Hu, Z., 1995. Evolution of the intragranular microstructure at moderate and large strains: modelling and computational significance. In: Shen, S., Dawson, P. R., (Eds.), Proceedings of Numiform’95 on Simulation of Materials Processing: Theory, Methods and Applications. Balkema, Rotterdam, pp. 173–182] and extended to strain rate sensitive one with applying the results of the thermal activation analysis. A satisfactory agreement has been achieved between model predictions and experimental results. # 2003 Elsevier Ltd. All rights reserved. Keywords: Constitutive Behaviour; Viscoplastic material; Finite elements; Dynamics plasticity
* Corresponding author at present address: Nippon Steel Research Laboratories, Nippon Steel Corp., 20-1 Shintomi, Futtsu-city, Chiba, 293-8511, Japan. Tel.: +81-4-39-803113; fax: +81-4-39-802743. E-mail address: [email protected] (A. Uenishi). 0749-6419/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2003.06.004
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1. Introduction The demand for increased vehicle safety has become a matter of considerable concern of users and tends to increase the weight of the car body. On the other hand, responding to the world-wide consciousness of environmental protection, the reduction of fuel consumption becomes a very important challenge. Thus, it is essential to reduce the car body weight. Clearly, meeting these contradictory requirements, i.e. designing a lightweight car with enhanced safety, requires an optimum combination of body structure and material, which necessitates, in turn, an adequate physical and numerical modelling of the mechanical behaviour both at low and high strain rates. Several points should be taken into account for this modelling. The materials used in the energy absorbing parts of a car are deformed at relatively low strain rates during press forming and eventually subjected during a crash to high strain rate deformations along different strain paths. It is, therefore, necessary to take into account the effects of the deformation history and of the eventual changes in strain path. In addition, the results of the high strain rate tests should be carefully analysed, in order to identify the specific effects of adiabatic heating and to separate the temperature and strain rate effects. In the present study, the thermo-mechanical analysis of a comprehensive set of high strain rate tensile tests has been conducted by using the finite element code MARC. Based on these data and on the thermal activation analysis of the deformation processes, a new constitutive model taking into account the deformation history has been proposed and identified.
2. Experimental procedure and results 2.1. Materials and high strain rate tensile tests The material studied is a 0.8 mm thick sheet of a deep-drawing quality IF steel (mass%: 0.006 C, 0.16 Mn, 0.038 Ti, 0.024 Al, 0.0015 Si, 0.005 P and 0.004 S), hot and cold rolled and subsequently subjected to recrystallization and annealing. This material is extensively used in car parts that require high formability. The mechanical behaviour including complex strain-path changes of the same material has been thoroughly investigated by Nesterova et al. (2001), including a detailed analysis of the evolution of texture and microstructure. The work hardening behaviour is greatly influenced by strain-path changes and displays peculiar cross and Bauschinger effects. These characteristics have been related to the microstructural evolution and have been successfully expressed in terms of the dislocation-based intragranular hardening model proposed by Teodosiu and Hu (1995). The stress–strain curves at strain rates higher than 100 s1 have been measured by using the so-called one-bar technique, whereas a conventional Instron screw-driven test machine has been used at strain rates below 0.1 s1. In addition to the monotonic tests, strain rate jump tests were also carried out in order to study the effect of
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a prestrain on the subsequent mechanical behaviour. The specimen was first subjected to uniaxial tension along the rolling direction at a strain rate of 0.001 s1 and subsequently to uniaxial tension in the same direction at 1000 s1. The usual way of characterizing the mechanical behaviour of a material by a uniaxial tensile test is to measure the loading force associated with the change of the length of the specimen. At a conventional strain rate, the load cell is considered to deform homogeneously and the loading force is measured by a strain gauge attached to it. As the strain rate increases, the time needed to attain the homogeneity of elastic deformation within the load cell approaches the testing time, and this leads to the necessity of considering the wave propagation within the load cell. At strain rates higher than about 10 s1, the signal of the loading force is greatly perturbed by multiple passages of waves reflected within the load cell if a conventional configuration is being used. This could be eventually avoided by increasing the length of the load cell, in order to finish the measurement before the arrival of the elastic wave reflected at the other end. This technique is usually called a split Hopkinson pressure bar method. For the tensile tests of sheet metal, a one-bar technique has been developed by Kawata et al. (1985), based on the Hopkinson bar method. As shown in Fig. 1, the testing system consists of a hammer, an impact block, a specimen and an output bar. When the impact block is given an impact by the hammer, the specimen is deformed in tension. At the instant of the impact, a transmitted wave starts to propagate in the output bar, its amplitude being proportional to the stress in the specimen. This wave is recorded by strain gauges attached to the output bar at section C, situated at a distance a from section B. In addition, an electro-optical extensometer is used to measure the velocity V(t) of the impact block, which is integrated to give the displacement of section A ðt uA ¼ Vð Þd: ð1Þ 0
Fig. 1. Schematic representation of the one-bar method high strain rate tensile tests.
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An analysis of the propagation of the elastic waves in a bar enables to derive the displacement uB of section B, from the strain of the transmitted wave "g ðtÞ at section C. The propagation of waves in a bar is considered as a one-dimensional problem, when the lateral inertia can be neglected. Considering the delay of the wave propagation, uB can be expressed in terms of "g as ðt ðt uB ¼ c "B ðtÞ dt ¼ c "g ð þ a=cÞd: ð2Þ 0
0
where c is the sound velocity in the bar. The elongation of the specimen is the difference between the displacements at sections A and B. By using Eqs. (1) and (2), the engineering strain and the engineering strain rate of the specimen may be expressed as ð uA uB 1 t eðtÞ ¼ ¼ VðÞ c"g ð þ a=cÞ d; ð3Þ L0 0 L0 1 : e ðtÞ ¼ VðtÞ c "g ðt þ a=cÞ ; L0
ð4Þ
where L0 is the length of the specimen. The axial force F(t) at section B can also be determined from the amplitude of the transmitted wave "g ðtÞ as FðtÞ ¼ Ebar Abar "B ðtÞ ¼ Ebar Abar "g ðt þ a=cÞ:
ð5Þ
Thus, the engineering stress in the specimen becomes n ðtÞ ¼
FðtÞ Abar Ebar ¼ "g ðt þ a=cÞ: A0 A0
ð6Þ
2.2. Stress–strain curves at high strain rates Stress–strain curves at different strain rates are shown in Fig. 2 as diagrams of engineering stress vs. engineering strain. The flow stress, the upper yield limit and the difference between the upper and lower yield limits increase with strain rate. However, the apparent work hardening rate decreases at high strain rates. Similar behaviour has been reported also for other bcc metals by Khan and Liang (1999). The results of strain rate jump tests are also shown in Fig. 2. The flow stress level after a strain rate jump is higher than the entirely constant strain rate curves for all tested conditions. There are several results in the literature concerning the mechanical behaviour of steels after a jump to a higher strain rate. Wilson et al. (1979) reported that the flow stress after the jump does not increase instantaneously to the value corresponding to the higher constant strain rate, but it approaches asymptotically this value for lowcarbon hot-rolled steels. This behaviour is similar to that generally observed in fcc
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Fig. 2. Stress–strain curves of an IF steel at different strain rates, including strain rate jump tests: (1) monotonic tensile test at 1000 s1; (2) tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) after a tensile prestrain of 16% at 0.001 s1; (4) monotonic tensile test at 0.001 s1.
metals (Chiem and Duffy, 1983; Duffy, 1979). On the other hand, it has been reported that the flow stress after a jump in strain rate exceeds that of a constant strain rate test at the higher strain rate for a hot rolled steel that contained 0.25% C (Klepaczko and Duffy, 1982). The results obtained in the present study are indeed closer to this latter finding. One way to solve such apparently contradictory results may be to take into account the thermal softening caused by adiabatic heating at high strain rates. Indeed, since the flow stress of bcc metals, including steels, is much more strain-rate sensitive than that of fcc metals, the dissipated plastic work is expected to increase more rapidly with strain rate in bcc than in fcc metals. Consequently, for a material that has both a high strain-rate sensitivity and a high temperature sensitivity of the flow stress, the adiabatic heating accelerates the onset of plastic instability at high strain rates, thus causing a necking to occur in an early stage of deformation. As a result, the apparent flow stress calculated by assuming a uniform deformation in the tensile zone of the specimen decreases rapidly with strain at high strain rates. Thus, in order to understand the results of strain rate jump tests, it is necessary to follow the necking process by a thermo-mechanical analysis. Furthermore, the heat induced by the plastic work causes thermal softening and changes the global response of the specimen. Therefore, for the correct assessment of mechanical tests involving strain rate jump tests and for the development of a constitutive model, it is essential to evaluate the effects of the thermal softening and to obtain isothermal stress–strain curves.
