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Modeling mechanical effects on promotion and retardation of martensitic transformation
Materials Science and Engineering A 528 (2011) 1318–1325
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Modeling mechanical effects on promotion and retardation of martensitic transformation Mehran Maalekian a,∗ , Ernst Kozeschnik b a b
Department of Materials Engineering, The University of British Columbia, 309-6350 Stores Road, Vancouver, B.C. V61Z4, Canada Christian Doppler Laboratory for ‘Early Stages of Precipitation’, Institute of Materials Science and Technology, Vienna University of Technology, Austria
a r t i c l e
i n f o
Article history: Received 9 September 2010 Received in revised form 7 October 2010 Accepted 11 October 2010
Keywords: Martensite Stress assisted transformation Mechanical stabilization Modeling Eutectoid steel
a b s t r a c t The influence of compressive stress and prior plastic deformation of austenite on the martensite transformation in a eutectoid steel is studied both experimentally and theoretically. It is demonstrated that martensite formation is assisted by stress but it is retarded when transformation occurs from deformed austenite. With the quantitative modeling of the problem based on the theory of displacive shear transformation, the explanation of the two opposite roles of mechanical treatment prior to or simultaneously to martensite transformation is presented. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The martensitic transformation in steels is usually athermal, i.e. a given fraction of austenite becoming martensite on cooling to a particular temperature below the martensite start temperature (Ms ) is time independent. The amount of transformation is a function of temperature and increases on subsequent undercooling below Ms [1]. The martensitic transformation is also characterized as displacive, since the transformation occurs by a co-operative movement of many atoms involving a shear-dominant shape change. A sufficiently high shear–strain (distortion) energy in the process dominates the kinetics and morphology during the martensitic transformation [2]. Thus, in general, it is natural to expect an interaction between any externally applied stress and the progress of the transformation, in a way which is uniquely related to the transformation mechanism. Scheil [3], first, stated that the martensitic transformation can be promoted by applied shear stress above Ms . Based on Scheil’s concept, in which the martensitic transformation is a mode of deformation, a quantitative analysis of the effects of stress on the Ms temperature was made by Patel and Cohen [4] and Fisher and Turnbull [5]. As illustrated in Fig. 1, if the Ms temperature is characterized by a critical value of the driving force Gs , the work done by an external stress compensates for the shortfall in the driv-
∗ Corresponding author. Tel.: +1 604 822 2610; fax: +1 604 822 3619. E-mail address: [email protected] (M. Maalekian). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.10.030
ing force (Gmech ) for transformation to occur above Ms (e.g. T1 in Fig. 1). The total driving force is partitioned into the chemical (Gchem ) and mechanical components. The chemical part required for the transformation can decrease if the mechanical component is increased by stressing. As Gchem is approximately linear in temperature [2,4] (Fig. 1), the Ms temperature will rise approximately linearly with stress as shown schematically in Fig. 2. When the austenite is deformed at temperatures above M , plastic strain precedes transformation. This lowers the stress required for transformation, as shown in Fig. 2. Regarding the role of plastic straining of austenite on strain-induced martensitic transformation, different hypotheses have been proposed [6]. One is based on stress concentrations at obstacles (e.g. grain boundaries, twin boundaries, etc.), which assist the transformation [7]. The other is based on the creation of new nucleation sites by plastic straining [8]. However, in contrast to the Ms temperature, where a martensitic transformation will begin at a critical driving force (Gs in Fig. 1), Md is the temperature above which the chemical driving force becomes so small that nucleation of martensite cannot be mechanically induced and austenite only deforms plastically (see Fig. 2). Thus, above Md , the stress-assisted or strain-induced martensitic transformation will not take place. Instead, above Md the martensitic transformation can be retarded by straining of austenite as illustrated in the left part of Fig. 2. It has been emphasized that displacive transformations involve the coordinated movement of atoms and that such movements cannot be sustained against strong defects such as grain boundaries.
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Fig. 1. Schematic diagram showing the free energy change for a martensitic transformation. T0 is the temperature at which the austenite and martensite of identical chemical composition have the same Gibbs free energy and Ms is the temperature at which transformation starts upon cooling. Gs is the critical driving force for the onset of the martensitic transformation, which can be either lowered (G1 ) or increased (G2 ) as a result of mechanical effects described in the text.
Thus, martensite plates, which form by a displacive mechanism, cannot traverse austenite grain boundaries. Defects, such as dislocations, also hinder the translation of the austenite/martensite (␥/␣) interface. Plastic deformation of austenite above Md prior to its transformation hinders the growth of martensite and reduces the transformed fraction [8–18,20–23]. This retardation of transformation by plastic deformation is known as mechanical stabilization and can be explained in terms of the structure of the ␣/␥ interface. The glissile interface of ␣/␥ can be rendered sessile when it encounters dislocation debris. Thus, whereas an elastic stress can promote the martensitic transformation in the same way that it enables normal deformation, mechanical stabilization retards the decomposition of austenite [24]. Fig. 1 shows schematically that the martensitic transformation starts at a lower temperature (T2 < Ms ) as a result of an increase in the required driving force (U2 ), which is related to the high defect density generated by the plastic deformation of austenite.
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To study the effect of pre-deformation on subsequent martensitic transformation kinetics, Breedis [9] conducted a transmission electron microscopy investigation of the substructure in deformed metastable stainless steel. It was shown that a small pre-strain leads to the development of planar dislocation pile-ups with longrange stress fields, which can assist nucleation and stimulate the martensitic transformation. However, with increasing level of prestrain it was found that the substructure changes from planar pile-up to dislocation forests that hinder the growth of martensite laths. Analytical models were developed to describe the kinetics of martensitic transformations using a dislocation model of interfacial structure [10]. The kinetic theory of dislocation motion was adapted to predict the mobility of martensitic interfaces. The effect of prior plastic strain of austenite on the activation energy for subsequent isothermal martensitic nucleation was measured by Ghosh and Raghavan [11]. Axisymmetric pre-straining of Fe–Ni–Mn alloy at 20 ◦ C showed a retarding effect on the isothermal transformation kinetics at all strain levels (0.005–0.05). It was found that the strain hardening of the matrix results in an increase in activation energy for isothermal martensitic nucleation. This increase in activation energy, which causes a reduction in driving force of the martensitic transformation, was attributed to the increase in athermal stress for the interfacial motion. Using the activation energy data for isothermal martensitic nucleation, and based on the assumption of linear superposition of forest hardening on the thermal and athermal contributions to solution hardening, a quantitative estimate of frictional work was made for martensitic interfaces [12,13]. Despite a plethora of literature about the mechanical effects on the kinetics of martensitic transformation, most work was concentrated on low temperature isothermal martensite (e.g. alloys with high nickel content and subzero Ms temperature) and little effort to investigate the influence of high temperature deformation on subsequent athermal martensitic transformation. Therefore, the present work deals with the mechanical effects on the kinetics of athermal martensite in high carbon steel. We examine both aspects of the problem; the promotion of the martensite transformation under axial compressive stresses, and the retardation of the martensite formation due to the externally imposed mechanical effects. Quantitative analyses for the two phenomena are presented and examined by the experimental observation on eutectoid steel.
