Accounting for the effect of heterogeneous plastic deformation on the formability of aluminium and steel sheets

Forming limit curves (FLCs) characterise the failure of metallic sheet materials. They are frequently used for material selection and for the simulation of the sheet metal forming process. With the increasing adoption of high-strength materials with smaller forming windows, engineers are motivated to measure their FLCs as accurately as possible and to identify individual safety margins for their materials.

The concept of the forming limits along varying strain paths was introduced by SP Keeler and G Goodwin in 1968 [1]. The concept has since been standardised in ISO 12004-2:2008 [2] and characterises formability from the uniaxial to the balanced biaxial strain path. Initial versions of the concept characterised failure in terms of “no failure”, “near failure” and “failure”. However, the standard method characterises failure as the onset of necking. It assumes continuum behaviour and therefore averages out any heterogeneity resulting from plastic deformation. It covers three aspects of the test: the method to deform the material, the technique for measuring deformation of the material and the method for determining the limit strains at the localised neck. The material can be deformed by following either the Nakajima or Marciniak method. Deformation of the material may be measured either by etching a grid on the sample surface or more commonly, with state-of-the-art digital image correlation (DIC). With DIC, the deformation of the sample is measured for the duration of the test with a set of optical cameras. The forming limits of the material are then identified with a “position-dependent” analysis of the strains just before the crack becomes visible. Practically, identifying the onset of cracking is subjective as it is a gradual process. DIC measurements show deformation to be “noisy” for most materials, making the onset of failure ambiguous to determine. The formability of a sheet material may also be characterised with other tests. Some, such as the limiting-dome height test and the Swift cupping test, are simulative tests which are appropriate for empirical measurements. Others, such as the bulge test, characterise material properties for a limited number of strain paths [1].

Amongst efforts to improve the accuracy of forming limits, one route is to reduce subjectivity by using “time-dependent” analysis, where the chronology of deformation measured by DIC of an FLC test is used to determine the onset of necking. Figure 1 shows a schematic of a material deformation parameter, such as major strain, that is tracked until failure. The onset of necking is associated with the inflection of the monotonically increasing curve. Three aspects to time-dependent methods have to be established:

When the broken-stick regression is used, a pair of linear curves is fitted to the data on either side of the inflection of the material deformation parameter (Fig. 1). The intersection of the linear curves marks the time or the frame at which necking occurs.

Table 1 summarises six time-dependent approaches that have been developed recently. Merklein et al. [3] monitored major strain rate in a circular area to failure in a HX260 steel and an AA6016 aluminium. They associated the onset of necking with the occurrence of a maximum in the major strain rate and the derivative of major strain rate respectively. Their technique produced a similar FLC for HX260 steel compared with the ISO method but a higher FLC for AA6016, particularly in the right-hand side.

Volk and Hora [4] monitored thinning acceleration along a line perpendicular to the eventual cracks in their samples of HC220YD steel at 0.8 mm, 1.1 mm and 1.6 mm. The onset of necking was identified with broken-stick regressions of the thinning acceleration. They did not compare their results against the ISO method but found agreement between experiments carried out between two laboratories.

Hotz et al. [5] monitored thinning rate in a circular area measuring 2 mm across but did not specify the material they tested their materials on. They proposed a broken-stick regression to identify the onset of necking. Min et al. [6] carried out plane strain Marciniak tests on 1.2-mm MP980 steel, 1.0-mm DP600 steel and 1.0-mm AA6022-T4 aluminium and compared Hotz et al. 2013’s method with the ISO method. They found that the DP600 and MP980 forming limits were measured to be higher using Hotz et al.’s [5] method compared with the ISO method while the AA6022 was lower.

Huang et al. [7] monitored the second derivative of major strain but track just a single point in the neck. They identified the onset of necking on the increase of strain. When Min et al. [6] compared Huang et al.’s [7] method with the ISO method, they found that the forming limits of DP600 and MP980 were measured to be higher, while AA6022 was measured to be lower.