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3. Finite element analysis of the high strain rate tests 3.1. Finite element model of the one-bar method As seen in the previous section, high strain rate properties are measured by using a special technique, called the one-bar method. For high strain rate tensile tests, the specimen geometry has to satisfy various requirements. In order to assure a homogeneous deformation of the specimen, the length of the deformed zone of the specimen should be short enough in order for the elastic wave to propagate before the plastic yielding of the material. Furthermore, since it is experimentally difficult to measure large strains at high strain rates, a small radius has to be used at the shoulder of the specimen, in order to reduce the constraint of the deformation around the shoulder and to simplify the strain analysis for the part of the specimen of uniform cross-section. All these conditions require a special geometry of the specimen, which is rather different from the one used at conventional strain rates. One of the main drawbacks of the high strain rate tensile test is that the results concern the global response of the specimen and do not directly characterize the material behaviour in the tensile zone. Although it has been pointed out that the geometry of specimens greatly influences experimental stress–strain relations tested at high strain rates in shear (Rusinek and Klepaczko, 2001) and in torsion (Gilat and Cheng, 2002), its effects are more complicated in tension due to the phenomenon of necking and an eventual fracture of specimen. Accurate evaluation of the stress is usually difficult after the occurrence of the diffuse necking, as the reduction of the cross-section of the specimen becomes non-uniform. A numerical study is thus necessary to take into account the geometry of the specimen, the necking and the thermal softening. To this end, a thermo-mechanical analysis of the high strain rate tensile test has been made, by using the finite element code MARC, in order to convert the experimental force–elongation data into true stress–true strain curves. By considering the symmetry of the problem in the thickness and the tensile directions, only one quarter of the specimen has been analysed. As shown in Fig. 3, the FE model consists of two parts, a tensile specimen and its reinforcement. In the experiment, the tensile force is applied through a rod that penetrates a hole of the
Fig. 3. Finite element mesh.
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specimen. A reinforcement plate is spot-welded onto the specimen to strengthen the hole. In the analysis, the spot weld is modelled by sharing the nodes of the specimen with those of the reinforcement at the position of the spot welds. The rod is modelled as a rigid body and its movement controls the process. The other edge of the half-specimen is clamped in the loading direction. The FE model has 2730 eight-node solid elements and 4887 nodes. The elements have eight integration points with reduced integration for the volumetric part of the deformation. The thickness of the specimen is divided into two layers of solid elements. 3.2. Adiabatic heating during plastic deformation Large plastic deformation induces an increase in temperature of the material, which is proportional to the plastic work. The plastic dissipation is strain dependent and also strain rate dependent in the case of a strain rate sensitive material. The heating of the specimen due to the dissipated plastic work can be derived from the equation C
dT : ¼ ij "ij ; dt
ð7Þ
where denotes the mass density, C the specific heat, T the temperature, t the time, the heat conversion ratio taken equal to unity in case of an adiabatic heating, ij : the Cauchy stress tensor, "ij the strain rate tensor. The calculation has been carried out by taking =7.874103 kg/m3 and C=452 J/kg K, both assumed to be independent of the temperature. 3.3. Numerical identification of a simplified thermo-viscoplastic constitutive model In order to simulate the high strain rate tensile test, it is convenient to take into account the effects of both temperature and strain rate by first using a simplified constitutive model. In the code MARC, it is possible to use to this end the JohnsonCook law " : ! #
: " T Troom m n "; " ; T ¼ ðA þ B" Þ 1 þ Clog : ð8Þ 1 "0 Tref Troom : where , " and " are the equivalent tensile stress, the equivalent tensile strain and the equivalent tensile strain rate, respectively. The first term represents the work hard: ening behaviour at the reference strain rate " 0 and at the room temperature. The second and third terms express the effects of strain rate and temperature, respectively. The parameters A, B, n, C and m are material parameters and T denotes the absolute temperature. Troom and Tref are the absolute room temperature and melting temperature of the material, in our case 293 and 1809 K, respectively. In fact, it is not possible to cover a wide range of strain rates and temperatures by using Eq. (8), because the real material behaviour is much too complex. However, it
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is possible to use this simplified law in order to give an appropriate description of the material response over a limited range of strain rates and temperatures. Identification of the parameters in Eq. (8) is complicated by the fact that the experimentally obtained stress–strain curves are not isothermal. The parameters related to the strain rate and temperature dependence of the flow stress have been determined by using the results of the thermal activation analysis, which will be presented in Section 4. The work hardening parameters A, B and n have been determined in an iterative way, by comparing the engineering strain and the engineering stress between the experiment and the calculation. In the tensile experiment the engineering strain e and the engineering stress n are calculated by e¼
L L0 P ; n ¼ ; A0 L0
ð9Þ
where L is the current length of the constant-cross section part of the specimen, L0 the initial length, P the tensile force and A0 the initial cross section of the specimen. The engineering strain vs. the engineering stress curve obtained by a numerical analysis has been compared to the experimental one (Fig. 4). Except for the initial peak, which is not considered in the Johnson–Cook model, the comparison shows a good agreement. The beginning of the diffuse necking is well reproduced. At higher
Fig. 4. Comparison of the engineering strain vs. engineering stress diagrams between the experiment (dashed line) and the FEM calculation (solid line).
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strains, an increasing discrepancy between the two curves was observed. There could be several reasons for this: (i) The strain rate and temperature sensitivity of the flow stress is not correctly expressed by the Johnson–Cook constitutive law at large strains. (ii) The damage accumulated during deformation causes a decrease in the flow stress, which becomes apparent at later stages of the deformation. (iii) A coarse FE mesh postpones the localization of the deformation. However, this effect may be considered negligible in our case, since very fine meshes were used in the FE model. Although we could not simulate the experimental curve perfectly, we considered the results to be sufficient for our purpose and tried to analyse the calculated results in terms of the stress and strain distributions. As discussed before, the specimen for a high strain rate tensile test is different from conventional tensile specimens. Thus, the level of homogeneity in the specimen has been checked. Fig. 5 shows a longitudinal distribution of the components of the Cauchy stress tensor in the middle of the specimen. The stress distribution is nonuniform at both ends of the specimen. However, in the central part of the specimen, a uniaxial tensile stress state is confirmed. It may be concluded that the peculiarity of the specimen geometry does not significantly affect the stress distribution in the central part of the specimen.
Fig. 5. Variation along the longitudinal axis of the specimen of the Cauchy stress components in a frame x1x2x3, with the x1-axis along the tensile direction and the x2-axis along the transverse direction for an average true strain of 20%.
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In the tensile experiment, the engineering stress is evaluated by Eq. (9) and then the true stress and the true strain " are calculated, under the assumption that the specimen deforms uniformly, by using the formulas
¼
P P A0 A0 L0 L ¼ ¼ n ¼ n ð1 þ eÞ; A A0 A A L L0
" ¼ ln
L : L0
ð10Þ
ð11Þ
First, the method of the strain evaluation is examined. As shown by Eq. (11), the strain is evaluated from the current and initial lengths of the constant-section of the specimen. The FEM value of the true strain at the centre of the specimen has been compared with the values obtained by using Eq. (11) and different gauge lengths L0. At the beginning of the deformation the difference is small, but it increases rapidly after necking. In our case, as shown by Fig. 6, the best agreement with the FEM simulation is obtained by using a gauge length of about 4 mm, while the maximum gauge length used in Eq. (11) was L0=10 mm.
Fig. 6. Variation of the longitudinal logarithmic strain obtained by different evaluations: (a) strain calculated by FEM at the centre of the specimen; (b) average strain evaluated using a gauge length of 7 mm; (c) average strain evaluated using a gauge length of 4 mm. The abscissa values are the corresponding strains evaluated by using as gauge length the total length L0 of the constant cross-section of the specimen. When decreasing the gauge length, the evaluated strain approaches the FEM result and coincides with the latter for gauge lengths less than 4 mm.
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After the occurrence of a diffuse necking, the stress evaluation by Eq. (10) is not accurate, for the decrease of the cross section at the centre is more rapid than that calculated under the assumption of a uniform decrease of the specimen width. The real true stress in the centre becomes larger than that calculated by Eq. (10). However, the evaluation of the stress at the centre of the specimen from the measured tensile load is possible even after necking, provided that the change of the cross section is correctly obtained by a FE simulation. Therefore, the global experimental data have been converted to local data at the centre of the specimen in the following way. Using the results of FE analysis shown in Fig. 6, the logarithmic strain at the centre of the specimen is evaluated as a function of lnðL=L0 Þ, say "centre ¼ fðlnðL=L0 ÞÞ: 11
ð12Þ
Then, the true stress at the centre is obtained by
¼
P P A0 ¼ ¼ n exp "centre ; 11 Acentre A0 Acentre
ð13Þ
where Acentre is the cross-sectional area of the specimen at its centre. Rigorously, the relationship Eq. (12) should be evaluated for each experimental condition. However, since a slight change of the material behaviour does not affect significantly this relation, the same master curve of the monotonic high strain rate test has been used also for the experiments involving a jump in the strain rate. The FE analysis has been made under the assumption that the process is adiabatic. The increase in temperature at the centre of the specimen is observed to be some tens of degrees, thus producing a significant thermal softening of material. However, by using the identified Johnson–Cook law, Eq. (8), it has been possible to separate the effect of the thermal softening in a numerical analysis and to obtain isothermal stress–strain relations. The results of strain rate jump tests (Fig. 2) show that the flow stress levels after a strain rate jump are higher than the entirely curve obtained at the higher constant strain rate. However, after compensations for the decrease of cross-section and the increase in temperature (Fig. 7), the flow stress level after a strain rate jump is comparable to the isothermal curve at the higher constant strain rate. Actually, these results resemble the behaviour reported for fcc metals (Chiem and Duffy, 1983; Wilson et al., 1979; Duffy, 1979), although a complete quantitative assessment is difficult, due to the already mentioned limitations of the FE analysis at large strains. Thus, the apparently different behaviour reported for some steels in strain rate jump tests, can indeed be ascribed to the thermal softening and the associated acceleration of strain localization caused by the adiabatic heating, which can be explained, in turn, by the higher strain-rate sensitivity of steels with respect to that of fcc metals.
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Fig. 7. Corrected stress–strain curves including strain rate jump tests: (1) monotonic tensile test at 1000 s1; (2) tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) tensile test at 1000 s1 after a tensile prestrain of 16% at 0.001 s1; (4) monotonic tensile test at 0.001 s1; (5) stress–strain relation at the centre of the specimen calculated by FEM for the monotonic tensile test at 1000 s1.