2. Experimental
Fig. 2. Schematic illustration showing two opposite effects of external stress or strain on the martensite start temperature. The right hand side of the diagram shows the stress or strain assisted martensite formation. The left hand side illustrates the retardation of the transformation by prior straining of austenite (i.e. mechanical stabilization). The plastic strain should be sufficiently large to observe the mechanical stabilization phenomenon.
A push-rod BÄHR DIL805 A/D high-speed quenching deformation dilatometer was used for measuring the transformation temperature. The chemical composition of the eutectoid steel studied is 0.75C–1.02Mn–0.28Si–0.11Cr–0.05Ni–0.08Cu–0.01S–0.009P (wt%). Cylindrical dilatometer samples of diameter 5 mm and length 10 mm were machined for thermo-mechanical simulation. The sample temperature was measured by a thermocouple welded to its surface using a precision welder and jig supplied by the dilatometer manufacturer. Each sample was heated at a rate of 100 K s−1 to 1100 ◦ C and austenitized for 20–30 s. In case of plastic deformation, the samples were cooled to 900 ◦ C in 1 s (or shorter time) and deformed in compression along the longitudinal direction. After a compressive deformation of 2.5–5 mm applied in 1 s, the samples were quenched to room temperature. Moreover, the effect of deformation temperature of austenite on the Ms temperature was also investigated by two complementary tests at deformation temperatures of 1100 ◦ C and 800 ◦ C. In another case, during cooling, the longitudinal compressive stress up to 250 MPa was applied at 600 ◦ C and held constant down to room temperature. The samples were kept at 600 ◦ C for 2 s in order to make sure that the applied stress has reached the prescribed value. The yield strength of the steel as a function of tem-
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Fig. 3. Schematic illustration of thermo-mechanical cycles: (a) hot plastic deformation of austenite and (b) transformation under stress. (c) Yield stress of the steel as a function of temperature showing that the applied stresses in (b) at 600 ◦ C are in the elastic regime and well below the yield strength of the steel.
perature (Fig. 3) shows that the applied stresses were in the elastic regime without plastic deformation. The thermo-mechanical simulation is also schematically illustrated in Fig. 3. All the dilatometry experiments were carried out in vacuum of 5 × 10−5 mbar. The typical cooling diagram of a specimen in Fig. 4a shows that the quench time from the holding temperature to the lowest deformation temperature (i.e. 800 ◦ C) is 1 s, and the average cooling rate from 1100 ◦ C to 200 ◦ C is about 180 ◦ C/s (i.e. cooling time t = 5 s). It should be emphasized that the rate of heating and the soaking time during thermal and thermo-mechanical cycles has a profound influence on carbide dissolution (homogenization of austenite) and the martensite transformation temperature. The continuous heating time–temperature-austenitizing diagram for similar steel composition [19] suggests that full homogenization is achieved under the present experimental conditions. Moreover, the cooling curve superimposed on the continuous cooling transformation (CCT) diagram of the steel (Fig. 4b) clearly suggests that the final microstructure, as shown in Fig. 4c, is fully martensitic and no precipitation occurs at such a fast cooling rate.
ative to the thermal contraction of the parent austenite. In this method a transformation strain corresponding to 2 vol.% martensite is set as the value of the offset at which Ms is measured. The corresponding 2 vol.% of martensite forming at room temperature is calculated using the lattice parameters of austenite and martensite [26]. However, since the method is based on a positive deviation resulting from a fixed strain, it is not possible to use this method for transformations taking place under compressive stress, because the compressive stress shifts the dilatometry curve in the negative direction (downward). Thus, the fixed positive deviation line, which is above the dilatometry curve, will not intersect the curve. Therefore, the Ms temperature of the sample under pressure is determined at the first deviation of the dilatometry curve from the linear extrapolation of the thermal contraction of austenite.
4. Analysis 4.1. Effect of stress
3. Determination of the Ms temperature Martensite formation from austenite is accompanied by a volume expansion (thus a change in length), which can be measured using dilatometry. The martensite start temperature Ms can be determined as the point when the dilatometry curve departs from the pure thermal contraction of austenite. For evaluation of the dilatometry experiments with plastic deformation of austenite, the offset method [25] is used to identify the Ms temperature from the experimental curves. In the offset method, the onset of martensite formation is defined at a critical strain (length change), which is achieved rel-
Fig. 1 shows schematically the change in free energies of austenite and martensite as a function of temperature. T0 is the temperature at which the stress-free austenite and martensite possess the same free energy. Gs is the critical driving force for the onset of the martensitic transformation at Ms temperature. A quantitative prediction of the effects of a given stress can be made by adding the mechanical work Gmech done by the external stress to the chemical free energy change G1 in order to obtain the total free energy difference: Gs = G1 + Gmech
(1)
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Fig. 4. (a) Typical cooling diagram shows that after isothermal holding at 1100 ◦ C the specimen is cooled to 800 ◦ C in 1 s, and the average cooling rate from 1100 ◦ C to 200 ◦ C is about 180 ◦ C/s. (b) Continuous cooling transformation (CCT) diagram of the present steel with an austenitizing temperature of 1100 ◦ C for 1 s against the cooling curve, and (c) light optical micrograph showing a fully martensitic microstructure after cooling.
