Citation: Claas Hüter, Mingxuan Lin, Diego Schicchi, Martin Hunkel, Ulrich Prahl, Robert Spatschek. A multiscale perspective on the kinetics of solid state transformations with application to bainite formation[J]. AIMS Materials Science, 2015, 2(4): 319-345. doi: 10.3934/matersci.2015.4.319
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The discovery of bainite by Bain in 1930 [1] rose global interest due to the very attractive combination of strength, toughness and fatigue properties. In spite of its industrial relevance, the fundamental dispute whether bainite formation is mainly displacive, diffusive or both remains unsolved [2]. The formation of bainite requires the displacive transformation of austenite into supersaturated bainite, carbon diffusion into the austenite and the diffusion controlled formation of carbides. If the displacive process controls the transformation, it is expected that bainite nucleation, similar to martensite, governs the process. If diffusion limits the transformation, the bainite growth is expected to exhibit a growth mechanism comparable to those observed in Widmanstätten ferrite formation. Investigations corresponding to the former perspective on bainite formation [3,4,5,6,7] often anticipate autocatalytic nucleation, while research conducted in the spirit of the latter perspective is typically based on Johnson-Mehl-Avrami-Kolmogorov models [8,9].
As described in [7], the autocatalytic nucleation shows a clear dependence on the prestrained state of the austenite, and additional stresses lead to an acceleration of the bainite formation [10,11]. Apart from the activation of intergranular nucleation sites due to pre-transformation strain and applied stresses, also the size of the bainite units changes drastically under large stresses [12,13,14], and the bainite needle orientation is stress dependent as well [15]. Additionally, it has been found for a medium carbon steel that large stresses change the orientation via variant selection [16] and prestrained austenite leads to much finer bainite [17,7]. Furthermore, the bainite phases show increased strength due to a higher carbide density, and similar effects were found also in the press hardening steel 22MnB5 [18,19].
Press hardening is one of the key production technologies for high strength steels specifically in the automotive industry [20]. Parts as A- and B-pillars, roof rails or bumpers are fabricated using production facilities based on this method. Currently, it is of specific interest to further develop advanced process technologies based on press hardening, where bainitic press hardening is one of the most promising candidates. This method, using a combination of interrupted cooling and quasi-isothermal holding in the bainite region, offers extremely attractive strength-failure-strain combinations [21]. In general, controlling the process tool temperature partially (partial press hardening) offers a vast amount of possible advances in process technology which can yield work pieces with locally varying thermoelastic properties for a wide range of applications, known as tailored tempering.
Apparently, such advanced concepts of steel design require the solution of various theoretical challenges accompanying this promising class of materials. The resulting microstructure, which can be obtained with such tailored tempering methods, exhibits a huge variety of complex process dependencies. One example for these intricate aspects of the specific material behavior is the prediction of transformation plasticity in quenching from the prestrained austenite phase. In the case of 22MnB5 press hardening, the process demands a quenching within 5-10 seconds for the transformation from the austenite to martensite, where the mechanical properties are mainly controlled by the carbon and boron content, while manganese and chromium have minor influence only on the final strength. The transformation plasticity, which appears during the rapid quenching, then needs to be controlled in agreement with the process-determined requirements, while the transformation from the prestrained austenite shows an elasto-plastic response to the applied stresses depending strongly on the precise strain state of the austenite. Also, the transformation plasticity in a post-yield-strength regime exhibits a strong influence on the applied stresses which are modified due to internal stresses. This specific part of the entire process chain exhibits the need for a scale bridging description coupling thermochemical and mechanical modeling.
