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Influence of non-metallic inclusions on the austenite-to-ferrite phase transformation
Materials Science and Engineering A365 (2004) 291–297
Influence of non-metallic inclusions on the austenite-to-ferrite phase transformation E. Gamsjäger a,b , F.D. Fischer a,∗ , J. Svoboda c a Institut für Mechanik, Montanuniversität Leoben, Franz Josef Street 18, A-8700 Leoben, Austria Christian Doppler Laboratory Functionally Oriented Material Design, Franz Josef Street 18, A-8700 Leoben, Austria Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic b
c
Abstract During continuous casting of low-alloy steels non-metallic inclusions (e.g. aluminium nitride (AlN)) may precipitate and grow inside the austenite phase at temperatures above 1200 K. At lower temperatures austenite (␥) will partly transform to the ferrite (␣) phase, which significantly influences the properties of the cast product. The kinetics of the ␥/␣ phase transformation depends on the mobility of the interface as well as on the diffusion coefficients of interstitial as well as substitutional components. The driving force for the ␥/␣ phase transformation can be separated in a chemical and a mechanical term. The actual chemical compositions in the ␥- and in the ␣-phase very near to the interface determine the chemical contribution to the actual driving force. The mechanical driving force of the ␥/␣ phase transformation is directly related to the strain energy and plastic work generated by the transforming spherical shell. Therefore, AlN-inclusions, producing large strains and stresses in their surroundings, may affect the ␥/␣ phase transformation. Finite element (FE) calculations show that AlN-inclusions are able to trigger the ␥/␣ phase transformation. This leads to the formation of ferrite at a higher temperature compared to a steel grade without AlN-precipitates. Finally, the kinetics of the ␥/␣ phase transformation is determined numerically by a finite difference method. © 2003 Elsevier B.V. All rights reserved. Keywords: Diffusion; Phase transformation; Steel; Aluminium nitride; Transformation kinetics; Driving force
1. Introduction Hot tensile tests have shown that many steel grades exhibit a minimum in hot ductility in a temperature region, where ␥/␣ transformation occurs. Review papers by Maehara et al. [1] and Mintz et al. [2] explain the high probability of intergranular fracture—experimentally detected by a minimum in hot ductility—by the presence of thin ferritic films at the austenitic grain boundaries. It is also stated that non-metallic precipitates, especially small ones, enhance the deterioration of the material [2]. The austenite grain boundaries are energetically preferred sites for the nucleation of non-metallic inclusions as well as for the precipitation of ferritic islands. While the non-metallic precipitates remain rather small (the grain size of AlN precipitates is typically in the order ∗ Corresponding author. Tel.: +43-3842-402-476; fax: +43-3842-46048. E-mail address: [email protected] (F.D. Fischer).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.09.038
of some few nanometers) the ferrite islands tend to grow together into thin films, and the ferrite volume fraction is a few percent. Both, the growth of an AlN-nucleus in the ␥-phase and the ␥/␣ phase transformation, are accompanied by a volume dilatation. The precipitation of AlN in austenitic steel results in a strain/stress state due to the accommodation of the volume strain of AlN by the surrounding medium ␥. Austenite is considered to be an elasto-plastic material. It is important to know whether the existence of non-metallic inclusions, treated as nucleation sites for ferrite, facilitates or impedes the growth of ferrite in the material. A decision is possible by comparing the total mechanical work performed by a transforming ␣-shell. The mechanical work can be calculated by a finite element (FE) program [3]. For a special case and some simplifying assumptions an analytical solution is provided by Fischer and Oberaigner [4]. Thus, the aims of the paper are: • To calculate the elastic energy and plastic work performed during the growth of a second phase particle in an infinite
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matrix, and to compare the numerical results with the analytical solutions; • To analyse numerically more complicated phase arrangements with different material properties; • To investigate the growth kinetics of a spherical ferrite shell under consideration of the mechanical contribution to the driving force.
2. Total strain energy and plastic work Franciosi et al. [6] studied the microplasticity induced by the accommodation of a dilating sphere embedded in a metallic matrix. They combined aspects of crystal plasticity with flow plasticity. The result of the previously cited work is that a dislocation based model leads to a higher amount of plastic deformation compared to a flow plasticity based model. However, flow plasticity is an easier approach to investigate the strain and stress field caused by the transformation of a sphere in a metallic matrix and, indeed, much work has been done in this research field. Two different models exist in literature. The first is an inclusion with constant radius subjected to a dilational volume change, the second concerns the transformational volume strain is applied to a growing sphere. As pointed out in [4] the plastic deformation and the dissipated plastic work during phase transformation can only be evaluated accurately, if the actual radius of the growing inclusion instead of the initial radius is taken into account in case of a large eigenstrain. Sen et al. [7] used an FE analysis to calculate the total elastic and the plastic work terms in a system, where a spherical NbH-precipitate is embedded in a finite Nb-matrix. They applied the transformational strain element by element and, thus, solved the problem by changing the material properties of the transforming element. The Nb/NbH-transformation is accompanied by a dilational volume strain as it is also the case for transforming AlN-spheres. However, the implementation of the problem into the FE program, the FE mesh and the way to change the material properties in the transforming sphere are different in our approach. Furthermore, the coexistence and the evolution of three phases (austenite, ferrite and aluminium nitride (AlN)) make the problem more complicated. In addition our FE calculations were checked with analytically obtained results for cases of two different phases with the same elastic properties. The transforming sphere is embedded in an infinite medium with elastic ideally-plastic material properties. It is very difficult or even impossible to derive analytical solutions for transformations in a more realistic material. However, if the FE calculations agree with the analytically obtained results in the described case, numerical solutions for different material properties can be found by simply changing the input parameters, and realistic results can be expected.
