Investigating fracture failure in origami-based sheet metal bending

Origami-based sheet metal (OSM) bending is a promising new die-free folding technique for sheet metal. OSM bending principle is based on deforming the material along a pre-defined fold line, which is determined using material discontinuity (MD) produced by laser or waterjet cutting. The objective of this work is to study and evaluate the fracture in OSM bending under the influence of various MD types, kerf-to-thickness (k/t) ratios, and sheet thicknesses. The research goal is to provide information on selecting an optimized k/t ratio and type of MD that allows for fracture-free bending. Four different ductile fracture criteria (DFC) are used and calibrated from experimental data to forecast fracture. The DFC calibration is used to produce a set of critical damage values (CDV) for assessing the possibility of fracture in the OSM bending. In addition, the study provides fracture evaluation using finite element analysis (FEA) integrated with experimental cases for a broader range of OSM bending parameters and MDs. The results demonstrated that an MD with a higher k/t ratio is less likely to fracture during the OSM bending, whereas a higher sheet thickness increases the possibility of fracture. Furthermore, the study identifies the k/t ratio limit that ensures successful bending without fracture and categorizes MD types into two groups based on fracture likelihood. The fracture in the first group is dependent on the limiting k/t ratio, whereas the possibility of fracture in the second group is independent of the k/t ratio due to its topology.


Origami-based sheet metal bending
Origami is a paper-folding art in Japanese culture where a three-dimensional (3-D) structure is formed from a flat twodimensional (2-D) sheet of paper. Folding or bending is the only operation in this process. Origami principles inspire applications in many fields including mechanical engineering [1]. As the origami concept is being used in more fields, it opened new research directions while presenting challenges as well. One of such challenges is implementing the origami concept for thick, rigid material to form 3-D structures since, in general, material thickness in origami is assumed to be zero. To overcome the thickness accommodation, researchers proposed various techniques each with its strengths and applicability [2]. One of the techniques highlighted, for instance, is the offset panel technique presented by Edmondson et al. [3], in which all the rotational axes and the zero-thickness model lie on the same plane in both unfolded and flat-folded states. This allows panels to move individually without intersecting other panels. However, this technique requires adding additional geometry for the offset. Crampton et al. [4] also demonstrated the viability of using sheet metal with the thickness accommodation techniques. They showed how offset panel technique and hinge shift technique were used to create the sheet metal square-twist mechanism. Wu and You [5] proposed a two-layer solution to the challenge of flat foldability of a shopping bag with a rectangular base using rigid materials. It demonstrated the application of origami folding using the non-monolithic flat pattern.
Origami-based sheet metal (OSM) bending is an extension of origami principles to sheet metal forming. It is a bending process that uses a pre-determined bending line to construct a 3-D structure. The OSM bending is enabled by creating material discontinuities (MD). MD can take various forms and shapes and can be fabricated by various means such as progressive stamping, laser, or waterjet cutting. Many researchers proposed different types of MD entities [6][7][8][9].  Fig. 2. It shows that a metal sheet with MD is held between the support and the blank holder. The support is fixed, and pressure is applied on top of the blank holder to hold the sheet. The punch is located at a distance (g) away from the edge of the blank holder (or support). Bending occurs with the motion of punch in the X and negative Y directions, simultaneously. This process also eliminates the need for a die, which is a significant advantage compared to conventional sheet metal bending techniques that require a costly die/mold set [10].
The OSM bending has been investigated from the viewpoint of efficiency of folding (unfolding) metal sheets to a given 3-D structure. This problem is addressed by developing an algorithm to determine the best flat pattern from which the 3-D structure can be constructed [11]. The work developed an approach to evaluate a flat pattern unfolding in terms of a topological analysis, geometrical analysis, stressbased analysis, and potential application of fold forming in the mass production of sheet metal parts. Qattawi et al. [12] presented flat pattern tools derived from zero thickness sheet that can be applied to sheet metal products. The analysis identified all possible flat patterns that transform into the same 3-D structure. It also selected the most optimal flat pattern among all possible patterns as a next step. Qattawi et al. [13] also developed a metric that assists in selecting the most optimal flat pattern in terms of material utilization, cost, and ease of manufacture and handling.
Another direction of research in the OSM bending has been replacing conventional manufacturing approaches and reducing assembly time compared to traditional manufacturing processes where structures are constructed by welding multiple components. The objective of the studies in this direction is to achieve die-free forming with lower manufacturing energy requirements. In this regard, Venhovens et al. [14] showcased the viability of replacing metal stamping with the OSM bending and reducing the cost of vehicle body-in-white structure. Shi et al. [15] presented an alternative approach to manufacture conventional thin-walled steel structures using the OSM bending. The study concluded that folding with OSM bending can avoid the complexity encountered in folded structure analysis. Shi et al. [16] also characterized rotational stiffness for OSM bending operation. The study proposed a rotational stiffness    [7][8][9] response prediction model under moment loading for steel structures constructed by OSM bending. Ablat and Qattawi [7] investigated the bending force requirement, the accuracy of bending, and the stress magnitude when an OSM sheet is placed in the wiping die bending process. The results showed that MDs reduced bending force requirement and that MDs reduced springback after bending. Further, Ablat and Qattawi [8] conceptualized OSM bending process. They studied parameters associated with the OSM bending process and design of MDs. It was found that the stress level along the bending line is dependent on the size of the curvature radius at the end of MD. Hence, a larger kerf-to-thickness (k/t) ratio is recommended to mitigate stress levels. Further, Ablat et al. [9] presented an experimental and analytical study for modeling the OSM bending force. In this study, a prediction model for OSM bending force was proposed. The prediction model revealed a shape factor representing the effect of MD topology.