4. Thermal activation analysis Here and in what follows, we shall make use of the previously obtained isothermal stress-strain relations for discussing the temperature and strain-rate sensitivity of the flow stress. Campbell and Ferguson (1970) have performed double shear tests covering a wide range of strain rates (0.001–10,000 s1) and temperatures (195–713 K) for a 0.12% carbon steel. In order to systematize and explain the results obtained, these authors have divided the domain of the yield stress vs. strain rate diagrams into regions associated with various deformation processes, as proposed by Rosenfield and Harn (1966). As shown in Fig. 8, three of these regions, labelled by I, II and IV, may interest the deformation of steels around the room temperature. The curves in region I are characterized by a small, nearly constant slope and the deformation is controlled by athermal mechanisms. In region II, the flow stress has a logarithmic dependence on the strain rate and the deformation is governed by thermally activated process. At very high strain rates (region IV), above 5000 s1, a dislocation drag mechanism, probably due to the interaction of the gliding dislocations with thermal phonons and conduction electrons, starts to contribute to the increase of the flow stress. During crash events, the maximum strain rate is expected to be of the order of 1000 s1. Thus, to model the crash behaviour of structural steels, it is important to
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Fig. 8. Variation of the lower yield stress with strain rate in a 0.12% C steel, after Campbell and Fergusson (1970). Three characteristic regions of the flow stress sensitivity may be distinguished: (I) small temperature and strain rate sensitivity; (II) greater temperature and strain rate sensitivity; (IV) rapid increase of the strain rate sensitivity.
describe the material behaviour at strain rates below 1000 s1 at about the room temperature. This corresponds to regions I and II in Fig. 8, and hence our analysis will be restricted to athermal and thermally activated processes, and will be aimed at evaluating the activation parameters from the measurements of the strain rate sensitivity of the flow stress. At larger strains, the effects of the thermal softening are not negligible, especially at high strain rates, as shown in the preceding section. Thus, it is better to use the flow stress at the beginning of deformation. On the other hand, all internal variables change rapidly at the yield point and the deformation is not yet quasi-stationary. In addition, the measurement of the upper yield limit is difficult at high strain rate tests, because it is influenced by the stiffness of the machine, specimen size, loading conditions, the rising time of the impact, etc. Recently, it has been pointed out that the micro-vibration of the output bar at the interface with the specimen, which is caused by a slight misalignment of the specimen and the impact block along the tensile direction, superposes on the measured signal and enhances the yield phenomena (Yoshida, 2001). Taking into account all these circumstances, we will use in our analysis the strain rate dependence of the flow stress at 5% tensile strain, which seems to be the lowest strain value at which the transitory yield point phenomena can be considered as negligible.
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The thermally-activated theory of the cold plastic deformation has been developed almost 50 years ago (Seeger, 1955) and has known since then a successful development for various types of crystalline materials and deformation conditions. Here we limit ourselves to a somewhat simplified version of this theory, referring to the review papers by Gibbs (1969), Frost and Ashby (1971), Teodosiu (1975), and Kocks et al. (1975) for more general settings. A gliding dislocation, subjected to a resolved shear stress ; encounters during its motion two kinds of obstacles: extended obstacles generating a long-range stress field, whose resolved shear stress is denoted by , and local obstacles, which act only over a few atomic distances and, unlike the formers, can be overcome with the help of thermal fluctuations, under the action of the local effective stress ¼ :
ð14Þ
The long-range stress has been denoted by because it is proportional to some effective shear modulus , 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. For a dislocation segment to surmount the strongest obstacle at zero absolute temperature, should equal some maximum slip resistance, say 0 ; called mechanical threshold (Kocks et al., 1975). However, at some temperature above 0 K, thermal fluctuations can assist the applied stress, and the dislocation can glide at a stress < 0 : As the temperature increases, can decrease until it becomes equal to the amplitude of the long-range stress field. Further increase in temperature does not give any additional decrease in the applied stress, since the energy barrier is too extended for thermal fluctuations to make a significant contribution. Alternatively, if the applied stress grows beyond 0 the dislocation segment can overcome the local obstacles with no thermal aid: local obstacles become penetrable to the dislocation motion. By using the rate theory it is possible to estimate the mean waiting time of a dislocation segment in front of a local obstacle and thus to correlate the average velocity of a mobile dislocation to the structure, applied stress, and temperature, by using some statistical assumptions concerning the distribution and strength of the local obstacles. However, for our present context, it is sufficient to retain only the essential features of the analysis, by adopting the following simplifying hypotheses: (i) the local obstacles have the same strength and are distributed uniformly in the glide plane; (ii) the frequency of the backward jumps, i.e. opposite to the effective stress, can be neglected with respect to that of the forward jumps; (iii) the flight time of the mobile dislocations between the local obstacles can be neglected with respect to the waiting time in front of the obstacles. With these simplifications, it may be shown that the average velocity of a gliding dislocation may be expressed as v ¼ 0 ‘F expðDG =kTÞ;
ð15Þ
where 0 is the attempt frequency of overcoming local obstacles, which is proportional to the Debye frequency and increases with decreasing distance between the pinning points along the dislocation line, ‘F is the average flight distance between two successive obstacles, while DG denotes the difference in the free enthalpy
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between the configuration of the dislocation segment in the saddle point and in the ground state for forward jumps, which is called the Gibbs free energy of activation. The slip rate on any active slip system is given by Orowan’s relation : ¼ b m v; ð16Þ ehere b is the magnitude of the Burgers vector, m is the density of the mobile dislocations and v is the average dislocation velocity in the direction of the slip vector for the considered slip system. Introducing Eq. (15) into Eq. (16) and assuming that the activation free enthalpy DG is the same for all dislocation segments of the slip system, yields : : ¼ 0 expðDG =kTÞ; ð17Þ : where 0 ¼ b m 0 ‘F : The activation free enthalpy DG depends on the interaction between the dislocation segments and the local obstacles. A detailed analysis of the various forms of this dependence has been given by Gibbs (1969) and by other authors. Kocks et al. (1975) have summarized these results into the formula
p q DG ¼ ðDF Þ 1 ; ð18Þ 0 with the parameters p and q lying within the ranges 0 4 p 4 1, 1 4 q 4 2. In Eq. (18), the quantity 0 characterizes the average resistance to slip of the local obstacles, while DF represents the average activation free energy required to overcome the local obstacles in the absence of an applied shear stress. 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, whereas the tensile strain rate results by volume averaging the contributions of all slip systems and grains. While such averages can be effectively calculated for a given texture, the quantitative result obtained depends critically on the simplifying hypotheses adopted in the micro-macro transition (see, e.g. Clausen, 1997). We shall, therefore, adopt an alternative approach, which is typical for the internalvariable formalism: Eqs. (14), (17) and (18) governing the thermally-activated plastic slip will be formally transcribed at the macroscopic scale,1 by simply replacing the : shear rate and the shear stress by the tensile flow stress and, respectively, the : tensile strain rate ": ¼ ;
ð19Þ
: : " ¼ "0 expðDG=kTÞ;
ð20Þ
1 The so-obtained one-dimensional model of the polycrystalline behaviour will be extended in the next : section to a three-dimensional model, : by further replacing and " by some equivalent stress and by its p power-conjugate plastic strain rate " , respectively.
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DG ¼ ðDFÞ
1
p q : 0
ð21Þ
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, i is related to the intensity of the long-range obstacles to : dislocation glide; " 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; DG is an apparent activation free enthalpy, which depends on the effective tensile stress and on the apparent free energy of interaction DF between the gliding dislocations and the local obstacles. By definition, the activation volume V is expressed as @ðDGÞ V ¼ : ð22Þ @ T¼const
Solving Eq. (20) for DG and introducing the result obtained into Eq. (22) yields : @ðln"Þ V ¼k T : ð23Þ @ T¼const This alternative expression has been used to 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 (Harding, 1969; Tanaka and Watanabe, 1971; Spitzig and Leslie, 1971) have in general arbitrarily taken ¼ ð1=2Þ ; and represented the variation of the activation volume : @ðln"Þ V ¼ k T ¼ 2V ð24Þ @ T¼const with respect to . Fig. 9 shows our results for IF steel, converted for the sake of comparison to V data, together with the results obtained by Harding (1969) and Spitzig and Leslie (1971) for similar materials. The good agreement of these three sets of data is by no means trivial. Indeed, the results of Harding (1969) were obtained from experiments at a fixed temperature from constant strain rate tests, at high strain rates, as in the present work. On the other hand, the results of Spitzig and Leslie (1971) were derived from jumps in strain rate at relatively low strain rates. The activation volume obtained from a strain rate jump tests is called the ‘‘true’’ activation volume. When it is calculated from stress–strain relations at several constant strain rates, it is called the ‘‘apparent’’ activation volume. It is known that for fcc metals these two values may show large differences (Chiem and Duffy, 1983). However, they seem to be rather closed for iron.
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Fig. 9. Variation of the activation volume with the effective stress.
By eliminating DG between Eqs. (20) and (21), we obtain "
1q #1p T ¼ 1 ; T0 0 where T0 ¼
DF
: "0 k ln : "
ð25Þ
ð26Þ
is the highest temperature for which, at a given strain rate, the thermal-activation analysis is valid. Smidt (1969) observed the temperature dependence of the flow stress of iron by temperature increment tests and found a relation, which can be written, with the present notation, as
T 2 ¼ 1 : ð27Þ T0 0 where 0 and T0 are constants. By comparing Eqs. (25) and (27), we find that the formula of Kocks et al. (1975) is satisfied, provided that we take p=1/2 and q=1. For these values of the parameters, Eq. (21) reduces to "
12 # DG ¼ ðDFÞ 1 : ð28Þ 0
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The strain rate dependence of the flow stress may be derived now, by first introducing Eq. (28) into Eq. (20), solving for and introducing the result obtained into Eq. (19). The result leads
: 2 k T "0 ln : ¼ þ ¼ þ 0 1 : ð29Þ DF " Fig. 10 shows the variation of the flow stress at 5% tensile strain with the strain rate. The strain rate sensitivity of the flow stress of IF steel is well reproduced by Eq. (29), which is shown in Fig. 10 by a solid line.