The analytical expressions for the elastic strain energy U and the plastic work Wp from [4] are repeated here as U=
σf2 4πR3 (1 − ν)(2κ − 1), E 3
σf2 4πR3 2(1 − ν)κ ln κ; E 3 κ is given by Eε0 κ= (1 − ν)σf
Wp =
(1) (2)
(3)
E denotes the Young’s modulus, ν the Poisson’s ratio, ε0 the linear transformational strain, being δ/3, δ is the transformation volume strain, and σ f denotes the yield stress. The total mechanical work performed by a growing ferrite sphere follows as Vα W= (dU + dWp ), (4) 0
where V␣ denotes the volume of ferrite. If the total work W is evaluated analytically or numerically, the thermodynamic force, acting per mole, can be calculated as dW fmech = − Vm (5) dV α Vm is an average value consisting of the molar volumes of the parent and the product phase. It should be noted that it is much easier and more precise to calculate W in FE programs and infer fmech from W than to calculate fmech from the stresses at the interface (see e.g. [5]), since the FE output quantities are averaged over the elements.
3. Material properties The elastic modulus EAlN in aluminium nitride is set to 300 GPa (this estimation is based on the values of the shear modulus µ = 128 GPa and Poisson’s ratio ν = 0.25 given in [8]). Elastic-ideally plastic material properties were assigned to austenite and ferrite. In order to compare the results of the FEM-analysis with the analytical solutions, Eqs. (1) and (2), the elastic moduli have been set equal in austenite and ferrite; E␥ and E␣ are set to 100 GPa. In reality austenite is in general somewhat stiffer than ferrite at the temperatures of γ ␥/␣ phase transformation. The yield stresses σy in austenite α and σy in ferrite amount to 60 and 10 MPa, respectively. Poisson’s ratio is set to 0.3 in both cases. A transforming ferrite sphere in an austenitic steel behaves elastically due to a purely hydrostatic stress state. The situation is similar in case of a precipitating and transforming AlN-sphere in a steel matrix. Again the AlN sphere remains elastic, whereas the steel shell surrounding the sphere deforms also plastically, if κ > 1. An incremental concept, as carried out in [7] and stated verbatim in [4], has to be followed to investigate the development of the plastic strain and the plastic work in case of a
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growing spherical inclusion. With respect to the total plastic work it must be considered that also the material inside the final radius of the transformed sphere has plastified during the progress of the phase transformation.
4. Volume dilatation In a review paper by Wilson and Gladman about AlN in steels [9] the crystal structure of AlN in steel is reported being a hexagonal unit cell of wurtzite structure. However, in a steel matrix also AlN with a cubic structure of the NaCl-type has been detected by X-ray diffraction. In case of cubic AlN a coherent interface between the precipitate and the steel matrix can be expected, thus, the cubic NaCl-type structure has been assumed for the AlN-crystals. Four AlN molecules can be found in each unit cell in case of an NaCl-structure, i.e. the number of atoms/molecules per unit cell Nuc equals four. The lattice constant a is 0.4071 nm, taken from [9]. Therefore, the molar volume VmAlN equals 1.016×10−5 m3 mol−1 . In fcc-crystals four atoms are located within a unit cell, whereas in bcc crystals two atoms are located within one unit cell. The temperature-dependent lattice parameter a␥ in ␥-Fe equals 0.3639 nm at 1073 K, the lattice parameter a␣ in ␣-Fe amounts to 0.290 nm at 1073 K, see e.g. the textbook [10] and the Ph.D. Thesis by Onink [11]. The molar volume γ Vm of ␥-Fe amounts to 7.255 × 10−6 m3 mol−1 , Vmα equals 7.344 × 10−6 m3 mol−1 . The dependence of the lattice parameters of ␥- and ␣-steel on the composition has not been considered. The time-dependent variation of the composition of steel during the diffusional ␥/␣-phase transformation certainly plays a subordinate role with respect to changes in the lattice parameter, since low-alloy steels are investigated. The volume strain δ due to a phase transformation can be calculated by dividing the difference of the molar volumes by the average molar volume at the interface. The volume strain due to ␥/AlN-phase transformation ␦␥/AlN results in 0.33. Therefore, a volume strain of more than thirty percent has to be accommodated during precipitation and growth of AlN in ␥-steel. In an analogous manner, the volume strain due to ␥/␣-phase transformation can be calculated. A volume strain δ␥/␣ of 0.012 occurs during ␥/␣-phase transformation at 1073 K.
5. Finite element model It is sufficient to mesh a sector of a circle with axially symmetric elements, when a spherically symmetric problem is considered. In order to guarantee the spherical symmetry the nodes of the upper and the lower edge of the sector can only be displaced in radial direction. This is ensured by boundary conditions depicted in Fig. 1. An FE mesh consisting of axisymmetric elements has been applied to the sector of the sphere. The generated element row, starting
Fig. 1. Ferrite has been nucleated on the surface of an AlN-sphere and grows at the expense of the surrounding austenitic matrix. The spherical symmetry is completely taken into account by meshing a sector of a circle with axisymmetric elements.
from the centre of the circle, is completed by an infinite element. The evolution and the growth of the spherical AlN- and ferrite shells are modeled by a fictitious thermal expansion, being δ = 3α T
(6)
α is the thermal expansion coefficient. The temperature change T can be chosen arbitrarily, if the temperaturedependent variables can be set to a constant value, e.g. if isothermal conditions prevail. Otherwise, T has to be set to a sufficiently small value, compensated by an appropriate higher value of α. During the transformation the thermal expansion has been applied element by element. The material properties change during transformation, since different properties are assigned to ferrite as well as austenite and in particular to AlN. The evaluation of the elastic strain energy in the system and the dissipated plastic work requires the positioning of the ␥/␣ phase boundary after each step exactly between two adjacent elements. FE calculations average the temperature jump T at the interface. Therefore, a temperature-independent material called “Gamma”(approximating the mechanical properties of austenite at the temperature investigated, see the chapter “Material properties”) is assigned to elements that have not been transformed after the transformation process (see Fig. 2). Several FE-input files were generated in order to take different volume fractions of the material “Gampha”, able to transform from ␥ to ␣, into account. (“Gampha” approximates the mechanical properties of austenite before transformation (T = 0) and that of ferrite after transformation T = 1.) • Input 1: The material “Gampha” is not assigned to any element in the model. Therefore, a zero ferrite fraction is assumed. The ␥/AlN phase transformation may occur.