Kerf-to-thickness (k/t) ratio
While the MD enables the bending and deformation of the sheet metal along the desired bend line, the effect of the k/t ratio on MD performance is not completely understood. More specifically, the effect of the k/t ratio on the bending process and fracture has not been studied.
In this study, the term "k/t ratio" is defined as the ratio of kerf to the thickness of the sheet. The kerf is the width of the material being removed by the cutting tool along the MD geometrical outline. The kerf and other parameters on an OSM sheet are illustrated in Fig. 3.
The previous work [8] concluded that the k/t ratio of the MD is critical for achieving successful bending without fracture. It was shown that the stress magnitude in the OSM bending is affected by the k/t ratio of MD, where a larger k/t ratio is recommended to have a lower magnitude of stress in the OSM bending. Hence, smaller k/t ratio results in a higher magnitude of stress, which can cause fracture and tear during the bending process. For instance, the smallest k/t ratio investigated in [8] was 0.2, which is 0.32 mm on a 1.6-mm-thick sheet of aluminum-2036-T4. The Von-Mises stress magnitude generated on MD-2 type (see Fig. 1) is 12.4% higher on a 0.2 k/t ratio compared to a k/t ratio of 0.3. Conversely, if a large k/t ratio of MD is used, it may undermine the purpose of OSM bending, and bending may not occur at the desired bending line. The study carried out by Nikhare [17] showed that a large k/t ratio of MD lead to an undesired bending that was not along the pre-determined bending line. Further, an experimental and analytical OSM bending force model was developed to predict the required bending force [9]. In this work, the fracture of OSM samples occurred during the concept testing phase. From the observations, it can be hypothesized that the proper selection of the k/t ratio for a given MD is a critical factor that influences the success of the OSM bending operation in terms of bending accuracy and avoiding failure by fracture. However, previous studies did not provide a quantitative approach to evaluate the possibility of a fracture during the OSM bending process. Investigating the desired k/t ratio of MD for OSM bending enables the higher success of the forming operation. Hence, this study quantitatively identifies the effect of the k/t ratio on sheet metal fracture that occurs during bending using ductile fracture criteria (DFC).

