A New Method for Determining Necking of Sheet Metal Based on Main Strain Topography

Abstract

There are various methods to evaluate the forming limit of a sheet, and these criteria can be classified as position-dependent, time-dependent, and position-time dependent according to the basis of judgment. However, these criteria have a single function and can only find the forming limit of the sheet and cannot determine the strain distribution, strain change, or fracture location during the sheet forming process. This paper introduces a time–location-dependent method, i.e., the spatial strain rate method, which is used to detect the onset of necking of a sheet. The spatial strain rate is directly based on the strain and can not only find the forming limit of the sheet but also depict the strain distribution and strain variation during the two phases of the experimental process—distributed instability and concentrated instability—as well as predict the location of sheet fracture. The spatial strain rate of AA5083 aluminum alloy of different widths was analyzed and verified in detail via Nakazima experiments using digital image correlation techniques and compared with the guidelines published in the literature in recent years.

1. Introduction

Formability assessment of metal sheets is important for optimal design and development of advanced forming technologies. Necking is the most common failure phenomenon in the forming and manufacturing of sheet metal [1]. Before 2008, there was no clear method to determine the forming limit of metal materials [2]. Due to demand from the industry, a generalized standard was developed internationally to guide industry production: ISO 12004-2:2008 [3]. This method has too many limitations and has not been unanimously accepted internationally. Eberle et al. [4] proposed a method for determining the change in strain with time; this method improves the sensitivity of the measurement, but it is still in the standardization stage. Based on this situation, researchers in various countries have proposed their own guidelines for experimental determination, but none of them have found a universal method for determining a material’s forming limit diagram (FLD). How to accurately determine the molding limit of a material has developed into a hot issue in today’s research. Hill [5] pioneered the criterion of local necking under planar stress conditions. This criterion was applied to obtain the left side of the FLC. For the right side, Swift [6] proposed that instability occurs when the principal stress reaches a maximum and predicted the critical strain for diffuse necking. The most widely used analytical tool is the Marciniak–Kuczynski (M-K) model [7]. This is based on the hypothetical theory that necking is caused by the initial defect. In biaxial tension, the onset of necking is associated with the establishment of a local plane strain state at the defect perpendicular to the principal strain direction. Later, Hutchinson and Neale [8] improved the M-K model to cover the entire range of FLCs. Eyckens et al. [9,10] extended the M-K model to include full-thickness shear (TTS) for isotropic and anisotropic metal sheets. With the rapid development of optical and digital imaging techniques, time-based judgment criteria have emerged. For example, Merklein et al. [11] developed a criterion involving the principal strain rates away from and close to the region of damage and the region. Situ et al. [12] developed a method to analyze the time evolution of the principal strain and its first and second time derivatives (called the strain rate and strain acceleration) at a point in the fracture region. A.J. Martínez-Donaire [13,14] proposed a new method based on the time location: the “flat valley method” [15]. This is based on the direct visualization and analysis of the displacement of the outer surface of the sheet during necking development. Lin et al. [16] developed a new biaxial test system using a linkage mechanism capable of transferring the input uniaxial force to the output biaxial force of the tensile cross specimen. Shao et al. [17,18] used this new biaxial tensile test system in an attempt to study a 1.5 mm thick AA6082 sheet and the ultimate pressure difference under hot stamping conditions. Zhang et al. [19,20] successfully applied the digital image correlation (DIC) technique to strain field measurements. Some scholars used the DIC technique to measure the primary and secondary strains based on the volume conservation assumption to derive the sheet thinning [21,22,23]. With the rapid development of the finite element method (FEM), some scholars have carried out research evaluating the formability of metals using the FEM method. However, Zhang et al. [24] concluded that the finite element method was effective in evaluating the formability of metals, but the results were in poor agreement with experimental results. Heidari et al. [25] also found that the simulation results showed large deviations on the right side of the FLD when the formability of alloy AA6063 was evaluated using the ductile fracture criterion. Marrapu et al. [26] used numerical and experimental methods to evaluate the formability of DP780 steel; the results showed that the simulation results of the strain localization criterion in the principal strain gradient method [27] were in good agreement with the experimental results. Lumelskyj et al. [28] performed Nakajima tests on DC04 steel to verify the accuracy of the numerical results under two different necking criteria. Pavel Hora et al. [29] proposed a new method to determine necking and fracture failure that can be directed to describe the localization of the sheet. This method is now designated as the localized horizontal forming limit diagram (LL-FLD). Zhaoxuan Hou and Zhigang Liu [30] further studied each anisotropy on the forming limit and found that the forming limit curve in the rolling direction was much lower than that in the transverse direction. Xiangrui et al. [31] compared the AA5086 FLC under four different necking criteria in their study. It was found that the FLCs of the obtained materials were very different for different criteria. Han Fei et al. [32] review and analyze the research progress in forming limit theory and experiments, and show that the forming limit criterion for complex loading paths is more convenient and practical. Junying Min et al. [33] used a new experimental/theoretical approach to generate stress-based forming limit diagrams for aluminum alloy 5182-O for a two-stage forming technique with an intermediate annealing step. In order to more accurately capture the onset of localized necking and to obtain the necking limit strain, Min, J. et al. [34] proposed a method to detect the onset of localized necking using curvature. K. Yu et al. [35] proposed an improved M-K method based on pre-assigned and non-evolving notch orientation for the constant orientation of the defective deformation band during stretching. Z. Wang et al. [36] conducted in-plane biaxial tensile tests for the first time using two different cruciform specimens in order to construct the FLC from shear to isobiaxial tension of AA6061-T4 sheets. Ruiqiang Zhanga et al. [37] proposed a new method in time and space for measuring fracture and necking failure of materials. The failure analysis of different material sheets using this new method was found to be simpler, more stable and more accurate in determining the local necking strain.

