A method for evaluating the formability of tailor rolled blank (TRB) by means of the forming limit margin field graph in forming process

Tailor rolled blank (TRB) with graded thickness has shown great potential in the automobile field. Using traditional forming limit diagrams (FLDs) to evaluate TRB formability is challenging due to thickness variations. In this paper, a 3D forming limit surface (FLS) considering the influence of thickness was obtained. A numerical model was developed to predict final strains. Moreover, a forming margin was denoted and calculated to generate the forming limit margin field graph for quantitative evaluation of the TRB formability. Results showed that as the punch travel increased, the forming margin value decreased. As the travel changed from 35.2 mm to 37.4 mm, the corresponding forming margin value changed from 0.002 to -0.024. The formability declined, and the specimen eventually cracked on the thinner side. Besides, the deformation and strain paths predicted by simulation agreed well with those measured from formed part, which indicated that the final strains used in formability evaluation were reliable. The method was suitable for quantitative evaluation of the formability and predicting the cracking position in TRB forming.


Introduction
With the increasing demand for high-performance materials in the automobile and aerospace industries, the application of lightweight materials such as aluminum alloys and high-strength steels has become more and more extensive. Advanced manufacturing technologies open up new possibilities for the application of lightweight materials [1][2][3]. Tailor rolled blank (TRB) with graded thickness obtained by rolling was regard as a potential lightweight metal sheet [4]. As shown in Fig. 1, the TRB rolling process was continuous. The constant thickness zone and thickness transition zone were made by adjusting the roll gap online according to actual needs. There was no abrupt change in thickness along the rolling direction, so that TRB could have good surface quality and excellent formability [5][6][7]. Due to the remarkable advantages of TRB, some well-known automotive manufacturers have introduced TRB in their products to reduce the vehicle weight [8][9][10]. The prediction of material formability has been a key requirement in the forming process of TRB applications. The formability prediction aimed to increase productivity by reducing the number of failures. To predict the cracking of TRB in forming process, criteria for the evaluation of TRB formability should be established. One way to evaluate the formability of a formed part was by thinning. In industrial applications, the formed part with a thinning rate of more than 15% was generally considered scrap.
However, the evaluation only through thinning was not sufficient, it was also inaccurate without considering the sheet material parameters. The forming limit diagram (FLD) worked as another effective limiting criterion to describe the deformation degree of materials without cracking [11,12]. It represented the largest major and minor strains that a blank could withstand before necking or cracking occurred in all possible combinations of strain paths during forming. Theoretically, the FLD used in TRB forming should contain an infinite number of forming limit curves (FLCs).
To obtain the FLCs, empirical methods and experiments could be used. Ghazanfari et al. [13] proposed an empirical law in terms of sheet thickness to determine FLD without experimental data. Abspoel et al. [14] derived predictive equations of the FLCs from the statistical relations between the measured FLC points and the mechanical properties. Kim et al. [15] used various constitutive models to predict the formability of high strength steels. Although the cost of experiments was higher than that of empirical methods, the experimental results were more realistic and direct. The Nakazima test was a standard experiment that provided the sheet formability information. Holmberg et al.
[16] developed a test method which can quickly determine the forming limit in plane. The tests were carried out in a tensile testing machine.
Previous studies on the formability of TRB were mainly focused on the forming limit in deep drawing process. Mayer et al. [8] used TRB to increase the maximum deep drawing depth. They evaluated the results through two-dimensional FLD of the sheet with constant thickness. Zhang et al. [9] discussed forming limit of TRB square box during the drawing process. The formability evaluation criteria they used were maximum drawing depth and thinning. These studies did not take into account the continuous thickness variation in the thickness transition zone. Moreover, there was little quantitative evaluation of formability in these studies. The thickness of TRB was continuously changing. The mechanical properties and formability of different thickness were different [17]. Quantitative assessment of the safety degree and the cracking degree can contribute to the design and production of TRB applications.
Therefore, it is necessary to establish a criterion by considering the thickness effect in the TRB forming process. The criterion should have the ability to evaluate the formability quantitatively.
In this paper, the forming limit margin field graph was proposed to quantify the formability and predict cracking. The method could be divided into four steps. Firstly, a 3D forming limit surface (FLS) was established based on the FLCs in the constant thicknesses zone. Secondly, uniaxial tensile tests and the Lagrange interpolation method were used to obtain the constitutive relation of the TRB. Thirdly, a numerical model was established to obtain major and minor strains of all elements. Finally, forming margins were calculated based on major strains, minor strains and the FLS.
The forming limit margin field graph was established and verified by the experiment.