Gmech is the mechanical driving force necessary for the stress induced martensitic transformation at T1 (see Fig. 1) and it is obtained by the work done by the external stress A in generating a macroscopic shape deformation [1,4,6,24]: Gmech = + N ε
When stressing the parent austenite, a martensite plate, whose orientation yields a maximum value of mechanical driving force Gmech , will be formed. The maximum value of Gmech is obtained when ˛ = 0 and d(Gmech )/d = 0.
(2)
where is the shear stress resolved along the habit plane, which is parallel to the direction along which the shear displacements of the shape deformation takes place. is the transformation shear strain, N is the normal stress on the habit plane and ε is the dilatational component of shape deformation (Fig. 5). The work done by the shear stress is expected to be positive, whereas the dilatational component may have either signs depending on the sign of N (i.e. compressive or tension). For an applied uniaxial tensile or compressive stress A , as shown in Fig. 5, the associated mechanical driving force can be expressed as [4,5]: Gmech =
1 1 a sin 2 cos ˛ ± a ε(1 + cos 2) 2 2
(3)
where is the angle between the applied stress axis and the normal to the habit plane, and ˛ is the angle between the transformation shear direction () and the maximum shear direction ( m ) on the invariant plane. Since the shear component of the shape change is large and the shear stress remains positive, irrespective of whether the sample is pulled or compressed, a uniaxial stress will always give rise to an increase in the transformation temperature for displacive transformations.
Fig. 5. Relation between the applied stress A and the resolved normal N and shear stresses acting on the habit plane.
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straining of austenite can be expressed quantitatively by balancing the force driving the transformation interface against the resistance from dislocation debris created by the deformation of austenite. The theory for mechanical stabilization [20,21] is based on the assumption that the stress driving the austenite/martensite interface through a forest of dislocations originates from the chemical free energy change G˛ of transformation =
Fig. 6. Change in free energy of the austenite–martensite reaction as a function of temperature for the eutectoid steel studied.
Using = 0.22 and ε = 0.03 as values for the macroscopic transformation product [29], the orientation of the first martensite plates to form under applied stress upon cooling ( = 41◦ ) and the maximum value of Gmech for applied compressive stresses a in MPa are obtained as: Gmech = 0.65a J mol−1
(4)
In Eq. (4), Gmech (the mechanical work per unit volume) is expressed in terms of J mol−1 by using the molar volume Vm ≈ 7 × 10−6 m3 mol−1 for unit conversion. Fig. 6 shows the calculated free energy diagram of the steel studied using MatCalc software version 5.31 and the corresponding mc steel database [30]. Although the diagram is not linear, however, for the temperature range of our interest, a linear portion of the curve can be taken for the calculation. Using Fig. 6 and Eq. (4), the increase in Ms due to the applied compressive stress is obtained as dMs Gmech 0.65a = ≈ 0.1 K MPa−1 = 6.79 d ∂G˛ /∂T
(5)
Eq. (5) suggests that the critical applied compressive stress for the start of the martensite transformation in the present steel is expected to increase linearly with an increase in stressing temperature. It should be mentioned that a general approach to account for the increase in Ms with elastic stress is to apply a distribution of nucleation-site orientations, in which the effective site-potency distribution is expressed in terms of the applied stress [32]. Such a model accounts for the non-linear dependence of Ms on the applied stress and the grain size. However, the Patel–Cohen approach that corresponds to the prediction for a fully biased orientation distribution, in which all nucleation sites are of the optimum orientation, has been found to be robust and it has even been extended to include grain size effect [33,34].
G˛
(6)
where is a constant assumed to be close to unity. The deformed samples, when compared to the undeformed ones, transform at lower temperatures (e.g. T2 in Fig. 1) as a result of mechanical stabilization. The magnitude of the change in driving force (U2 in Fig. 1) due to the presence of dislocations in the austenite and the corresponding reduction in Ms temperature can be obtained by using Eq. (6), the chemical free energy of the steel (Fig. 6), and the necessary shear stress to force dislocations past each other [31] √ √ Gb [ − 0 ] = U2 = 6.79T 8 (1 − )
(7)
where G ≈ 60 GPa is the shear modulus of the austenite at 800 K [35], = 0.27 is the Poisson’s ratio and the dislocation density has a value of 0 at zero plastic strain. It follows that the depression of transformation temperature can be calculated as a function of the change in the dislocation density of austenite. Therefore, it is necessary to derive a relationship between plastic strain ε and dislocation density. A physical approach based on the stored energy in the material is followed. It is assumed that approximately 5% of the plastic work is stored in the material [1,36], both owing to changes in the austenite grain surface per unit volume, Sv , and owing to the expected change in dislocation density. By balancing the stored plastic work against the energy of defects created, the following equation is obtained 0.05y ≈ Sv +
(8)
where y is the yield stress of austenite (measured to be 40 MPa at 900 ◦ C), is the austenite grain boundary energy per unit area, taken to be 0.6 J m−2 [37] and ϕ ≈ 0.5 Gb2 is the dislocation energy per unit length. The value of Sv can be calculated from an equation for axisymmetric compression, given in [38], by taking d = 80 m, which is the mean equivalent area diameter of the austenite grain size. The change in dislocation density due to the deformation is simply obtained from Eqs. (7) and (8). Using Eq. (8) and the corresponding parameters explained above, an estimated value for d /dε of 8 × 1014 m−2 is obtained for the eutectoid steel deformed in axisymmetric compression at 900 ◦ C. This inferred level of dislocation production is comparable to 9.2 × 1014 m−2 estimated for Fe–Ni–Mn alloy [11] and, also, the levels of ≈2 × 1015 m−2 directly measured in polycrystalline Ag [39] and Cu [40] after axisymmetric deformation at room temperature.