The criteria for judging necking proposed by scholars vary, but in general, they can be classified into two categories: time-based criteria and location-based criteria.

• Time-based criteria are based on the characteristics of the change in a physical quantity over time to determine the necking of the sheet, such as strain rate criteria, strain path criteria, etc. These criteria are based on the selection of several stages in the sheet and the comparison of the changes in stress or strain at these stages. These methods are highly sensitive, but selecting a large number of stage points and processing the data for comparison are not only complex and time-consuming, but they also neglect the distance between points and cut the continuity of the sheet and data, which will cause certain errors.
• Position-based criteria are based on a physical quantity on the change in position characteristics to determine sheet necking, such as ISO, the flat valley method and surface curvature. In this type of criterion, the ISO criterion is used for fractures before the frame frequency part of the data for secondary fitting, taking the maximum value as the limit strain. This approach is very controversial because necking failure occurs prior to fracture, but there is no clear boundary between the two. Although fitting the value with the fracture frame frequency data ensures that the sheet does not fracture, it is not reasonable and is a questionable method to determine the necking failure of the sheet. With the exception of the ISO guidelines, the rest of the position-dependent guidelines are based on the occurrence of depressions on the surface form of the sheet to determine necking; these methods are physically meaningful, but are poorly sensitive and only applicable to softer materials where necking is evident.
• Time- and position-related methods: These methods are both time- and position-related and combine the advantages of both time and position judgment criteria, and the results are more reliable. The above three types of judgment criteria have their own advantages and disadvantages, but they have a single function, can only determine the forming limit of the sheet under specific circumstances, and cannot describe the strain distribution, strain change or sheet fracture location of a sheet during the experimental process.

Based on the above analysis, this paper proposes a spatial strain rate method for detecting the beginning of the necking of a plate. The spatial strain rate is directly based on the strain, which can not only be used to determine the forming limit of the plate, but also to depict the strain distribution and strain change in the plate during the two stages of dispersive instability and concentrated instability during the experiment and to predict the location of the plate fracture.

2. Methods

2.1. Basis of the Main-Strain-Based Shape Judgment

The inspiration for the method based on the principal strain topography comes from the “flat valley method” [15] and the “surface curvature” method. These two methods are based on the surface depression of the sheet when necking occurs, where the ultimate strain of the sheet failure is determined by the variation in the height with the length of the selected stage line or the area of the surface. In the course of the experiment, we found that the principle of using the valley method and surface curvature method is applicable to smaller-radius and softer materials, and that the valley method and surface curvature method are not applicable under high hardness and large radius test conditions. In order to solve the above problems, a judgment method was proposed based on the main strain morphology.

