key: cord-0427637-3gfdz82z authors: Ablat, Muhammad Ali; Qattawi, Ala; Jaman, Md Shah; Alafaghani, Ala’aldin; Yau, Curtis; Soshi, Masakazu; Sun, Jian-Qiao title: An experimental and analytical model for force prediction in sheet metal forming process using perforated sheet and origami principles date: 2020-12-31 journal: Procedia Manufacturing DOI: 10.1016/j.promfg.2020.05.063 sha: 54c3da5cad8ffa3af91833194f05835e2efdd4c3 doc_id: 427637 cord_uid: 3gfdz82z Abstract In this work, we present a model for origami-based sheet metal (OSM) folding that can predict the required bending force. OSM folding is enabled by introducing bending features along the bending line, named material discontinuity (MD). Firstly, the introduced analysis in this work developed a bending setup for OSM that does not require a die then introduced a method to analytically derive the bending force for a regular sheet without any added folding features. The analytical approach is followed using an experimental setup identical to the developed OSM bending configuration. Secondly, the OSM bending experiment is conducted to measure the required bending force and to identify which parameters influence the OSM bending force significantly. Finally, an empirical model is proposed considering the analytical force in the first step and the influencing parameters defined in the second step. Regression analysis is carried out using one portion of experimental force measurement to determine the coefficients of the proposed empirical model. The other measurement set is used to validate the prediction model. The validation results of the proposed prediction model show good agreement between the developed prediction model and the experiment results. The prediction model provided a shape factor that represents the topology of each bending feature, as well. / Web-to-width ratio Effective width of the sheet Bending is one of the standard fabrication processes for sheet metal parts [1] , [2] . It is used to produce various structures such as braces, brackets, frames, and channels. Sheet metal bending has wide applications in automotive, aerospace, construction, and shipbuilding industries. With the emerging needs for reduced cost and low energy fabrication processes, new sheet metal forming techniques have been developed that do not require high press tonnage, and complex shape dedicated dies to reshape the final threedimensional part [3] , [4] . Origami-based sheet metal (OSM) bending is a new forming process that utilizes the concept of origami with sheet metal [4] - [7] . OSM bending aims to bring forward a new bending technique as an alternative approach to construct a three-dimensional structure from a twodimensional sheet of metal with no need for a shape dedicated dies. In OSM, a sheet is prepared by introducing material discontinuities (MD) with predefined shapes, geometries, and scales. Fig. 1 illustrates 12 possible configurations of MD. MD can be induced by material removal processes such as laser slitting or by material deformation such as progressive stamping. Before bending, a metal sheet is held between a support and a blank holder. The support is fixed, and a force is applied on the blank holder in downward (negative Y) direction to hold the sheet firmly. The MDs determine the bending line, and it is located on an offset distance of from the edge of the blank holder. Fig. 2 illustrates the initial setup of OSM bending. The bending takes place with the downward motion of the punch. The movement of the punch is determined by the punch placement, the offset distance, the sheet thickness in the initial setup, and the desired bending angle. Therefore, the MDs play the role of a die in the bending process. They also control the material deformation during bending by localizing the strain along the bending line. The advantage of OSM bending can overcome the shortcomings of conventional fabrication since OSM bending eliminates the need for heavy die machinery, and therefore it saves manufacturing energy and cost. Despite the potential of OSM, the state-of-the-art still lacks the understanding of the force requirements for sheet metal folding and its correlation to the MD geometry. The ability to correctly predict the required force to fold a sheet with a certain thickness and MD features enables the successful implementation of this manufacturing technique. It can also reveal the operation limits of the OSM bending tool. In addition, estimating the bending force is necessary for design consideration and safety concerns during the fabrication. Literature shows various approaches to predict the necessary manufacturing force requirements for sheet metal forming, including bending. The approaches are based on experimental methods, analytical analysis, regression techniques such as Response Surface Method (RSM), or a combination of these techniques. Wang et al. [8] reported a detailed mathematical force prediction model for the air bending process. The inputs required for the model are the material properties, bending tool dimensions, and bending process parameters. The bending force is one of the four outputs of the model. Zhang et al. [9] presented an analytical model to predict the bending force for plane-strain bending of a metal sheet. The model is very general and can be used for any deformation where both bending and stretching exist. Marciniak et al. [2] discussed the sheet metal bending process in detail, including load type, springback, and choice of material model. Singh et al. [10] showed an analytic force prediction model for the three-point bending process. The model was refined and exhibited better agreement with an experiment reported in [11] . Boljanovic [12] showcased bending force empirical prediction models for wiping die Author name / Procedia Manufacturing 00 (2019) 000-000 3 bending, U-die bending, and air bending. However, the derivation process of these models is not detailed. Narayanasamy et al. [13] reported a method that uses RSM regression to predict the necessary bending force in air bending. The RSM regression model takes five process parameters as inputs. More specifically, the inputs are the punch travel, strain hardening exponent, punch radius, punch velocity, and width of the sheet. Malikov et al. [14] reported a bending force requirement for the air bending process on structured sheet metal. They developed an analytical model