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Fig. 2. Schematic representation of the FE-mesh (The original mesh consists of about 100 elements.) and the phase arrangement in the model. The natural numbers name the elements.
• Input 2: The material “Gampha” is only assigned to element 6 in the model. • Input 3: The material “Gampha” is assigned to the elements 6 and 7 in the model (see Fig. 2). • Input m: The material “Gampha” is assigned to the elements 6, 7, . . . , m in the model. The temperature change T of 1 K has been applied linearly with respect to time t in each element successively in order to change the material properties as well as to apply the transformational volume strain δ. This fictitious “heating-procedure” was applied to each input file. The expansion coefficient ␣␥/AlN for ␥-to-AlN transformation (T increases from 0 to 1 K in “Gamaln-elements”) is set to 0.11 K−1 , the coefficient ␣␥/␣ associated with ␥/␣ phase transformation (T increases from 0 to 1 K in “Gampha-elements”) is set to 4 × 10−3 K−1 , according to the section “Volume dilatation” and Eq. (6).
6. Results 6.1. Comparison of the analytical and the numerical solutions In a first step the results of the already mentioned analytical solutions are compared with the FE analysis. The growth of a spherical ferrite inclusion in an elastic-plastic austenite medium is considered (see Fig. 1, with RAlN = 0). Analytical solutions have been obtained under the following limitations, see [4]. • The transforming sphere is embedded in an infinite medium. • The elastic moduli and Poisson’s ratios are the same in the parent and the product phase. (A feasible assumption during ␥/␣-transformation, certainly not true when ␥/AlN-transformation is considered.) • The parent phase is assumed to be elastic-ideally plastic.
Fig. 3. Elastic strain energy U and the plastic work Wp during growth of a ferrite sphere in austenite.
In order to compare the analytical solution with the FE results an infinite space is taken into account. For further FE calculations finite model dimensions could be considered easily. However, an infinite elastic-plastic body is quite a realistic assumption with respect to this problem. The ferrite precipitates and the spherical ferrite shells surrounding an AlN-core are very small compared with the grain size of austenite. Thus, the beginning of the ferrite growth is discussed in this model. At later stages of transformation the spherical ferrite shell will grow together to form a thin film. In order to check the FE analysis, the elastic constants are the same in austenite and ferrite, see section “Material properties”. The elastic strain energy and the plastic work during transformation of a ferrite sphere in an elastic-plastic medium are depicted in Fig. 3. The results of the FE analysis and of the analytical solutions Eqs. (1) and (2) are in good agreement. In the next step the same FE-mesh has been used to investigate the influence of AlN-precipitates. 6.2. Influence of AlN on the γ/α phase transformation The FE model is used to investigate three different situations always starting from the same geometrical configuration. A spherical ferrite shell grows on the surface of an inner sphere filled with AlN or ferrite. The outer space is filled with austenite. The total work W, consisting of the elastic energy U and the plastic work Wp , performed during the growth of a ferrite sphere shell was calculated for three cases: • Case A: A spherical AlN-inclusion nucleates and grows to a sphere with a final radius RAlN = 11.192 nm. The un-accommodated AlN-sphere serves as nucleation site for ferrite. A spherical ferrite shell grows on the surface of the un-accommodated AlN-sphere. The elastic energy and the plastic work are investigated during the growth of the ferrite shell. • Case B: Again, a spherical AlN-inclusion nucleates and grows to the same size RAlN = 11.192 nm. The material
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between the driving force f and the interface velocity v(t) is implied as, v(t) = (Vm−1 M f).
(7)
The total driving force f consists of a chemical part (fchem ) and a mechanical one (fmech ), f = fchem + fmech .
Fig. 4. Total work W performed by the growing ferritic sphere shell depending on its volume V␣ .
has had enough time to relax the stresses performed by the AlN-growth process (e.g. AlN has been formed at a higher temperature). Ferrite nucleates and grows on the surface of the accommodated AlN-sphere. • Case C: In this case the AlN-inclusion is replaced by ferrite. The ferrite sphere grows to the same diameter as the AlN-inclusion in case A and case B. The stresses are not relaxed. The ferrite shell investigated grows on the ferrite sphere formed initially. In Fig. 4 the total work performed during the growth of a ferrite shell has been plotted versus the volume of the ferrite shell in the three cases investigated. Fig. 4 shows that the mechanical work performed during the growth of spherical ␣-shells has the highest value in case A. Therefore, ferrite growth on un-accommodated AlN-spheres is not realistic, because other nucleation sites (accommodated AlN-spheres, grain boundaries) are energetically advantageous. Ferrite nucleation is more probable on accommodated AlN-spheres or any other nucleation sites. The question arises, whether accommodated AlN-spheres enhance or impede the growth kinetics of the ␥/␣ phase transformation. The work performed during the growth of ferrite shells is studied in case B (ferrite grows from accommodated AlN) and case C (ferrite growth). Fig. 4 indicates that the mechanical work performed during ␣-growth is slightly lower in case B compared to case C. Ferrite growth on an accommodated ferrite sphere was not been considered, since it is not likely that ␥/␣ transformation stops, the stresses are relaxed, and transformation keeps on later. 6.3. γ/α phase transformation kinetics Whereas the previous calculation allows to find the preferred arrangement with respect to ␥/␣ transformation, a theoretical prediction of the kinetics of ␥/␣ transformation requires a different mathematical approach, see Svoboda et al. [12]. The interface motion is determined by carbon diffusion as well as by the mobility of the interface. A linear relation
(8)
The aim of this work is to evaluate the influence of the mechanical driving force on the phase transformation and, thus, only a binary Fe–C alloy is investigated. The chemical driving force fchem can be calculated as the difference of the chemical potentials of iron in both phases, γ
fchem = µFe − µαFe ,
(9)