The principal strain topography, as the name implies, is an image of the principal strain of the sheet, which is formed by drawing a stage line at an appropriate position on the sheet and extracting the principal strain at each frame of the stage line with the length of the stage line. Figure 1a shows the evolution of the principal strain deformation over time for the stage line selected from the vertical damage zone in the Nakazima test (punch radius of 50 mm), where the curve in the figure represents the magnitude of the principal strain at each moment on the stage line, which is referred to here as the principal strain topography. Figure 1b shows a three-dimensional view of the evolution of the principal strain topography over time projected in the x-z plane, so that the changes in the principal strain topography at different moments can be more clearly observed. The principal strain increases uniformly over time for the entire section at the beginning of the experiment, resulting in a principal strain topography very close to the convex die topography. After reaching a certain point, the principal strain increases rapidly in a certain region. Figure 2a is identical to Figure 1 in terms of the experimental conditions and data selection, but the curves in the figure represent the evolution of the principal strains with time at different points on the stage line. Figure 2b shows a three-dimensional view of the evolution of the principal strains at different points with time projected into the y-z plane, so that an image of the evolution of the principal strains at each point with time can be obtained. It is worth noting that the changes in the principal strains at different points at different locations show different characteristics: the principal strains at points near the edge of the stage line increase slowly at the beginning of the experiment and increase very slowly or not at all before sheet fracture at the later stage of the experiment, while the principal strains at points near the main strain at the point near the center of the stage line increase at a high rate at the beginning of the experiment and increase very fast before sheet fracture at the end of the experiment.

The main strain morphology is analyzed separately as shown in Figure 3. Each frame represents the main strain at a different position at the corresponding time during the experiment; each frame together constitutes an image of the strain of the sheet at all times during the experiment with respect to the position, so that not only the main strain magnitude and state changes between positions on the stage line of the sheet but also the strain magnitude and state changes at different times at the same position can be seen from the image of the main strain topography. This makes it possible to find the limit strain at which the sheet fails based on the characteristics of the principal strain with position and time. The FLC curve is composed of the failure limit strains under different stresses, and it is not only convenient and simple to use the main strain as the failure criterion, but also more intuitive and reasonable.

Based on the above analysis, a method was developed to determine the forming limit of the sheet based on the main strain topography: the spatial strain rate determination method.

2.2. Spatial Strain Rate Judgment Criterion Based on the Main Strain Topography

Although the main strain topography can indicate the change in strain in the sheet with respect to position and time, it is not possible to obtain a precise quantitative description by visual observations alone to accurately determine the forming limit. To solve these problems, we propose the spatial strain rate method. The spatial strain rate is a first-order derivative of strain with respect to position, and the change in strain with respect to position is described by the change in the derivative of the same frame frequency. The change in strain with respect to time is described by the change in the derivative of different frame frequencies. The principles of the spatial strain rate are described below.

According to the characteristics of the main strain topography and the evolution of the main strain at each point with time, as described in Section 2.1, the sheet goes through two stages during the experiment: the dispersive instability stage and the aggregated instability stage. As shown in Figure 4, part a is the early stage of the experiment when the sheet deformation is small, and the strain cloud is light in color and the distributions of different colors str intertwined and are not easy to distinguish. This indicates that the overall strain of the sheet belongs to a low level and the strain size at this time shows an irregular distribution with large fluctuations, which indicates the dispersive instability stage. Because the middle part of the sheet is in direct contact with the steel punches, the strain is higher but the strain difference with the edge is not particularly large and the strain transition between the two areas is smooth. The phase line topography image at this time is shown in Figure 5a as a circular arc with a slightly higher middle, lower sides and a small change in slope. Because the overall strain of the sheet in the dispersive instability phase is low and does not change much, the whole area of the sheet in this phase is classified as a low-strain zone. At this point, the derivative of the main strain with respect to position is found to find the rate of change in the main strain with respect to position. Figure 5b shows what we call the spatial strain rate. Because the strain difference between the middle part and the edge part of the sheet is not particularly large, the strain transition is smooth, so the spatial strain gradually increases from the edge position to the maximum value at this time. As the strain enters the middle part of the sheet, the strain increase in this region decreases further, which also leads to a gradual decrease in the spatial strain rate, which decreases to 0 when the strain in the phase line reaches its maximum. Since the main strain topography of the stage line is axisymmetric, the other half of the spatial strain rate image is centrosymmetric, with the first half of the spatial strain rate image about the point where the strain rate is 0.