to predict the maximum bending force in air bending. On the other hand, Farsi et al. [15] presented a bending force prediction model for air bending of sheet metal with holes. This model considers the effective width of the sheet, the radius of the hole, and the angle factor as parameters to predict the bending force magnitude. Srinivasan et al. [16] presented a bending force prediction model for air bending by using RSM regression. Inputs to this model include the sheet metal strain-hardening exponent, coating thickness, die opening, die radius, punch radius, punch travel, and punch velocity. The outputs of the model are bending force and bending angle. Chudasama et al. [17] - [19] developed a mathematical model to predict the maximum bending force in the 3-roller bending process, while Vorko et al. [20] - [22] presented a bending force prediction model for air bending taking multi-breakage effect during the forming process into account. In the literature presented above, the bending force is predicted using a traditional punch-die bending configuration in addition to using a sheet with continuous material, i.e., without MDs. However, in OSM forming analysis, the introduction of MDs and the new bending setup without the punch-die configuration can alter the bending force requirements [23] , [24] . Therefore, further investigations need to be carried out to estimate the bending force in the OSM process. These investigations need to consider the parameters of the MDs and the OSM bending process factors such as the offset distance, sheet thickness, and punch placement. The objective of this study is to present a prediction model for OSM bending force that can assist the evaluation of manufacturing needs for certain sheet metal thickness and a specific MD design. The bending force can be a function of the mechanical properties of the sheet, the geometrical configuration of the bending setup, and features of the MD pattern. Springback is a critical issue associated with the accuracy of bending and remains to be a challenging aspect of sheet metal forming [1] . Springback can undoubtedly affect the accuracy of OSM bending operation. Previously published work [24] , has shown that introducing MD into a wiping die process has decreased the springback after bending, which was in agreement with results in [25] . However, the presented work in this paper is confined to modeling OSM bending force. Therefore, analyzing springback is considered for future research. This research develops a prediction model to estimate the force requirements for OSM bending by combining information from an analytical analysis and experimental data. The procedure to obtain the prediction model is as following: 1) an analytical force analysis is performed on a sheet assuming no MD is applied. Bending force is derived with moment equilibrium condition on a cross-section of the sheet placed in the OSM bending setup. 2) OSM bending experiment is conducted to measure the bending force under combinations of different bending configuration and parameters. The goal is to collect bending force measurement data for regression analysis, in addition to identifying which parameters influence the bending force significantly. 3) a regression model is purposed to find OSM bending force, which considers OSM bending parameters that were identified as significant using the experimental data. The purposed OSM bending force formulation contains variables corresponding to MD parameters, material properties, OSM bending configuration, and regression coefficients. The OSM bending force measurement during the experiment is divided into two sets. One set is used to determine the regression coefficients, and the other set is used to validate the OSM bending force regression model. For the analytical force analysis, the OSM bending force is derived using moment equilibrium applied on a sheet with no MDs in an OSM bending configuration. Under these conditions, a cross-section of the sheet parallel to the plane is selected, as shown in Fig. 3 . A general bending state where the sheet is bent to an angle θ is considered and a segment of the sheet cross-section is analyzed. The free-body diagram of the analyzed section of the sheet is illustrated in Fig. 3 . Initially, the punch contacts the blank at point , while the punch is positioned at a distance from the blank holder (or support). The bending line indicated with a dashed line is crossing point and has an offset distance from the blank holder and support. At the final bending or folding stage, when the sheet is bent to an angle of θ, the punch is in contact with the sheet at point ′. Up to this stage, the punch has traveled , along the positive − and negative − axis, respectively. The magnitude of the bending force exerted by the punch on the blank at point ′ is equal to , at an angle of ɣ from the −axis. The following assumptions are made during the analysis: (a) During bending, the sheet material is assumed to undergo isotropic hardening. Therefore, the yield surface keeps its shape but expands uniformly with increasing stress. The nonlinearity of the sheet material obeys the Von Mises criterion and associated flow rule; (b) The sheet is sufficiently wide in the dimension along the Z-axis, which is perpendicular to the sheet cross-section. Hence, the change in the width can be neglected, which implies plane-strain condition; (c) The stress component normal to sheet surface is zero, i.e., the plane stress condition holds; (d) The stressstrain characteristics of the sheet material are the same in tension and compression; (e) The neutral axis is always in the middle of the sheet thickness. The cross-section plain remains plane and normal to the neutral axis during bending; (f) The sheet does not undergo any thinning; hence the thickness remains constant during the bending; (h) Bauschinger effect is not considered. Under the assumptions above, the OSM bending process occurs under deformation mode with plane strain [ 1 ; 2 = 1 = 0; 3 = −(1 + ) 1 ] and plane stress [ 1 ; 2 = α 1 = 1/2 1 ; 3 = 0], where the subscript 1, 2, 3 denote principle directions, α and are stress ratio and strain ratio, (1) For the analysis, the Von Mises yield criteria, which relates the flow stress and equivalent stress ̅, takes the form as in Eqn. (2), Under plane stress condition where the principal stresses are 1 ≠ 0, 2 = α 1 , 3 = 0, α = 1/2, