3.1. Minor and major strains
Figure 6 illustrates a qualitative comparison for deformed specimens with a hole diameter of 2 mm in both square and triangular layouts, and a comparison for specimens with a diameter of 4 mm in square and triangular layouts is shown in Figure 7.
After obtaining the minor and major strains, the data points per each test should be correspondingly placed on a diagram where the horizontal and vertical axes represent the minor and major strain values, respectively. To obtain an FLD, a curve must be fit through the points to indicate the boundary between the safe and failure zones.
3.2. Obtained FLDs
The resulted FLDs are shown in Figure 8 and 9 for parts with 2 mm holes with square and triangular arrangements, respectively. Also, Figure 10 and 11 shows the FLD of parts with 4 mm hole with square and triangular layouts, respectively. In FLDs, generally, the major strain corresponding to the minor strain of zero (the y-intercept) represents the least amount of formability. Therefore, to compare the formability of specimens with different hole size or layout, the value of y-intercept of FLDs can be looked at. According to Figures 8 and 9, for parts with a diameter of 2 mm, the minimum formability for the triangular layout with major true strain of 0.27 is greater than that of a square layout with major true strain of 0.22, indicating the greater formability of the triangular arrangement.
Also, for parts with 4 mm diameter holes according to Figures 10 and 11, the minimum formability for the triangular arrangement with major true strain of 0.26 is greater than that of a square arrangement with major true strain of 0.12, which follows the results obtained for parts with hole size of 2 mm. Therefore, it can be concluded that the formability of the perforated sheets in triangular layout of holes is higher; this is because the ligament ratio between the holes in triangular layout is larger than that of the square layout.
To compare the effect of diameter size on formability, according to the FLD of square arrangement shown in Figures 8 and 10, the 0.22 minimum strain for a diameter of 2 mm is greater than that of a 4 mm diameter with minimum strain of 0.12. Similarly, for a triangular layout according to Figures 9 and 11, the minimum strain for a diameter of 2 mm (0.27) is greater than the corresponding value for a diameter of 4 mm (0.26). The mentioned cases show more formability of the sheet with a triangular layout and holes with diameter size of 2 mm since the combination of triangular layout and smaller holes created larger ligament ratios.
3.3. Part’s Extension at Failure
Another criterion for investigating the formability of samples is by measuring the depth of deformed specimens at the moment of failure (appearance of cracking around the holes), which is commonly referred to as the dome height. The formability of perforated sheets with a diameter of 2 mm was found to be higher than that of perforated specimens with a diameter of 4 mm due to existence of greater amount of bulk material between the 2 mm holes. To avoid interaction of specimens’ dimensions when comparing the dome height for different hole dimeters and layouts, the specimens’ dimensions were kept constant at 80x80 mm, and the force-displacement diagrams of experiments (three replications per sample) were compared as shown in Figure 12; the dome height (extension depth) of the specimen with 4 mm hole in square layout is 6.3 mm and can withstand a force of 110 N at failure while the same specimen with a triangular layout has a dome height of 6.7 mm and endures a force of 130 N at failure. Therefore, the triangular arrangement resulted in higher formability. Also, the sample with a hole diameter of 2 mm in a square layout after formability test shows a dome height of 7.7 mm and can bear a force of 220 N at failure while the same model with a triangular layout has a dome height of 9.5 mm and tolerates a force of 320 N at failure. The jaw of tensile test machine moves a longer distance in the case of triangular layout. Also, by keeping all variables constant, the formability of sheets with 2 mm holes is higher than the sheets with 4 mm holes.
3.4. Analysis of Variance
To evaluate whether the observed formability results are statistically significant different, an ANOVA was run on the results with significance level of 0.05 [18]. As mentioned before, the hole size (2 and 4 mm) and hole layout (square and triangular) were considered as factors and response variables are the observed major strains, the ultimate tensile force at failure, and the extension (dome height) of parts.
