Modeling the thermomechanical effects on baking behavior of low carbon steels using response surface methodology
Introduction
The car industries require a low strength steel before press forming whereas a high strength one after paint-baking. The former provides a good formability as well as lower damage and cost regarding the dies, etc. whilst the latter results in reducing the car weight, saving in energy and increasing the dent resistance [1], [2], [3]. Bake hardenable steels are excellent choices to meet the mentioned requirements for automotive industries. In general, baking a material should lead to decreasing the strength or to bake softening. However, in case of bake hardening the situation is revered as it results in a higher strength after baking rather than lowering the strength. On the other hand, bake hardenability is a unique feature that should be provided by producing the bake hardenable steels. Regarding the hardening mechanism during baking, then the paint is baked at the temperature range of 100–250 °C [4], the dissolved carbon migrates to free dislocations introduced during the forming. Locking the dislocations result in significant increment in strength through the Cottrell atmosphere formation. Therefore, bake hardening takes advantages of strain aging when the interstitial atoms segregate on the dislocations. However, a significant portion of the solute carbon can also segregate on the grain boundaries during cooling followed by reheating which may substantially affect the subsequent baking behavior [5]. The bake hardening amount is determined by subtracting the flow stress after pre-strain from the final yield stress after baking (Fig. 1).
In general, in case of a process such as bake hardening, the experimental procedures are not only time consuming but also requires special care as many parameters are interacting simultaneously. Besides, the contribution of errors when measuring and controlling many parameters such as carbon content, annealing temperature, cooling rate after annealing, pre-strain percent, baking time, and baking temperature cannot be ignored. Thus, developing an appropriate model can be of significant interest to simulate and predict the parameters involved in a complex process such as pain-baking or bake hardening. Among the modeling approaches, RSM is a powerful technique in optimizing the industrial processes. Dehghani et al. [6] used successfully the RSM to model the baking behavior of Al7075. Naceur et al. [7] and Chirita [8] used RSM to design the sheet metal forming parameters involved in spring-back phenomenon. Also, Bahloul et al. [9] investigated the optimization of sheet metal bending using RSM. An appropriate process design for drilling the metal matrix composites was developed by Basavarajappa et al. [10].
Using RSM, one is able to find out the best combinations of process parameters when there are many factors interacting simultaneously. Besides, RSM is a promising method to lowering the cost and time involved in laboratory simulations when studying the industrial processes. Response surface methodology (RSM) can be very helpful in designing the experiments and to figure out the optimum correlations among the many variables of any industrial process. This method can be therefore used to investigate the responses of any industrial process and to determine the relation between the inputs and outputs of a process by developing a statistical model. In order to design the experiments for a statistical method, it is necessary first to estimate the parameters that exhibit significant influence on an industrial process such as bake hardening here. Then, the experimental procedure should be designed so that to take into account all the important parameters at several levels. This is followed by analyzing the experimental results using the analysis of variance (ANOVA) to determine which parameters show the strongest interactions and/or which ones exhibit significant influence on the outputs of the process. The process responses are then represented as statistical developed models. The developed models are then validated by testing at the optimum conditions predicted by RSM [11], [12], [13], [14]. Thus, RSM can be also used to design the experiments so as to optimize the process parameters. The other significant feature of RSM, is offering the contour plots. Using the contour plots, one is able to determine the significance of each parameter individually or in combination with other process parameters.
The aim of present work was to study the baking behavior (BH) of low carbon (LC) steels after employing various thermomechanical treatments. Response surface methodology based on central composite design (CCD) is then used for optimizing the baking behavior of these steels. This is followed by optimizing the variables of BH process to predict the mechanical properties of baked steels. Finally, using RSM, an appropriate experimental procedure was designed for bake hardening response of low carbon steels. Besides, two models were then developed to predict the BH amount and final yield stress (FYS) after baking the LC steels.
Section snippets
Material and thermomechanical treatments
The LC steels with the carbon content ranging from 0. 01 to 0.13 wt.% were used to optimize their bake hardening behaviors. Samples for laboratory tensile testing were taken from the cold-rolled LC sheets having the thickness of 0.7 mm. This a thickness normally used for car-body productions. Tensile samples were prepared from the steels sheets in the rolling direction according to ASTM-E8. All the samples were reheated to the annealing temperatures of 500, 600, 700, 800 and 900 °C in order to
Modeling by response surface methodology
To develop the RSM models for BH and FYS, three process variables were considered as cooling rate, annealing temperature and carbon content. The processing parameters were then coded according to the following equation:where xi is the coded value; Xi is the actual value of the ith test variable; is the value of Xi at the center point of the investigated area; and ΔXi is the step size.
To develop the models for BH and FYS, a set of experiments should be designed and carried out
Response surface of BH
Using the numerical approaches to develop the regression models, one can readily determine the optimum conditions for BH and FYS responses. The developed models for the bake hardenability (BH) and final yield stress (FYS) responses as the function of baking parameters are as follows:
The regression coefficients obtained by the model are summarized in Table 3.
Conclusions
RSM can be an excellent tool to design the experimental procedures and model when many parameters are interacting simultaneously in an industrial process such as bake hardening. Therefore, the experimental design based on RSM results in minimizing the number of tests to be performed. This is done by predicting the interactions among these parameters to determine the significance of each parameter as well as to predict the best combinations of these parameters to attain the maximum BH and final
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