α in ferrite is negliassuming that the amount of carbon XC gible. The mechanical driving force fmech for a transforming (growing) ferrite inclusion in an elastic ideally-plastic medium is available in analytic form, following from Eq. (5), and see e.g. [4,5] as 1 fmech = − Vm ε0 σf 2 − + 2lnκ , κ ≥ 1, (10) κ
with an average molar volume Vm of austenite and ferrite at 1073 K being 7.3 × 10−6 m3 mol−1 . Note that fmech is a negative quantity meeting the fact that both the production of elastic strain energy and plastic work consumes energy which is not available for the transformation process. As already remarked the interface migration and carbon diffusion must be treated simultaneously. This goal has been achieved with a finite difference program. The transformation kinetics has been solved for a transforming ␣-sphere in an austenitic steel with elastic-ideally plastic properties. The kinetics is strongly dependent on the mobility M and on the diffusion coefficient D, both being thermally activated quantities as −QM M = M0 exp , (11) RT −QD D = D0 exp . (12) RT Van Leeuwen et al. [13] stated a value of M0 ≈ 5.8 × 10−6 m2 s kg−1 and QM = 147 kJ mol−1 in an Fe lattice. The pre-exponential factor D0 = 2.343 × 10−5 m2 s−1 and the activation energy QD = 148 kJ mol−1 for C-diffusion in ␥, see Ågren [14]. A constant temperature T of 1073 K is considered during the transformation process. Before the ␥/␣ phase transformation starts, the mole fraction of C in the infinite austenite medium is set to 0.01. In Fig. 5 the kinetics of a transforming ␣-sphere shell is depicted. Models of ␥/␣ phase transformation in low alloy steels usually do not take the mechanical driving force into account. The dashed line shows the transformation kinetics, if the mechanical term (Eq. (10)) is ignored (case “0”). The
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Fig. 5. Transformation kinetics of a growing ferrite shell.
situation is fictitious, because the stress-state during transformation will influence the kinetics. However, when the chemical driving force exceeds the mechanical driving force, case “0” is an appropriate approximation. According to Eq. (5) the mechanical driving force fmech is directly proportional to the slope of the curves plotted in Fig. 4 and always negative in these cases, which means that the growth kinetics is retarded. The slope of the “Accommodated-AlN”-curve (Fig. 4, case B) yields a driving force fmech = −10.1 J mol−1 . The estimation of fmech for case C (␣-growth) amounts to −11.8 J mol−1 . These values have been used to calculate numerically the transformation kinetics for case B and C (Fig. 5). Thus, the ␥/␣ phase transformation is stimulated by fully accommodated AlN-spheres. The solid line shows the kinetics for case B (ferrite grows from an accommodated AlN-sphere). Fig. 5 shows that the difference in the growth kinetics comparing case B (solid line) and C (dotted line) is small due to a small difference in the driving force. However, the result points in the expected direction that the ferrite shells grow faster on the surface of accommodated AlN-inclusions. When case “0” is compared with the cases B and C, it is evident that the mechanical driving force influences the transformation kinetics. Fig. 6 presents the total driving force f during transformation. The absolute value of the chemical driving force fchem ≈ 50 J mol−1 is much higher than that of the mechanical driving force fmech ≈ 10 J mol−1 . Case B and C lead to a difference in the mechanical driving force of only ≈2 J mol−1 . As the driving forces in case B and case C are quite similar, its effect on the kinetics is small, but still AlN-precipitates stimulate the ferrite growth kinetics. The total driving force f in the fictitious case “0” differs from the total driving forces in case B and C. It has been observed that the mechanical driving force influences the transformation kinetics (see Figs. 5 and 6). The activation energy of nucleation is not considered in this work. However, due to a distorted lattice the activation energy is assumed to be smaller for ferrite nucleating
Fig. 6. Driving force during phase transformation.
on AlN-spheres. This is an additional argument for ferrite growth on AlN-spheres. The dashed line (Fig. 6) has been obtained by ignoring the mechanical driving force (case “0”). The solid line represents the total driving force f with fmech = −10.1 J mol−1 for case B (␣-growth from accommodated AlN). The dotted line shows f in case C (␣-growth on an accommodated ␣-sphere), fmech = −11.8 J mol−1 . Although the value of fmech is assumed to be constant throughout the phase transformation in the cases investigated, the distances between the curves B or C and “0” in Fig. 6 becomes smaller with increasing transformation time. Since the transformation is slowed down in case B and C, the chemical driving force has a higher value in these cases (lower amount of carbon at the interface) compared to case A at a certain transformation time t. Therefore, the difference in the kinetics and the total driving forces, consisting of a chemical and a mechanical part, is small, if again the cases B and C are compared. A higher absolute value of the negative mechanical driving force will lead to a higher chemical driving force at the same time, nearly compensating the retarding effect of the mechanical driving force. As expected from the previous results, the total driving force is slightly higher in case B compared to case C. Comparison of the cases B and C with case A (neglecting the mechanical driving force) reveals that the mechanical term is important for the evolution of the microstructure during ␥/␣ phase transformation. In the case of a massive transformation, e.g. ␥/␣ transformation in pure iron, a constant mechanical driving force would lead to a constant shift in the total driving force at a given temperature.
7. Conclusion The impact of AlN precipitates on the ␥/␣ phase transformation is studied. Accommodated AlN-nuclei in a steel matrix turn out to be a favourable material arrangement for
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ferrite growth compared to a purely austenitic matrix. The small AlN-precipitates deteriorate the material properties by facilitating ferrite nucleation and growth. The kinetics of ␥/␣ phase transformation has been investigated numerically. The mechanical term of the total driving force retards the transformation kinetics and its influence achieves a maximum at the beginning of the ferrite growth. At later stages of the transformation the chemical driving force will be higher in the case of an additional mechanical driving force, since the solute of the ␥-side of the interface is less enriched compared with the fictitious case of zero mechanical driving force.