At the later stage of the experiment, the strain distribution on the plates is hierarchical, and the strain maps show different colors and are easily distinguishable, as shown in Figure 4b. This indicates that the overall strain of the sheet is at a high level, and the strain magnitude shows a regionalized distribution, with less strain fluctuations in each region but huge strain variations in the over-region from region to region, which indicates the aggregated instability stage. The sheet enters the aggregated instability phase, which will occur in two regions: the aggregated instability region and the non-aggregated instability region. At this time, the strain in each region will show different characteristics: the strain increase in the non-aggregative instability region will gradually slow down after the sheet reaches the aggregative instability stage and finally stabilize in a certain range, as shown in Figure 2b—the strain variation characteristics of part A with time. Part C in Figure 2 is located in the aggregated instability region. The strain in this region increases with time in both dispersive and aggregated instability regions, but the strain in this region increases sharply when it enters the aggregated instability phase. Part B in Figure 2 is the transition zone between the two regions, and the strain at this location has a increase rate between that of Part A and C after entering the aggregated instability phase. The strain level of the sheet is divided into the following three regions according to the characteristics of strain changes in different regions during the aggregative instability phase, as shown in Figure 4b:

• A low-strain region—all of this region is located within the dispersive instability region, where the overall degree of strain is small and the strain levels within the region are similar and less variable.
• The medium-strain region, which is partly located in the aggregated instability region and partly located in the non-aggregated instability region and has a medium overall strain level, but the strain level in this region exhibits large differences and large variations.
• High-strain region—all of this region is located within the aggregated instability region, where the degree of strain is greater overall, but the strain levels within the region are similar and less variable.

At this time, the stage line morphology image shown in Figure 5a (C) is a bell shape with a high middle, low sides and a large change in slope, which corresponds to three regions of high strain, medium strain and low strain. The spatial strain rate changes in each of the three strain regions are characterized as follows: The low-strain region has a low degree of strain and a slow change, and the spatial strain rate gradually increases with a gentle slope from the edge position, as shown in Figure 5b (C)—the part located in the yellow region. The strain level is highest in the high-strain region, but the strain increase at adjacent locations gradually decreases, and the spatial strain rate gradually decreases from the maximum to 0 in this region, as shown in Figure 5b (C)—the part located in the red region. The first half of the graph corresponding to the spatial strain rate described above is centrosymmetric with the first half of the spatial strain rate image about the point where the strain rate is 0 because the main strain morphology of the phase line is axisymmetric.

Figure 5b shows a schematic diagram of the spatial strain rate at different moments. Based on this characteristic variation in the spatial strain rate, it is possible to not only quantify the variation in the width of the strain concentration region from the beginning to the end of the experiment, but also to determine the location of the fracture of the sheet.

This leads to a jump in the width of the spatial strain rate between the two frame frequencies. The maximum strain at the jumped frame frequency is considered to be the forming limit of the sheet.

3. Experimental Procedure

3.1. Material and Method

In this study, a thin sheet of domestic 5083 aluminum alloy with a thickness of 1.5 mm was used as the experimental material. At present, there are two mainstream methods for sheet metal forming limit measurements: one is the Nakazima [38] experiment, as shown in Figure 6, and the other is the Marciniak [7] experiment. The Marciniak test avoids the friction effect and bending effect to obtain the pure expansion experimental state; however, this state is too ideal for actual processing, and the FLC of the material determined with this experimental method is often too large. In this experiment, the more widely used Nakazima test combined with the PMLAB DIC-3D measurement system was chosen. In the determination of the material forming limit, three types of stresses were simulated: unidirectional tension, plane strain and equal double tension. In this experiment, five widths of 20 mm, 60 mm, 100 mm, 120 mm and 180 mm were used according to ISO 12004-2:2008 [3] to simulate the unidirectional tension, plane strain and equal double tension. Laser cutting was used to process the formation; the shape, size and position of the sheet in the forming limit are shown in Figure 7.

The experimental equipment used in this experiment consists of a PMLAB DIC-3D three-dimensional strain measurement system, jointly developed by Nanjing Zhongxun Micro (PMLAB) Sensing Technology Company and the University of Science and Technology of China and Southeast University; the BCS-30A [30] sheet formability limit testing machine; and two sets of experimental equipment developed by Beijing University of Aeronautics and Astronautics.

