From High Accuracy to High Efficiency in Simulations of Processing of Dual-Phase Steels

The classification of phase transformation models with respect to predictive capabilities and computing time is shown in Figure 1. Models in the left bottom corner are used mainly for basic simulations and optimization of processes or whole manufacturing cycles. Historically, the JMAK equation[1‐3] and its modifications[4] were commonly used for fast simulations of phase transformations.

The models in the center of Figure 1 are based on the FE or phase-field method, and they are used in the technology design and optimization of processes. The latter approach has emerged as one of the most powerful methods for modeling various microstructure evolution processes, including the austenite decomposition. A description of this method was presented in References 5 and 6. Briefly, the phase-field model treats a polycrystalline system, containing both bulk and boundary regions, in an integral manner. A set of continuous phase-field variables, each representing an individual grain, is defined to have a constant value inside the grains and will change continuously over a diffuse boundary. The driving force is calculated from the local carbon composition within the interface, controlled by the diffusion in the austenite. A specific feature of the phase-field approach is that the boundary between two distinct regions is diffused. Therefore, the computational complexity of the phase-field model is comparable to the deterministic model based on the direct solution of the diffusion equation.

In the 1990s, connection of the JMAK equation to FE codes created a possibility to predict the distribution of microstructural parameters in the volume of products.[7] The need for simulation of microstructural phenomena accounting explicitly for the structure of polycrystals led to the development of multiscale models located in the top right corner in Figure 1.

The microstructure can also be explicitly accounted for in modeling phase transformation when the FE solution of the diffusion equation with moving boundary is applied.[8] This solution supplies information about carbon distribution in the austenite. In consequence, predicting the hardness distribution in martensite islands becomes possible.

In multiscale modeling, usually FE codes are connected with such discrete methods as CA, MD, or MC. A review of multiscale modeling methods can be found in Reference 9. The problem of computing time is the challenge, which is particularly important when optimization of the process is performed. Multiscale methods give better insight in the phenomena occurring at lower dimensional scales, but they require long computing times.

This article presents the results of research on the improvement of the accuracy of the JMAK equation, as well as on a decrease of the computing time for multiscale modeling techniques. The former was achieved by introduction of relations of coefficients on the temperature. The latter was achieved by introduction of the statistically similar representative volume element (SSRVE).

The model is based on the JMAK equation:where t is the time and X is the transformed volume fraction.

Equation [1] is combined with the Scheil additivity rule, which accounts for the temperature variations. Theoretical considerations show that a constant value of n in Eq. [1] can be used. This coefficient is introduced further in the vector a of the model coefficients containing a ₄, a ₁₆, and a ₂₄ for ferritic, pearlitic, and bainitic transformations, respectively. On the contrary, the value of the coefficient k must vary with temperature. The formalism of the function k = f(T) must be carefully chosen to describe properly the temperature dependence of transformation kinetics. This problem was thoroughly discussed in Reference 10. A sensitivity analysis was performed and the best functions k = f(T) were selected. All equations of the final version of the model are given in Tables I and II.

The simulation of ferritic transformation begins with Eq. [1] when the temperature drops below Ae₃. The transformed volume fraction X _f is calculated with respect to the maximum volume fraction of ferrite X _f0 at the current temperature calculated with ThermoCalc (Thermo-Calc Software, Stockholm, Sweden). Thus, this volume fraction with respect to the whole volume of the body is F _f = X _f X _f0. During simulation in the varying temperature, the current value of X _f calculated from Eq. [1] has to be corrected to account for the change of the maximum volume fraction of ferrite X _f0, which according to equation in Table I is a function of temperature.

Because continuous annealing in the intercritical region was selected as a case study in the current work, transformation of the ferritic-pearlitic microstructure into the austenite has to be also considered. The kinetics of the austenitic transformation during heating is described by Eq. [1] with the coefficient n = a ₃₀ and the coefficient k defined as:

The incubation time for the ferrite-austenite transformation is calculated from the equation:

The full model contained 30 coefficients grouped in the vector a. After the sensitivity analysis,[10] the model was simplified and 26 coefficients remained. These coefficients are determined using an inverse analysis of dilatometric tests.

