A DISLOCATION BASED MODEL FOR THE WORK HARDENING BEHAVIOUR OF DUAL PHASE STEELS

A DISLOCATION BASED MODEL FOR THE WORK HARDENING BEHAVIOUR OF DUAL PHASE STEELS INTRODUCTION The properties of DP steels are characterized by a low yield strength due to the absence of Lüders bands and a high rate of work hardening, which results in a high tensile strength and good formability (1). DP steels also show a high energy absorbing ability which implies a good crashworthiness (2 5). The explanation to the materials behaviour is to be found in the microstructure, which in DP steels mainly consists of two phases: ferrite and martensite. Over the years many attempts have been made to describe the stress-strain behaviour of DP steels (6-10). In most cases empirical relationships have been applied but also physically based models have been developed. One frequently used concept is the Ashby-model (11) which is based on the assumption that the dislocations may be divided into two types: statistically stored (SSD) and geometrically necessary dislocations (GND) with pile ups of dislocations in arrays. The latter type of dislocations are supposed to eliminate stress concentrations and strain gradients originating from differences in hardness between the ferrite and martensite phases. Unfortunately, these arrays have never been observed in high stacking fault materials like ferrite suggesting that this type of explanation is not applicable to steel (12). A large number of strain gradient plasticity (SGP) theories have also been proposed with moderate success. It is quite obvious, therefore, that a new simple physically based dislocation theory for the stress-strain behaviour of DP steels is needed. It is also obvious that such a theory must be based on the in-homogeneous behaviour of the plastic deformation process involved in this type of materials. Such a model was recently presented by Bergström et al (13) and we will below give a short presentation of it and apply it to a DP800 steel tested at room temperature and a strain rate of 10-2 s -1. THEORY (13) Background Quite often uniaxial stress-strain curves are used to investigate the plastic deformation behaviour in metals and it is customary to assume that the whole testing volume of the tensile test specimen takes part in the plastic deformation process. This is a reasonable assumption for single phase materials but it is definitely a wrong one for DP steel consisting of a soft ferritic phase and a hard martensitic phase. In fact, experiments clearly indicate that the martensite phase in a DP steel normally does not undergo plastic deformation for strains below necking. Hence the plastic deformation process is localised to the ferrite implying that the plastic deformation is indeed inhomogeneous and gives rise to large local strains in the specimen. a) b) c) Strain Töjning the i αferrite 0 l 0 2 = l / 2 l 0 0 2 = l / 2 l 0 M α M l 0 α α l 0 /2 Makroskopisk Macroscopic strain töjn Fig.1: a) an elongation l of a specimen of length, l 0, containing 100% ferrite b-c) an elongation l of a specimen containing 50% ferrite and 50% martensite distributed in two different ways. l 0 l 0 l 0 1

This is illustrated in Fig.1 where the macroscopic, or global strain and the corresponding local strain in the ferrite is compared for three different cases. Specimen a/ in Fig. 1 contains 100% ferrite and, since the plastic deformation is homogeneous i.e. containing 100% ferrite, we can l see that the macroscopic strain,, after an elongation l equals the local strain in the l0 ferrite. In specimens b/ and c/ consisting of 50% ferrite and 50% martensite distributed in two different ways we notice that the local strain in the ferrite, for the same amount of elongation, is two times larger than the macroscopic strain. The explanation is of course that the martensite does not take part in the plastic deformation and that all plastic deformation is localised to the ferrite. This simple example therefore explains why the rate of strain hardening is higher in a DP steel than in a single-phase steel. It has also been experimentally verified that the plastic deformation in the ferrite of a DP steel is in-homogeneous. This can be seen from the EBSD-pictures in Fig. 2 (13). The blue colour indicates small miss-orientations (small plastic deformation) and red colour large missorientations(large plastic deformation). The colours green and yellow lie in between. The results in Fig. 2b emanate from an un-deformed specimen of a DP 800 steel. The green colour close to the martensite particles suggests that the stresses caused by the transformation of austenite to martensite gives rise to a local plastic deformation in the ferrite close to the martensite surfaces. Inside the ferritic grains the colour is blue indicating small or negligible plastic deformation. Fig. 2b. shows result from the same DP steel being strained 8%. In this case we can see that the colours in the ferrite, close to the martensite, have changed from green to yellow and even red. Inside the ferrite grain the colour have changed from dark blue to red, yellow, green and light blue. The results also indicate the possibility that small volume fractions ferrite, inside the ferrite grains, have not undergone plastic deformation at all almost dark blue spots. Hence, the results suggest that there are pronounced strain gradients also inside the ferrite grains with large miss-orientations close to the martensite particles and smaller ones in the centre of the grains. It is therefore reasonable to assume that local volume fractions of the ferrite controls the plastic deformation process in DP steels. It is also obvious that these volume fractions are strain dependent. Fig. 2a; EBSD-analysis of undeformed DP800. The colours indicate degree of plastic deformation (13). Blue is small deformation and red is large deformation. Green and yellow represent deformations in between. See text. 2