Ductile fracture
Generally, failure in sheet metal forming is assessed using forming limit diagrams (FLD) [18][19][20][21]. The process of constructing a forming limit curve is shown using the strain distributions obtained from 0.2-in. inter-locking circle grid patterns on blanks. Often, the strain values on the sheet are compared with an empirical failure curve to indicate the proximity of stamping to failure. This approach has become the basis for the FLD. The application of FLD is extended to various problems. For instance, an inter-locking circle grid on sheet metal is used to predict fracture forming limit diagram (FFLD) [22], and FLD is used to evaluate mechanical properties and formability of produced DP steels with different martensite morphologies [23].
However, assessing the failure in OSM bending using FLD raises the question of suitability due to limitations in the FLD such as the difficulty to determine the onset of necking. In the FLD, the strain is measured until necking with different strain paths, and the strain magnitude causing failure is recorded [24]. The FLDs are widely used because they conveniently offer a way to evaluate the maximum amount of deformation that can take place during a sheet forming operation before the material becomes plastically unstable. However, FLDs are not suitable for applications where the material does not exhibit necking before fracture [25], as observed in a biaxial test [26,27] and incremental sheet forming process [28]. Because OSM bending does not exhibit necking and thinning of the sheet, FLDs are not a suitable choice to predict fracture in OSM bending.
In bending-dominated processes such as hemming and air bending, the failure mechanism is not controlled by necking [29]. This is because it is not possible to form a neck in the pure bending process due to the compression stress at the concave side of the sheet, which is below the limit for initiation of a necking instability. Therefore, the ductile fracture should be used as the failure criteria [30,31]. Because OSM bending is a bending dominant process, this study examines various DFC to predict the occurrence of a fracture during OSM bending.

Fracture prediction method using DFC
The available DFC on sheet metal fracture can be grouped into two general categories [30], coupled and uncoupled. The coupled DFC models focus on microscopic damage evolution to model fracture. The models are based on nucleation, growth, and coalescence of the crack until fracture. The uncoupled DFC is a phenomenological relation that models the fracture macroscopically. This approach has gained popularity due to its simplicity, and because fewer coefficients are needed than for a coupled DFC.
Most of the uncoupled DFC models use the function form of ∫ f 0 f ( , )d , which represents plastic work. The function f involves various stress invariants. The uncoupled fracture criteria state that fracture occurs when accumulated plastic strain reached a critical value.
Based on these facts, the uncoupled DFC proposed by Cockcroft and Latham [32,33], Brozzo et al. [34,35], Ayada [36,37], and Rice and Tracy [38,39] are used in this study. The specific forms of these DFC are given in Eqs. where D i (i = CL, B, A, RT) is the damage value corresponding to the selected four DFC, j ( j = 1, 2, 3) are the principal stresses, eq is the equivalent stress, h is the hydrostatic stress, p is the equivalent plastic strain, and f p is the equivalent plastic strain at fracture.
From Eqs. (1)-(4), Cockcroft and Latham criterion (D CL ) [32,33] argued that that fracture is associated with maximum principal stress 1 , and the equivalent stress eq . The criterion was proposed considering a tensile cylindrical specimen. Therefore, the logic behind the suggestion is that maximum principle stress 1 acting at a centerline of cylindrical tensile specimen initiates fracture. The Brozzo criterion (D B ) [34,35] depends on the hydrostatic stress. Brozzo criterion (D B ) is an empirically modified Cockcroft and Latham (D CL ) to fit experimental results because Cockcroft and Latham criterion (D CL ) predicted the strain at fracture lower than experimental values. The Ayada criterion (D A ) [36,37] also depends on the hydrostatic stress h , and it was proposed to predict fracture during the extrusion process. Since the stress state is 3-D in the extrusion process, especially at the location where the center burst occurs, it modified Cockcroft and Latham criterion (D CL ) and introduced hydrostatics stress. The Rice and Tracy criterion (D RT ) [38,40] has an exponential relation with the hydrostatic stress h . The Rice and Tracy criterion (D RT ) is proposed based on investigation of ductile fracture for porous materials under various stress triaxialities.
These DFC are chosen because one parameter is needed, leading to a relatively straightforward calibration process. For all these DFC, fracture happens when the parameter D i reaches a critical damage value (CDV), which can be identified by uniaxial tension test and corresponding finite element analysis (FEA).
The complete workflow developed to predict the fracture in OSM bending in this work is shown in Fig. 4. The workflow consists of three main steps, which are DFC calibration, FEA of OSM bending, and OSM bending experiment. The proposed approach relies on the calibration of DFC and using computational modeling through FEA and finally experimental approach to provide a comparative analysis of the failure strain. Figure 4 shows the three interconnected steps that provide a prediction using computational and experimental studies.
In the first step, the four DFC are calibrated to compute the CDVs that represent each of the four DFC. The CDVs are then used as a benchmark when predicting the possibility of a fracture. During this process, the CDVs are determined by the entire stress-strain characteristics of the material from a uniaxial tensile test. The stress-strain characteristics until necking are determined from the uniaxial tensile test on a standard sheet type sample. Because the deformation mode changes after necking in the uniaxial tensile test, the stress-strain relationship beyond necking is determined with the help of an iterative finite element process of the same tensile test.
In the second step, the FEA of OSM bending is performed to calculate the damage values (DVs) for different types of MDs, different k/t ratios, and different sheet metal thicknesses. The DVs represent the damage value of each case considered in this study. Then, DVs are compared against CDVs to determine the possibility of fracture during the OSM bending process.
Finally, the FEA of OSM bending is experimentally validated by comparing the equivalent strain results from FEA to experimental equivalent strain measurement using digital image correlation (DIC) at the top surface of the OSM sheet. This step assesses the accuracy of the FEA performed in the second step.
In this study, four different MDs are selected for the investigation. They are MD-1, MD-2, MD-3, and MD-5, respectively, shown in Fig. 1. These MD types are chosen for the study since they are the basis for other MD types. Previous work [7][8][9] implied that each MD may fracture at a different k/t ratio for the same sheet material type and thickness. Hence, a different set of k/t ratios is chosen for each MD as listed in Table 1. Three-level thicknesses are taken into consideration to determine if the same k/t ratio causes fracture at a different thickness. The material used for this study is aluminum alloy 6061 temper O, denoted as AA-6061-O below.