5. Constitutive modelling Based on the results shown in the preceding section, the dislocation model of intragranular hardening proposed by Teodosiu and Hu (1995) has been selected as a basis for the extension to a strain-rate sensitive constitutive model.2 The original model successfully describes various anisotropic hardening phenomena occurring in mild steels under strain path changes. Due to the clustering of dislocations at moderate and large strain, the material becomes to develop an intragranular microstructure, whose effects on the work-hardening are clearly seen
Fig. 10. Variation of the flow stress at 5% strain with the strain rate in an IF steel. The solid curve is a result of data fitting by Eq. (29). 2
For an extension of this model coupling the dislocation model of intragranular hardening with a texture-fitted anisotropic plastic potential, see also Hiwatashi et al. (1997, 1998) and Saiyi Li et al. (2003).
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on the stress–strain curves after Bauschinger and orthogonal strain path changes. The influence of the deformation history is represented by a set of internal state variables, which are denoted by R, X, S, P. The scalar variable R describes the isotropic hardening beyond the yield limit produced by statistical (i.e. non structured) distributions of dislocations. The second-order tensor variable X describes the rapid changes in stress following a strain-path change and is a kind of generalized back stress. The fourth-order tensor variable S describes the directional strength of planar dislocation structures. Finally, the second-order tensor variable P is associated with the polarity of these planar dislocation structures. The yield condition is supposed of the form ¼ Y0 þ R þ fkSk;
ð30Þ
where is the equivalent tensile stress, Y0 is the initial yield stress, f is a material parameter, whereas R and fkSk denote, respectively, the contributions of the randomly distributed dislocations and of the dislocation structures to the isotropic hardening. The evolution equations of the internal state variables describe the anisotropic hardening of the material under continuous or sharp strain-path changes. In particular, the evolution law of R is governed by the Voce-type differential equation : : R ¼ CR ðRsat RÞ" p ; ð31Þ with the initial condition R(0)=0. The proposed extended model assumes that: (i) the flow stress is additively composed of the effective stress and the internal stress and (ii) after a jump in strain rate, the flow stress tends asymptotically to the values corresponding to the isothermal curve at the new strain rate, while the excess of the flow stress after a jump is merely an effect of the thermal softening at high constant strain rates and large strains. Furthermore, a complementary TEM analysis, whose results will be published elsewhere, has shown that an increase in strain rate causes a delay in the formation of organized dislocation structures and thus induces a more uniform distribution of dislocations at a given strain, whereas the difference in the microstructures produced at various strain rates seems to gradually disappear with increasing strain. Based on the above considerations, the model has been extended to the strain-rate sensitivity of steels, as follows. The internal state variables Y0 and R are let to depend on the strain rate, in order to take into account the influence of the strain rate on the effective stress and on the delay in the formation of the dislocation structures, respectively. More precisely, the parameters Y0 and Rsat, which were supposed constant in the original model are let now depend on strain rate, according to the equations
: 2 k T " Y0 ¼ Y^ 0 þ Y0 1 ln : 0 ; ð32Þ DF0 " p
Rsat ¼ Rsat;0
: !
" p ; : "0
ð33Þ
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: : where Y^ , Y0 , DF0 , "0 , Rsat;0 and are material parameters, and " 0 is a reference 1 strain rate, in our case 0.001 s . The parameters of the proposed model have been identified by using the following set of mechanical tests : (1) uniaxial tensile tests at different strain rates, (2) monotonic simple shear tests, (3) Bauschinger tests carried out in simple shear, (4) simple shear following uniaxial tensile tests in the same direction (orthogonal strain path change), (5) strain rate jump tests in tension from 0.001 s1 to 1000 s1. The values of the material parameters are given in Table 1, including those describing the behaviour at conventional strain rates, which have been taken from Bouvier et al. (2000). Inspection of Fig. 11, which compares the predictions of the new constitutive model with the experiments, reveals that the new model describes quite satisfactorily the material response under complex changes of strain path and strain rate. Table 1 Material parameters of the studied IF steel CP CR
CSD CSL CX
3.5 50.8 6.9
f
n
np
r
Y^ 0
Y0
: DF0 (eV) "0 (s1)
6.67 145.1 0.859 1.0 48.9 8.5 114.9 465.1 0.627
Rsat,0
1.81106 29.7
Ssat
X0
0.02 245.8 16.0
Fig. 11. Comparison between experiments (dotted line) and the generalized constitutive model (solid line): (1) monotonic tensile test at 0.001 s1; (2) monotonic tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) monotonic simple shear test at 0.001 s1; (4) reversed simple shear after at 10, 20 and 30% forward shear at 0.001 s1 (Bauschinger test); (5) simple shear test after a tensile true prestrain of 10%, both in the same direction and at 0.001 s1 (orthogonal test). The strains in the abscissa are true strains for the tensile tests and amounts of shear strain for the simple shear tests.
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6. Conclusion A new constitutive model based on the dislocation model of intragranular hardening proposed by Teodosiu and Hu (1995) has been developed to describe the high strain rate properties of IF steel for the crash analysis. A special emphasis has been placed on the consideration of the effects of the predeformation to which the part is generally subjected during press forming. A procedure for obtaining isothermal stress–strain curves has been proposed with the aid of finite element analysis. Interpretation of strain rate jump tests has been made based on thermal activation analysis. The isothermal behaviour of IF steel reveals that the apparent excess of the flow stress after a jump, which is frequently observed in bcc metals, is in fact due to the thermal softening at large strains, and that the flow stress after a jump in strain rate tends asymptotically to the values corresponding to the curve at the new strain rate. Thus, strain rate history effects are similar to those of fcc metals, except for one distinct feature, namely that the instantaneous increase in flow stress following a jump in strain rate is almost insensitive to the amount of prestrain. The proposed constitutive model could successfully predict the mechanical behaviour of IF steel under rather complex strain path and/or strain rate changes.
Acknowledgements This work has been supported by Nippon Steel Corporation. The authors are grateful to Dr. H. Yoshida of Nippon Steel Corp. for experimental work.
References Bouvier, S., Haddadi, H., Teodosiu, C., 2000. Identification of the Microstructural Model. Report of the Brite-Euram Contract No. BRPR-CT97-0492. Campbell, J.D., Ferguson, W.G., 1970. The temperature and strain-rate dependence of the shear strength of mild steel. Philos. Mag. 21, 63. Chiem, C.Y., Duffy, J.G., 1983. Strain rate history effects and observations of dislocation substructure in aluminum single crystals following dynamic deformation. Mater. Sci. Eng. 57, 233. Clausen, B., 1997. Characterization of Polycrystal Deformation by Numerical Modelling and Neutron Diffraction Measurements. PhD Thesis, Risø National Laboratories, Roskilde, Denmark. Duffy, J., 1979. The J. D. Campbell memorial lecture: testing techniques and material behaviour at high rates of strain. Inst. Phys. Conf. Ser. No. 47, p. 1. Frost, H.J., Ashby, M.F., 1971. Motion of a dislocation acted on by a viscous drag through an array of discrete obstacles. J. Appl. Phys. 42, 5273. Gibbs, G.B., 1969. Thermodynamic analysis of dislocation glide controlled by dispersed local obstacles. Mater. Sci. Eng. 4, 313. Gilat, A., Cheng, C.S., 2002. Modeling torsional split Hopkinson bar tests at strain rates above 10,000 s1. Int. J. Plasticity 18, 787. Harding, J., 1969. The effect of grain size and strain rate on the lower yield stress of pure iron at 288 K. Acta Metall. 17, 949. Hiwatashi, S., Van Bael, A., Van Houtte, P., Teodosiu, C., 1997. Modelling of plastic anisotropy based on texture and dislocation structure. Comp. Mater. Sci. 9, 274.