Dual-phase (DP) steel containing 0.1 pct C, 1.51 pct Mn, 0.23 pct Si, 0.41 Cr, 0.05 pct Mo, 0.045 pct V, 0.006 pct Ti, 0.007 pct P, 0.007 pct S, 0.037 pct Al, and 0.009 pct N was investigated. Experiments involving dilatometric tests during heating and cooling were performed to supply data for identification of the models. Samples measuring 1.4 × 2.2 × 7 mm were cut from a cold-rolled strip. In the first set of tests, samples were heated to the selected temperature at a rate of 3 K (3  °C)/s and fast cooled at a rate of approximately 3 K (180  °C)/s with helium. The microstructure was analyzed with Neophot 2 light microscope and Inspect field emission gun electron microscope, and volume fractions of phases were evaluated using the Metilo program for automatic image analysis. In the second set of experiments, the samples were austenitized at 1163 K (890  °C) and cooled with various cooling rates. An inverse analysis preceded by the sensitivity analysis was applied for the identification of the coefficients in the model. Details of the sensitivity analysis for the phase transformation model are given in Reference 10. The inverse algorithm proposed in Reference 11 and applied to phase transformation model in Reference 12 was used in the current work. Briefly, a mathematical model of an arbitrary process or physical phenomenon can be described by a set of equations:where d = {d ₁,…,d _r } is the vector of output variables of the model (start and end of transformations temperatures, volume fractions), a = {a ₁,…,a _k } is the vector of coefficients of the model, and p = {p ₁,…,p _l } is the vector of the known process parameters (cooling rates). When vectors p and a are known, the solution of the Eq. [4] is called a direct solution. The inverse solution of Eq. [1] is defined as determination of the components of the vector a for known vectors d (from dilatometric tests) and p:

When the problem is linear, the inverse function can be usually found and the problem can be solved analytically. In the investigated problem, this relation is nonlinear and the problem is transformed into the optimization task. Thus, the objective of the inverse analysis is to determine the optimum components of vector a by searching for the minimum, with respect to the components of this vector, of the objective function defined as a square root error between measured and calculated components of the vector d:where \( {\mathbf{d}}_{i}^{m} \) is a vector containing the start and end of transformations temperatures and volume fractions of phases measured in dilatometric tests, \( {\mathbf{d}}_{i}^{c} \) is a vector containing these parameters calculated by the model, β _i is the weights of the points, (i = 1,…,n), and n is the number of measurements. The values of the coefficients given in Table III were obtained from the inverse analysis of dilatometric tests.

The model with the coefficients in Table III was validated. All performed dilatometric tests were simulated with the developed model. Figure 2 shows a comparison of the measured (filled symbols) and predicted (open symbols) start and end temperatures for the phase transformations and volume fractions of phases. The shape of the symbol refers to the temperature or volume fraction in the legend. An analysis of the results shows that the model predicts well the start and end temperatures for the transformations. The accuracy is worse for the bainite end temperature only, possibly because of difficulties with a very accurate evaluation of this temperature from the dillatometric data. In the experiment, the martensite start temperature was interpreted as the end of bainitic transformation, while the model predicts the end of this transformation at higher temperature.

The volume fractions of phases are properly predicted for slower cooling rates (below 25 K [25  °C]/s). There were some difficulties in distinguishing between bainite and martensite for high cooling rates, but good agreement between measurement and calculations was observed for the sum of the volume fractions of hard components. This is because the transition between bainite and martensite is not predicted accurately with the developed model. One may assume that this is an inherited feature of the conventional models based on the JMAK model. There is also a significant influence of the inaccuracies of the ferrite and pearlite transformation predictions on the further start and kinetics of bainite and martensite transformation because in the model the mean carbon content at the end of diffusional transformations is calculated to predict B _s and M _s temperatures. One must also consider the fact that in a real process, the carbon may not be evenly distributed in the austenite as is assumed in the current model.

