Traditionally, the mechanical properties of materials have been characterized using the uniaxial tension test. This test is considered adequate for simple forming operations where single axis loading is dominant. Previous studies, however, have noted that the data acquired from this type of testing are not enough and additional details in other axes under simultaneous deformation conditions are important. To analyze the biaxial strain, some studies have suggested using the limiting dome height test and bulge test. However, these tests limit the extent of using multi-axial loading and the resulting stress pattern due to contact surfaces. Therefore, researchers devised the biaxial machine which is designed specifically to provide biaxial stress components using multiple and varying loading conditions. The idea of this work is to evaluate the relationship between the dome test data and the biaxial test data. For this comparison, cruciform specimens with a diamond shaped thinner gage in the center were deformed with biaxial stretching on the biaxial testing machine. In addition, the cruciform specimens were biaxially stretched with a hemispherical punch in a conventional die-punch setting. Furthermore, in each case, the process was simulated using a three-dimensional (3D) model generated on abaqus. These models were then compared with the experimental results. The forces on each arm, strain path, forming, and formability were analyzed. The differences between the processes were detailed. It was found that biaxial tests eliminated the pressurization effect which could be found in hemispherical dome tests.
Modern day society has an ever growing need for lighter weighing, compactly designed vehicle parts. This need has steadily increased due to recent economic and environmental changes. Stricter environmental regulations and the cost of fuel have made the need for these light weight parts to be considered of the upmost importance. Explorations realized that either the higher density material to be stretched uniformly to a thinner parts or implement the lower density material for lighter part manufacturing. In both cases, the material needs the highest deformation without losing the functionality or crash properties. Thus, characterization methods, which could successfully test these materials to high deformation and provide data for analysis in multi-axial differential force direction, are needed. In this study, one such method, called a biaxial test, will be examined when applied to 5083 aluminum alloys. The 5083 aluminum was employed since other research has proved this material to be a promising material that provides significant deformation when annealed.
While the aluminum 5083 alloy is characterized by a high strength to density ratio, the previous tests have shown that it can also be brittle [1,2]. This brittleness, unfortunately, limits the material's formability. For the research discussed herein, the testing of this material will be completed on Penn State Behrend's newly developed high-capacity biaxial machine. The data collected from this machine will be compared to the data collected from a hemispherical punch machine for various reasons. One concern is that the results of both testing apparatus can vary for reasons derived from the process of the testing on each machine. Biaxial testing machines are a very viable way to depict the behavior of a sample without the effect of outside or uncontrolled forces . The biaxial machine at Penn State Behrend is able to test the biaxial behavior by pulling in tension equally or variably on both the horizontal and vertical axis.
The second test employed involves a hemispherical shaped punch which is brought into contact with the cruciform specimen at a rate equal to that used of the biaxial machine due to contact condition with the hole; it is difficult to experience the pure biaxial strain mode, and predictions are challenging based on these tests [4–6]. This contact condition can come in the form of either pressure or friction [7–8]. This friction becomes prevalent simply because the punch is in contact with the sample while manipulating it. This is an unwanted variable that can cause variations during testing and data collection [9–11]. Pressure is the other contact condition that can cause the sample to fail at higher forces or time because of how the pressure makes the material behave while under stress [12–20]. All referenced studies show that the through thickness normal stress influences the forming limit curve (FLC). FLC is the plot between minor and major strain of a point on a plane of a stretched metal when it is going to fail at that point [21,22]. Numerous studies were performed on finding the influences of various parameters on FLC [23–28]. Other than pressure, there is an effect that annealing has on samples during testing. It has been found that heating the samples to a constant temperature and time yields a sample that has a longer duration of testing and can withstand larger forces prior to their failure. Once annealed samples are tested, differences can be seen in the tests performed on the two machines. Speculations may vary with respect to the specific cause but this difference becomes very prevalent when the results from biaxial and hemispherical punch tests are compared. Discussion into this will follow but the cause is believed to reside in the contact forces of the hemispherical punch; namely, the pressure that the punch exerts on the sample. Simulations were created and compared with that of the real tests and based on the accuracy of simulations, conclusions, and comparisons can start to be drawn between the tests and the behavior of the aluminum AA 5083.