Determining the stress-strain curve beyond necking
The calibration process of DFC carried in this work refers to the process of determining the stress-strain relationship of AA-6061-O prior to and post-necking. The calibration also determines the CDVs for the chosen DFC listed in Eqs. (1)-(4). This procedure is carried out using a standard uniaxial tensile test as outlined in the ASTM testing standard [41] and iterative FEA studies of the same tensile test setup. The  During the uniaxial tensile test, the sheet specimen is prepared according to the ASTM testing standard [41] along the rolling direction of the material. The specimen length, width, gauge length, and width of the gauge section are 200 mm, 20 mm, 82 mm, and 12.5 mm, respectively. The specimen thickness is 14 gauge (1.6 mm). In the test, the elongation of the specimen is controlled by the crosshead speed. Before the yielding occurs, the speed of the crosshead was set to 1.23 mm/s. For the post-yielding phase, the crosshead speed was changed to 10 mm/s. The tensile test was done on an Instron 3369 universal testing machine to collect the load-extension data.
The FEA of the tensile test was performed using ANSYS. For simplicity, the bending was conducted along the rolling direction of sheet metal such that the ratio of strain in thickness and width direction stays constant. So, anisotropy was not considered in the FEA model. Note that anisotropy needs to be considered if the material exhibits strong anisotropy. When anisotropy is present, it affects the stress-strain curve, which subsequently alters values of D i in Eqs. (1)-(4). One can refer to the procedure presented by Hyun et al. [43] to determine the stress-strain curve for a sheet-type specimen when anisotropy is considered. The material model followed the Von-Mises yield criterion and the associated flow rule.
The plasticity model for AA-6061-O is elastic-plastic with multilinear isotropic hardening. To reduce computing time, one fourth of the sample was modeled, halfwidth and half-thickness. Due to the sample symmetry, the symmetry boundary condition was applied along the length and thickness of the specimen. The gripped surfaces from the uniaxial tension test were identified on the specimen geometry. At one end of the specimen geometry, the gripped surfaces were fixed, and a displacement load was applied on the gripped surfaces at the other end. The magnitude of the displacement load was 20.8 mm, which is the same displacement measured from the tensile test experiment at fracture. In the FEA of the uniaxial tensile test, the fracture is assumed to take place when the displacement reaches the value of 20.8 mm. To ensure the necking happens at the center of the specimen, a small defect on the two sides of the specimen is introduced [44,45], which ensured that a tapered uniaxial tension test specimen in the FEA can provide an effective stress-strain curve to capture the diffuse necking at the center of the specimen. The geometry of the specimen shown in Fig. 5 was discretized with 8-node hexa-linear elements. Mesh size at the center was much finer and it is gradually coarsened towards the two ends of the sample.
The stress-strain relationship prior to necking was obtained from the recorded load-displacement data in the uniaxial tensile test. The stress-strain relationship beyond necking was estimated using the weighted average method [42], which uses Eq. (5) iteratively to determine the stress-strain relationship beyond necking.
where is the true stress, σ u is the ultimate true stress, ϵ is the true strain, ϵ u is the ultimate true strain, and w is the weight factor.
The values of stress beyond necking were estimated based on the weight factor w using Eq. (5). The estimated stress values in addition to stress values until necking from the uniaxial tensile test were fed to the new FEA iteration of the uniaxial tensile test. The force-displacement data from the FEA of the tensile test and the experimental force-displacement curve were compared. The goodness of matching between the two curves was determined by the error sum of squares (SSE), and an adjustment was After the iterations were completed, the weight factor, w, is determined to be 0.865. The final force-displacement curve matched the tensile test closely and is shown in Fig. 6. The identified stress-strain data in Fig. 7 was then provided as an input for the multilinear isotropic hardening material model of AA-6061-O in the OSM bending FEA studies.