Vysochinskiy et al. [8] monitored the ratio of the thickness in the neck and its surrounding area, k, in 2 circular areas of their samples: an inner circle and an outer circle. The inner circle contained the neck while the area between the inner and outer circle containing un-necked material. The onset of necking was detected with a threshold value, k_limit. When compared with the ISO method, the method measured similar forming limits on the right-hand side for an AA6016 aluminium.

Martinez-Donnaire et al. [9] monitored major strain and tracked the strains along a line across the eventual crack. Failure was detected at the threshold of maximum strain rate. In their plain strain Marciniak tests, Min et al. [6] applied the Martinez-Donnaire et al.’s [9] technique and found that, when compared with the ISO method, the technique measured forming limits that were similar for DP600 and MP980 but was lower for AA6022.

To summarise, compared with the ISO method, time-dependent methods produced FLCs that were similar or higher for steels. For aluminiums, there was no clear trend. The level of a time-dependent FLC was higher or lower to the ISO method, depending on the technique used. As a result, defining the onset of necking more accurately with deformation parameters such as major strain rate has been mixed.

In this work, we drop the continuum assumption to investigate the effect of heterogeneity in plastic deformation on formability. This was carried out by including the statistical characteristics of deformation that are captured by DIC measurements. Figure 3b shows the DIC measurements of a Marciniak test (Fig. 3a) at the onset of necking. The topology of thickness strains, ε₃, shows the complexity of the strain distribution that is typically present at the onset of necking. The fluctuations imply that material is deformed to different extents in different areas, resulting in a heterogeneous strain distribution.

It was demonstrated, through a design of experiment, that these fluctuations were significant compared with the uncertainty of the DIC measuring system in terms of its accuracy and precision (“Establishing the measurement uncertainty of DIC” section). A series of Marciniak tests were then carried out on 2 aluminium grades, NG5754-O, AA6111-T4, a mild steel grade, MS3 and a dual-phase steel, DP600. The observed heterogeneity was modelled with Gaussian models to quantify the mean and dispersion of the distributions. Single distributions were observed during the uniform deformation phase. At the onset of necking, the single distribution became “multi-modal” or formed multiple distributions (“The strain distribution during plastic deformation” section). The Gaussian models were used to develop a framework to characterise formability as a probability to failure (“Characterising the formability of a material” section). The novelty of this work is the observation (1) that the DIC system is able to detect heterogeneity of plastic deformation at the macroscale, (2) that the presence of heterogeneity means that the forming limits of each measurement is uncertain and can only be stated probabilistically and (3) that the range over which failure can occur can be used as a basis to balance the desired forming window with the risk of failure.

The DIC system used in this work was a GOM Aramis DIC system that consisted of a pair of Vosskuhler 1300bg cameras, fitted with 50-mm lenses. The cameras had monochrome sensors with a resolution of 1280 × 1024 pixels. A monochromatic speckle pattern was applied to the gauge are of each sample.

Measurement uncertainty was determined under three conditions using the Nakajima test setup (Fig. 4): depth of the punch, blankholder force and orientation of the blank.

Depth of the punch (Fig. 4b) was identified as a factor because as the sample deforms, the sample is located in different areas of the measurement volume and the angle required to view all areas of the sample changes can cause errors in tracking the pattern. Blankholder force was a second factor because the camera system was mounted on an Erichsen 145-60 sheet forming test machine that used hydraulic actuators to operate the blankholder. As the camera structure was mounted on the casing of the machine, vibrations from the machine as it applied blankholder pressure on the sheet could have been transmitted to the cameras, causing blurring of the images and errors in tracking the pattern on the samples. Finally, the sample was rotated on the blankholder by 0°, 90°, 180° and 270° without application of load. This was done to determine the effect of the applied pattern on measurement uncertainty because Hild and Roux [15], in a review of DIC accuracy, identified this as an important factor. To test for the effect of these factors, a full factorial experiment was carried out. Table 2 lists the factors and the conditions of the test.