Consequently, the approaches pursued to tackle the problem cover calculations performed with various different methods. On the smallest scale, where a basic thermodynamic description is clearly available, phase field models [22,23] are an established tool to connect the morphology selection understanding obtained from sharp-interface descriptions to extended mesoscale regimes. More sophisticated phase field implementations as the multi-phase-field model used in the software MICRESS [24] allow the connection to process specific environmental conditions and material specific thermodynamic data. To capture the lath-like morphology of ferrite formed by displacive mechanism, many models assign anisotropic interfacial mobility and energy to the ferrite-austenite phase boundary. The resulting shape of the phase boundary comprises several facets which are kinetically or energetically favored. Loginova et al. [25] simulated the 2D growth of Widmanstätten ferrite and obtain a morphology similar to the realistic microstructure. A related approach has been applied by Song et al. [26,27] to the simulation of bainitic transformation in a 100Cr6 steel. Some other extended phase field models introduce a certain form of mechanical energy into the free energy functional. Arif and Qin [28,29,30] employed an approximated formula for the strain energy of each bainitic sub-unit proposed by Bhadeshia [31]. They also introduced autocatalytic nucleation of neighboring sub-units to simulate the growth of a complete bainite sheaf. In more sophisticated models, the mechanical response of both austenite and ferrite is also resolved by linear elastic [32] or elasto-plastic [33,34,35] models. An elasto-plastic phase-field model was developed by Kundin et al. [33,34] using the crystal plasticity framework [36]. The model is then applied to explore the interaction of the martensitic transformation with dislocation evolution.
To take into account process-specific complex plasticity phenomena on larger scales, finite element calculations on top of Johnson-Mehl-Avrami-models are an established method [37,38,8]. There are numerous papers dealing experimentally with the transformation plasticity for the bainite formation under small stresses [10], meaning stresses below half of the yield strength of austenite. Under these conditions, the transformation plasticity depends linearly on the stress. However, for higher stresses the transformation plasticity can be strongly nonlinear. During press hardening the transformation plasticity is responsible for a small spring-back compared with conventional cold or hot forming [39,40,41].
The present article intends to give an overview on several aspects mentioned above, starting from fundamental theoretical concepts, via phase field simulations of bainite formation, to macroscopic plastic effects during bainitic press hardening. It is organized as follows.
Section 2 briefly summarizes the materials and methods used for the work in this article. The central results and their discussion is contained in section 3. In detail, the analysis of the transformation processes from a fundamental thermodynamic perspective is developed in section 3.1. A further theoretical treatment of solid-solid transitions in the displacive and the diffuse-displacive regime is given in section 3.2. The onset of displacive transformations is described in terms of a Griffith-like elastic growth criterion, and the dominance of appearing morphologies is distinguished both by sharp interface predictions of the growth velocity and phase field simulations concerning the stability of the growth modes. The inclusion of diffusional transport as governing mechanism for the transformation also yields a strong dependence of the diffusional transformation on the relative orientation of growth direction and lattice strains.
In section 3.3 we develop our phase field approach to account for available thermodynamic material data to provide more insights into the modeling of bainite transformation. We focus on the high carbon steel 100Cr6 and simulate the microstructure evolution during an isothermal bainite transformation, starting from a sample which is austenitized at higher temperature. Here, the emphasis is on the carbon redistribution in the austenitic matrix. The results obtained here transfer the basic thermodynamic description to the process-governed environmental conditions required for the detailed understanding of the entire transformation. At this level the role of transformational plasticity is not yet accounted for, and we describe the approach to this aspect of the process in section 3.4.
In section 3.4 the focus is on the influence of high stresses, i.e. beyond the austenite yield strength, on the bainitic transformation in a 22MnB5 steel. Experimental and finite element simulations regarding the kinetics of the bainitic transformation are presented. The study is based on a Johnson-Mehl-Avrami model and shows the strong dependence of the transformation speed and the strain states on the applied stress. A strong nonlinearity of the anisotropic strain response to stresses has been found in the longitudinal and thickness directions, beginning in the range between 50-100 MPa, where the austenite yield strength is located for the relevant temperatures.
The different aspects of the article are summarised in Section 4, with an outlook to future activities to bring the different levels of modeling together.
The work in this article is based on both experimental and theoretical work.
On the experimental side, flat tensile samples were performed in a servo-hydraulic deformation dilatometer. The specimens are quenched to 500℃ after austenitization and held for 20 minutes. Length and width changes are determined by a laser extensometer. As materials the press hardening steel 22MnB5 and the high carbon steel 100Cr6 are used.