Calculating critical damage values for DFC
The CDVs for the four DFC were calculated from the FEA of the uniaxial tensile test. The CDVs correspond to the accumulated damage value of the material until fracture. For each increment of the load step in the FEA of the uniaxial tensile test, the necessary fields, such as the maximum principle stress, equivalent stress, hydrostatic stress, and equivalent plastic strain, were extracted using an Ansys APDL code. The extracted field data was then used to integrate according to Eqs. (1)-(4). The integration was performed using the trapezoid integration rule in MATLAB. The identified CDVs at fracture from the chosen DFC are tabulated in Table 2. Accumulation of CDV over simulation step increments is shown in Fig. 8. The curves are calculated for each simulation step based on the stress-strain fields extracted using Ansys APDL using Eqs. (1)-(4).

FEA of OSM bending
A series of OSM bending FEA is performed to calculate the DVs for each of the MDs under different k/t ratios and thicknesses. The FEA is set up in a configuration identical to the setup shown in Fig. 2. Taking all MD types, k/t ratios, and sheet thickness listed in Table 1 into account, there are 48 cases in total as tabulated in Table 3. The case name in Table 3 starts with the MD type followed by the thickness of the sheet, then the k/t ratio of the MD indicated by "k-t," and a number representing the k/t ratio. All of the OSM bending FEA cases use the same configuration parameters, i.e., g = 23 mm, s = 3 mm, and R p = 3 mm, where g is the gap between the punch and edge of the support (blank holder), s is the offset distance, and R p is the punch radius.
The sheet size is 50 (mm) × 50 (mm). The MDs are applied in the middle of the sheet. The dimensions for the sheet and MDs are illustrated for the k/t ratio of 0.3 in Fig. 9 as an example. OSM sheet samples with the same dimensions were made using waterjet cutting.
In the FEA of the OSM bending process, the punch, the support, and the blank holder were modeled as rigid bodies. The sheet was modeled as a deformable flexible body. A four-node rectangular element type was used for all rigid bodies in the FE model. The blank sheet was discretized with 20-node hexa-dominant elements. To save computing time, only half (for MD-3 and MD-5) or a quarter (MD-1 and MD-2) of the blank sheet geometry was simulated due to geometric symmetry. The meshing of the blank sheets is shown in Fig. 10.
The OSM bending process was simulated in two load steps. The first load step was to apply a pressure load on the blank holder to hold the sheet in the correct position during the bending process. The second load step was to apply a displacement load on the punch along with horizontal and vertical directions to bend it to 90°.