For tests where punch depth was 10, 20 and 30 mm, the speckle pattern was applied to each sample after it was deformed. This was done to measure the accuracy of a known strain value, zero strain, at the various punch heights. Assuming that the zero-strain measurements are normally distributed, the means of the measurements were used to describe the accuracy of the measurements (within a given tolerance, or standard error) and its standard deviation to describe precision.

The material tested was 1.2-mm thick DX54, a mild steel grade. Each sample was cut into a 220-mm sided square. The speckle pattern applied to the samples consisted of a matt black background followed by a white speckle pattern (50:50 mixture of white emulsion paint to water) using an airbrush held at a distance of approximately 0.3 m away from the sample. The camera system was calibrated and the camera angle was found to be 14.7°, the measurement volume was 115 × 90 × 70 mm, measuring distance was 800 mm and the calibration deviation was 0.015 pixels. Each measurement contained about 3700 facets measuring 1 mm² each.

The mean major strain, ε₁, from the 48 experiments (Appendix Table 3) was less than 0.0005 and standard deviation was 0.0006. The standard deviation was at least an order of magnitude lower than the plastic strains measured in the rest of this work. An analysis of variance found that depth had the greatest effect on measured strains and this was due to the increased dispersion from curvature. Blank holder force, sample orientation or factor interactions did not show a significant effect on the accuracy of the measured strains and will be considered negligible in this work.

The forming limit curve tests were carried out in accordance with the test procedure described in ISO12004-2 [1]. Marciniak tooling (Fig. 5) was chosen to minimise bending in the gauge area of the samples and to reduce measurement errors due to curvature of tooling described in the “Establishing the measurement uncertainty of DIC” section. Specimen geometries with 5 different widths in the gauge area were machined to test forming limits between the uniaxial and equibiaxial strain paths (60 mm, 100 mm, 120 mm, 160 mm and 220 mm, Fig. 5). Three repeats were carried out. Carrier blanks for the Marciniak test were manufactured from ductile DX-54 mild steel and mirrored the specimen geometries with the inclusion of a central 33-mm diameter hole. As before, the samples were tested on an Erichsen 145-60 machine and monochromatic speckle patterns were applied to the gauge are of each sample and its deformation was tracked with the GOM Aramis DIC system. Two aluminium grades, NG5754-O (1.5 mm), AA6111-T4 (1.2 mm), a mild steel grade, MS3 (0.9 mm) and a dual-phase steel, DP600 (1.6 mm) were tested.

The observation that necking causes the GMM components to diverge was used as a characteristic material parameter for a time-dependent method to identify the onset of necking. In particular, the difference in the means of the components of the GMM was tracked, and the onset of necking was identified with the broken-stick regression method. However, the presence of a spread of strains in the sample meant that it was not possible to associate the necking event with a single value of strain. Instead, the strains in the sample were evaluated for the probability that it belonged to the necked component.

The method was implemented in a Matlab 2012a program to automate the process. To identify the onset of necking:

Since a dispersion of strains always exists in a material, the formability of the material is better described in terms of a risk to failure. At the onset of necking:

The posterior probability that each strain point belonged to the necked component was overlaid on each strain point.

where a, b, c and d are the fitting parameters for the model, P₂ is the probability of component 2 and e is the residual error of the fit for the model.

The resulting curves assign probabilities to the strains that were classified as necking strain to represent a map of the probable formability of the material (Fig. 13). These curves were compared with forming limits calculated with the ISO method [1].

When compared with ISO-based FLC data, the P(failure) = 1 contour was generally conservative or lower, apart from AA5754. This is due to the ability of the time-dependent procedure to identify the onset of necking by tracking the difference of the means between components 1 and 2, ∣μ₂ − μ₁∣ (Fig. 12). The gap reflects the sensitivity of this method compared with the ISO method. The gap is particularly prominent for MS3 (Fig. 13d). Figure 14 shows that the rate of separation of the GMM components does not display as sharp an inflection as that exhibited by 1.2-mm AA6111T4 (Fig. 12). This implies that the development of a localised neck is more gradual in MS3 than in AA6111T4. The large discrepancy between the probability map and the ISO-based FLC suggests that the inflection point in Fig. 14 is due to the onset of diffuse necking rather than localised necking, resulting in a lower forming limit.