Figur 2b: EBSD-analys av 800DP strained to 8%. The colours indicate degree of plastic deformation (13). Blue is small deformation and red is large deformation. Green and yellow represent deformations in between. See text. We will now take these two types of plastic in-homogeneity into account and derive a simple physically based dislocation theory for the stress-strain behaviour of DP steels (13). The theory is discussed in the light of results from experimental uniaxial testing, light optical and SEM/EBSD-studies of the microstructure. Since the new theory is developed from the original Bergström dislocation theory, it is appropriate to start with a short summary. The original Bergström theory: a summary (14) It is well established that the true flow stress, σ(ε), in metals and alloys is related to the total dislocation density ρ(ε) as σ ( ε) = σ + σ ( ε) = σ + α ρ( ε) (1) i0 def i0 Gb also known as the Taylor equation(15), whereσ i0, is the friction stress, σ def ( ε ), is the work hardening component of the flow stress, α is a dislocation strengthening constant, G is the shear modulus, b is the nominal value of the Burgers vector, and ε is the true strain. If the validity of eqn(1) is accepted the main problem in formulating a dislocation theory for the true stress-true strain behaviour of metals and alloys is to derive a relationship for ρ(ε). For single-phase metals and alloys Bergström (14) derived the following expression for the strain dependence of ρ(ε): dρ m = Ω ρ( ε ) dε b s( ε) (2) where ρ(ε) is the total dislocation density, s(ε) is the mean free path of dislocation motion, b is the nominal value of the dislocation Burger s vector, Ω is a strain independent material constant representing the remobilisation (dynamic recovery) of immobile dislocations and m is the Taylor constant. The following relationship for s(ε) has proved to give an excellent description of the strain dependence of the dislocation mean free path, s, in single phase metals (16) s( ε ) = s + ( s s ) e (3) 0 1 0 k ε 3

where s 1 and s 0 are the initial and final values of s. k is the rate constant determining the rate at which s(ε) goes from s 1 to s 0. A dislocation theory for the uniaxial stress-strain behaviour of DP-steel.(13) As mentioned in the introduction there are strong experimental evidences indicating that the plastic deformation process in DP steel is in-homogeneous. In deriving a dislocation theory for this type of metal we will therefore make the following assumptions: 1. The theory is valid for uniaxial strain and for tensions up to necking 2. The work hardening process starts in the ferrite close to the phase interface of ferrite and martensite, partly as a result of the presence of high residual stresses caused by the transformation from austenite to martensite, partly due to the incompatibility conditions which give rise to stress concentrations in these areas. The work hardening process is thus initially located in the ferrite close to the martensite surfaces. As a result of increasing work hardening in these areas, the deformation process is forced to proceed inwards the ferrite grains due to a lower hardening resistance there. 3. The investigated DP steels consist of the following volume fractions: (i) martensite, f m, which deforms elastically but not plastically (ii) active ferrite, f, which deforms elastically and plastically (iii) non-active ferrite, f undef, which deforms elastically and successively deforms plastically 4. A non-homogeneity factor f is introduced which is defined as de active part of the total volume fraction of ferrite, f 0, participating in the plastic deformation process. At small strains, that is, in the beginning of the deformation process, f << f 0. With increasing strain, f is assumed to grow towards f 0. As a result of the continuously increasing volume fraction of active ferrite, the rate of dislocation creation and work hardening decreases with increasing strain. 5. The un-deformed fraction of ferrite, that is, the continuously decreasing amount f 0 - f, may be considered as hard, not participating in the deformation process and can in that respect be compared with the martensite fraction. Therefore, stress concentrations may arise in between the deformed and undeformed ferrite as well. This reminds of the Lüder-phenomenon where a front separates deformed ferrite from undeformed. 6. The plastic deformation process does not necessarily start in all ferrite grains at the same time but may randomly and continuously be initiated at different locations during the plastic deformation process. 7. Although the dislocation density will show local variations due to the inhomogeneous plastic deformation, eqn(1) is assumed to hold for the average dislocation density in the active ferrite volume. 8. The mean free path of mobile dislocations is assumed to decrease exponentially with strain in accordance with eqn(3). The plastic strain dependence of the dislocation density, ρ, in the active ferrite may according to the chain-rule be written dρ dρ dε = (4) dε dε dε f f where ε f is the local strain in the active ferrite and ε is the global strain. Since the global strain, ε, is related to the local strain, ε f, as 4