Considering the major strains as the response variable, the results were found to be statistically significant different since the P value of ANOVA was found to be less than 0.05 (<.0001). Also, no interaction was found between the hole size and hole layout although both were found to be significant main effects. Figure 13(a) and (b) shows the least square means plot for hole size and hole layout, respectively, and evaluation of interaction between hole size and hole layout is illustrated in Figure 13(c) where no interaction can be observed (the lines do not intersect). As can be seen in Table 1, a Tukey test was also performed on the results, indicating that all the parameter sets are significantly different where 2 mm hole in triangular layout and 4 mm hole in square layout resulted in the highest and lowest major strain values, respectively. For Tukey test it should be noted that the levels that are not connected by the same letter are significantly different.
Table 1
Results of a Tukey test performed on the major strain values
Level | | | | | Least Sq Mean |
Triangle,2 | A | | | | 0.389 |
Triangle,4 | | B | | | 0.336 |
Square,2 | | | C | | 0.285 |
Square,4 | | | | D | 0.191 |
Considering the specimens’ ability to withstand the ultimate tensile force at failure, different samples showed a statistically significant difference in the ANOVA results (P value less than 0.05). No interaction was found between variables (the hole size and the hole layout) as shown in the least square means plot in Figure 14(c) although both variables were found to have significant main effects (Figure 14(a) and (b)). Table 2 shows the results of the Tukey test where the 2 mm hole in triangular layout was found to withstand the highest tensile force at failure (294.7 N) while the 4 mm hole in square layout endures the least force (67.6 N), among other parameter sets. The endurance forces at failure for 2 mm hole in square layout (179.9 N) and 4 mm hole in triangular layout (152.2) were not found to be statistically significant different based on the Tukey test (the same letter B for both parameter sets), meaning that the plastic behavior of these two types of perforated sheets is similar since the 2 mm hole in square layout results in similar ligament ratio as the 4 mm hole in triangular layout. Thus, if the sample preparation cost for these two specimen types (2 mm hole in square layout and 4 mm hole in triangular layout) is different, the cheaper method of part fabrication could be used.
As mentioned earlier, the dome height can be a good metric for assessing the specimen’s formability. Running an ANOVA on the dome height of formed parts shows that the results are significantly different (P value=0.0002). As the least squares means plots in Figure 15 show, no interaction was found between the hole size and hole layout, but both were significant main effects. Performing a Tukey test on the dome height (see Table 3) shows similar results to that of the ultimate tensile force at failure, i.e., the 2 mm hole in triangular layout has the highest dome height (12.6 mm) while the 4 mm hole in square layout has the least dome height (5.8 mm), among the rest parameter sets. The dome heights for 2 mm hole in square layout (8.7 mm) and 4 mm hole in triangular layout (9.2) were not found to be statistically significant different based on the Tukey test.
Table 2
Results of a Tukey test performed on the tensile force at failure
Level | | | | Least Sq Mean |
Triangle,2 | A | | | 294.772 |
Square,2 | | B | | 179.974 |
Triangle,4 | | B | | 152.269 |
Square,4 | | | C | 67.611 |
Table 3
Results of a Tukey test performed on the dome heights
Level | | | | Least Sq Mean |
Triangle,2 | A | | | 12.678 |
Triangle,4 | | B | | 9.292 |
Square,2 | | B | | 8.709 |
Square,4 | | | C | 5.825 |
3.4. Investigation of Failure Fractographs
The ruptured surface of the specimens was observed using scanning electron microscopy (SEM). Since the specimen with 2 mm hole in triangular pattern led to the highest formability, the fracture surface at failure of one of its replications is shown as an instance in Figure 16. During the formability test of specimens, the micro-voids would merge to each other and lead to the progress of ductile fracture through crack propagation, creating surface fracture patterns. Having a closer look at the surface patterns in Figure 16, several dimples in various shapes and sizes can be observed. These dimples are features of a ductile fracture; each dimple is attributed to a crack location during the plastic deformation process. By advances of void propagation, the unfractured material is strained until the ultimate fracture occurs [19]. The variation of dimple size is affected by the void’s stress threshold and growth rate, which are influenced by the physical characteristics of the microstructural features [20].