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Hiwatashi, S., Van Bael, A., Van Houtte, P., Teodosiu, C., 1998. Prediction of forming limit strains under strain-path changes: application of an anisotropic model based on texture and dislocation structure. Int. J. Plasticity 14, 647. Khan, A.S., Liang, R., 1999. Bahaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling. Int. J. Plasticity 15, 1089. Kawata, K., Hashimoto, S., Takeda, N., Sekino, S., 1985. On high-velocity brittleness and ductility of dual-phase steel and some hybrid fiber reinforced plastics. In: Vinson, J.R., Taya, M. (Eds.), Recent Advances in Composites in the United States and Japan (ASTM STP 864). Am. Soc. for Testing and Materials, Philadelphia, p. 700. Klepaczko, J., Duffy, J., 1982. History effects in polycrystalline BCC metals and steel subjected to rapid changes in strain rate and temperature. Arch. Mech. 34, 419. Kocks, U.F., Argon, A.S., Ashby, M.F., 1975. Thermodynamics and kinematics of slip. Progress in Material Science 19, 230. Li, S., Hoferlin, E., Van Bael, A., Van Houtte, P., Teodosiu, C., 2003. Finite element modelling of plastic anisotropy induced by texture and strain-path change. Int. J. Plasticity 19, 647. Nesterova, E.V., Bacroix, B., Teodosiu, C., 2001. Microstructure and texture evolution under strain-path changes in low-carbon interstitial-free steel. Metall. Mater. Trans. 32A, 2527. Rosenfield, A.R., Harn, G.T., 1966. Numerical description of the ambient low-temperature, and highstrain rate flow and fracture behaviour of plain carbon steel. Trans. Am. Soc. Metals 59, 962. Rusinek, A., Klepaczko, J.R., 2001. Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress. Int. J. Plasticity 17, 87. Seeger, A., 1955. The generation of lattice defects by moving dislocations, and its application to the temperature dependence of the flow-stress of F.C.C. crystals. Phil. Mag. 46, 1194. Smidt Jr., F.A., 1969. An analysis of thermally activated flow in a-iron based on T-t* considerations. Acta Metall. 17, 381. Spitzig, W.A., Leslie, W.C., 1971. Solid-solution softening and thermally activated flow in alloys of Fe with 3 at.% Co, Ni or Si. Acta Metall. 19, 1143. Tanaka, T., Watanabe, S., 1971. The temperature dependence of the yield stress and solid solution softening in Fe-Ni and Fe-Si alloys. Acta Metall. 19, 991. Teodosiu, C., 1975. A physical theory of the finite elastic–viscoplastic behaviour of single crystals. Engineering Transactions 23, 151. Teodosiu, C., Hu, Z., 1995. Evolution of the intragranular microstructure at moderate and large strains: modelling and computational significance. In: Shen, S., Dawson, P.R. (Eds.), Proceedings of Numiform’95 on Simulation of Materials Processing: Theory, Methods and Applications. Balkema, Rotterdam, pp. 173–182. Wilson, M.L., Hawley, R.H., Duffy, J., 1979. Technical Report (NSF-18532/8). Division of Engineering, Brown University. Yoshida, H., 2001. Exact identification of strain rate dependency of steel properties by one bar method. In: Proceedings of the 52nd Japanese Joint Conference for the Technology of Plasticity, Fukui, pp. 175– 176.
Constitutive modelling of the high strain rate behaviour of interstitial-free steel A. Uenishi*, C. Teodosiu LPMTM-CNRS, University Paris 13, 93430 Villetaneuse, France Received in final revised form 27 June 2003
Abstract A physically based modelling and experimental investigation of the work hardening behaviour of IF steel covering a wide range of strain rates including complex strain path and/or strain rate changes are presented. In order to obtain isothermal stress–strain curves at high strain rates, a procedure has been proposed with the aid of finite element analysis. The result reveals that the apparent excess of the flow stress after a jump in strain rate, which is frequently observed in bcc metals, is in fact due to the thermal softening at large strains, and that the flow stress after a jump in strain rate tends asymptotically to the values corresponding to the curve at the new strain rate. The strain rate affects not only the short-range stress but also the long-range stress via the strain-rate dependant evolution of dislocation structures. The proposed model is based on the dislocation model of intragranular hardening proposed by Teodosiu and Hu [Teodosiu, C., Hu, Z., 1995. Evolution of the intragranular microstructure at moderate and large strains: modelling and computational significance. In: Shen, S., Dawson, P. R., (Eds.), Proceedings of Numiform’95 on Simulation of Materials Processing: Theory, Methods and Applications. Balkema, Rotterdam, pp. 173–182] and extended to strain rate sensitive one with applying the results of the thermal activation analysis. A satisfactory agreement has been achieved between model predictions and experimental results. # 2003 Elsevier Ltd. All rights reserved. Keywords: Constitutive Behaviour; Viscoplastic material; Finite elements; Dynamics plasticity
* Corresponding author at present address: Nippon Steel Research Laboratories, Nippon Steel Corp., 20-1 Shintomi, Futtsu-city, Chiba, 293-8511, Japan. Tel.: +81-4-39-803113; fax: +81-4-39-802743. E-mail address: [email protected] (A. Uenishi). 0749-6419/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2003.06.004
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1. Introduction The demand for increased vehicle safety has become a matter of considerable concern of users and tends to increase the weight of the car body. On the other hand, responding to the world-wide consciousness of environmental protection, the reduction of fuel consumption becomes a very important challenge. Thus, it is essential to reduce the car body weight. Clearly, meeting these contradictory requirements, i.e. designing a lightweight car with enhanced safety, requires an optimum combination of body structure and material, which necessitates, in turn, an adequate physical and numerical modelling of the mechanical behaviour both at low and high strain rates. Several points should be taken into account for this modelling. The materials used in the energy absorbing parts of a car are deformed at relatively low strain rates during press forming and eventually subjected during a crash to high strain rate deformations along different strain paths. It is, therefore, necessary to take into account the effects of the deformation history and of the eventual changes in strain path. In addition, the results of the high strain rate tests should be carefully analysed, in order to identify the specific effects of adiabatic heating and to separate the temperature and strain rate effects. In the present study, the thermo-mechanical analysis of a comprehensive set of high strain rate tensile tests has been conducted by using the finite element code MARC. Based on these data and on the thermal activation analysis of the deformation processes, a new constitutive model taking into account the deformation history has been proposed and identified.
2. Experimental procedure and results 2.1. Materials and high strain rate tensile tests The material studied is a 0.8 mm thick sheet of a deep-drawing quality IF steel (mass%: 0.006 C, 0.16 Mn, 0.038 Ti, 0.024 Al, 0.0015 Si, 0.005 P and 0.004 S), hot and cold rolled and subsequently subjected to recrystallization and annealing. This material is extensively used in car parts that require high formability. The mechanical behaviour including complex strain-path changes of the same material has been thoroughly investigated by Nesterova et al. (2001), including a detailed analysis of the evolution of texture and microstructure. The work hardening behaviour is greatly influenced by strain-path changes and displays peculiar cross and Bauschinger effects. These characteristics have been related to the microstructural evolution and have been successfully expressed in terms of the dislocation-based intragranular hardening model proposed by Teodosiu and Hu (1995). The stress–strain curves at strain rates higher than 100 s1 have been measured by using the so-called one-bar technique, whereas a conventional Instron screw-driven test machine has been used at strain rates below 0.1 s1. In addition to the monotonic tests, strain rate jump tests were also carried out in order to study the effect of
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a prestrain on the subsequent mechanical behaviour. The specimen was first subjected to uniaxial tension along the rolling direction at a strain rate of 0.001 s1 and subsequently to uniaxial tension in the same direction at 1000 s1. The usual way of characterizing the mechanical behaviour of a material by a uniaxial tensile test is to measure the loading force associated with the change of the length of the specimen. At a conventional strain rate, the load cell is considered to deform homogeneously and the loading force is measured by a strain gauge attached to it. As the strain rate increases, the time needed to attain the homogeneity of elastic deformation within the load cell approaches the testing time, and this leads to the necessity of considering the wave propagation within the load cell. At strain rates higher than about 10 s1, the signal of the loading force is greatly perturbed by multiple passages of waves reflected within the load cell if a conventional configuration is being used. This could be eventually avoided by increasing the length of the load cell, in order to finish the measurement before the arrival of the elastic wave reflected at the other end. This technique is usually called a split Hopkinson pressure bar method. For the tensile tests of sheet metal, a one-bar technique has been developed by Kawata et al. (1985), based on the Hopkinson bar method. As shown in Fig. 1, the testing system consists of a hammer, an impact block, a specimen and an output bar. When the impact block is given an impact by the hammer, the specimen is deformed in tension. At the instant of the impact, a transmitted wave starts to propagate in the output bar, its amplitude being proportional to the stress in the specimen. This wave is recorded by strain gauges attached to the output bar at section C, situated at a distance a from section B. In addition, an electro-optical extensometer is used to measure the velocity V(t) of the impact block, which is integrated to give the displacement of section A ðt uA ¼ Vð Þd: ð1Þ 0
Fig. 1. Schematic representation of the one-bar method high strain rate tensile tests.
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An analysis of the propagation of the elastic waves in a bar enables to derive the displacement uB of section B, from the strain of the transmitted wave "g ðtÞ at section C. The propagation of waves in a bar is considered as a one-dimensional problem, when the lateral inertia can be neglected. Considering the delay of the wave propagation, uB can be expressed in terms of "g as ðt ðt uB ¼ c "B ðtÞ dt ¼ c "g ð þ a=cÞd: ð2Þ 0
0
where c is the sound velocity in the bar. The elongation of the specimen is the difference between the displacements at sections A and B. By using Eqs. (1) and (2), the engineering strain and the engineering strain rate of the specimen may be expressed as ð uA uB 1 t eðtÞ ¼ ¼ VðÞ c"g ð þ a=cÞ d; ð3Þ L0 0 L0 1 : e ðtÞ ¼ VðtÞ c "g ðt þ a=cÞ ; L0
ð4Þ
where L0 is the length of the specimen. The axial force F(t) at section B can also be determined from the amplitude of the transmitted wave "g ðtÞ as FðtÞ ¼ Ebar Abar "B ðtÞ ¼ Ebar Abar "g ðt þ a=cÞ:
ð5Þ
Thus, the engineering stress in the specimen becomes n ðtÞ ¼
FðtÞ Abar Ebar ¼ "g ðt þ a=cÞ: A0 A0
ð6Þ
2.2. Stress–strain curves at high strain rates Stress–strain curves at different strain rates are shown in Fig. 2 as diagrams of engineering stress vs. engineering strain. The flow stress, the upper yield limit and the difference between the upper and lower yield limits increase with strain rate. However, the apparent work hardening rate decreases at high strain rates. Similar behaviour has been reported also for other bcc metals by Khan and Liang (1999). The results of strain rate jump tests are also shown in Fig. 2. The flow stress level after a strain rate jump is higher than the entirely constant strain rate curves for all tested conditions. There are several results in the literature concerning the mechanical behaviour of steels after a jump to a higher strain rate. Wilson et al. (1979) reported that the flow stress after the jump does not increase instantaneously to the value corresponding to the higher constant strain rate, but it approaches asymptotically this value for lowcarbon hot-rolled steels. This behaviour is similar to that generally observed in fcc
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Fig. 2. Stress–strain curves of an IF steel at different strain rates, including strain rate jump tests: (1) monotonic tensile test at 1000 s1; (2) tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) after a tensile prestrain of 16% at 0.001 s1; (4) monotonic tensile test at 0.001 s1.
metals (Chiem and Duffy, 1983; Duffy, 1979). On the other hand, it has been reported that the flow stress after a jump in strain rate exceeds that of a constant strain rate test at the higher strain rate for a hot rolled steel that contained 0.25% C (Klepaczko and Duffy, 1982). The results obtained in the present study are indeed closer to this latter finding. One way to solve such apparently contradictory results may be to take into account the thermal softening caused by adiabatic heating at high strain rates. Indeed, since the flow stress of bcc metals, including steels, is much more strain-rate sensitive than that of fcc metals, the dissipated plastic work is expected to increase more rapidly with strain rate in bcc than in fcc metals. Consequently, for a material that has both a high strain-rate sensitivity and a high temperature sensitivity of the flow stress, the adiabatic heating accelerates the onset of plastic instability at high strain rates, thus causing a necking to occur in an early stage of deformation. As a result, the apparent flow stress calculated by assuming a uniform deformation in the tensile zone of the specimen decreases rapidly with strain at high strain rates. Thus, in order to understand the results of strain rate jump tests, it is necessary to follow the necking process by a thermo-mechanical analysis. Furthermore, the heat induced by the plastic work causes thermal softening and changes the global response of the specimen. Therefore, for the correct assessment of mechanical tests involving strain rate jump tests and for the development of a constitutive model, it is essential to evaluate the effects of the thermal softening and to obtain isothermal stress–strain curves.
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3. Finite element analysis of the high strain rate tests 3.1. Finite element model of the one-bar method As seen in the previous section, high strain rate properties are measured by using a special technique, called the one-bar method. For high strain rate tensile tests, the specimen geometry has to satisfy various requirements. In order to assure a homogeneous deformation of the specimen, the length of the deformed zone of the specimen should be short enough in order for the elastic wave to propagate before the plastic yielding of the material. Furthermore, since it is experimentally difficult to measure large strains at high strain rates, a small radius has to be used at the shoulder of the specimen, in order to reduce the constraint of the deformation around the shoulder and to simplify the strain analysis for the part of the specimen of uniform cross-section. All these conditions require a special geometry of the specimen, which is rather different from the one used at conventional strain rates. One of the main drawbacks of the high strain rate tensile test is that the results concern the global response of the specimen and do not directly characterize the material behaviour in the tensile zone. Although it has been pointed out that the geometry of specimens greatly influences experimental stress–strain relations tested at high strain rates in shear (Rusinek and Klepaczko, 2001) and in torsion (Gilat and Cheng, 2002), its effects are more complicated in tension due to the phenomenon of necking and an eventual fracture of specimen. Accurate evaluation of the stress is usually difficult after the occurrence of the diffuse necking, as the reduction of the cross-section of the specimen becomes non-uniform. A numerical study is thus necessary to take into account the geometry of the specimen, the necking and the thermal softening. To this end, a thermo-mechanical analysis of the high strain rate tensile test has been made, by using the finite element code MARC, in order to convert the experimental force–elongation data into true stress–true strain curves. By considering the symmetry of the problem in the thickness and the tensile directions, only one quarter of the specimen has been analysed. As shown in Fig. 3, the FE model consists of two parts, a tensile specimen and its reinforcement. In the experiment, the tensile force is applied through a rod that penetrates a hole of the
Fig. 3. Finite element mesh.
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specimen. A reinforcement plate is spot-welded onto the specimen to strengthen the hole. In the analysis, the spot weld is modelled by sharing the nodes of the specimen with those of the reinforcement at the position of the spot welds. The rod is modelled as a rigid body and its movement controls the process. The other edge of the half-specimen is clamped in the loading direction. The FE model has 2730 eight-node solid elements and 4887 nodes. The elements have eight integration points with reduced integration for the volumetric part of the deformation. The thickness of the specimen is divided into two layers of solid elements. 3.2. Adiabatic heating during plastic deformation Large plastic deformation induces an increase in temperature of the material, which is proportional to the plastic work. The plastic dissipation is strain dependent and also strain rate dependent in the case of a strain rate sensitive material. The heating of the specimen due to the dissipated plastic work can be derived from the equation C
dT : ¼ ij "ij ; dt
ð7Þ
where denotes the mass density, C the specific heat, T the temperature, t the time, the heat conversion ratio taken equal to unity in case of an adiabatic heating, ij : the Cauchy stress tensor, "ij the strain rate tensor. The calculation has been carried out by taking =7.874103 kg/m3 and C=452 J/kg K, both assumed to be independent of the temperature. 3.3. Numerical identification of a simplified thermo-viscoplastic constitutive model In order to simulate the high strain rate tensile test, it is convenient to take into account the effects of both temperature and strain rate by first using a simplified constitutive model. In the code MARC, it is possible to use to this end the JohnsonCook law " : ! #
: " T Troom m n "; " ; T ¼ ðA þ B" Þ 1 þ Clog : ð8Þ 1 "0 Tref Troom : where , " and " are the equivalent tensile stress, the equivalent tensile strain and the equivalent tensile strain rate, respectively. The first term represents the work hard: ening behaviour at the reference strain rate " 0 and at the room temperature. The second and third terms express the effects of strain rate and temperature, respectively. The parameters A, B, n, C and m are material parameters and T denotes the absolute temperature. Troom and Tref are the absolute room temperature and melting temperature of the material, in our case 293 and 1809 K, respectively. In fact, it is not possible to cover a wide range of strain rates and temperatures by using Eq. (8), because the real material behaviour is much too complex. However, it
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is possible to use this simplified law in order to give an appropriate description of the material response over a limited range of strain rates and temperatures. Identification of the parameters in Eq. (8) is complicated by the fact that the experimentally obtained stress–strain curves are not isothermal. The parameters related to the strain rate and temperature dependence of the flow stress have been determined by using the results of the thermal activation analysis, which will be presented in Section 4. The work hardening parameters A, B and n have been determined in an iterative way, by comparing the engineering strain and the engineering stress between the experiment and the calculation. In the tensile experiment the engineering strain e and the engineering stress n are calculated by e¼
L L0 P ; n ¼ ; A0 L0
ð9Þ
where L is the current length of the constant-cross section part of the specimen, L0 the initial length, P the tensile force and A0 the initial cross section of the specimen. The engineering strain vs. the engineering stress curve obtained by a numerical analysis has been compared to the experimental one (Fig. 4). Except for the initial peak, which is not considered in the Johnson–Cook model, the comparison shows a good agreement. The beginning of the diffuse necking is well reproduced. At higher
Fig. 4. Comparison of the engineering strain vs. engineering stress diagrams between the experiment (dashed line) and the FEM calculation (solid line).
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strains, an increasing discrepancy between the two curves was observed. There could be several reasons for this: (i) The strain rate and temperature sensitivity of the flow stress is not correctly expressed by the Johnson–Cook constitutive law at large strains. (ii) The damage accumulated during deformation causes a decrease in the flow stress, which becomes apparent at later stages of the deformation. (iii) A coarse FE mesh postpones the localization of the deformation. However, this effect may be considered negligible in our case, since very fine meshes were used in the FE model. Although we could not simulate the experimental curve perfectly, we considered the results to be sufficient for our purpose and tried to analyse the calculated results in terms of the stress and strain distributions. As discussed before, the specimen for a high strain rate tensile test is different from conventional tensile specimens. Thus, the level of homogeneity in the specimen has been checked. Fig. 5 shows a longitudinal distribution of the components of the Cauchy stress tensor in the middle of the specimen. The stress distribution is nonuniform at both ends of the specimen. However, in the central part of the specimen, a uniaxial tensile stress state is confirmed. It may be concluded that the peculiarity of the specimen geometry does not significantly affect the stress distribution in the central part of the specimen.
Fig. 5. Variation along the longitudinal axis of the specimen of the Cauchy stress components in a frame x1x2x3, with the x1-axis along the tensile direction and the x2-axis along the transverse direction for an average true strain of 20%.
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In the tensile experiment, the engineering stress is evaluated by Eq. (9) and then the true stress and the true strain " are calculated, under the assumption that the specimen deforms uniformly, by using the formulas
¼
P P A0 A0 L0 L ¼ ¼ n ¼ n ð1 þ eÞ; A A0 A A L L0
" ¼ ln
L : L0
ð10Þ
ð11Þ
First, the method of the strain evaluation is examined. As shown by Eq. (11), the strain is evaluated from the current and initial lengths of the constant-section of the specimen. The FEM value of the true strain at the centre of the specimen has been compared with the values obtained by using Eq. (11) and different gauge lengths L0. At the beginning of the deformation the difference is small, but it increases rapidly after necking. In our case, as shown by Fig. 6, the best agreement with the FEM simulation is obtained by using a gauge length of about 4 mm, while the maximum gauge length used in Eq. (11) was L0=10 mm.