On the theoretical side, various methods were used, starting from thermodynamics, including linear elasticity. Sharp interface formulations are used to express the microstructure evolution equations for diffusionless and diffusion limited processes. Proper phase field methods are used for more flexible and convenient formulation, being employed for the simulations. The phase field code MICRESS is used for the simulation of bainite formation, including a coupling to ThermoCalc databases for free energy curves. On the macroscopic level Johnson-Mehl-Avrami models are used to describe the transformation kinetics.
For further details we refer to the following sections.
In this section we inspect several aspects of the equilibrium properties of solid state transformations. Since the focus of this section is to gain generic insights, we use simplified models to illustrate fundamental concepts. This means that we restrict ourselves for the purpose of clarity to elastic deformations and do not consider plastic effects. This constraint will be released in section 3.4. Moreover, we use only linear elasticity, and more specifically only isotropic linear elasticity. This has the advantage that one needs only two elastic constants, and for further simplification we assume that they are identical in all phases. These assumptions have central consequences: First, without the anisotropy in the elastic constants there is no preferred spatial orientation. Nevertheless, anisotropy can still be induced by transformation strains. As we will see in detail in the following, this effect is important, as for the diffusion limited processes only this aspect leads to selection of microstructure scales and transformation velocities. It is therefore important to understand the role of fundamental symmetries starting from the most symmetrical situation of isotropic elasticity. In Section 3.3 we will then generalise the developed concepts to bainitic transformations. This will require in particular to use anisotropic formulations, which we leave out here for clarity.
We begin the analysis of the transformation processes from a thermodynamic perspective. In an isothermal situation we consider the transition of a metastable phase, which we will in the following denote as γ phase into a more stable phase α. These two phases could for example be the austenite γ and the bainitic ferrite α of an iron based alloy. For the moment we ignore concentration variations inside the phases, as we first have displacive transformations in mind, which are controlled by the interface kinetics. However, the following considerations are of thermodynamic nature and serve as a background for the model formulation steps in the following sections.
In the framework of the approximations mentioned above the free energy densities of the two phases are
fγ=f0γ+12λϵ2ii+μϵ2ik, | (1) |
fα=f0α+12λ(ϵii−ϵ0ii)2+μ(ϵik−ϵ0ik)2. | (2) |
For the solution of the elastic problem we need to consider specific geometries. In view of later applications to press hardening processes the situation of given volumetric constraints, i.e. prescribed displacements at the sample boundaries, is of special interest. We consider a strip like geometry (see Fig. 1), where at the upper and lower boundary of the strip displacements shall vanish, ui = 0. In perpendicular direction z the sample is assumed to be thick, such that a plain strain assumption, uz = 0, ux = ux(x; y) and uy = uy(x; y) is legitimate. We assume that the constraints at the boundary are chosen such that the pure parent γ phase is stress free. We expect later in a dynamical situation the α phase to invade the γ phase in the direction of the sample, as depicted in Fig. 1. This will be the case if the thermodynamic driving force, which is related to the difference f0γ−f0α is sufficiently high. Far ahead of the growing front the material is stress free, hence the energy density is f0γ there. Behind the tip, we usually find a two-phase configuration, which is homogeneous far away. This implies that the only non-vanishing strain component is there ϵyy, which is different in the two phases, but spatially homogeneous in each of them. The volumetric constraint of given displacement at the upper and lower surface of the sample implies the “lever rule” (1−χ)ϵγyy+χϵαyy=0, where χ is the volume fraction (which is formally equivalent to a driving force) of the α phase in the tail region. Equilibrium demands that the free energy per area, F=(1−χ)fγ+χfα is minimised with respect to the degrees of freedom χ and ϵαyy. The latter condition is equivalent to a stress balance at the interface between α and γ. From this we obtain the equilibrium volume fraction of the α phase in the tail,