In this paper, cruciform specimens are tested using hemispherical dome and biaxial tests to evaluate the relationship between these processes, if any. The cruciform specimen with a thinner diamond gage at the center was stretched on a biaxial machine using equibiaxial conditions. The force data from both axes were captured, and the strain pattern (using a digital image correlation (DIC)) was acquired. For comparison, specimens were equibiaxially stretched using a hemispherical punch on a conventional die punch setting. The punch force–displacement during the forming was analyzed. In addition, both processes were simulated and compared via experiments. The arm forces, arm strain, center strain, strain path, and failure were analyzed and compared between the two processes. The differences found between the processes are detailed below.
AA5083 alloy is considered as a part of this study. Dog-bone shaped specimens with dimensions as shown in Fig. 1(a) were prepared along the rolling direction. The sheet metal's thickness was 2 mm. All specimens for all tests were annealed at 500 °C for 5 min to produce softness as the recieved material showed brittle behavior. The 500 °C was chosen because that is the max annealing temperature that AA5083 would typically be annealed to . Once the specimen reached temperature, an MTS tensile testing machine (as shown in Fig. 1(b)) was used to conduct the uniaxial tests. The actuator speed was kept constant as 5 mm/min until failure was reached. The force–displacement data were collected at 100 Hz in order to clearly visualize the tensile behavior. The test was repeated for a set of three specimens in order to establish repeatability.
The force–displacement data obtained from the machine were converted to an engineering stress–strain curve in order to identify the yield stress, tensile stress, and total elongation. Furthermore, the data were also examined with respect to the true stress–strain curve in order to identify if the material was constant for parameters such as the strength coefficient and strain-hardening coefficient (which is further used for numerical simulation—material curve only in rolling direction with isotropic hardening law).
Figure 2 shows the cruciform geometry with a diamond shaped center which was used to perform the equibiaxial test on a NSF-funded biaxial machine. Figure 2 also provides the critical dimensions of a cruciform sample. To produce the maximum deformation at the center, the specimen was milled to a diamond shape from both sides such that the remaining thickness was 0.762 mm. A smooth radius was made in the diamond profile in order to reduce the stress concentration at those locations. Furthermore, the strain distribution during deformation was captured using DIC. In order to accomplish this, the test specimens were painted with an appropriately sized speckled pattern.
The specimens were then mounted on the biaxial machine such that the two jaws of both of the horizontal and vertical axis were holding the end of cruciform spokes. Further the specimens were preloaded to 100 N manually to make sure that there was sufficient contact between the specimen and the jaws. Once confirmed, the horizontal and vertical axes were set to pull at 5 mm/min through the controller. The DIC software was set to capture 7 frames/s. DIC cameras were calibrated using 5 mm spacing plate from correlated solutions prior to testing. The load data channel from the load cell was also connected to the same data acquisition system as the DIC which synchronized the load data, time, and DIC data. The biaxial machine setup with a specimen and with the DIC is shown in Fig. 3.
Hemispherical Dome Test.
Same specimen dimensions were used for the dome test; except, in this case, the specimens were electrochemically etched with 2.54 mm circles in order to measure the strain after deformation. Three successful dome tests were conducted. The Nakajima hemispherical dome test setup is shown in Fig. 4. The specimen was placed between the die and blankholder and was held in place using tightened bolts. A lock-bead (not shown in the figure) was used to prevent material from feeding inward during the test. The hemispherical punch was set to move with a constant speed of 5 mm/min.
Biaxial Test Model.
To simulate the biaxial tension test on the sample, four rigid plates were created to pull the specimen in horizontal and vertical directions. A three-dimensional modeling approach was used to simulate this test (Fig. 5(a)). The interaction between the rigid plates and the specimen was kept surface to surface contact with a coefficient of friction of 0.12. The specimen was kept as a deformable body with S4R shell plane stress elements (four-node quadrilateral, reduced integration). Finer mesh (element size = 0.5 mm) was applied at the center portion of the sample, while the spokes were meshed with coarse elements with size = 2 mm (Fig. 5(b)). Five integration points were used through thickness to accurately predict the necking.