Fig. 6. Variation of the longitudinal logarithmic strain obtained by different evaluations: (a) strain calculated by FEM at the centre of the specimen; (b) average strain evaluated using a gauge length of 7 mm; (c) average strain evaluated using a gauge length of 4 mm. The abscissa values are the corresponding strains evaluated by using as gauge length the total length L0 of the constant cross-section of the specimen. When decreasing the gauge length, the evaluated strain approaches the FEM result and coincides with the latter for gauge lengths less than 4 mm.
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After the occurrence of a diffuse necking, the stress evaluation by Eq. (10) is not accurate, for the decrease of the cross section at the centre is more rapid than that calculated under the assumption of a uniform decrease of the specimen width. The real true stress in the centre becomes larger than that calculated by Eq. (10). However, the evaluation of the stress at the centre of the specimen from the measured tensile load is possible even after necking, provided that the change of the cross section is correctly obtained by a FE simulation. Therefore, the global experimental data have been converted to local data at the centre of the specimen in the following way. Using the results of FE analysis shown in Fig. 6, the logarithmic strain at the centre of the specimen is evaluated as a function of lnðL=L0 Þ, say "centre ¼ fðlnðL=L0 ÞÞ: 11
ð12Þ
Then, the true stress at the centre is obtained by
¼
P P A0 ¼ ¼ n exp "centre ; 11 Acentre A0 Acentre
ð13Þ
where Acentre is the cross-sectional area of the specimen at its centre. Rigorously, the relationship Eq. (12) should be evaluated for each experimental condition. However, since a slight change of the material behaviour does not affect significantly this relation, the same master curve of the monotonic high strain rate test has been used also for the experiments involving a jump in the strain rate. The FE analysis has been made under the assumption that the process is adiabatic. The increase in temperature at the centre of the specimen is observed to be some tens of degrees, thus producing a significant thermal softening of material. However, by using the identified Johnson–Cook law, Eq. (8), it has been possible to separate the effect of the thermal softening in a numerical analysis and to obtain isothermal stress–strain relations. The results of strain rate jump tests (Fig. 2) show that the flow stress levels after a strain rate jump are higher than the entirely curve obtained at the higher constant strain rate. However, after compensations for the decrease of cross-section and the increase in temperature (Fig. 7), the flow stress level after a strain rate jump is comparable to the isothermal curve at the higher constant strain rate. Actually, these results resemble the behaviour reported for fcc metals (Chiem and Duffy, 1983; Wilson et al., 1979; Duffy, 1979), although a complete quantitative assessment is difficult, due to the already mentioned limitations of the FE analysis at large strains. Thus, the apparently different behaviour reported for some steels in strain rate jump tests, can indeed be ascribed to the thermal softening and the associated acceleration of strain localization caused by the adiabatic heating, which can be explained, in turn, by the higher strain-rate sensitivity of steels with respect to that of fcc metals.
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Fig. 7. Corrected stress–strain curves including strain rate jump tests: (1) monotonic tensile test at 1000 s1; (2) tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) tensile test at 1000 s1 after a tensile prestrain of 16% at 0.001 s1; (4) monotonic tensile test at 0.001 s1; (5) stress–strain relation at the centre of the specimen calculated by FEM for the monotonic tensile test at 1000 s1.
4. Thermal activation analysis Here and in what follows, we shall make use of the previously obtained isothermal stress-strain relations for discussing the temperature and strain-rate sensitivity of the flow stress. Campbell and Ferguson (1970) have performed double shear tests covering a wide range of strain rates (0.001–10,000 s1) and temperatures (195–713 K) for a 0.12% carbon steel. In order to systematize and explain the results obtained, these authors have divided the domain of the yield stress vs. strain rate diagrams into regions associated with various deformation processes, as proposed by Rosenfield and Harn (1966). As shown in Fig. 8, three of these regions, labelled by I, II and IV, may interest the deformation of steels around the room temperature. The curves in region I are characterized by a small, nearly constant slope and the deformation is controlled by athermal mechanisms. In region II, the flow stress has a logarithmic dependence on the strain rate and the deformation is governed by thermally activated process. At very high strain rates (region IV), above 5000 s1, a dislocation drag mechanism, probably due to the interaction of the gliding dislocations with thermal phonons and conduction electrons, starts to contribute to the increase of the flow stress. During crash events, the maximum strain rate is expected to be of the order of 1000 s1. Thus, to model the crash behaviour of structural steels, it is important to
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Fig. 8. Variation of the lower yield stress with strain rate in a 0.12% C steel, after Campbell and Fergusson (1970). Three characteristic regions of the flow stress sensitivity may be distinguished: (I) small temperature and strain rate sensitivity; (II) greater temperature and strain rate sensitivity; (IV) rapid increase of the strain rate sensitivity.
describe the material behaviour at strain rates below 1000 s1 at about the room temperature. This corresponds to regions I and II in Fig. 8, and hence our analysis will be restricted to athermal and thermally activated processes, and will be aimed at evaluating the activation parameters from the measurements of the strain rate sensitivity of the flow stress. At larger strains, the effects of the thermal softening are not negligible, especially at high strain rates, as shown in the preceding section. Thus, it is better to use the flow stress at the beginning of deformation. On the other hand, all internal variables change rapidly at the yield point and the deformation is not yet quasi-stationary. In addition, the measurement of the upper yield limit is difficult at high strain rate tests, because it is influenced by the stiffness of the machine, specimen size, loading conditions, the rising time of the impact, etc. Recently, it has been pointed out that the micro-vibration of the output bar at the interface with the specimen, which is caused by a slight misalignment of the specimen and the impact block along the tensile direction, superposes on the measured signal and enhances the yield phenomena (Yoshida, 2001). Taking into account all these circumstances, we will use in our analysis the strain rate dependence of the flow stress at 5% tensile strain, which seems to be the lowest strain value at which the transitory yield point phenomena can be considered as negligible.
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The thermally-activated theory of the cold plastic deformation has been developed almost 50 years ago (Seeger, 1955) and has known since then a successful development for various types of crystalline materials and deformation conditions. Here we limit ourselves to a somewhat simplified version of this theory, referring to the review papers by Gibbs (1969), Frost and Ashby (1971), Teodosiu (1975), and Kocks et al. (1975) for more general settings. A gliding dislocation, subjected to a resolved shear stress ; encounters during its motion two kinds of obstacles: extended obstacles generating a long-range stress field, whose resolved shear stress is denoted by , and local obstacles, which act only over a few atomic distances and, unlike the formers, can be overcome with the help of thermal fluctuations, under the action of the local effective stress ¼ :
ð14Þ
The long-range stress has been denoted by because it is proportional to some effective shear modulus , 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. For a dislocation segment to surmount the strongest obstacle at zero absolute temperature, should equal some maximum slip resistance, say 0 ; called mechanical threshold (Kocks et al., 1975). However, at some temperature above 0 K, thermal fluctuations can assist the applied stress, and the dislocation can glide at a stress < 0 : As the temperature increases, can decrease until it becomes equal to the amplitude of the long-range stress field. Further increase in temperature does not give any additional decrease in the applied stress, since the energy barrier is too extended for thermal fluctuations to make a significant contribution. Alternatively, if the applied stress grows beyond 0 the dislocation segment can overcome the local obstacles with no thermal aid: local obstacles become penetrable to the dislocation motion. By using the rate theory it is possible to estimate the mean waiting time of a dislocation segment in front of a local obstacle and thus to correlate the average velocity of a mobile dislocation to the structure, applied stress, and temperature, by using some statistical assumptions concerning the distribution and strength of the local obstacles. However, for our present context, it is sufficient to retain only the essential features of the analysis, by adopting the following simplifying hypotheses: (i) the local obstacles have the same strength and are distributed uniformly in the glide plane; (ii) the frequency of the backward jumps, i.e. opposite to the effective stress, can be neglected with respect to that of the forward jumps; (iii) the flight time of the mobile dislocations between the local obstacles can be neglected with respect to the waiting time in front of the obstacles. With these simplifications, it may be shown that the average velocity of a gliding dislocation may be expressed as v ¼ 0 ‘F expðDG =kTÞ;
ð15Þ
where 0 is the attempt frequency of overcoming local obstacles, which is proportional to the Debye frequency and increases with decreasing distance between the pinning points along the dislocation line, ‘F is the average flight distance between two successive obstacles, while DG denotes the difference in the free enthalpy
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between the configuration of the dislocation segment in the saddle point and in the ground state for forward jumps, which is called the Gibbs free energy of activation. The slip rate on any active slip system is given by Orowan’s relation : ¼ b m v; ð16Þ ehere b is the magnitude of the Burgers vector, m is the density of the mobile dislocations and v is the average dislocation velocity in the direction of the slip vector for the considered slip system. Introducing Eq. (15) into Eq. (16) and assuming that the activation free enthalpy DG is the same for all dislocation segments of the slip system, yields : : ¼ 0 expðDG =kTÞ; ð17Þ : where 0 ¼ b m 0 ‘F : The activation free enthalpy DG depends on the interaction between the dislocation segments and the local obstacles. A detailed analysis of the various forms of this dependence has been given by Gibbs (1969) and by other authors. Kocks et al. (1975) have summarized these results into the formula
p q DG ¼ ðDF Þ 1 ; ð18Þ 0 with the parameters p and q lying within the ranges 0 4 p 4 1, 1 4 q 4 2. In Eq. (18), the quantity 0 characterizes the average resistance to slip of the local obstacles, while DF represents the average activation free energy required to overcome the local obstacles in the absence of an applied shear stress. 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, whereas the tensile strain rate results by volume averaging the contributions of all slip systems and grains. While such averages can be effectively calculated for a given texture, the quantitative result obtained depends critically on the simplifying hypotheses adopted in the micro-macro transition (see, e.g. Clausen, 1997). We shall, therefore, adopt an alternative approach, which is typical for the internalvariable formalism: Eqs. (14), (17) and (18) governing the thermally-activated plastic slip will be formally transcribed at the macroscopic scale,1 by simply replacing the : shear rate and the shear stress by the tensile flow stress and, respectively, the : tensile strain rate ": ¼ ;
ð19Þ
: : " ¼ "0 expðDG=kTÞ;
ð20Þ
1 The so-obtained one-dimensional model of the polycrystalline behaviour will be extended in the next : section to a three-dimensional model, : by further replacing and " by some equivalent stress and by its p power-conjugate plastic strain rate " , respectively.