χ=1−2ν1+ν[1−νEϵ2(f0γ−f0α)−1], |
(3) |
(f0γ−f0α)nuc=Eϵ21−ν. |
(4) |
To judge the possibility of a phase transformation we also need to consider the free energy change. Only if the free energy decays during the process, the transformation will proceed as anticipated, otherwise the tip of the α phase will convert back to the γ phase. From the difference between the free energy ahead of the tip and in the tail (both far away from the advancing front) we get
ΔF:=W(f0γ−F)−2γs=2γs(χ2χ2W−1), |
(5) |
W∗=4(1−2ν)(1−ν)γsEϵ2(1+ν), |
(6) |
We can also consider more complex transformation strains, in particular involving shear components. This can for example occur in hexagonal-orthorhombic transitions in ferroelastic materials [46]. We inspect the transformation in the basal plane, where the transformation induces a symmetry reduction from C6 of the γ phase to C2 for the α phase. The new phase can appear in three variants, and the transformation strains are given by
ϵ0xx=−ϵ0yy=ϵcos2θ,ϵ0xy=ϵsin2θ, |
(7) |
χ=11−2ν[4(1−ν2)Eϵ2(f0γ−f0α)−12]. |
(8) |
W∗=8(1−ν2)(2γs+γgb)(1−2ν)Eϵ2. |
(9) |
So far, we have only looked at the asymptotic regions far away from the transformation front. At the front itself equilibrium demands for stationary fronts the condition
˜fγ−˜fα−γκ=0, |
(10) |
˜f=f−σnnϵnn−2σnτϵnτ, |
(11) |
Whenever we have a disbalance of energy in the spirit of a violation of Eq. (10) at an interface point between the different phases, kinetics will lead to front propagation in order to reduce the total free energy. In this section we inspect such situations both for diffusionless and diffusion-limited transformations as limiting cases. Also, we introduce both sharp interface and phase field formulations to simulate these microstructure evolution processes.
In the framework of linear non-equilibrium thermodynamics we therefore expect as local equation of motion for solid-state transformations which are limited by interface kinetics,
vn=ˉμαγ(˜fγ−˜fα−γκ), |
(12) |
vv0∼χ−χW, |
(13) |
Whereas such a sharp interface description is feasible and useful for theoretical inspections, a proper phase field formulation is useful for generalisations of the model and complex morphologies. We illustrate this here for the isotropic eigenstrain, as we have to distinguish only between two phases and not between variants of the secondary phase. This allows to discriminate the two phases by a single order parameter ϕ, which we normalise to be ϕ=1 in the γ and ϕ=0 in the α phase. The generating free energy functional for the phase field evolution equation is given by
F=∫(fγϕ+(1−ϕ)fα+fs+fdw)dV |
(14) |
fs=3γsξ2(∇ϕ)2 |
(15) |
fdw=6γsξϕ2(1−ϕ)2. |
(16) |
∂ϕ∂t=−μαγδFδϕ. |
(17) |
δFδui=0. |
(18) |
For the transformations involving shear, which lead to bicrystal patterns, require in general to distinguish also the variants and to introduce the twin boundary energy, which can be realized using multi-order parameter phase field models, and this will be explored in the following section. For small grain boundary energies γgb≪γs we can invoke symmetry along the plane y = 0 and proper boundary conditions for the phase field to reflect the correct contact angles at the triple junction. This allows to describe the dynamics with only minor modifications of the above single-order parameter phase field model, as explained in more detail in [42]. The resulting microstructures are shown in Fig. 2,depending on the layering of the variants. The two twin structures with a single or two separate tips differ in the steady state velocity, as shown in Fig. 3. Altogether, these examples show that the combination of sharp interface, phase field and thermodynamic modeling allows to obtain important insights into the statics and dynamics of displacive solid-state transformations. The analytic predictions are most useful in simple geometries and provide exact predictions for asymptotic regions. The phase field simulations are more flexible concerning complicated patterns than sharp interface implementations. Further examples of phase field models for more complex interactions of microstructure and free interface evolution in martensitic transformations can be found e.g. in [49,50,22,23]. Applications to bainite formation will be discussed below in Section 3.3.
Pearlitic or bainitic transformations are examples for diffusional or mixed diplacive-diffusional solid-solid transitions which are strongly influenced by elastic effects. The coupling of displacive and diffusional transitions has been investigated recently emphasising engineering aspects, see [32,51], and will also be discussed in the next section. In order to separate these effects for a fundamental understanding, we continue with the discussion of diffusional transformations alone.