dε f dε (5) f the global strain dependence of the total dislocation density for a DP steel may be expressed as (compare with eqn(2) dρε ( ) 1 m = ρ( ε ) dε f( ε) Ω b s( ε) (6) We will now assume that the strain dependence of f(ε) may be written, see (16), f( ε ) = f + ( f f ) e (7) 0 1 0 r ε where r is a material parameter that controls the rate of this progress, f 1 is the initial active volume fraction of the ferrite taking part in the deformation process and f 0 is the total amount of ferrite. f(ε) thus starts from the value of f 1 and hereafter grows with increasing strain towards f 0. Using eqn(3) and (7) in eqn(6) we finally achieve an expression for the inhomogeneous deformation behaviour of ferrite in DP steel (13) dρε ( ) 1 m = ρ( ε ) dε rε Ω kε f0 + ( f1 f0) e b ( s0 + ( s1 s0) e ) (8) Eqn(8) gives the dislocation strain dependence for a DP steel expressed in the ferrite phase parameters, that is, a description of the continuous localisation process in the DP steel. A comparison between eqn(8) and eqn(2), which is the original Bergström expression for the homogeneous ρ-ε behaviour in pure ferrite, shows that eqn(8) transcends into eqn(2) if f 0 =f 1 =1, that is, if we assume that f m =0 and that the deformation is homogeneous. EXPERIMENTS, RESULTS AND DISCUSSION Experiments In order to demonstrate the possibilities of the dislocation model presented above we shall analyse a true stress-strain curve from a DP800 steel recorded at room temperature and at a strain rate of 10-2 s -1. The martensite volume fraction in the investigated steel varies in the range 25-30%. Fitting procedure A special Matlab subroutine, based on the Matlab Curve Fitting Toolbox, was designed for the purpose of this type of study. In the fitting procedure the following parameters were kept constant: α = 0.5, G = 80000 MPa, b = 2.5 10-10 m, m = 2. The following parameters were allowed to vary freely within certain physical limits: Ω, σi0, ρ 0, f 1, f 0, r, s 1, s 0 and k. The start values were chosen on the basis of the experience obtained from previous analyses of this type of DP steel. 5

Results The results from this fitting procedure are presented in Fig. 3, 4 and 5. The left graph in Fig. 3 shows the experimental and theoretical stress-strain curves. The theoretical red curve is covering the experimental data. The red cross represents the theoretically calculated true strain to necking and the corresponding true flow stress. The error of fit is shown in the right graph where medium and running medium values are given. We can see that the fit is god over the whole strain interval. At smaller strains, however, the accuracy of fit is slightly worse. This is presumably due to the presence of residual stresses caused by the austenite martensite transformation. It seems that these stresses have been released after ~ 2% strain which seems reasonable. Fig 3. Experimental and theoretical σ ε curves for the investigated DP 800 steel left, and corresponding accuracy of fit right. For details, see the text. Fig 4. A summary of the parameter values obtained in the fit left, and the theoretical ρ-ε curve The parameter values recorded in the fit are presented to the left in Fig. 4 and to the right the corresponding theoretically calculated ρ-ε curve is shown. The variations of f and s with strain are depicted in Fig. 5. The left graph indicates that the effective volume fraction of ferrite increases from an initial value f 1 =0.15 towards a 6