The probabilistic forming limits may be compared more broadly with other time-dependent methods. Table 1 shows deformation parameters used by past authors for time-dependent methods to identify the onset of necking and calculate formability. For steels, these deformation parameters have measured forming limits to be either similar or higher than the ISO method. Since the ISO method relies on the visual identification of a crack to characterise the formability of a material, these deformation parameters either equal or overestimate the forming limits measured with the ISO technique. In contrast, the probabilistic forming limit curve is consistently lower than the ISO method and is sensitive enough to identify the onset of diffuse necking in MS3. For the AA6xxx grade aluminiums, Table 1 shows that there are no clear trends for the deformation parameters used in past time-dependent methods compared with the ISO method. In contrast, the probabilistic forming limit method measures forming limits to be lower for AA6111T4. For NG5754-O, the probabilistic forming limits are measured to be higher than the ISO method. This may be due to the presence of Luders bands in NG5754-O during deformation that results in highly heterogeneous distribution of strains on the surface of the samples. This may have affected the forming limits calculated by both the probabilistic method outlined here and the ISO method.

The characterisation of measured strains as PDFs showed that deformation during the “uniform plastic deformation” and “necking” phases is heterogeneous in all four materials. In particular, a range of strain deformation existed in the neck. When this uncertainty is incorporated into the description of forming limits, the width of the probability contours conveys the idea of the level of heterogeneity that occurs in the material. The widths were material dependent and reflected the range of strains in the neck that had P(g₂(ε₃)|x) > 0.5. NG5754-O had the greatest width (~ 0.07 strain) in the plane strain path and MS3 had the lowest (~ 0.03 strain). By accounting for the range of strains in the neck, the width of the probability band is a logical starting point for balancing the required forming windows with a risk to failure for individual grades.

There are several sources of this heterogeneity, such as the anisotropy of grains and deformation that is localised to slip systems within grains. However, the scale of this structure is of the order of 10 μm, which is several magnitudes smaller to the size of the size of the gauge area of the FLC sample (30 mm). Small [19] speculated that heterogeneity may persist at the macroscale if it is caused by surface roughness because this can come about through collective granular distortions. The resulting macroscopic-scale roughness may influence failure through variations in the sheet’s effective thickness.

Forming limit curves are an important component for the simulation of sheet metal forming processes. However, they average out the heterogeneity in plastic deformation to define the threshold of failure in a binary or deterministic manner. In reality, it is acknowledged that the deformation and failure of materials are heterogeneous. This work shows that this heterogeneity can be detected at the macroscale using DIC and proposes a framework to account for it so that formability can be seen more appropriately as a risk to failure.

An investigation was carried out on 2 aluminium grades, NG5754-O, AA6111-T4, a mild steel grade, MS3 and a dual-phase steel, DP600. DIC measurements of the plastic deformation of all 4 materials were found to be heterogeneous or “noisy” and were significant compared with measurement uncertainty of the DIC.

This behaviour was incorporated in the new framework for formability by modelling it with 2-component GMMs. At the start of plastic deformation, single-mode PDFs were apparent. As deformation proceeded, the dispersion in the strains increased until the occurrence of necking, when the single mode transitioned became bimodal. The modes with higher mean strains were associated with the necking material of the sample. The presence of a range of strains in the neck confirmed that formability cannot be described by a single value or a forming limit.

The statistical parameters of the 2-component GMM (|μ₂−μ₁|) were used to identify the onset of necking. At the onset of necking, the strains measured were characterised in terms of the probability that they lay in the neck. The resulting probability distribution was represented as a logistic regression that formed a probability map of forming limits of the material. The widths of the map were material dependent and indicated the range of strains in the neck. Because the widths were characteristic to each material, they are logical starting point for balancing the requirements for part design with the risk to failure for individual materials.