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Influence of non-metallic inclusions on the austenite-to-ferrite phase transformation E. Gamsjäger a,b , F.D. Fischer a,∗ , J. Svoboda c a Institut für Mechanik, Montanuniversität Leoben, Franz Josef Street 18, A-8700 Leoben, Austria Christian Doppler Laboratory Functionally Oriented Material Design, Franz Josef Street 18, A-8700 Leoben, Austria Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic b
c
Abstract During continuous casting of low-alloy steels non-metallic inclusions (e.g. aluminium nitride (AlN)) may precipitate and grow inside the austenite phase at temperatures above 1200 K. At lower temperatures austenite (␥) will partly transform to the ferrite (␣) phase, which significantly influences the properties of the cast product. The kinetics of the ␥/␣ phase transformation depends on the mobility of the interface as well as on the diffusion coefficients of interstitial as well as substitutional components. The driving force for the ␥/␣ phase transformation can be separated in a chemical and a mechanical term. The actual chemical compositions in the ␥- and in the ␣-phase very near to the interface determine the chemical contribution to the actual driving force. The mechanical driving force of the ␥/␣ phase transformation is directly related to the strain energy and plastic work generated by the transforming spherical shell. Therefore, AlN-inclusions, producing large strains and stresses in their surroundings, may affect the ␥/␣ phase transformation. Finite element (FE) calculations show that AlN-inclusions are able to trigger the ␥/␣ phase transformation. This leads to the formation of ferrite at a higher temperature compared to a steel grade without AlN-precipitates. Finally, the kinetics of the ␥/␣ phase transformation is determined numerically by a finite difference method. © 2003 Elsevier B.V. All rights reserved. Keywords: Diffusion; Phase transformation; Steel; Aluminium nitride; Transformation kinetics; Driving force
1. Introduction Hot tensile tests have shown that many steel grades exhibit a minimum in hot ductility in a temperature region, where ␥/␣ transformation occurs. Review papers by Maehara et al. [1] and Mintz et al. [2] explain the high probability of intergranular fracture—experimentally detected by a minimum in hot ductility—by the presence of thin ferritic films at the austenitic grain boundaries. It is also stated that non-metallic precipitates, especially small ones, enhance the deterioration of the material [2]. The austenite grain boundaries are energetically preferred sites for the nucleation of non-metallic inclusions as well as for the precipitation of ferritic islands. While the non-metallic precipitates remain rather small (the grain size of AlN precipitates is typically in the order ∗ Corresponding author. Tel.: +43-3842-402-476; fax: +43-3842-46048. E-mail address: [email protected] (F.D. Fischer).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.09.038
of some few nanometers) the ferrite islands tend to grow together into thin films, and the ferrite volume fraction is a few percent. Both, the growth of an AlN-nucleus in the ␥-phase and the ␥/␣ phase transformation, are accompanied by a volume dilatation. The precipitation of AlN in austenitic steel results in a strain/stress state due to the accommodation of the volume strain of AlN by the surrounding medium ␥. Austenite is considered to be an elasto-plastic material. It is important to know whether the existence of non-metallic inclusions, treated as nucleation sites for ferrite, facilitates or impedes the growth of ferrite in the material. A decision is possible by comparing the total mechanical work performed by a transforming ␣-shell. The mechanical work can be calculated by a finite element (FE) program [3]. For a special case and some simplifying assumptions an analytical solution is provided by Fischer and Oberaigner [4]. Thus, the aims of the paper are: • To calculate the elastic energy and plastic work performed during the growth of a second phase particle in an infinite
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matrix, and to compare the numerical results with the analytical solutions; • To analyse numerically more complicated phase arrangements with different material properties; • To investigate the growth kinetics of a spherical ferrite shell under consideration of the mechanical contribution to the driving force.
2. Total strain energy and plastic work Franciosi et al. [6] studied the microplasticity induced by the accommodation of a dilating sphere embedded in a metallic matrix. They combined aspects of crystal plasticity with flow plasticity. The result of the previously cited work is that a dislocation based model leads to a higher amount of plastic deformation compared to a flow plasticity based model. However, flow plasticity is an easier approach to investigate the strain and stress field caused by the transformation of a sphere in a metallic matrix and, indeed, much work has been done in this research field. Two different models exist in literature. The first is an inclusion with constant radius subjected to a dilational volume change, the second concerns the transformational volume strain is applied to a growing sphere. As pointed out in [4] the plastic deformation and the dissipated plastic work during phase transformation can only be evaluated accurately, if the actual radius of the growing inclusion instead of the initial radius is taken into account in case of a large eigenstrain. Sen et al. [7] used an FE analysis to calculate the total elastic and the plastic work terms in a system, where a spherical NbH-precipitate is embedded in a finite Nb-matrix. They applied the transformational strain element by element and, thus, solved the problem by changing the material properties of the transforming element. The Nb/NbH-transformation is accompanied by a dilational volume strain as it is also the case for transforming AlN-spheres. However, the implementation of the problem into the FE program, the FE mesh and the way to change the material properties in the transforming sphere are different in our approach. Furthermore, the coexistence and the evolution of three phases (austenite, ferrite and aluminium nitride (AlN)) make the problem more complicated. In addition our FE calculations were checked with analytically obtained results for cases of two different phases with the same elastic properties. The transforming sphere is embedded in an infinite medium with elastic ideally-plastic material properties. It is very difficult or even impossible to derive analytical solutions for transformations in a more realistic material. However, if the FE calculations agree with the analytically obtained results in the described case, numerical solutions for different material properties can be found by simply changing the input parameters, and realistic results can be expected.