An overview of diffusional transformations with elastic effects can be found e.g. in [52,49,22]. We focus here on situations with coherent interfaces [36,42,47,53,54]. As for displacive transformation studies, the combination of classical sharp interface descriptions and phase field modeling turns out to be a fruitful approach. Central aspects are related to the interaction of diffusive transport, transformational lattice strains, grain boundaries and triple junctions. Component diffusion will be considered in Section 3.3, and here we complementary discuss the effect of heat diffusion and its absorption and release at propagating phase fronts.
In a dimensionless formulation we introduce w=cP(T−T∞)/L, where T is the local temperature, cP the specific heat capacity, L the latent heat for the γ−α transition, and T∞ is the temperature in the γ-phase far away from the interface. The heat diffusion in the bulk is described by
∂wα,γ∂t=D∇2wα,γ |
(19) |
vn=D→n⋅(∇wα|int−∇wγ|int) |
(20) |
wα|int=wβ|int=w|int=Δ−d0κ+TeqcPδfel/L2 |
(21) |
δfel=σ0ikϵ(γ)ik−E[(ϵ0ττ)2+(ϵ0ss)2+2ν(ϵ0ττ)(ϵ0ss)+2(1−ν)(ϵ0sτ)2]2(1−ν2), |
(22) |
The physical picture emerging from Eqs. (19)-(22) is clear: The propagation of the interface releases latent heat proportional to the normal component of the growth velocity vn, while the local transition temperature itself depends on the elastic state at the interface and the curvature. The transport of the latent heat from the interface limits the propagation velocity.
In accordance with studies on displacive transformations, which are discussed above, we choose the hexagonal-orthorhombic transition here as example. Emerging bicrystal patterns, as they result in the steady state regime, are shown in Fig. 4. They are obtained from a sharp interface modeling for phase transformations in an infinite sample.
One of the central features of the elastic effects during solid state transformations is their strong influence on the growth velocities. In comparison to other diffusion limited processes e.g. during solidification, where elastic effects play only a minor role, they lead here to significantly higher transformation velocities. This experimentally well known phenomenon is correctly captured by the above model. Moreover, it is known that anisotropy is important for selection of the tip scale and the solidification velocity during dendritic growth. Here it turns out that the elastic anisotropies dominate the selection compared to the much weaker surface tension anisotropy effects, which therefore can be neglected here. It is therefore worthwhile to study the role of anisotropies of the transformation strain ϵ0ij in more detail. Hence we use additionally the more complicated eigenstrain ϵ0±ik=(1−η)ϵs±ηϵdik as a weighted superposition of a hydrostatic lattice strain ϵdik=ϵδik and the shear lattice strain ϵsik given by Eq. (7) with angles θ=0,±2π/3. For η=1 and η=0 we recover the pure shear and pure hydrostatic stress free strains, respectively. η is therefore a measure for the deviatoric part of the eigenstrain. This model allows to consider the drastic influence of the orientation of lattice strains relative on the growth direction of the new phase. In Fig. 5 the eigenvalue σ∗=d0/pρ of the steady state problem is shown as function of the mixing parameter η between the two types of mismatch strains. Here, p is the Peclet number and ρ the radius of the asymptotic Ivantsov parabola, see [55] for details. Therefore, the eigenvalue σ∗ is a measure for the growth velocity, to which it is directly proportional at a fixed value of the driving force. We find a strong difference in the growth velocity for different orientations of the bicrystal microstructure. In fact, the growth velocity can easily differ by two orders of magnitude. Hence, the elastic anisotropy strongly affects the selection of the growth direction. Moreover, there is a strong dependence of the growth velocity on the type of transformation strain, and some steady state modes can only be found in a limited regime of the parameter η. Therefore, we can expect to find also discontinouos transitions between different microstructures.