Experimental validation of FEA for OSM bending
An OSM bending device, as shown in Fig. 11, was designed in this study to validate the OSM bending FEA results. The bending device consists of a base, sheet holder, bending bar, mounting column, and a pair of rotating links. The base is connected to the mounting column, which is used to position the whole device in a vise. On top of the base, there is a 1-mm-deep slot that precisely matches the width of the OSM sheet specimen. The slot also ensures that the OSM sheet specimen is placed with a 3-mm offset distance(s). The base also has a step at the front that guarantees the bending stops at 90°. The sheet holder is placed on top of the sheet sample. The sheet holder fixes the sheet sample through four screws connected to the base. A bending bar is transversely connecting a pair of rotating links that   can rotate freely. The bending takes place with the guide of rotating links once the bending bar starts pressing the sheet. Because the experiment device is non-standard equipment specifically designed for this study, it was hard, if not possible, to make the angular motion of the rotating link controlled automatically. So, the angular rate of rotation in the rotating link was controlled manually at a slow pace. The samples for the OSM bending experiment were prepared by micro waterjet cutting to ensure the dimensional accuracy within 0.025 mm because dimensions of MD topology are significantly important for the accuracy of the experiment. The effect of micro waterjet cutting was assumed to be negligible according to the study by Arola and Ramulu [46], where analysis on the residual stress on six different metals cut by waterjet showed that no clear trends between the mechanical properties of the metal and the residual stress field.
In the OSM bending experiment, the equivalent strain that occurred on the surface of the OSM sample was analyzed using DIC. The trend of equivalent strain was evaluated over the entire duration of 90° bending. The trend of equivalent strain variation over the bending angle was then compared to the trend obtained from the FEA of OSM bending.
For the experiment validation, six different cases were selected from the 48 cases listed in Table 3, which are case nos. 5, 10, 18, 21, 36, and 40. These cases are chosen based on two rules. First, they cover different thicknesses. The second is to select the largest k/t ratio among all k/t ratios investigated for each MD to increase testing data acquisition accuracy and the field of view for DIC.
Images of the OSM sheet specimen surface were captured by the DIC system with a frequency of four images per second. The equivalent strain field from the images Fig. 9 Dimensions of the sheet and MDs with a k/t ratio of 0.3 included in this study was then computed on the surface of the specimens. Then the maximum equivalent strain on the surface was evaluated with respect to the bending angle. A comparison is made between the experiment data and FEA in terms of equivalent strain magnitude variation over the bending angle, as shown in Figs. 12, 13, 14, 15, 16, and 17. The trend of maxim equivalent strain and bending angle appears to be a linear relationship. The linear relationship was also observed and captured in the experiment, where the curves are mostly linear except for some deviations due to noise and equipment error.
The comparison analysis indicates that FEA is in good agreement with the experiment in general. There are two sources of the discrepancy that caused the experimental curves could not follow the FEA curves effectively, for instance, in Figs. 13, 15 and 17. First, the noise in the DIC system caused fluctuation in the magnitude of the equivalent strain. The noise depends on stochastic paint quality on the sheet and ambient light. Second, the FEA strains appear larger because the boundary of manually created DIC surface components is slightly smaller than the real surface of the sheet sample. Because the points at the edge Good repeatability was observed when the same case was tested twice. The results can be seen in Figs. 18 and 19, which showed that the same trend is followed with certain fluctuations. The higher deviation between the two repetitions in Fig. 19 might be due to the error resulting from the noise and manual bending process.

Fracture prediction in OSM bending
The possibility of fracture occurring was evaluated from the OSM bending FEA. Related stress-strain fields are extracted, and DVs were calculated for all the cases following the same approach used to calculate CDVs for the tensile test. Table 3 lists the DVs for all 48 cases based on the selected four DFC. Fracture is predicted to occur when the DV of an OSM bending case is equal to or greater than CDVs obtained from the tensile test. In Table 3, the DVs are in boldface for the cases in which a fracture is predicted. Bar charts in Figs. 20, 21, 22, and 23 show the prediction case by case. In the bar chart, a horizontal line representing the CDV based on the selected DFC is added to indicate the CDV, which aids in identifying the fracture cases conveniently.
As a next step in the experimental process, the bent samples were analyzed by optical microscopy to assess cracks. The speckle patterns were removed by acetone before the analysis. Seven samples in total were analyzed. They are case no's 1, 5, 10, 18, 21, 36, and 40. Figure 24 shows the results. Case no. 1 showed a crack, whereas the other six cases were free of any cracks, which is consistent with the prediction in Table 3 and Figs. 20, 21, 22 and 23.