4.2. Effect of plastic strain 5. Results and discussion Prior deformation of austenite above the Md temperature (Fig. 2) leads to a retardation of the martensitic transformation. Usually, in this situation, Ms is lowered, i.e. the stability of the austenite is increased, which is attributed to the additional resistance to the motion of the austenite/martensite interface resulting from the high defect densities generated by the deformation. This phenomenon is known as mechanical stabilization. The mechanical stabilization can be explained in terms of the structure of the transformation interface. Martensitic transformations occur by the advance of glissile interfaces, which can be rendered sessile when they encounter strong defects, such as grain boundaries or dislocation debris. Thus, the change in Ms due to the
5.1. Stress assisted transformation The experimental dilatometric curves of the specimens during cooling under various uniaxial compressive stresses are shown in Fig. 7. It is apparent that, with the increase of applied compressive stress, the length increase of the specimens due to the martensitic transformation lessens (e.g. for 50 MPa) or even entirely vanishes for higher stresses (e.g. 150 MPa). This strain, which remains after phase transformation under an externally applied stress lower than the yield stress of the material, is known as transformation plasticity [41,42]. It should be mentioned that the transformation
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with the applied uniaxial stress. It is concluded that the theory of promotion of martensitic transformation under uniaxial stresses, developed on the basis of macroscopic shape deformation by Patel and Cohen [4] and Fisher and Turnbull [5], can reasonably well predict the change in kinetics of martensite transformation for the eutectoid steel studied. 5.2. Plastic straining of austenite
Fig. 7. Experimental dilatometric curves during quenching under various uniaxial compressive stresses.
strain becomes anisotropic when martensite forms under the external stress. Therefore, as the axial strain decreases with stress (or vanishes at high stresses) the radial strain (change in diameter) increases instead. Due to this phenomenon, the Ms temperature was determined using the first deviation of the dilatometry curve from the extrapolated austenite dilatation. The calculated increase in the martensite start temperature due to the uniaxial compressive stresses (Eq. (6)) and the corresponding experimental findings are compared in Fig. 8. The calculated Ms is in good accordance with the measured values. The theory successfully predicts the increase of Ms as a result of compressive stresses. The quantitative agreement found between theory and experiment implies that the Ms temperature of eutectoid steel increases linearly
Fig. 8. Increase of the Ms temperature with compressive stress.
Fig. 9a shows the effect of plastic deformation of austenite at high temperature (2.5 and 5 mm reduction in height at 900 ◦ C) on the Ms temperature. A comparison of the two methods (the offset method and the first deviation of the dilatometry curve) used to determine Ms temperature shows that the two methods deliver comparable results. The depression in Ms becomes more pronounced with larger deformation. The fact that the retardation of transformation at small plastic deformation is not remarkable (i.e. Ms = 1 ◦ C for 2.5 mm reduction in height shown in Fig. 9) indicates that mechanical stabilization only manifests itself if sufficient dislocation debris is generated by severe plastic deformation of austenite, thus causing a reduction in the Ms temperature. The change in Ms with respect to deformation temperature depicted in Fig. 9b shows that, at high temperature (1100 ◦ C), the deformation does not influence the Ms temperature. This is attributed to the dynamic recrystallization taking place during plastic deformation. By lowering the deformation temperature to 900 ◦ C and 800 ◦ C, Ms decreases as a result of mechanical stabilization. The predicted depression of Ms as a function of plastic strain is shown in Fig. 10. Since the compressive deformation is heterogeneous, the variation of effective strain through the specimen was calculated using the commercial finite element (FE) analysis software DEFORM [43] as illustrated in Fig. 11. The FE analysis shows that the effective local strain for 5 mm compressive deformation varies from 0.04 to 1.4. As illustrated in Fig. 11, the maximum strain zone (ε > 1) comprises only a small volume fraction (∼10 vol.%) of the whole specimen. However, since the measurement of Ms is over the entire specimen volume, an average uniform strain can be taken for the analysis. The calculated average strains for 2.5 and 5 mm deformation are 0.3 and 0.7, respectively. The heterogeneous strain also gives rise to the fact that the transformation occurs over a greater range of temperatures when compared with the undeformed specimen [21]. The predicted depression in the Ms temperature for the eutectoid steel studied, as well as for 22MnB5 steel [44] deformed in the austenite region, is compared against the measured values in Fig. 10. The theory of mechanical stabilization predicts well the suppression of the Ms temperature as martensite grows in deformed austenite. The model explains quantitatively how the plastic deformation of austenite above Md increases the resistance to the interface (␣/␥) motion in martensite transformation as a
Fig. 9. (a) Measured variation in the Ms temperature of the eutectoid steel determined from dilatometric data. (b) The effect of deformation temperature on the Ms temperature. The amount of deformation is 5 mm reduction in height.
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2. It is demonstrated experimentally that the large plastic deformation (εave ≈ 70%) of austenite before transformation retards the formation of martensite by lowering the Ms temperature. However, at high temperature (1100 ◦ C) the deformation does not influence the Ms temperature, which is attributable to dynamic recrystallization. Using a theory for mechanical stabilization, developed by balancing the force driving the transformation interface against the resistance from deformation defects in the deformed austenite, the depression in the martensite start temperature is quantitatively estimated. The theory explains, how the reduction in the chemical free energy difference between austenite and martensite, due to the severe plastic deformation of austenite above Md , decreases Ms . Acknowledgment
Fig. 10. Suppression of Ms as a function of plastic deformation of austenite prior to transformation for the eutectoid steel studied in the present work and a boron steel plate.
The experimental part of this work was performed at the Institute for Materials Science and Welding, Graz University of Technology, Austria. One of the authors (MM) gratefully acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
result of high defect density. Therefore, for initiation of the martensite transformation, a higher driving force is required (Gmech in Fig. 1), which is reflected in a lower Ms temperature, as illustrated in Figs. 2, 9 and 10.
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6. Summary The role of compressive stress and prior plastic deformation of austenite in the martensitic transformation of eutectoid steel is quantitatively analyzed. It is demonstrated, both experimentally and theoretically, that stress promotes martensite formation at higher temperature, whereas straining of austenite retards martensitic transformation. The two opposite roles of mechanical deformation on the martensite transformation analyzed in the present paper can be summarized as follows: 1. Using the theory of promotion of martensitic transformation, based on the work done by shear and dilatational components of uniaxial external stress on the habit plane, which adds to the chemical free energy difference, the change in the Ms temperature is successfully estimated for the eutectoid steel studied.
Fig. 11. Distribution of strain in uniaxial compressive deformation with 5 mm reduction in height.