The phase transformation model for heating was validated next. Figure 3 shows comparison of measured and calculated volume fraction of the austenite after heating to various temperatures. The measurement of the austenite volume fraction was performed using a dilation curve obtained in the dilatometric test. Selected examples of microstructure after heating for these tests are shown in Figure 4.

Simulations of the continuous annealing process were conducted for testing the models. Four arbitrary annealing thermal cycles (dotted lines and open symbols in Figure 5(a)) were simulated and the calculated volume fractions are shown in Figure 5(b). It was concluded that although these cycles gave good results for other chemistries of DP steels, see for example Reference 10, the ferritic transformation was too fast for the chemistry investigated in this study. After numerical tests for various cycles, it was possible to select the one that gives less than 80 pct of the ferrite in the microstructure, the martensite as the main hard constituent and small amount of bainite (cycle A5 in Figure 5).

This thermal cycle was reproduced in the DIL 805 dilatometer. The microstructure of the sample is shown in Figure 6. It is composed of ferrite and hard constituents. The volume fraction of the hard constituent is approximately 0.32, of which around 0.07 can be identified as bainite and the rest is martensite or auto-tempered martensite. The examples of martensite and bainite are shown in Figure 7. The result is in good qualitative agreement with the developed schedule using the phase transformation model. The volume fraction of martensite was predicted quantitatively well.

Numerical tests and physical simulations confirmed good predictive capability of the model based on the modified JMAK equation as far as volume fractions of phases are considered. The capabilities and computing times for multiscale models are discussed below.

Multiscale techniques are now commonly used for modeling of phase transformations; see Authors Cellular Automata model for DP steels.[13] However, even for the small size of CA spaces, this approach requires long computing times. The possibility of simplification of the multiscale model is discussed below.

The process of initial stamping during the crash box manufacturing cycle was selected for a deeper analysis by using multiscale simulations. Two different SSRVEs (presented in Figure 10), obtained for HTC600 DP steel, were applied as a basis for microscale simulations. The results shown in Figures 12 and 13 present a distribution of effective strains and stresses. However, it can be seen that for different shapes of SSRVEs, different values of maximum strain and stress are obtained. This suggests that an additional factor in the form of strain and stress distributions should be incorporated into an objective function during SSRVE procedure. It will allow the determination of the best unambiguous solution and will lead to more reliable simulations.

The computing times were measured for all simulations in microscale and the following results were obtained dependently on the number of finite elements:

The measured time of calculations is strongly related to the number of finite elements. Therefore, numerical simulations using real microstructure were performed to compare obtained timings with its related to SSRVE. The extended version of microstructure presented in Figure 9 was used for this purpose. Its segmented and meshed versions are presented in Figures 14(a) and (b), respectively.

This computational characteristic of applied meshed microstructure are as follows:

It is clearly visible that even SSRVE composed of many NURBS control points is much faster than sophisticated microstructure. Simultaneously the quality of the obtained results is maintained.

An analysis of the computing times and predictive capabilities of the models allowed to conclude that the modified Avrami equation gives good results as far as the prediction of volume fractions after annealing is needed. Optimization of the laminar cooling and continuous annealing for DP steels is possible using a connection of this equation with the FE solution for the macro temperature field. In contrast, a multiscale model with microstructure taken explicitly into consideration is needed to analyze distributions of strains in the ferritic-martensitic microstructure. This approach is computationally very expensive. However, it is shown in this article that application of the statistically similar representative element allows to decrease the computing times even by more than an order of magnitude, while the accuracy and predictive capabilities remain almost unchanged.

The results of simulation of the cycle composed of cold rolling, continuous annealing, and stamping to manufacture parts for the automotive industry are presented in this article.