3D forming limit surface
The FLD obtained by experiments or empirical methods has proven to be a good failure criterion. In order to better analyze the deformation process of TRB, the traditional 2D FLCs were replaced by the 3D FLS. The process of obtaining the FLS was divided into two steps. Firstly, the FLCs in the constant thicknesses zone were obtained by the forming limit test. Secondly, upon the basis of the FLCs, the FLS was constructed by polynomial fitting.
The FLCs of HC340LA under thicknesses of 1.2 mm, 1.4 mm, 1.6 mm and 1.8 mm were selected [17]. Nine types of geometrics in each group were applied to obtain different strain paths and the FLC. The forming limit tests combined with the 3D Digital Image Correlation (DIC) technology were carried out. The limit strains were determined by the combination of position dependent method and DIC technology. The FLCs were obtained by fitting.
To obtain the forming limit in the thickness transition zone, the polynomial interpolation method was adopted. The 3D FLS of TRB was obtained by fitting the four FLCs. In the method, the minor strain was defined as x, the thickness was defined as y and the major strain was defined as z. The relationship between the fitting result and the actual value can be defined by the following equation: where zfitting (x, y) represents fitted polynomial, and ε represents synthetic error. For polynomial fitting surface, the goodness of fit was used to evaluate the surface fitting accuracy. The coefficient of determination (R-square) and the root mean squared error relative error (RMSE) were important statistics to measure goodness of fit.
The fitted 3D FLS is shown in Fig. 2. The polynomial fitting coefficients of the 3D FLS are listed in Table 1. R-square = 0.998 and RMSE = 0.002564 indicate that the fitting result has a good accuracy. Thus, the 3D FLS could be used to determine the forming limit of any specific thickness in the TRB.

Forming margin field graph
Cracking of the formed part could be judged qualitatively by the traditional FLD, while the safety degree of the non-cracking area and the cracking degree of the crack area could not be evaluated quantitatively. In previous studies, Naceur et al. [18] defined a constant as the "safety margin", which could be expressed as the difference between the major strain of the formed element and the corresponding major strain on the "secure FLC". Wei et al. [19] proposed an efficient method for controlling the forming quality, in which the distance between the strain state point and the FLC was regard as one of the constraints in the process of the blank metal forming optimization.
In order to evaluate the formability of blank quantitatively, the forming margin was proposed in this paper. The forming margin refers to the formability of the blank after a certain degree of plastic deformation.
The minimum distance between the final strain state point of the element and the FLS at the same initial special thickness was defined as absolute value of the forming margin. In detail, the strain state and initial thickness could be represented as P0 (x0, y0, z0) in the coordinate space. The strain state and initial thickness on the FLS could be represented as P (x, y, z), where, x, y and z represent the minor strain, the thickness and the major strain respectively.
The thickness y0 was brought into the fitting equation of the FLS to obtain the FLC corresponding to the thickness y0. Then the FLS could be simplified as: P (x, y0, z) was a point on the FLC. The distance between the point P0 and the point P was the minimum distance between the point P0 and the FLC. The point P0 (x0, y0, z0) and the point P (x, y0, z) satisfied the following conditions with on the x-z plane: where kp is the tangent slope of the FLC at the point P; kPP0 is the slope of the line PP0.
Eq. (6) can be equivalent to the following equation: By combining Eqs. (5) and (7), the point P (x, y0, z) could be obtained. Then the shortest distance between the point P0 and the FLC could be calculated by the following Eq. (8), which was expressed as d.
The forming margin could be expressed as: zfitting(x0, y0) -z0 > 0 indicates that the strain state point is below the FLS. As shown in Fig. 3, P01 is below the FLS, P1 is on the FLS. The distance of line P1P01 is the minimum distance between P01 and the FLC corresponding to the thickness y01. The distance of line P1P01 is defined as the forming margin at the thickness y01. As the d decreases, the forming margin decreases and the formability of the blank decreases.
zfitting(x0, y0) -z0 < 0 indicates that the strain state point is above the FLS. As shown in Fig. 3, P02 is above the FLS; P2 is on the FLS. The distance of line P2P02 is the minimum distance between P02 and the FLC corresponding to the thickness y02. The opposite value of the distance is defined as the forming margin at thickness y02, which represents that cracking occurs during the forming process at this position. As the d increases, the -d decreases, which represents the forming margin decreases and the cracking degree of the blank increases.
zfitting(x0, y0) -z0 = 0 indicates that the strain state point is on the FLS, the blank is 8 in a critical state of cracking. Then the color gradient was used to characterize the difference in forming margin of each element on the part. The forming margin filed graph was generated. The calculation process for all elements on the part is shown in Fig. 4.