PMLAB DIC-3D uses non-contact optical measurements to measure the three-dimensional deformation of the surface of a specimen. In the experiment, a specimen was mounted on the upper end of the BCS-30A sheet formability limit tester, as shown in Figure 8. The displacement and deformation of the specimen at each point of the surface during the deformation were recorded by two high-speed cameras. The displacement and strain data of the sheet were analyzed and calculated in PMLAB system software (PMLAB 2017 V1.5.02). The whole measurement process does not require the installation of sensor components or human calculations, avoiding the human measurement error caused by using conventional methods.

Figure 8 (bottom) shown the BCS-30A sheet formability limit tester. It is suitable for forming property tests of ordinary steel sheets, ultra-high-strength steel sheets, aluminum alloys, titanium alloys, aluminum–lithium alloys, magnesium alloys and other sheets at room temperature and high temperatures.

3.2. Preparation of the Experimental Measurement System

The light source of the DIC and the exposure of the industrial camera were adjusted according to the experimental requirements, and different specifications of the calibration sheet were selected for equipment calibration. The significance of calibration is that the system performs a comparison calculation according to the reference after the experiment. In this experiment, the BCS-30A sheet formability limit testing machine was used for the expansion experiment; according to the properties of the experimental material—5083 aluminum alloy—the crimping force was set to 30 t, the punch speed was set to 15 mm/min, and the shooting frequency was set to 15 frames per second.

Scatter preparation is crucial in DIC data acquisition, and commonly used methods include laser printing, the galvanic corrosion grid method, transferring, printing, and paint or ink spraying. In this experiment, the paint spraying method was used by first spraying white primer and then spraying black spots. The main point of spraying is to master the control of the size, density and shape of the spots by pressure and spraying distances. The spot size was controlled at 10 pixels, as shown in Figure 9.

During the experiment, friction between the sheet and the punch will affect the force on the sheet, and if the force is not uniform, this will cause the sheet to break at an offset position and affect the accuracy of the experimental results. In this experiment, we further improved the lubrication method provided by ISO 12004-2:2008 [3] by using two layers of polyurethane, wrapping PTFE film around each polyurethane sample and brushing oil between the film and polyurethane for lubrication. The sheets after the special lubrication experiment are shown in Figure 9, and the cracks are all in the center of the expansion, which meets the experimental requirements.

3.3. Acquisition of Experimental Data

This experiment is based on the DIC experimental method, and the real-time stresses of the sheets will be displayed as strain clouds in the PMLAB DIC-3D computer software (PMLAB DIC-3D 2017 V1.5.02) at the end of the experiment. Figure 10 shows the strain clouds of the sheet from the beginning of the experiment to the fracture of the sheet in the Nakazima experiment for three states: approximate uniaxial tension (a), plane strain (b) and biaxial tension (c). It is clear from the figure that the sheets undergo three stages of dispersion destabilization, aggregation destabilization and fracture during the experiment.

The experimental data extraction method is shown in Figure 11, where a stage line is drawn in the middle of the stress cloud, and the length of the stage line runs through the whole strain cloud. All the stage line principal strain data were extracted and the principal strain morphology was plotted, as shown (partially) on the left in Figure 11 (left), showing the evolution of the principal strain topography with time for the three stress states: uniaxial tension (a), plane strain (c) and biaxial tension (e). On the right is the evolution of the principal strain topography with time for the three stress states corresponding to the spatial strain rate.

4. Results and Discussion

According to the experimental method and data extraction method described in Section 3, the experiments and data extraction were performed for 5083 sheets of different widths, and the corresponding principal strain morphologies were drawn, as shown in Figure 11 (left) for some of the 20 mm, 60 mm and 180 mm wide sheets. The corresponding spatial strain rate images are shown in Figure 11 (right) (for the sake of visibility, we present a selection of frame rates as an example). Based on the spatial strain rate characteristics corresponding to the dispersive instability, the aggregated instability and the necking phase, the frame frequency corresponding to the necking moment can be found and then the ultimate strain of the sheet can be found. The main strain topography of the sheet with a width of 20 mm shown in Figure 11 shows that there is a clear gap between the main strain topography images of 384 and 385 frames with adjacent frame frequencies, and to confirm and facilitate the observation of this phenomenon, we derived the corresponding spatial strain rate from the main strain topography. In the spatial strain rate image, it is easier to observe a clear gap between frames 384 and 385, which implies a sudden increase in sheet strain at frame 384. This is caused by the transition of the sheet from the aggregative instability phase to the necking phase, whereby the maximum strain at the frame frequency is the ultimate strain of the sheet. The main strain topography and spatial strain rate of the sheet at a width of 60 mm show a clear gap between the images of adjacent frame frequencies at frames 492 and 493, which means that the sheet strain suddenly increases at frame 492 and the sheet transitions from the aggregated instability phase to the necking phase. The main strain topography and spatial strain rate of the sheet at a width of 180 mm show a clear gap between the images of adjacent frame frequencies at frames 583 and 584, which means that the sheet strain suddenly increases at frame 583 and the sheet jumps from the aggregated instability phase to the necking phase. Based on the above analysis, the forming limit of the 5083 sheet is found.