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J.E. Bailey, Phil. Mag. 8 (1963) 223. G.W. Greenwood, R.H. Johnson, Proc. R. Soc. London 283A (1965) 403. E. Gautier, A. Simon, G. Beck, Acta Metall. 35 (1987) 1367. http://www.DEFORM.com. M. Naderi, A. Saeed-Akbari, W. Bleck, Mater. Sci. Eng. A 487 (2008) 445– 455.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Modeling mechanical effects on promotion and retardation of martensitic transformation Mehran Maalekian a,∗ , Ernst Kozeschnik b a b
Department of Materials Engineering, The University of British Columbia, 309-6350 Stores Road, Vancouver, B.C. V61Z4, Canada Christian Doppler Laboratory for ‘Early Stages of Precipitation’, Institute of Materials Science and Technology, Vienna University of Technology, Austria
a r t i c l e
i n f o
Article history: Received 9 September 2010 Received in revised form 7 October 2010 Accepted 11 October 2010
Keywords: Martensite Stress assisted transformation Mechanical stabilization Modeling Eutectoid steel
a b s t r a c t The influence of compressive stress and prior plastic deformation of austenite on the martensite transformation in a eutectoid steel is studied both experimentally and theoretically. It is demonstrated that martensite formation is assisted by stress but it is retarded when transformation occurs from deformed austenite. With the quantitative modeling of the problem based on the theory of displacive shear transformation, the explanation of the two opposite roles of mechanical treatment prior to or simultaneously to martensite transformation is presented. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The martensitic transformation in steels is usually athermal, i.e. a given fraction of austenite becoming martensite on cooling to a particular temperature below the martensite start temperature (Ms ) is time independent. The amount of transformation is a function of temperature and increases on subsequent undercooling below Ms [1]. The martensitic transformation is also characterized as displacive, since the transformation occurs by a co-operative movement of many atoms involving a shear-dominant shape change. A sufficiently high shear–strain (distortion) energy in the process dominates the kinetics and morphology during the martensitic transformation [2]. Thus, in general, it is natural to expect an interaction between any externally applied stress and the progress of the transformation, in a way which is uniquely related to the transformation mechanism. Scheil [3], first, stated that the martensitic transformation can be promoted by applied shear stress above Ms . Based on Scheil’s concept, in which the martensitic transformation is a mode of deformation, a quantitative analysis of the effects of stress on the Ms temperature was made by Patel and Cohen [4] and Fisher and Turnbull [5]. As illustrated in Fig. 1, if the Ms temperature is characterized by a critical value of the driving force Gs , the work done by an external stress compensates for the shortfall in the driv-
∗ Corresponding author. Tel.: +1 604 822 2610; fax: +1 604 822 3619. E-mail address: [email protected] (M. Maalekian). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.10.030
ing force (Gmech ) for transformation to occur above Ms (e.g. T1 in Fig. 1). The total driving force is partitioned into the chemical (Gchem ) and mechanical components. The chemical part required for the transformation can decrease if the mechanical component is increased by stressing. As Gchem is approximately linear in temperature [2,4] (Fig. 1), the Ms temperature will rise approximately linearly with stress as shown schematically in Fig. 2. When the austenite is deformed at temperatures above M , plastic strain precedes transformation. This lowers the stress required for transformation, as shown in Fig. 2. Regarding the role of plastic straining of austenite on strain-induced martensitic transformation, different hypotheses have been proposed [6]. One is based on stress concentrations at obstacles (e.g. grain boundaries, twin boundaries, etc.), which assist the transformation [7]. The other is based on the creation of new nucleation sites by plastic straining [8]. However, in contrast to the Ms temperature, where a martensitic transformation will begin at a critical driving force (Gs in Fig. 1), Md is the temperature above which the chemical driving force becomes so small that nucleation of martensite cannot be mechanically induced and austenite only deforms plastically (see Fig. 2). Thus, above Md , the stress-assisted or strain-induced martensitic transformation will not take place. Instead, above Md the martensitic transformation can be retarded by straining of austenite as illustrated in the left part of Fig. 2. It has been emphasized that displacive transformations involve the coordinated movement of atoms and that such movements cannot be sustained against strong defects such as grain boundaries.
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Fig. 1. Schematic diagram showing the free energy change for a martensitic transformation. T0 is the temperature at which the austenite and martensite of identical chemical composition have the same Gibbs free energy and Ms is the temperature at which transformation starts upon cooling. Gs is the critical driving force for the onset of the martensitic transformation, which can be either lowered (G1 ) or increased (G2 ) as a result of mechanical effects described in the text.
Thus, martensite plates, which form by a displacive mechanism, cannot traverse austenite grain boundaries. Defects, such as dislocations, also hinder the translation of the austenite/martensite (␥/␣) interface. Plastic deformation of austenite above Md prior to its transformation hinders the growth of martensite and reduces the transformed fraction [8–18,20–23]. This retardation of transformation by plastic deformation is known as mechanical stabilization and can be explained in terms of the structure of the ␣/␥ interface. The glissile interface of ␣/␥ can be rendered sessile when it encounters dislocation debris. Thus, whereas an elastic stress can promote the martensitic transformation in the same way that it enables normal deformation, mechanical stabilization retards the decomposition of austenite [24]. Fig. 1 shows schematically that the martensitic transformation starts at a lower temperature (T2 < Ms ) as a result of an increase in the required driving force (U2 ), which is related to the high defect density generated by the plastic deformation of austenite.