At this point it should be pointed out that the sharp interface method allows to track also metastable solutions. Without that, it would not have been possible to track slower growth modes. In phase field simulations, in contrast, typically only the fastest growing mode will be found in the steady state regime, unless it is suppressed by suitable boundary conditions. In turn, this implies that the combination of phase field and sharp interface methods allows not only to find potential steady state patterns, but also to decide about their stability. For that, we also have implemented a phase field formulation analogous to Eqs. (19)-(22). We refrain here from giving the model equations explicitly and refer to [53] instead. The phase field simulations can be compared to the sharp interface predictions in a regime of small widths of the transformation pattern relative to the (finite) width of the system. We note that the phase field simulations correspond to a finite-width channel geometry while the sharp-interface calculations use infinite systems. The comparison of phase field simulations and sharp-interface predictions in Fig. 6 shows excellent agreement. The obtained agreement for pattern widths relative to the channel size in the range from χ=0.2−0.65 is therefore very convincing concerning both the existence of steady transformation modes and their dynamical stability.
Based on the fundamental concepts discussed above, we generalize the description for the application to the austenite to bainite transformation. This involves the displacive deformation of the matrix, diffusion controlled growth of bainitic sheaves and the precipitation of carbides inside the ferrite (lower bainite) or inside the austenite (upper bainite). A successful prediction of the transformation kinetics requires consideration of the complex interaction of these processes: Elastic and plastic deformations, carbon partitioning and carbide precipitation. The local driving force for the transformation depends sensitively on the carbon concentration, stress tensor and temperature. During the transformation, the supersaturated carbon in the bainitic ferrite can either go to the surrounding austenite or the embedded carbides, which offer sites for the solute atoms. The multiphase field method proposed by Steinbach et al. [56,49] is most suitable for the simulation of such complex interactions. Here we briefly review the multi-order parameter phase field model and its application to the bainitic transformation of a high-carbon steel 100Cr6. For further details we refer to [26,57,27].
The starting point is the free energy of the system as a functional of the phase fields ϕα(→x,t), the concentration field c(→x,t) and the elastic strain field ϵ(→x,t),
F=∫Ωf(ϕ,c,ϵ)dV=∫Ω[fintf(ϕ)+fchem(ϕ,c)+felast(ϕ,ϵ)] dV, |
(23) |
fintf=N∑α,β=1,α≠β4γαβξ[ϕαϕβ−(ξπ)2∇ϕα⋅∇ϕβ], |
(24) |
fchem=N∑α=1ϕαfα(ciα)+˜μi(ci−N∑α=1ϕαciα), |
(25) |
felast=N∑α=1ϕα(ϵαij−ϵα,0ij)Cαijkl(ϵαkl−ϵα,0kl)2. |
(26) |
N∑α=1ϕα=1, |
(27) |
N∑α=1ciα=ci. |
(28) |
˙ϕα=N∑β=1μαβ{γαβ[ϕβ∇2ϕα−ϕα∇2ϕβ+π22ξ2(ϕα−ϕβ)]+πξ√ϕαϕβΔgαβ}, |
(29) |
˙cαi=∇(∑jDijα∇cjα), |
(30) |
The volume driving force Δgαβ comprises the differences of the bulk energy densities.
By assuming anisotropic interfacial mobility and tension we can reproduce the plate or needle like shape of the bainitic ferrite in the phase field simulations. As proposed in [26], both mobility and tension of the phase boundary are expressed as a function of the inclination θ, the angle between the interface norm and the facet norm,
γαβ(θ)=γαβ,0k2st,αβ(k2st,αβcos2θ+sin2θ)−3/2 |
(31) |
μαβ(θ)=μαβ,0[kkm,αβ+(1−kkm,αβ)tan(κtanθ)tanθκ], |
(32) |
The application of the multi-order parameter phase field model to the bainitic transformation in 100Cr6 has been reported in our previous publications [26,57,27]. A commercial implementation of the multi phase field model, MICRESS (MICRostructure Evolution Simulation Software) [24] is employed. The free energy functions fα(ciα) of each phase are obtained from the Calphad database TCFe7 [60] using ThermoCalc. In Fig. 7 we show the results reported by Song et al. [26]. The experimental curve was determined by a dilatometer measurement, where the sample was austenized at 850℃, quenched to 260℃ and then held isothermally. The corresponding simulation considered the diffusion controlled growth of the needle-shaped bainite. As shown in the figure, the volume fraction of bainite in the simulation agrees well with the measurement of the dilatometer.