Hemispherical Dome Test Model.
Similar to the biaxial model, hemispherical dome tests were also simulated using a three-dimensional model approach. The tooling was assumed as rigid surfaces while S4R shell elements (four-node quadrilateral, reduced integration) were used to mesh the specimen (Fig. 6). Lock beads were used in the experiment which prevents the material slippage during deformation, and thus, the blankholder was not included, and the specimen's boundary (at the lock bead location) was constrained in all degrees-of-freedom. The average sheet thickness measured experimentally for AA 5083 was detailed in the model. In experiments, the process was not lubricated and, thus, the interaction between the specimen and the tooling was assumed as surface to surface contact with different trial for coefficient of friction to match the experimental force–displacement curve. For good match, the friction coefficient came as 0.12 (refer to the Results and Discussion section).
Results and Discussion
Tensile Test Data.
Figure 7 shows the engineering stress–strain relationship of the uni-axially deformed samples. As can be seen, by comparing the data for all three tests, there was a good agreement between the tests. It can further be observed in Fig. 7 that the material exhibits Luder's banding. In order to further examine the tensile response of the material, the data for one of the curves were also converted to true stress–strain and fitted with the power law (Fig. 8). The fitted power law is best at from 2.5% to 7.5% strain. The material data which were used for numerical simulation were a combination of actual material data (initial constant stress for some plastic strain) plus the fitted power law. The mechanical properties characterized from testing are detailed in Table 1. K is the strength coefficient, and n is the strain-hardening exponent.
Biaxial Test Data.
A total of three specimens were tested using the biaxial machine under conditions such that equibiaxial deformation occurred. Figure 9(a) provides the intermediate strain pattern representation with regions (white dots) located (four on arm and one at center with numbering in white) for analysis. The points of strain data extraction in each arm were chosen based on their freedom from the effects of local stress concentrations, so that they would show as close a representation as possible to the axial strain experienced by each arm. Figures 9(b) and 9(c) provide the strain pattern just at the start (when the diamond started to strain) and near to failure. From these images, it can be seen that, near failure, the gage area has strained significantly more than the arms of the cruciform, and the specimen has also begun to favor the opposing corners where fracture will begin. Similar behavior can be observed in simulations (Fig. 10).
The engineering stress–strain curve, in a particular direction, through vic-3D for four arm locations of cruciform, is shown in Fig. 11. Similar to the tensile test, Luder's banding behavior can also be observed in these curves. Furthermore, in Fig. 12, the comparison of each arm's stress–strain curve was provided between experiments and simulation along with the uniaxial stress–strain curve. From this, it can be observed that the experiments and the model are in agreement in-between and with tensile test. However, they fail at different strains. The reason of straining more in the simulation is because the spoke ends are deformed a little. Also, this difference arises due to the shell elements, and the contact of edges with the plates was challenging.
Hemispherical Dome Test Data.
The experimentally and numerically deformed specimens are shown in Fig. 13. It can be seen that the specimen shows a similar failure location both in the experiments and simulations. The punch force-displacement curve provides a very good match for a friction value of 0.12 (Fig. 14). The punch-formed dome tests exhibited a similar degree of accuracy as the biaxially deformed cruciform specimens. The force–displacement comparison below indicates that the simulation expected a slightly slower force–displacement development after extreme thinning than was seen in the experiments. This is because the failure criterion was not used.
From both process and further comparison with the simulation, it can be concluded that the simulation had a good agreement with the experimental data and, therefore, the simulation alone can be used for detail comparisons between the processes. In the section “Process Comparison,” the processes were compared through numerical simulations.