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To study the effect of pre-deformation on subsequent martensitic transformation kinetics, Breedis [9] conducted a transmission electron microscopy investigation of the substructure in deformed metastable stainless steel. It was shown that a small pre-strain leads to the development of planar dislocation pile-ups with longrange stress fields, which can assist nucleation and stimulate the martensitic transformation. However, with increasing level of prestrain it was found that the substructure changes from planar pile-up to dislocation forests that hinder the growth of martensite laths. Analytical models were developed to describe the kinetics of martensitic transformations using a dislocation model of interfacial structure [10]. The kinetic theory of dislocation motion was adapted to predict the mobility of martensitic interfaces. The effect of prior plastic strain of austenite on the activation energy for subsequent isothermal martensitic nucleation was measured by Ghosh and Raghavan [11]. Axisymmetric pre-straining of Fe–Ni–Mn alloy at 20 ◦ C showed a retarding effect on the isothermal transformation kinetics at all strain levels (0.005–0.05). It was found that the strain hardening of the matrix results in an increase in activation energy for isothermal martensitic nucleation. This increase in activation energy, which causes a reduction in driving force of the martensitic transformation, was attributed to the increase in athermal stress for the interfacial motion. Using the activation energy data for isothermal martensitic nucleation, and based on the assumption of linear superposition of forest hardening on the thermal and athermal contributions to solution hardening, a quantitative estimate of frictional work was made for martensitic interfaces [12,13]. Despite a plethora of literature about the mechanical effects on the kinetics of martensitic transformation, most work was concentrated on low temperature isothermal martensite (e.g. alloys with high nickel content and subzero Ms temperature) and little effort to investigate the influence of high temperature deformation on subsequent athermal martensitic transformation. Therefore, the present work deals with the mechanical effects on the kinetics of athermal martensite in high carbon steel. We examine both aspects of the problem; the promotion of the martensite transformation under axial compressive stresses, and the retardation of the martensite formation due to the externally imposed mechanical effects. Quantitative analyses for the two phenomena are presented and examined by the experimental observation on eutectoid steel.
2. Experimental
Fig. 2. Schematic illustration showing two opposite effects of external stress or strain on the martensite start temperature. The right hand side of the diagram shows the stress or strain assisted martensite formation. The left hand side illustrates the retardation of the transformation by prior straining of austenite (i.e. mechanical stabilization). The plastic strain should be sufficiently large to observe the mechanical stabilization phenomenon.
A push-rod BÄHR DIL805 A/D high-speed quenching deformation dilatometer was used for measuring the transformation temperature. The chemical composition of the eutectoid steel studied is 0.75C–1.02Mn–0.28Si–0.11Cr–0.05Ni–0.08Cu–0.01S–0.009P (wt%). Cylindrical dilatometer samples of diameter 5 mm and length 10 mm were machined for thermo-mechanical simulation. The sample temperature was measured by a thermocouple welded to its surface using a precision welder and jig supplied by the dilatometer manufacturer. Each sample was heated at a rate of 100 K s−1 to 1100 ◦ C and austenitized for 20–30 s. In case of plastic deformation, the samples were cooled to 900 ◦ C in 1 s (or shorter time) and deformed in compression along the longitudinal direction. After a compressive deformation of 2.5–5 mm applied in 1 s, the samples were quenched to room temperature. Moreover, the effect of deformation temperature of austenite on the Ms temperature was also investigated by two complementary tests at deformation temperatures of 1100 ◦ C and 800 ◦ C. In another case, during cooling, the longitudinal compressive stress up to 250 MPa was applied at 600 ◦ C and held constant down to room temperature. The samples were kept at 600 ◦ C for 2 s in order to make sure that the applied stress has reached the prescribed value. The yield strength of the steel as a function of tem-
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Fig. 3. Schematic illustration of thermo-mechanical cycles: (a) hot plastic deformation of austenite and (b) transformation under stress. (c) Yield stress of the steel as a function of temperature showing that the applied stresses in (b) at 600 ◦ C are in the elastic regime and well below the yield strength of the steel.
perature (Fig. 3) shows that the applied stresses were in the elastic regime without plastic deformation. The thermo-mechanical simulation is also schematically illustrated in Fig. 3. All the dilatometry experiments were carried out in vacuum of 5 × 10−5 mbar. The typical cooling diagram of a specimen in Fig. 4a shows that the quench time from the holding temperature to the lowest deformation temperature (i.e. 800 ◦ C) is 1 s, and the average cooling rate from 1100 ◦ C to 200 ◦ C is about 180 ◦ C/s (i.e. cooling time t = 5 s). It should be emphasized that the rate of heating and the soaking time during thermal and thermo-mechanical cycles has a profound influence on carbide dissolution (homogenization of austenite) and the martensite transformation temperature. The continuous heating time–temperature-austenitizing diagram for similar steel composition [19] suggests that full homogenization is achieved under the present experimental conditions. Moreover, the cooling curve superimposed on the continuous cooling transformation (CCT) diagram of the steel (Fig. 4b) clearly suggests that the final microstructure, as shown in Fig. 4c, is fully martensitic and no precipitation occurs at such a fast cooling rate.
ative to the thermal contraction of the parent austenite. In this method a transformation strain corresponding to 2 vol.% martensite is set as the value of the offset at which Ms is measured. The corresponding 2 vol.% of martensite forming at room temperature is calculated using the lattice parameters of austenite and martensite [26]. However, since the method is based on a positive deviation resulting from a fixed strain, it is not possible to use this method for transformations taking place under compressive stress, because the compressive stress shifts the dilatometry curve in the negative direction (downward). Thus, the fixed positive deviation line, which is above the dilatometry curve, will not intersect the curve. Therefore, the Ms temperature of the sample under pressure is determined at the first deviation of the dilatometry curve from the linear extrapolation of the thermal contraction of austenite.
4. Analysis 4.1. Effect of stress
3. Determination of the Ms temperature Martensite formation from austenite is accompanied by a volume expansion (thus a change in length), which can be measured using dilatometry. The martensite start temperature Ms can be determined as the point when the dilatometry curve departs from the pure thermal contraction of austenite. For evaluation of the dilatometry experiments with plastic deformation of austenite, the offset method [25] is used to identify the Ms temperature from the experimental curves. In the offset method, the onset of martensite formation is defined at a critical strain (length change), which is achieved rel-
Fig. 1 shows schematically the change in free energies of austenite and martensite as a function of temperature. T0 is the temperature at which the stress-free austenite and martensite possess the same free energy. Gs is the critical driving force for the onset of the martensitic transformation at Ms temperature. A quantitative prediction of the effects of a given stress can be made by adding the mechanical work Gmech done by the external stress to the chemical free energy change G1 in order to obtain the total free energy difference: Gs = G1 + Gmech
(1)
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Fig. 4. (a) Typical cooling diagram shows that after isothermal holding at 1100 ◦ C the specimen is cooled to 800 ◦ C in 1 s, and the average cooling rate from 1100 ◦ C to 200 ◦ C is about 180 ◦ C/s. (b) Continuous cooling transformation (CCT) diagram of the present steel with an austenitizing temperature of 1100 ◦ C for 1 s against the cooling curve, and (c) light optical micrograph showing a fully martensitic microstructure after cooling.