The analytical expressions for the elastic strain energy U and the plastic work Wp from [4] are repeated here as U=
σf2 4πR3 (1 − ν)(2κ − 1), E 3
σf2 4πR3 2(1 − ν)κ ln κ; E 3 κ is given by Eε0 κ= (1 − ν)σf
Wp =
(1) (2)
(3)
E denotes the Young’s modulus, ν the Poisson’s ratio, ε0 the linear transformational strain, being δ/3, δ is the transformation volume strain, and σ f denotes the yield stress. The total mechanical work performed by a growing ferrite sphere follows as Vα W= (dU + dWp ), (4) 0
where V␣ denotes the volume of ferrite. If the total work W is evaluated analytically or numerically, the thermodynamic force, acting per mole, can be calculated as dW fmech = − Vm (5) dV α Vm is an average value consisting of the molar volumes of the parent and the product phase. It should be noted that it is much easier and more precise to calculate W in FE programs and infer fmech from W than to calculate fmech from the stresses at the interface (see e.g. [5]), since the FE output quantities are averaged over the elements.
3. Material properties The elastic modulus EAlN in aluminium nitride is set to 300 GPa (this estimation is based on the values of the shear modulus µ = 128 GPa and Poisson’s ratio ν = 0.25 given in [8]). Elastic-ideally plastic material properties were assigned to austenite and ferrite. In order to compare the results of the FEM-analysis with the analytical solutions, Eqs. (1) and (2), the elastic moduli have been set equal in austenite and ferrite; E␥ and E␣ are set to 100 GPa. In reality austenite is in general somewhat stiffer than ferrite at the temperatures of γ ␥/␣ phase transformation. The yield stresses σy in austenite α and σy in ferrite amount to 60 and 10 MPa, respectively. Poisson’s ratio is set to 0.3 in both cases. A transforming ferrite sphere in an austenitic steel behaves elastically due to a purely hydrostatic stress state. The situation is similar in case of a precipitating and transforming AlN-sphere in a steel matrix. Again the AlN sphere remains elastic, whereas the steel shell surrounding the sphere deforms also plastically, if κ > 1. An incremental concept, as carried out in [7] and stated verbatim in [4], has to be followed to investigate the development of the plastic strain and the plastic work in case of a
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growing spherical inclusion. With respect to the total plastic work it must be considered that also the material inside the final radius of the transformed sphere has plastified during the progress of the phase transformation.
4. Volume dilatation In a review paper by Wilson and Gladman about AlN in steels [9] the crystal structure of AlN in steel is reported being a hexagonal unit cell of wurtzite structure. However, in a steel matrix also AlN with a cubic structure of the NaCl-type has been detected by X-ray diffraction. In case of cubic AlN a coherent interface between the precipitate and the steel matrix can be expected, thus, the cubic NaCl-type structure has been assumed for the AlN-crystals. Four AlN molecules can be found in each unit cell in case of an NaCl-structure, i.e. the number of atoms/molecules per unit cell Nuc equals four. The lattice constant a is 0.4071 nm, taken from [9]. Therefore, the molar volume VmAlN equals 1.016×10−5 m3 mol−1 . In fcc-crystals four atoms are located within a unit cell, whereas in bcc crystals two atoms are located within one unit cell. The temperature-dependent lattice parameter a␥ in ␥-Fe equals 0.3639 nm at 1073 K, the lattice parameter a␣ in ␣-Fe amounts to 0.290 nm at 1073 K, see e.g. the textbook [10] and the Ph.D. Thesis by Onink [11]. The molar volume γ Vm of ␥-Fe amounts to 7.255 × 10−6 m3 mol−1 , Vmα equals 7.344 × 10−6 m3 mol−1 . The dependence of the lattice parameters of ␥- and ␣-steel on the composition has not been considered. The time-dependent variation of the composition of steel during the diffusional ␥/␣-phase transformation certainly plays a subordinate role with respect to changes in the lattice parameter, since low-alloy steels are investigated. The volume strain δ due to a phase transformation can be calculated by dividing the difference of the molar volumes by the average molar volume at the interface. The volume strain due to ␥/AlN-phase transformation ␦␥/AlN results in 0.33. Therefore, a volume strain of more than thirty percent has to be accommodated during precipitation and growth of AlN in ␥-steel. In an analogous manner, the volume strain due to ␥/␣-phase transformation can be calculated. A volume strain δ␥/␣ of 0.012 occurs during ␥/␣-phase transformation at 1073 K.
5. Finite element model It is sufficient to mesh a sector of a circle with axially symmetric elements, when a spherically symmetric problem is considered. In order to guarantee the spherical symmetry the nodes of the upper and the lower edge of the sector can only be displaced in radial direction. This is ensured by boundary conditions depicted in Fig. 1. An FE mesh consisting of axisymmetric elements has been applied to the sector of the sphere. The generated element row, starting
Fig. 1. Ferrite has been nucleated on the surface of an AlN-sphere and grows at the expense of the surrounding austenitic matrix. The spherical symmetry is completely taken into account by meshing a sector of a circle with axisymmetric elements.
from the centre of the circle, is completed by an infinite element. The evolution and the growth of the spherical AlN- and ferrite shells are modeled by a fictitious thermal expansion, being δ = 3α T
(6)
α is the thermal expansion coefficient. The temperature change T can be chosen arbitrarily, if the temperaturedependent variables can be set to a constant value, e.g. if isothermal conditions prevail. Otherwise, T has to be set to a sufficiently small value, compensated by an appropriate higher value of α. During the transformation the thermal expansion has been applied element by element. The material properties change during transformation, since different properties are assigned to ferrite as well as austenite and in particular to AlN. The evaluation of the elastic strain energy in the system and the dissipated plastic work requires the positioning of the ␥/␣ phase boundary after each step exactly between two adjacent elements. FE calculations average the temperature jump T at the interface. Therefore, a temperature-independent material called “Gamma”(approximating the mechanical properties of austenite at the temperature investigated, see the chapter “Material properties”) is assigned to elements that have not been transformed after the transformation process (see Fig. 2). Several FE-input files were generated in order to take different volume fractions of the material “Gampha”, able to transform from ␥ to ␣, into account. (“Gampha” approximates the mechanical properties of austenite before transformation (T = 0) and that of ferrite after transformation T = 1.) • Input 1: The material “Gampha” is not assigned to any element in the model. Therefore, a zero ferrite fraction is assumed. The ␥/AlN phase transformation may occur.