To complement this approach, we apply a stress-coupled multi-phase-field-model in which the plate-like morphology results from the displacive essence of the transformation rather than the anisotropy of the interfaces. The deformation gradient tensor for the displacively transformed phase is defined as FF=II+→m⊗→n. Here the vectors →m and →n denote the shear strain vector and the norm of the habit plane. The deformation gradient tensor is zero in the directions parallel to the habit plane and is an invariant-plane strain. To demonstrate the model, we run a simulation of the isothermal bainitic transformation of a carbon-manganese steel. The model is implemented in MICRESS, the equilibrium thermodynamic data for ferrite and austenite is calculated from the TCS steels/Fe-alloy database and the mass diffusion data from the TCS alloys mobility database. Due to the lack of plasticity, we scale down the transformation strain to 1/5 of the realistic value, which is 0.03 dilatational strain and 0.22 shear strain [61]. This prevents a premature frustration of the transformation due to artificially high elastic energy. The stress equilibrium is in agreement with a parallel-expansion boundary condition.
The results of stress-coupled multi phase field model are presented in Figs. 8 and 9. The visible domain in the figures is approximately 1/9 of the simulation domain. The stress arising from the elastic accommodation of the austenite matrix significantly affects the growth of the bainitic ferrite (αB). Fig. 8 shows the evolution of the phase boundary and the partitioning of dissolved carbon during isothermal holding at 450℃. The growth of the bainite grain is constrained parallel to the habit plane by the trend of minimization of elastic energy in the matrix. The resulting stress field is shown in Fig. 9. The stress is certainly overestimated since the constitutive relation in this model is purely elastic. The simulated growth of bainitic ferrite is accompanied by carbon diffusion from the supersaturated ferrite to austenite, while substitutional elements, i.e. Cr, Si, Mn, are effectively frozen due to their low diffusivity. The bainitic ferrite exhibits partially a supersaturation of carbon.
Plastic effects on bainite formation, in particular under large external stresses, are not yet taken into account in the simulations, and this will be a future activity. Some open issues, which require to be understood on a mesoscale simulation level, are summarized in the next section.
The comparison of the multi-phase field simulations and the experiments have shown convincing agreement for 100Cr6 in several aspects. However, for applications of press hardening, where 22MnB5 is used, specifically anisotropic plastic response (transformation plasticity), remains to be incorporated in future considerations. Here we give a short overview on this issue.
Transformation plasticity is meant to play a major role for bainitic transformations in partial presshardening processes [41]. In Fig. 10 the simulated von Mises stress of a formed hat-profile of 22MnB5, shortly after the bainitic transformation has started, is shown. The simulation is started from an austenitized state, followed by a rapid cooling up to the process temperature (500℃). In the rounded edge regions the von Mises stress is significantly higher than the austenite yield strength of around 80MPa. These findings suggest to develop an understanding of the material behavior and the kinetics of the bainite transformation under high stresses, i.e. higher than the yield strength, and also under the influence of pre-strain states.