maximum value f 0 = 0.73. The rate constant r is equal to 21.6. In the right graph we can see that the mean free path s of dislocation motion decreases from the initial value s 1 = 0.86 µm to a final value s 0 = 5.7 µm. The rate constant k is equal to 297.8 implying that s reaches s 0 after around 1% of strain. In Fig. 5 The corresponding theoretically calculated f(ε)-ε curve is shown to the left while the corresponding s(ε)-ε curve is depicted to the right. Discussion Let us now discuss the reasonableness of the parameter values obtained in the fitting procedure presented above, see Fig. 4. The dislocation remobilisation parameter, Ω, takes a value of 5 in the fit. This is a typical room temperature Ω - value for ferrite (14) and supports the assumption that the plastic deformation process is localised to the ferritic phase in the investigated DP steel. The high value of the grown-in dislocation density ρ 0 = 1.12 10 14 m -2 is not surprising since the austenite-martensite transformation gives rise to large stresses in the ferrite. The above figure is an average value and it is reasonable to believe that the grown-in dislocation density is even higher close to the martensite surfaces. f 1 is the initial volume fraction of active ferrite and the low value of 0.15 tells us that initially only 15% of the ferrite takes part in the plastic deformation process. At necking f has reached a value of approximately 0.6, see Fig 5. If we neglect the small effect of a strain dependent mean free path s this means that the rate of dislocation generation has decreased by a factor of ~4 and the rate of work hardening by a factor of ~2 after 7% global strain. The high initial rate of work hardening is thus caused by a low active volume fraction ferrite. It is also obvious from Fig. 5 that f does not reach the value of f 0 at necking and that therefore there may exist ~ 10% un-deformed ferrite in the testing specimen. This may then explain the blue dots observed in the centre of the ferrite grains in Fig 2b/. The mean free path, s, of dislocation motion decreases very rapidly (due to a high k value) from s 1 = 0.86 µm to s 0 = 0.57 µm within 1% of deformation. This means that the dislocation cell diameter attains a value of approximately 0.6 µm after this strain. These s-values are in fair agreement with the dislocation cell diameters measured by Korzekwa et al (17) in a DP 7

steel by using transmission electron microscopy. They are also in agreement with the distance measured between the precipitates observed in the ferrite matrix in the investigated DP steel. A simple and useful way of checking the reasonableness of the friction stress, σ i0, and, ρ 0, is to proceed from eqn(1) and (8) and assume that the global strain is small. The flow stress may then approximately be written m ε σε ( ) σi0 + α Gb ρ0 + (9) b s f 1 1 where the involved parameters are defined above. At a global strain of 0,002 and the parameter values presented in Fig. 4. the corresponding true flow stress is calculated to be ~675 MPa which is in good agreement with the corresponding experimentally observed flow stress value, see Fig. 4. It is also interesting to note that the actual local strain in the ferrite at this stage is approximately equal to 0.002/0.15=0.013 which also explains the high initial rate of work hardening in DP steels. In a recently published paper (13) the following four types of DP steels were investigated with the aid of the present dislocation theory: DP500, DP600, DP800 and DP 980. The objective of that study was to investigate the impact of martensite content and ferrite grain size on the work hardening behaviour. The theory proved to be very useful for this purpose. CONCLUSIONS The above discussed dislocation theory accurately describes the plastic deformation behaviour of DP steel. Results from microstructure investigations as well as results from analysis of the stress-strain behaviour in terms of the present theory, tell us that the plastic deformation process in DP steel is inhomogeneous By introducing the concept of a non-homogeneity parameter f (ε), that specifies the volume fraction of ferrite that takes active part in the plastic deformation process, it is possible to give a precise physical description of the deformation behaviour until necking. REFERENCES (1) C. Federici, S.Maggi, and S. Rigoni, The Use of Advanced High Strength Steel Sheets in the Automotive Industry, Fiat Auto Engineering & Design, Turin, Italy, 2005. (2) J. R. Fekete, J. N. Hall, D. J. Meuleman, and M. Rupp, Progress in implementation of advanced high-strength steels into vehicle structures, Iron & Steel Technology, vol. 5, no. 10, pp. 55 64, 2008. (3) R. Koehr, ULSAB-AVC Advenced Vehicle Concepts Overview Report, in Safe, Affordable, Fuel Efficient Vehicle Concepts for the 21st Century Designed in Steel, Porsche Engineering Services, Detroit, Mich, USA, 2002. (4) P. Tsipouridis, E. Werner, C. Krempaszky, and E. Tragl, Formability of high strength dual-phase steels, Steel Research International, vol. 77, no. 9-10, pp. 654 667, 2006. 8

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