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DG ¼ ðDFÞ
1
p q : 0
ð21Þ
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, i is related to the intensity of the long-range obstacles to : dislocation glide; " 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; DG is an apparent activation free enthalpy, which depends on the effective tensile stress and on the apparent free energy of interaction DF between the gliding dislocations and the local obstacles. By definition, the activation volume V is expressed as @ðDGÞ V ¼ : ð22Þ @ T¼const
Solving Eq. (20) for DG and introducing the result obtained into Eq. (22) yields : @ðln"Þ V ¼k T : ð23Þ @ T¼const This alternative expression has been used to 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 (Harding, 1969; Tanaka and Watanabe, 1971; Spitzig and Leslie, 1971) have in general arbitrarily taken ¼ ð1=2Þ ; and represented the variation of the activation volume : @ðln"Þ V ¼ k T ¼ 2V ð24Þ @ T¼const with respect to . Fig. 9 shows our results for IF steel, converted for the sake of comparison to V data, together with the results obtained by Harding (1969) and Spitzig and Leslie (1971) for similar materials. The good agreement of these three sets of data is by no means trivial. Indeed, the results of Harding (1969) were obtained from experiments at a fixed temperature from constant strain rate tests, at high strain rates, as in the present work. On the other hand, the results of Spitzig and Leslie (1971) were derived from jumps in strain rate at relatively low strain rates. The activation volume obtained from a strain rate jump tests is called the ‘‘true’’ activation volume. When it is calculated from stress–strain relations at several constant strain rates, it is called the ‘‘apparent’’ activation volume. It is known that for fcc metals these two values may show large differences (Chiem and Duffy, 1983). However, they seem to be rather closed for iron.
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Fig. 9. Variation of the activation volume with the effective stress.
By eliminating DG between Eqs. (20) and (21), we obtain "
1q #1p T ¼ 1 ; T0 0 where T0 ¼
DF
: "0 k ln : "
ð25Þ
ð26Þ
is the highest temperature for which, at a given strain rate, the thermal-activation analysis is valid. Smidt (1969) observed the temperature dependence of the flow stress of iron by temperature increment tests and found a relation, which can be written, with the present notation, as
T 2 ¼ 1 : ð27Þ T0 0 where 0 and T0 are constants. By comparing Eqs. (25) and (27), we find that the formula of Kocks et al. (1975) is satisfied, provided that we take p=1/2 and q=1. For these values of the parameters, Eq. (21) reduces to "
12 # DG ¼ ðDFÞ 1 : ð28Þ 0
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The strain rate dependence of the flow stress may be derived now, by first introducing Eq. (28) into Eq. (20), solving for and introducing the result obtained into Eq. (19). The result leads
: 2 k T "0 ln : ¼ þ ¼ þ 0 1 : ð29Þ DF " Fig. 10 shows the variation of the flow stress at 5% tensile strain with the strain rate. The strain rate sensitivity of the flow stress of IF steel is well reproduced by Eq. (29), which is shown in Fig. 10 by a solid line.
5. Constitutive modelling Based on the results shown in the preceding section, the dislocation model of intragranular hardening proposed by Teodosiu and Hu (1995) has been selected as a basis for the extension to a strain-rate sensitive constitutive model.2 The original model successfully describes various anisotropic hardening phenomena occurring in mild steels under strain path changes. Due to the clustering of dislocations at moderate and large strain, the material becomes to develop an intragranular microstructure, whose effects on the work-hardening are clearly seen
Fig. 10. Variation of the flow stress at 5% strain with the strain rate in an IF steel. The solid curve is a result of data fitting by Eq. (29). 2
For an extension of this model coupling the dislocation model of intragranular hardening with a texture-fitted anisotropic plastic potential, see also Hiwatashi et al. (1997, 1998) and Saiyi Li et al. (2003).
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on the stress–strain curves after Bauschinger and orthogonal strain path changes. The influence of the deformation history is represented by a set of internal state variables, which are denoted by R, X, S, P. The scalar variable R describes the isotropic hardening beyond the yield limit produced by statistical (i.e. non structured) distributions of dislocations. The second-order tensor variable X describes the rapid changes in stress following a strain-path change and is a kind of generalized back stress. The fourth-order tensor variable S describes the directional strength of planar dislocation structures. Finally, the second-order tensor variable P is associated with the polarity of these planar dislocation structures. The yield condition is supposed of the form ¼ Y0 þ R þ fkSk;
ð30Þ
where is the equivalent tensile stress, Y0 is the initial yield stress, f is a material parameter, whereas R and fkSk denote, respectively, the contributions of the randomly distributed dislocations and of the dislocation structures to the isotropic hardening. The evolution equations of the internal state variables describe the anisotropic hardening of the material under continuous or sharp strain-path changes. In particular, the evolution law of R is governed by the Voce-type differential equation : : R ¼ CR ðRsat RÞ" p ; ð31Þ with the initial condition R(0)=0. The proposed extended model assumes that: (i) the flow stress is additively composed of the effective stress and the internal stress and (ii) after a jump in strain rate, the flow stress tends asymptotically to the values corresponding to the isothermal curve at the new strain rate, while the excess of the flow stress after a jump is merely an effect of the thermal softening at high constant strain rates and large strains. Furthermore, a complementary TEM analysis, whose results will be published elsewhere, has shown that an increase in strain rate causes a delay in the formation of organized dislocation structures and thus induces a more uniform distribution of dislocations at a given strain, whereas the difference in the microstructures produced at various strain rates seems to gradually disappear with increasing strain. Based on the above considerations, the model has been extended to the strain-rate sensitivity of steels, as follows. The internal state variables Y0 and R are let to depend on the strain rate, in order to take into account the influence of the strain rate on the effective stress and on the delay in the formation of the dislocation structures, respectively. More precisely, the parameters Y0 and Rsat, which were supposed constant in the original model are let now depend on strain rate, according to the equations
: 2 k T " Y0 ¼ Y^ 0 þ Y0 1 ln : 0 ; ð32Þ DF0 " p
Rsat ¼ Rsat;0
: !
" p ; : "0
ð33Þ
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: : where Y^ , Y0 , DF0 , "0 , Rsat;0 and are material parameters, and " 0 is a reference 1 strain rate, in our case 0.001 s . The parameters of the proposed model have been identified by using the following set of mechanical tests : (1) uniaxial tensile tests at different strain rates, (2) monotonic simple shear tests, (3) Bauschinger tests carried out in simple shear, (4) simple shear following uniaxial tensile tests in the same direction (orthogonal strain path change), (5) strain rate jump tests in tension from 0.001 s1 to 1000 s1. The values of the material parameters are given in Table 1, including those describing the behaviour at conventional strain rates, which have been taken from Bouvier et al. (2000). Inspection of Fig. 11, which compares the predictions of the new constitutive model with the experiments, reveals that the new model describes quite satisfactorily the material response under complex changes of strain path and strain rate. Table 1 Material parameters of the studied IF steel CP CR
CSD CSL CX
3.5 50.8 6.9
f
n
np
r
Y^ 0
Y0
: DF0 (eV) "0 (s1)
6.67 145.1 0.859 1.0 48.9 8.5 114.9 465.1 0.627
Rsat,0
1.81106 29.7
Ssat
X0
0.02 245.8 16.0
Fig. 11. Comparison between experiments (dotted line) and the generalized constitutive model (solid line): (1) monotonic tensile test at 0.001 s1; (2) monotonic tensile test at 1000 s1 after a tensile prestrain of 8% at 0.001 s1; (3) monotonic simple shear test at 0.001 s1; (4) reversed simple shear after at 10, 20 and 30% forward shear at 0.001 s1 (Bauschinger test); (5) simple shear test after a tensile true prestrain of 10%, both in the same direction and at 0.001 s1 (orthogonal test). The strains in the abscissa are true strains for the tensile tests and amounts of shear strain for the simple shear tests.
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6. Conclusion A new constitutive model based on the dislocation model of intragranular hardening proposed by Teodosiu and Hu (1995) has been developed to describe the high strain rate properties of IF steel for the crash analysis. A special emphasis has been placed on the consideration of the effects of the predeformation to which the part is generally subjected during press forming. A procedure for obtaining isothermal stress–strain curves has been proposed with the aid of finite element analysis. Interpretation of strain rate jump tests has been made based on thermal activation analysis. The isothermal behaviour of IF steel reveals that the apparent excess of the flow stress after a jump, which is frequently observed in bcc metals, is in fact due to the thermal softening at large strains, and that the flow stress after a jump in strain rate tends asymptotically to the values corresponding to the curve at the new strain rate. Thus, strain rate history effects are similar to those of fcc metals, except for one distinct feature, namely that the instantaneous increase in flow stress following a jump in strain rate is almost insensitive to the amount of prestrain. The proposed constitutive model could successfully predict the mechanical behaviour of IF steel under rather complex strain path and/or strain rate changes.
Acknowledgements This work has been supported by Nippon Steel Corporation. The authors are grateful to Dr. H. Yoshida of Nippon Steel Corp. for experimental work.
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