Gmech is the mechanical driving force necessary for the stress induced martensitic transformation at T1 (see Fig. 1) and it is obtained by the work done by the external stress A in generating a macroscopic shape deformation [1,4,6,24]: Gmech = + N ε
When stressing the parent austenite, a martensite plate, whose orientation yields a maximum value of mechanical driving force Gmech , will be formed. The maximum value of Gmech is obtained when ˛ = 0 and d(Gmech )/d = 0.
(2)
where is the shear stress resolved along the habit plane, which is parallel to the direction along which the shear displacements of the shape deformation takes place. is the transformation shear strain, N is the normal stress on the habit plane and ε is the dilatational component of shape deformation (Fig. 5). The work done by the shear stress is expected to be positive, whereas the dilatational component may have either signs depending on the sign of N (i.e. compressive or tension). For an applied uniaxial tensile or compressive stress A , as shown in Fig. 5, the associated mechanical driving force can be expressed as [4,5]: Gmech =
1 1 a sin 2 cos ˛ ± a ε(1 + cos 2) 2 2
(3)
where is the angle between the applied stress axis and the normal to the habit plane, and ˛ is the angle between the transformation shear direction () and the maximum shear direction ( m ) on the invariant plane. Since the shear component of the shape change is large and the shear stress remains positive, irrespective of whether the sample is pulled or compressed, a uniaxial stress will always give rise to an increase in the transformation temperature for displacive transformations.
Fig. 5. Relation between the applied stress A and the resolved normal N and shear stresses acting on the habit plane.
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straining of austenite can be expressed quantitatively by balancing the force driving the transformation interface against the resistance from dislocation debris created by the deformation of austenite. The theory for mechanical stabilization [20,21] is based on the assumption that the stress driving the austenite/martensite interface through a forest of dislocations originates from the chemical free energy change G˛ of transformation =
Fig. 6. Change in free energy of the austenite–martensite reaction as a function of temperature for the eutectoid steel studied.
Using = 0.22 and ε = 0.03 as values for the macroscopic transformation product [29], the orientation of the first martensite plates to form under applied stress upon cooling ( = 41◦ ) and the maximum value of Gmech for applied compressive stresses a in MPa are obtained as: Gmech = 0.65a J mol−1
(4)
In Eq. (4), Gmech (the mechanical work per unit volume) is expressed in terms of J mol−1 by using the molar volume Vm ≈ 7 × 10−6 m3 mol−1 for unit conversion. Fig. 6 shows the calculated free energy diagram of the steel studied using MatCalc software version 5.31 and the corresponding mc steel database [30]. Although the diagram is not linear, however, for the temperature range of our interest, a linear portion of the curve can be taken for the calculation. Using Fig. 6 and Eq. (4), the increase in Ms due to the applied compressive stress is obtained as dMs Gmech 0.65a = ≈ 0.1 K MPa−1 = 6.79 d ∂G˛ /∂T
(5)
Eq. (5) suggests that the critical applied compressive stress for the start of the martensite transformation in the present steel is expected to increase linearly with an increase in stressing temperature. It should be mentioned that a general approach to account for the increase in Ms with elastic stress is to apply a distribution of nucleation-site orientations, in which the effective site-potency distribution is expressed in terms of the applied stress [32]. Such a model accounts for the non-linear dependence of Ms on the applied stress and the grain size. However, the Patel–Cohen approach that corresponds to the prediction for a fully biased orientation distribution, in which all nucleation sites are of the optimum orientation, has been found to be robust and it has even been extended to include grain size effect [33,34].
G˛
(6)
where is a constant assumed to be close to unity. The deformed samples, when compared to the undeformed ones, transform at lower temperatures (e.g. T2 in Fig. 1) as a result of mechanical stabilization. The magnitude of the change in driving force (U2 in Fig. 1) due to the presence of dislocations in the austenite and the corresponding reduction in Ms temperature can be obtained by using Eq. (6), the chemical free energy of the steel (Fig. 6), and the necessary shear stress to force dislocations past each other [31] √ √ Gb [ − 0 ] = U2 = 6.79T 8 (1 − )
(7)
where G ≈ 60 GPa is the shear modulus of the austenite at 800 K [35], = 0.27 is the Poisson’s ratio and the dislocation density has a value of 0 at zero plastic strain. It follows that the depression of transformation temperature can be calculated as a function of the change in the dislocation density of austenite. Therefore, it is necessary to derive a relationship between plastic strain ε and dislocation density. A physical approach based on the stored energy in the material is followed. It is assumed that approximately 5% of the plastic work is stored in the material [1,36], both owing to changes in the austenite grain surface per unit volume, Sv , and owing to the expected change in dislocation density. By balancing the stored plastic work against the energy of defects created, the following equation is obtained 0.05y ≈ Sv +
(8)
where y is the yield stress of austenite (measured to be 40 MPa at 900 ◦ C), is the austenite grain boundary energy per unit area, taken to be 0.6 J m−2 [37] and ϕ ≈ 0.5 Gb2 is the dislocation energy per unit length. The value of Sv can be calculated from an equation for axisymmetric compression, given in [38], by taking d = 80 m, which is the mean equivalent area diameter of the austenite grain size. The change in dislocation density due to the deformation is simply obtained from Eqs. (7) and (8). Using Eq. (8) and the corresponding parameters explained above, an estimated value for d /dε of 8 × 1014 m−2 is obtained for the eutectoid steel deformed in axisymmetric compression at 900 ◦ C. This inferred level of dislocation production is comparable to 9.2 × 1014 m−2 estimated for Fe–Ni–Mn alloy [11] and, also, the levels of ≈2 × 1015 m−2 directly measured in polycrystalline Ag [39] and Cu [40] after axisymmetric deformation at room temperature.