Fig. 4 Flow chart of forming margin calculation for all elements
In the forming margin field graph, the different margins of elements on the part were expressed in different colors. The same color gradient was applied to the strain state points of the corresponding elements in the 3D FLD. The 3D FLD was used together with the forming margin field graph for TRB forming. The combination of two was called the forming limit margin field graph. By the forming limit margin field graph, the formability of the TRB could be quantified and the cracking in the forming process could be predicted.

Materials
The investigated TRB was made of the HC340LA cold rolled steel. Some mechanical properties [17] were listed in Table 2. The thickness of the investigated TRB was 1.2/1.6 mm. properties should be considered in the different zones [20]. One way to obtain the mechanical properties in the thickness transition zone was by using the interpolation method based on the uniaxial tensile data in the constant thickness zone [9,21].
The mechanical properties in the constant thickness zone were obtained by uniaxial tensile tests [17]. The true stress-true strain curves in the constant thickness zone were shown in Fig. 5(a). It was seen that the mechanical properties of two thicknesses had difference. The flow stress of the blank with the thickness of 1.2 mm was higher than the flow stress of the blank with the thickness of 1.6 mm. To obtain the mechanical properties in the thickness transition zone, the Lagrange polynomial interpolation method was adopted based upon the mechanical properties in the constant thickness zone. Lagrange interpolation polynomial can be expressed as [22]: where li(x) is Lagrange primary function, it can be expressed by following equation: li(x) is an n-degree polynomial and has the following rule: l0 (

Numerical modeling
The TRB bulging model was developed by using the HyperMesh, Dynaform and Matlab software. The schematic of TRB bulging modeling is shown in Fig. 6. The blank was modeled using four-node quadrilateral Belytschko-Tsay shell elements, which had 5 integration points in the thickness direction. In the model, the keyword *ELEMENT_SHELL_THICKNESS was used to change the thickness of the shell elements at the four nodes. As shown in Fig. 8, T1 -T4 represented the thicknesses at nodes N1 -N4, respectively. For elements in the constant thickness zone, the thicknesses were same at the four nodes (T1 = T2 = T3 = T4). For elements in the thickness transition zone, the interpolation method was employed to obtain the thicknesses according to the coordinate of the nodes (T1 = T2, T3 = T4). Each part was assigned the corresponding mechanical properties (see Fig. 5).

Fig. 6 Schematic of bulging modeling
In an ideal model, the thickness transition zone should be discretized into countless parts, the element size would be small. However, the small element size would result in high computational cost and numerical instability [21]. It was found that an element size of 4 mm × 4 mm was sufficient in the thickness transition zone, while the thickness transition zone was discretized into 10 parts. Within the constant thickness zone, the 1.6 mm zone and the 1.

TRB bulging experiment
The TRB bulging experiment was performed on the setup shown in Fig. 7. The TRB bulging experiments combined with the 3D DIC technology were carried out [23,24]. The axisymmetric setup consisted of a pair of cameras, lights, a laser generator, a die, a holder and a hemispherical punch. The inner diameter of the die and the blank holder were both 105 mm, the punch diameter was 100 mm.

Simulation model verification
The cracking position and shape of the formed specimens were used as two simple indexes to evaluate the accuracy of the model. The comparison between the simulation result and the experimental result is illustrated in Fig. 9. The cracking position of the TRB specimen was on the thinner side in the experiment in the simulation, the most severe thinning area was found on the thinner side. As shown in the Fig. 9, the red area (marked area in Fig. 9) was the most severely thinned area in the blank. In addition, the deformation shape of the formed specimen was basically the same in the experimental and simulation result. Therefore, it was initially indicated that the simulation model was accurate.

Fig. 9 Deformation comparison of experimental and simulation result
To prove the accuracy of the final strain distributions, the strain paths at three typical points were obtained by the simulation and the experiment, respectively. The 14 first point was near the edge of cracking. The other two points and the first point were on a line. The distance between the points was 8 mm. A comparison was made and the results are shown in Fig. 10. The strain paths at the P1 and P2 showed that the deviation between the major strains obtained by the simulation and those obtained through the experiment was large when the minor strain was small. Then with the increase of the minor strain, the deviation decreased gradually and coincided in a certain position. After that, with the increase of the minor strain, the deviation increased. At the final minor strain, the deviation between the final major strain obtained by the simulation and that obtained through the experiment reached 9.97%, 5.86%, respectively. At the P3, the deviation between the major strain obtained by the simulation and that obtained through the experiment was 3.75% at the final minor strain. In the calculation of the forming limit margin, the final strains were used. Therefore, it was concluded that final strains obtained by simulation could be used. In summary, the deformation of the specimens and the strain paths at the three typical points in simulation were proved by the experiment. Thus, the final major and minor strains could be used in the forming limit margin field graph to evaluate the formability of the TRB.