In order to verify the reliability of the spatial strain rate method for determining the necking of the sheets, the T-D method based on time and the flat valley method based on position were additionally used to predict the necking failure of sheets of 5083 aluminum alloy as reference values, as shown in

Table 1. Forming limit data by different methods.

Method		Space Strain Rate Method		T-D Method		Flat Valley Method
Sheet Width	Number of Experiments	Minor Strain	Major Strain	Minor Strain	Major Strain	Minor Strain	Major Strain
wide20	1	0.036476539	0.124505529	−0.011369701	0.12424656	0.011201164	0.1171638
	2	−0.04345199	0.149014732	−0.045819861	0.167106059	0.039860501	0.134659137
	3	−0.03737116	0.159841732	−0.03979905	0.171893834	−0.03905048	0.168062695
	average	0.039099896	0.144453997	−0.032329537	0.154415484	0.030037382	0.139961878
wide60	1	−0.01434684	0.153256637	−0.015213876	0.152603543	−0.01483906	0.148081124
	2	0.014767866	0.154879479	−0.013910533	0.14799986	0.012004909	0.132480289
	3	0.012868338	0.149245275	−0.01301867	0.153391101	0.011847663	0.141863957
	average	0.013994348	0.152460463	−0.014047693	0.151331501	0.012897211	0.140808457
wide100	1	0.015135523	0.144302924	0.015028344	0.142782338	0.015228287	0.142116574
	2	0.015835353	0.138690188	0.015312571	0.139975283	0.015674182	0.116654937
	3	0.018578544	0.136522113	0.01895961	0.134384487	0.019170094	0.134453057
	average	0.016516473	0.139838408	0.016433508	0.139047369	0.016690854	0.131074856
wide120	1	0.029633455	0.129931186	0.033115859	0.127610476	0.031916686	0.110900275
	2	0.029549416	0.1294918	0.034916783	0.13608088	0.032045297	0.101176126
	3	0.028705891	0.146314005	0.028729853	0.149846722	0.02495014	0.117210857
	average	0.029296254	0.135245664	0.032254165	0.137846026	0.029637375	0.109762419
wide180	1	0.187692871	0.215057325	0.198686267	0.231029141	0.176226649	0.199157627
	2	0.205491423	0.222483037	0.235601357	0.254189705	0.194409041	0.204843259
	3	0.160965112	0.188399407	0.17162379	0.202850342	0.113011632	0.123065819
	average	0.184716469	0.20864659	0.201970472	0.229356396	0.161215774	0.175688902

. From Figure 12, we can see that the spatial strain rate method is similar to the T-D and flat valley methods, and both show a small variation in their results. The maximum difference for the T-D method is found to be in the range of 0.69%–3.17%, and the maximum difference for the flat valley method is found to be in the range of 0.207%–6.63%.

Figure 10 shows the strain clouds from the beginning of the experiment to the frame frequency of sheet fracture for the three approximate states of uniaxial tension (a), plane strain (b) and biaxial tension (c) in the Nakazima test, from which the strain distribution in the sheet during the experiment can be seen. The strains are widely distributed and dispersed at the beginning for the 20 mm and 60 mm diameter sheets, and higher strains are concentrated in the middle area of the sheet, which is about the same width as the contact area between the sheet and the die. As the experiment proceeds, the distribution of strain in the sheet is further reduced, the width of the high strain zone is further reduced to become concentrated in a narrow strip, and fracture occurs in this region. The strains in the 180 mm wide sheet were mainly distributed in a stepwise manner in the middle part of the sheet from the beginning of the experiment, and this area was highly coincident with the contact area between the sheet and the punch. As the experiment proceeded, the strains in the sheets were further increased, but the distribution area remained the same as the contact area between the sheets and the punch, and there was no obvious narrow strip of high strains as was observed in the 20 mm and 60 mm sheets until the frame before fracture. However, there was a sudden collapse in a circular high-strain area, which is closely related to the stress state of the sheets. The phenomenon shown in Figure 10 can be better illustrated in the main strain graph shown in Figure 11a. The strains of the sheets under the three stresses are uniformly distributed throughout the contact area between the sheet and the convex die in the early stage of the experiment, while a narrow convex peak appears in the main strain topography in the later stage and fractures at the peak. The whole process of the evolution of the main strain topography with time is shown in Figure 1a as a three-dimensional figure and Figure 1b as a two-dimensional figure, and this phenomenon can be easily detected by comparing the two figures.