4.2. Effect of plastic strain 5. Results and discussion Prior deformation of austenite above the Md temperature (Fig. 2) leads to a retardation of the martensitic transformation. Usually, in this situation, Ms is lowered, i.e. the stability of the austenite is increased, which is attributed to the additional resistance to the motion of the austenite/martensite interface resulting from the high defect densities generated by the deformation. This phenomenon is known as mechanical stabilization. The mechanical stabilization can be explained in terms of the structure of the transformation interface. Martensitic transformations occur by the advance of glissile interfaces, which can be rendered sessile when they encounter strong defects, such as grain boundaries or dislocation debris. Thus, the change in Ms due to the
5.1. Stress assisted transformation The experimental dilatometric curves of the specimens during cooling under various uniaxial compressive stresses are shown in Fig. 7. It is apparent that, with the increase of applied compressive stress, the length increase of the specimens due to the martensitic transformation lessens (e.g. for 50 MPa) or even entirely vanishes for higher stresses (e.g. 150 MPa). This strain, which remains after phase transformation under an externally applied stress lower than the yield stress of the material, is known as transformation plasticity [41,42]. It should be mentioned that the transformation
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with the applied uniaxial stress. It is concluded that the theory of promotion of martensitic transformation under uniaxial stresses, developed on the basis of macroscopic shape deformation by Patel and Cohen [4] and Fisher and Turnbull [5], can reasonably well predict the change in kinetics of martensite transformation for the eutectoid steel studied. 5.2. Plastic straining of austenite
Fig. 7. Experimental dilatometric curves during quenching under various uniaxial compressive stresses.
strain becomes anisotropic when martensite forms under the external stress. Therefore, as the axial strain decreases with stress (or vanishes at high stresses) the radial strain (change in diameter) increases instead. Due to this phenomenon, the Ms temperature was determined using the first deviation of the dilatometry curve from the extrapolated austenite dilatation. The calculated increase in the martensite start temperature due to the uniaxial compressive stresses (Eq. (6)) and the corresponding experimental findings are compared in Fig. 8. The calculated Ms is in good accordance with the measured values. The theory successfully predicts the increase of Ms as a result of compressive stresses. The quantitative agreement found between theory and experiment implies that the Ms temperature of eutectoid steel increases linearly
Fig. 8. Increase of the Ms temperature with compressive stress.
Fig. 9a shows the effect of plastic deformation of austenite at high temperature (2.5 and 5 mm reduction in height at 900 ◦ C) on the Ms temperature. A comparison of the two methods (the offset method and the first deviation of the dilatometry curve) used to determine Ms temperature shows that the two methods deliver comparable results. The depression in Ms becomes more pronounced with larger deformation. The fact that the retardation of transformation at small plastic deformation is not remarkable (i.e. Ms = 1 ◦ C for 2.5 mm reduction in height shown in Fig. 9) indicates that mechanical stabilization only manifests itself if sufficient dislocation debris is generated by severe plastic deformation of austenite, thus causing a reduction in the Ms temperature. The change in Ms with respect to deformation temperature depicted in Fig. 9b shows that, at high temperature (1100 ◦ C), the deformation does not influence the Ms temperature. This is attributed to the dynamic recrystallization taking place during plastic deformation. By lowering the deformation temperature to 900 ◦ C and 800 ◦ C, Ms decreases as a result of mechanical stabilization. The predicted depression of Ms as a function of plastic strain is shown in Fig. 10. Since the compressive deformation is heterogeneous, the variation of effective strain through the specimen was calculated using the commercial finite element (FE) analysis software DEFORM [43] as illustrated in Fig. 11. The FE analysis shows that the effective local strain for 5 mm compressive deformation varies from 0.04 to 1.4. As illustrated in Fig. 11, the maximum strain zone (ε > 1) comprises only a small volume fraction (∼10 vol.%) of the whole specimen. However, since the measurement of Ms is over the entire specimen volume, an average uniform strain can be taken for the analysis. The calculated average strains for 2.5 and 5 mm deformation are 0.3 and 0.7, respectively. The heterogeneous strain also gives rise to the fact that the transformation occurs over a greater range of temperatures when compared with the undeformed specimen [21]. The predicted depression in the Ms temperature for the eutectoid steel studied, as well as for 22MnB5 steel [44] deformed in the austenite region, is compared against the measured values in Fig. 10. The theory of mechanical stabilization predicts well the suppression of the Ms temperature as martensite grows in deformed austenite. The model explains quantitatively how the plastic deformation of austenite above Md increases the resistance to the interface (␣/␥) motion in martensite transformation as a
Fig. 9. (a) Measured variation in the Ms temperature of the eutectoid steel determined from dilatometric data. (b) The effect of deformation temperature on the Ms temperature. The amount of deformation is 5 mm reduction in height.
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2. It is demonstrated experimentally that the large plastic deformation (εave ≈ 70%) of austenite before transformation retards the formation of martensite by lowering the Ms temperature. However, at high temperature (1100 ◦ C) the deformation does not influence the Ms temperature, which is attributable to dynamic recrystallization. Using a theory for mechanical stabilization, developed by balancing the force driving the transformation interface against the resistance from deformation defects in the deformed austenite, the depression in the martensite start temperature is quantitatively estimated. The theory explains, how the reduction in the chemical free energy difference between austenite and martensite, due to the severe plastic deformation of austenite above Md , decreases Ms . Acknowledgment
Fig. 10. Suppression of Ms as a function of plastic deformation of austenite prior to transformation for the eutectoid steel studied in the present work and a boron steel plate.
The experimental part of this work was performed at the Institute for Materials Science and Welding, Graz University of Technology, Austria. One of the authors (MM) gratefully acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
result of high defect density. Therefore, for initiation of the martensite transformation, a higher driving force is required (Gmech in Fig. 1), which is reflected in a lower Ms temperature, as illustrated in Figs. 2, 9 and 10.
References
6. Summary The role of compressive stress and prior plastic deformation of austenite in the martensitic transformation of eutectoid steel is quantitatively analyzed. It is demonstrated, both experimentally and theoretically, that stress promotes martensite formation at higher temperature, whereas straining of austenite retards martensitic transformation. The two opposite roles of mechanical deformation on the martensite transformation analyzed in the present paper can be summarized as follows: 1. Using the theory of promotion of martensitic transformation, based on the work done by shear and dilatational components of uniaxial external stress on the habit plane, which adds to the chemical free energy difference, the change in the Ms temperature is successfully estimated for the eutectoid steel studied.
Fig. 11. Distribution of strain in uniaxial compressive deformation with 5 mm reduction in height.
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