Quantitative evaluation of the formability before cracking
Three forming limit margin field graphs under three different punch travels of 31.9 mm, 33.0 mm and 34.1 mm were established. As shown in Fig. 11, the 3D FLDs are on the left, and the forming margin field graphs are on the right.
In the 3D FLDs, as the punch travel increased, the strain state points were closer to the FLS. All strain state points were below the FLS. In the forming margin field graph, the forming margin distributions on the formed TRB part can be obtained intuitively. According to the cloud ruler, the minimum forming margin was 0.030 when the punch travel was 31.9 mm. When the punch travel was 33.0 mm, the minimum forming margin was 0.017. When the punch travel reached 34.1 mm, the minimum forming margin changed to 0.001. The minimum forming margins were positive, which indicated that the formed TRB specimen was not cracked. As the punch travel increased, the minimum forming margin decreased, indicating that the formability of the TRB specimen decreased. When the punch travel was 35.2 mm in the simulation, the forming limit margin field graph can be seen from Fig. 12(a). In the 3D FLD, all strain state points were below the FLS. Some of the strain state points near the FLS were displayed in red. The thickness coordinates of these red strain state points were within the range of 1.2 mm to 1.4 mm. In the forming margin field graph, the forming margin distributions on the formed part can be obtained intuitively. According to the cloud ruler, the forming margins of the red area were between 0.002 and 0.037, which indicated that the formed specimen was not cracked but the ability of material in the red area to continue to deform was insufficient. The red area was located on the thinner side.
When the punch travel reached 36.3 mm in the simulation, the forming limit margin field graph was shown in Fig. 12(b). In the 3D FLD, some strain state points marked red were located above the FLS. This indicated a cracking of the TRB specimen.
The thickness coordinates of these strain state points were between 1.2 mm and 1.4 mm.
In the forming margin field graph, negative values appeared in the forming margins of all elements. The element with the minimum margin was on the thinner side. The minimum margin value was -0.012, which meant that the formed specimen had cracked.
Some elements near the draw bead were marked red (see in Fig. 12(b)), but their margin values were greater than zero, and there was no cracking.
When the punch continued to move, the travel reached 37.4 mm in the simulation.
FLD, there were more strain state points marked red above the FLS. In the forming margin field graph. The minimum margin value was -0.024. The absolute value of the minimum margin became greater than its value at the 36.3 mm punch travel. The TRB part cracked more seriously. The cracking position was on the thinner side. Similarly, the elements marked red near the draw bead did not crack.  Table 3. As the punch travel increased, the minimum forming margin changed from 0.002, to -0.012, and to -0.024. The minimum forming margin value changed from positive to negative, and the absolute value of the forming margin increased. The number of cracking elements was increased from 0, to 8, and to 26. These results indicated that with the increase of the punch travel, the cracking degree increased.  [25]. The forming limit on the thinner side was smaller than that on the thicker side. Under the same load, the stretching stress on the thinner side is greater than that on the thicker side. As a result, the thickness thinning of the thinner side was severe, and cracking eventually occurred.

Conclusions
In this paper, the influence of blank thickness on the TRB constitutive relation and formability was considered. A calculation method of the forming margin was proposed to quantify the formability. The forming limit margin field graph of TRB was established. The major conclusions could be drawn as follows: (1) The deformation and strain paths predicted by simulation agreed well with that measured from experiments results, which indicated that the simulation model was reliable. The constitutive relation of the TRB established by means of the combination of the uniaxial tensile tests and the Lagrange polynomial interpolation method was credible.
(2) The forming margin was a quantification of the formability. During the TRB bulging process, the punch travel changed from 35.2mm, to 36.3mm, and to 37.4mm, the minimum forming margin value changed from 0.002, to -0.012, and to -0.024. The TRB could continue to deform until TRB cracked, and eventually the cracking became more serious. The cracking occurred on the thinner side.
(3) The simulation and experiments proved the forming limit margin field graph was efficient. It was suitable for studying the formability of TRB and the prediction of cracking in the forming processes. The 3D FLS obtained based on the forming limit tests and polynomial fitting was reasonable.
coupled design method for hot-stamped tailor rolled blank structure.      Table 2 Some mechanical properties of tested TRB Table 3 Forming margins and the number of cracking elements of three punch travels