In addition, we can also observe in Figure 13 that the width of the aggregated instability region decreases from 14.94 mm to 6.87 mm when the sheet fractures during the whole phase from the beginning of aggregated instability to the fracture of the sheet. From the variation in the width of the zone of aggregative instability, it can be concluded that the sheet does not fracture immediately after the onset of aggregative instability, but further intensifies the strain concentration. The width of the aggregated instability region can describe this strain intensification process; the greater the width, the smaller the degree of strain exacerbation. Although failure has occurred, at this time, there is almost no change in the appearance of the sheet and it is difficult to distinguish necking failure. More fracture will not occur. The smaller the width, the greater the strain increase and the more likely the sheet is to neck or fracture and fail. This evolution of the aggregative instability phase also leads to the inability to derive accurate ultimate strains for some methods that rely on the observation of surface features of the sheets to determine necking, such as the flat valley method, the 3D curvature method, etc.

The spatial strain rate is directly based on the strain as a judgment of the physical meaning. This not only makes it easier to identify the frame frequency of the occurrence of aggregated instability, but also can depict the change in the width of the aggregated instability region. The change in width proves that aggregation instability is a gradual process and does not occur overnight. Different degrees of aggregation instability can be depicted according to the width, which is not achievable with other methods. In addition, the T-D method and the flat valley method are local methods that can be applied in the case of less data, but their operation is complicated, they have poor continuity, and cannot be adapted to disadvantages. ISO 12004-2:2008 [3] is the method applied most commonly, and this method cannot be applied when there are less data or the results are highly inaccurate. Both methods have their advantages and disadvantages, and the newly proposed spatial strain rate has the advantages of both the local criterion and the overall criterion while avoiding both their disadvantages. It has the advantages of good continuity, high sensitivity, a clear physical meaning and simple operation, and can not only determine the forming limit simply and accurately, but can also depict the width and strain variation characteristics of the dispersed instability and aggregated instability regions, as well as the locate sheet fracture locations with a huge amount of information.

The first-order derivative of strain with respect to position depends on the strain and position, while the strain in each frame is time-dependent. The method can be considered as a combined time- and position-dependent method, which is defined here as the “spatial strain rate”.

5. Conclusions

This paper describes and analyzes in detail two new criteria based on the major strain topography to determine sheet necking failure, and the reliability of the new criteria is verified using the existing time correlation method and time position correlation method criteria. The following conclusions can be obtained as a result of this study.

(1)

The spatial strain rate is directly based on the strain, and can respond sensitively to the change in strain with position and time. When the spatial strain rate of the adjacent frame frequency suddenly increases, the sheet is judged to have entered the aggregated instability stage or necking area from the dispersed instability stage.

(2)

The sheet strain can be divided into low-, medium- and high-strain zones according to the change in spatial strain rate after the emergence of the aggregated instability region. In the low-strain zone, the strain is not only small but also increases weakly or even remains unchanged after entering the dispersive instability phase. The medium-strain zone is located between the high- and low-strain zones, and the strain increases monotonically and the strain difference between adjacent locations is large. The high-strain zone has the highest strain level and exhibits a monotonic increase, but the strain difference between adjacent locations is small.

(3)

The spatial strain rate method can depict the variation in the width of the aggregated instability region. This variation in width can be used to prove that aggregation instability is a gradual process and is not instantaneous. The width can not only depict the degree of aggregation instability but also determine the fracture location in the sheet.

(4)

The spatial strain rate method successfully predicted the forming limit of the sheet in the Nakazima experiment. This new method obtains results very similar to the results calculated by the T-D method and the flat valley method, and is an attractive alternative method for determining FLC.