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Fig. 2. Schematic representation of the FE-mesh (The original mesh consists of about 100 elements.) and the phase arrangement in the model. The natural numbers name the elements.
• Input 2: The material “Gampha” is only assigned to element 6 in the model. • Input 3: The material “Gampha” is assigned to the elements 6 and 7 in the model (see Fig. 2). • Input m: The material “Gampha” is assigned to the elements 6, 7, . . . , m in the model. The temperature change T of 1 K has been applied linearly with respect to time t in each element successively in order to change the material properties as well as to apply the transformational volume strain δ. This fictitious “heating-procedure” was applied to each input file. The expansion coefficient ␣␥/AlN for ␥-to-AlN transformation (T increases from 0 to 1 K in “Gamaln-elements”) is set to 0.11 K−1 , the coefficient ␣␥/␣ associated with ␥/␣ phase transformation (T increases from 0 to 1 K in “Gampha-elements”) is set to 4 × 10−3 K−1 , according to the section “Volume dilatation” and Eq. (6).
6. Results 6.1. Comparison of the analytical and the numerical solutions In a first step the results of the already mentioned analytical solutions are compared with the FE analysis. The growth of a spherical ferrite inclusion in an elastic-plastic austenite medium is considered (see Fig. 1, with RAlN = 0). Analytical solutions have been obtained under the following limitations, see [4]. • The transforming sphere is embedded in an infinite medium. • The elastic moduli and Poisson’s ratios are the same in the parent and the product phase. (A feasible assumption during ␥/␣-transformation, certainly not true when ␥/AlN-transformation is considered.) • The parent phase is assumed to be elastic-ideally plastic.
Fig. 3. Elastic strain energy U and the plastic work Wp during growth of a ferrite sphere in austenite.
In order to compare the analytical solution with the FE results an infinite space is taken into account. For further FE calculations finite model dimensions could be considered easily. However, an infinite elastic-plastic body is quite a realistic assumption with respect to this problem. The ferrite precipitates and the spherical ferrite shells surrounding an AlN-core are very small compared with the grain size of austenite. Thus, the beginning of the ferrite growth is discussed in this model. At later stages of transformation the spherical ferrite shell will grow together to form a thin film. In order to check the FE analysis, the elastic constants are the same in austenite and ferrite, see section “Material properties”. The elastic strain energy and the plastic work during transformation of a ferrite sphere in an elastic-plastic medium are depicted in Fig. 3. The results of the FE analysis and of the analytical solutions Eqs. (1) and (2) are in good agreement. In the next step the same FE-mesh has been used to investigate the influence of AlN-precipitates. 6.2. Influence of AlN on the γ/α phase transformation The FE model is used to investigate three different situations always starting from the same geometrical configuration. A spherical ferrite shell grows on the surface of an inner sphere filled with AlN or ferrite. The outer space is filled with austenite. The total work W, consisting of the elastic energy U and the plastic work Wp , performed during the growth of a ferrite sphere shell was calculated for three cases: • Case A: A spherical AlN-inclusion nucleates and grows to a sphere with a final radius RAlN = 11.192 nm. The un-accommodated AlN-sphere serves as nucleation site for ferrite. A spherical ferrite shell grows on the surface of the un-accommodated AlN-sphere. The elastic energy and the plastic work are investigated during the growth of the ferrite shell. • Case B: Again, a spherical AlN-inclusion nucleates and grows to the same size RAlN = 11.192 nm. The material
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between the driving force f and the interface velocity v(t) is implied as, v(t) = (Vm−1 M f).
(7)
The total driving force f consists of a chemical part (fchem ) and a mechanical one (fmech ), f = fchem + fmech .
Fig. 4. Total work W performed by the growing ferritic sphere shell depending on its volume V␣ .
has had enough time to relax the stresses performed by the AlN-growth process (e.g. AlN has been formed at a higher temperature). Ferrite nucleates and grows on the surface of the accommodated AlN-sphere. • Case C: In this case the AlN-inclusion is replaced by ferrite. The ferrite sphere grows to the same diameter as the AlN-inclusion in case A and case B. The stresses are not relaxed. The ferrite shell investigated grows on the ferrite sphere formed initially. In Fig. 4 the total work performed during the growth of a ferrite shell has been plotted versus the volume of the ferrite shell in the three cases investigated. Fig. 4 shows that the mechanical work performed during the growth of spherical ␣-shells has the highest value in case A. Therefore, ferrite growth on un-accommodated AlN-spheres is not realistic, because other nucleation sites (accommodated AlN-spheres, grain boundaries) are energetically advantageous. Ferrite nucleation is more probable on accommodated AlN-spheres or any other nucleation sites. The question arises, whether accommodated AlN-spheres enhance or impede the growth kinetics of the ␥/␣ phase transformation. The work performed during the growth of ferrite shells is studied in case B (ferrite grows from accommodated AlN) and case C (ferrite growth). Fig. 4 indicates that the mechanical work performed during ␣-growth is slightly lower in case B compared to case C. Ferrite growth on an accommodated ferrite sphere was not been considered, since it is not likely that ␥/␣ transformation stops, the stresses are relaxed, and transformation keeps on later. 6.3. γ/α phase transformation kinetics Whereas the previous calculation allows to find the preferred arrangement with respect to ␥/␣ transformation, a theoretical prediction of the kinetics of ␥/␣ transformation requires a different mathematical approach, see Svoboda et al. [12]. The interface motion is determined by carbon diffusion as well as by the mobility of the interface. A linear relation
(8)
The aim of this work is to evaluate the influence of the mechanical driving force on the phase transformation and, thus, only a binary Fe–C alloy is investigated. The chemical driving force fchem can be calculated as the difference of the chemical potentials of iron in both phases, γ
fchem = µFe − µαFe ,
(9)