On a macroscopic level, the transformation plasticity effect is modeled using a general approach [62]
dϵtpijdt=32KSijdf(p)dpdpdt, |
(33) |
f(p)=p(2−p), |
(34) |
f(p)=p−plnp |
(35) |
f(p)=p. |
(36) |
ϵtpij=t∫032KSij(τ)ddτf(p(τ))dτ, |
(37) |
The diffusion-based phase transformation from austenite to bainite is described combining the models of Johnson, Mehl and Avrami (JMA) [37,38] and Leblond [66]. This leads to the describing equation
˙p(t,T,˙T)=F(T,˙T)(ˉp(T)−p(t))n(T)τ(T)[−ln(1−p(t)ˉp(T))]1−1n(T), |
(38) |
In order to obtain values for the fit parameters n and τ, the effect of the applied stress on the bainitic transformation kinetics are studied experimentally. To understand the effect on the transformation behavior of an applied stress, several tests were performed in a servo-hydraulic deformation dilatometer Gleeble 3500. Flat tensile specimens are obtained from the sheet material by wire-cut electrical discharge machining with a total length of 120mm in accordance with ASTM E8/E8M-11 standard [67]. These tensile specimens were rapidly cooled down to 500℃ after austenitization and were held at this temperature for 20 minutes. Shortly above this temperature a stress in the range of -20 to 200MPa was applied. The temperature and load schedules utilized are illustrated in Fig. 11. The length and width changes of the specimens, in the L and T directions (see Fig. 11) respectively, are recorded by a laser extensometer. Finally, from the length change the kinetics of the bainite formation was calculated by fitting the above JMA equation. For these isothermal transformations, the bainite volume fraction evolution in time can be determined as
p(t)ˉp(T)=l(t)−l0lmax−l0, |
(39) |
log[−ln(1−p(t)ˉp(T))]=n(T)logτ(T)−n(T)log(t−ts). |
(40) |
From these tests another important effect can be observed when the transformation is performed under the influence of high stresses. This refers to the strong nonlinearity of the transformation plasticity or the anisotropic strain [68]. Such a behavior is not explained by the commonly used transformation plasticity models. In Fig. 13 the maximum measured anisotropic part of the length and the width strain after completion of the transformation are shown. The elastic and plastic strains due to the application of the external load are substracted, as well as the thermal strain up to the initiation of the bainitic transformation. During the tests, strains are measured both in longitudinal direction, ϵl, and in transversal direction, ϵt. Then, the volumetric transformation strain, ϵtr, is obtained as
ϵtr=ϵl+2ϵt. |
(41) |
ˉϵtr=13(ϵl+2ϵt) |
(42) |
ϵtpl=ϵl−ˉϵtr. |
(43) |
ϵtpt=ϵt−ˉϵtr. |
(44) |
The model of equation (33) gives a good approximation for the linear response of the transformation plasticity effect. The values of K for the tensile and compression cases have been determined as K+ = 11 · 10-5 MPa-1 and K- = 7.4 · 10-5, MPa-1 respectively. However, to model the nonlinear response another model must be introduced [69,70]. This aspect will be assessed in future works.
The experiments reveal that also another effect, namely the influence of pre-strained austenite, which is a consequence of the forming process, has substantial influence on the transformation kinetics. During the forming certain deformations can be reached in the austenitized component, even before the bainitic transformations starts. This topic is beyond the scope of the present paper, however, some initial results on this matter are reported in [71].
One of the main conclusions from the experimental observations is that applied stresses can accelerate the bainite formation during press hardening. It therefore remains an important task to understand this aspect also from a mesoscopic level, in particular using suitable extensions of phase field descriptions.
We have reviewed distinct approaches in the experimentally benchmarked modeling of different aspects of the bainite transformation during press hardening. To start with a basic thermodynamic model which allows to estimate the competitive behavior of different fundamental transformation morphologies, we have discussed combinations of sharp-interface and basic phase field models. The resulting insights about length scale and velocity regimes of appearing patterns in dependence of strain states and diffusional behavior of the material provides connections to multi phase field implementations, which are specifically adapted to the material data of 100Cr6 and 22MnB5. Taking into account also kinetic and surface stress anisotropies, simulations of the austenite-bainite transformation exhibit excellent agreement concerning the kinetics in the considered process regime with experimental data from scanning electron microscopy and dilatometric investigations. For the high stress regime, the conducted finite element simulations and the application of the Johnson-Mehl-Avrami model to 22MnB5 has allowed to extend these modeling approaches to the corresponding process regime. The results show a strong dependence of the plastic response on the applied stresses and the strain state of the material at austenitization. We note that a refined description of the coupling of nonlinear plastic response and the diffusional-displacive model is desirable for an advanced understanding of the later process regimes, especially for the austenite to bainite transformation in presence of a prestrain and high stresses.
This work has been supported by the Deutsche Forschungsgemeinschaft under the priority program SPP 1713 “Strong coupling of thermo-chemical and thermo-mechanical states in applied materials”, project M7 “Modeling bainitic transformations during press hardening”.
Authors state no conflicts of interest.
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