Effect of k/t ratio
Different numbers of k/t ratios are evaluated for each type of MD. The results exhibited two different trends. The first trend is that the larger the k/t ratio is, the less likely a fracture occurs. This can be seen from the predictions for MD-1 and MD-2 shown in Figs. 20 and 21. All four DFC followed the same pattern to the k/t ratio. It is seen that the increase of the k/t ratio results in a drop in the DVs, and gradually the DVs go under the safe zone below the CDVs. The second trend is exhibited on MD-3 and MD-5 as shown in Figs. 22 and 23, where the DVs stay constant regardless of changes in the k/t ratio. Even though there is some fluctuation on MD-3, it is not as large as the fluctuation observed on MD-1 and MD-2.
The reason for the two different trends is related to the curvature radius of the MDs, as shown in Fig. 25. On MD-1 and MD-2, the curvature radius is significantly smaller than that of MD-3 and MD-5. A small curvature radius causes high-stress concentration leading to fracture. Hence, a change in the k/t ratio of MD-1 and MD-2 generates a large shift in DVs, whereas, in MD-3 and MD-5, the k/t ratio does not result in a change of DVs.
The results also yield the limiting k/t ratio for the MDs, beyond which a fracture can occur. For sheets with an MD-1, a k/t ratio of 2 is the limit k/t ratio when the thickness is less than 2.3 mm because a fracture is predicted when the thickness is 3.2. For a sheet with MD-2, a k/t ratio of 0.7 is the safe k/t ratio when the thickness is equal to or less than 2.3 mm. The reason MD-2 can tolerate a much smaller k/t ratio is due to the opening at the end of the MD-2.
For MD-3 and MD-5, all k/t ratios investigated remain below the fracture limiting criterion regardless of the thickness. MD-3 and MD-5 can be applied with a k/t ratio as small as 0.2. These two MDs are good choices for small k/t ratios.

Effect of sheet thickness
The thicker the sheet thickness is, the more likely a fracture happens during the OSM bending. This trend is evident regardless of the MD type investigated in this study. This can be explained by the fact that the DVs are defined as plastic work done to the sheet. The thicker the sheet is, the more work it requires to be bent to the same angle. This indicates that a smaller k/t ratio can be combined with a thinner thickness to obtain a balanced choice for the design.

Conclusion
This work carried out a quantitative investigation on the possibility of fracture during OSM bending due to different MD k/t ratios, sheet thicknesses, and MD types. The fracture possibility is predicted using four different DFC to determine proper MD k/t ratios for the investigated MD types.
The study results reveal the following: • A higher ratio of kerf to the thickness of the sheet (i.e., referred to as k/t ratio) reduces the possibility of MD fracture because a small k/t ratio is associated with a higher stress concentration on MDs. • The MDs are more likely to fracture on a thicker sheet than it does on a thinner sheet. • The investigation also provided a safe k/t ratio for the MD studied in this work. For MD1, the safe k/t ratio is 2, and

Curvature Radius
Opening the safe k/t ratios for MD-2, MD-3, and MD-5 are 0.7, 0.2, and 0.2, respectively. • The fracture possibility is influenced by the topology of the MDs. • For some MD types such as MD-1 and MD-2, there is a limiting k/t ratio that induces a fracture, while no fracture occurs for other MD types such as MD-3 and MD-5 even with very small k/t ratios due to their unique topology.
The investigation can be extended to different types of materials, MDs, and thicknesses to cover various design requirements. It can also be expanded to identify a topology that can resist fracture more efficiently. When extending this method to other materials, we need to consider anisotropy. Thus, the process may need multiple uniaxial tests to determine the stress-strain curve and the critical damage values. If a different DFC is to be used, it also affects the number and type of experiments to calibrate DFC to obtain critical damage value.