α in ferrite is negliassuming that the amount of carbon XC gible. The mechanical driving force fmech for a transforming (growing) ferrite inclusion in an elastic ideally-plastic medium is available in analytic form, following from Eq. (5), and see e.g. [4,5] as 1 fmech = − Vm ε0 σf 2 − + 2lnκ , κ ≥ 1, (10) κ
with an average molar volume Vm of austenite and ferrite at 1073 K being 7.3 × 10−6 m3 mol−1 . Note that fmech is a negative quantity meeting the fact that both the production of elastic strain energy and plastic work consumes energy which is not available for the transformation process. As already remarked the interface migration and carbon diffusion must be treated simultaneously. This goal has been achieved with a finite difference program. The transformation kinetics has been solved for a transforming ␣-sphere in an austenitic steel with elastic-ideally plastic properties. The kinetics is strongly dependent on the mobility M and on the diffusion coefficient D, both being thermally activated quantities as −QM M = M0 exp , (11) RT −QD D = D0 exp . (12) RT Van Leeuwen et al. [13] stated a value of M0 ≈ 5.8 × 10−6 m2 s kg−1 and QM = 147 kJ mol−1 in an Fe lattice. The pre-exponential factor D0 = 2.343 × 10−5 m2 s−1 and the activation energy QD = 148 kJ mol−1 for C-diffusion in ␥, see Ågren [14]. A constant temperature T of 1073 K is considered during the transformation process. Before the ␥/␣ phase transformation starts, the mole fraction of C in the infinite austenite medium is set to 0.01. In Fig. 5 the kinetics of a transforming ␣-sphere shell is depicted. Models of ␥/␣ phase transformation in low alloy steels usually do not take the mechanical driving force into account. The dashed line shows the transformation kinetics, if the mechanical term (Eq. (10)) is ignored (case “0”). The
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Fig. 5. Transformation kinetics of a growing ferrite shell.
situation is fictitious, because the stress-state during transformation will influence the kinetics. However, when the chemical driving force exceeds the mechanical driving force, case “0” is an appropriate approximation. According to Eq. (5) the mechanical driving force fmech is directly proportional to the slope of the curves plotted in Fig. 4 and always negative in these cases, which means that the growth kinetics is retarded. The slope of the “Accommodated-AlN”-curve (Fig. 4, case B) yields a driving force fmech = −10.1 J mol−1 . The estimation of fmech for case C (␣-growth) amounts to −11.8 J mol−1 . These values have been used to calculate numerically the transformation kinetics for case B and C (Fig. 5). Thus, the ␥/␣ phase transformation is stimulated by fully accommodated AlN-spheres. The solid line shows the kinetics for case B (ferrite grows from an accommodated AlN-sphere). Fig. 5 shows that the difference in the growth kinetics comparing case B (solid line) and C (dotted line) is small due to a small difference in the driving force. However, the result points in the expected direction that the ferrite shells grow faster on the surface of accommodated AlN-inclusions. When case “0” is compared with the cases B and C, it is evident that the mechanical driving force influences the transformation kinetics. Fig. 6 presents the total driving force f during transformation. The absolute value of the chemical driving force fchem ≈ 50 J mol−1 is much higher than that of the mechanical driving force fmech ≈ 10 J mol−1 . Case B and C lead to a difference in the mechanical driving force of only ≈2 J mol−1 . As the driving forces in case B and case C are quite similar, its effect on the kinetics is small, but still AlN-precipitates stimulate the ferrite growth kinetics. The total driving force f in the fictitious case “0” differs from the total driving forces in case B and C. It has been observed that the mechanical driving force influences the transformation kinetics (see Figs. 5 and 6). The activation energy of nucleation is not considered in this work. However, due to a distorted lattice the activation energy is assumed to be smaller for ferrite nucleating
Fig. 6. Driving force during phase transformation.
on AlN-spheres. This is an additional argument for ferrite growth on AlN-spheres. The dashed line (Fig. 6) has been obtained by ignoring the mechanical driving force (case “0”). The solid line represents the total driving force f with fmech = −10.1 J mol−1 for case B (␣-growth from accommodated AlN). The dotted line shows f in case C (␣-growth on an accommodated ␣-sphere), fmech = −11.8 J mol−1 . Although the value of fmech is assumed to be constant throughout the phase transformation in the cases investigated, the distances between the curves B or C and “0” in Fig. 6 becomes smaller with increasing transformation time. Since the transformation is slowed down in case B and C, the chemical driving force has a higher value in these cases (lower amount of carbon at the interface) compared to case A at a certain transformation time t. Therefore, the difference in the kinetics and the total driving forces, consisting of a chemical and a mechanical part, is small, if again the cases B and C are compared. A higher absolute value of the negative mechanical driving force will lead to a higher chemical driving force at the same time, nearly compensating the retarding effect of the mechanical driving force. As expected from the previous results, the total driving force is slightly higher in case B compared to case C. Comparison of the cases B and C with case A (neglecting the mechanical driving force) reveals that the mechanical term is important for the evolution of the microstructure during ␥/␣ phase transformation. In the case of a massive transformation, e.g. ␥/␣ transformation in pure iron, a constant mechanical driving force would lead to a constant shift in the total driving force at a given temperature.
7. Conclusion The impact of AlN precipitates on the ␥/␣ phase transformation is studied. Accommodated AlN-nuclei in a steel matrix turn out to be a favourable material arrangement for
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ferrite growth compared to a purely austenitic matrix. The small AlN-precipitates deteriorate the material properties by facilitating ferrite nucleation and growth. The kinetics of ␥/␣ phase transformation has been investigated numerically. The mechanical term of the total driving force retards the transformation kinetics and its influence achieves a maximum at the beginning of the ferrite growth. At later stages of the transformation the chemical driving force will be higher in the case of an additional mechanical driving force, since the solute of the ␥-side of the interface is less enriched compared with the fictitious case of zero mechanical driving force.
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