In Fig. 15, the engineering stress–strain behavior is shown for biaxial and hemispherical simulated cruciform arms. It is observed that the hemispherical dome test process puts higher (i.e., approximately 6% more) strain as compared to the biaxial test. Similar behavior was found for the stress–strain response at the center location of the specimen (Fig. 16) and the strain path of center location (Fig. 17). In Fig. 17, it was also noted that the drawn forming limit diagram (from the literature)  is unable to capture the failure since, in the current study, the material was annealed, which increases its forming limits. Now, noting the difference in deformation, this may be due to the difference in the process. The main difference in the hemispherical dome test as compared to the biaxial test is that the punch is in contact with the sheet metal, and it forces downward in order to deform the material by pressurizing the sheet surface. It is assumed that this difference is making the material deform more during the hemispherical dome tests than was found with the biaxial tests. This finding means that the maximum stretching can be obtained from the hemispherical dome test.
Some previous research [12–20] has observed that the application of pressure through the thickness direction can soften the material and increase its forming. As investigated in these literatures, the effect of pressurization on formability in sheet metals is a case that an effective softening of the material occurs to varying degrees, depending upon the amount of pressure that is applied to a given material through thickness direction. This increased malleability has the effect of extending the forming limit of a material in a predictable manner. Figure 18 shows the relationship between the applied pressure, increased in increments by normal stress σy as a scalar multiple of constant α, and the true stress is manifested in material. Figure 19 reports the same data represented as measured stress, and both relationships show a linear increase in forming limit with relation to the α value of the applied pressure.
It was further concluded in these research that the forming limit diagram of a material is affected by predictable degrees following the above observation, and the formalized presentation of their data is shown below in Fig. 20. This figure shows that the strain paths of the tested material in uniaxial and biaxial tension as higher pressures have increasingly more effect on the material's ductility.
From these literatures [12–20], it is concluded that the observed difference in biaxial and hemispherical dome test failure points in the present research can be explained by the pressurization effect that the punch has on the specimen during hemispherical dome forming. This effect is not present in specimens pulled on the biaxial machine, and, therefore, the ductility is not affected, and a lower formability and failure is seen. Figure 21 shows the strain plot and the strain path of center location from both processes. It was assumed that increases in the softness of the material due to annealing increases the forming limits. The previously-mentioned comparison between experiments and simulation assumes that the biaxial test provides the limit of a material and, thus, shifted the extended as-received FLC to biaxial failure. However, this shifted FLC does not capture the failure in hemispherical dome test and, thus, further shifted this curve to realize the failure in hemispherical dome test, and assumed that this shift is due to pressurization on sheet metal by punch force. This shifted FLC is assumed as pressurized FLC of a material.
To understand the difference between the processes which provide the same deformation mode (i.e., biaxial test and hemispherical dome test), experiments were conducted using a cruciform specimen. For both tests, the specimens were annealed before deformation. Furthermore, in both processes, the parameters were kept as similar as possible. Both processes were further simulated on abaqus software. From the results, it was found that all four arms progress with a similar deformation and dropped force at a similar strain when the center region fractured. The comparison between the biaxial test and simulation were also in good agreement with the only difference being a difference in the prediction of higher strain at failure. Similarly, the hemispherical dome simulation was validated by demonstrating good agreement with the experiments. Once the simulations were validated, the simulation results were compared between the processes, and it was noted that the deformation in hemispherical dome test was higher than the biaxial tests.
This observation was made both in the arm region and in the center region. By analyzing the result and closely observing the process, it was noted that, in hemispherical dome test, the punch forces the sheet metal during deformation which eventually pressurizes the sheet metal and increases the softness in the material which increases the ductility. Overall, it was concluded that, due to pressurization by punch on the sheet metal, the material deforms more in a hemispherical dome test than in a biaxial test. This difference was not observed because, traditionally, most of the process used to manufacture sheet metal part were having contact tools and, thus, would fail in conjunction with forming limits of that material. However, the experimental tests during a biaxial test eliminate the pressurization factor and bring the pure forming limits. Thus, careful use of this forming limits needs to be understood. Considering this forming limit, various factors can be added (one of them is pressurization) and can build the process forming limits.
Author would like to thank Penn State Erie, The Behrend College for research facilities and resources and Mr. Glenn Craig for specimen fabrication.
National Science Foundation (CMMI-1100356).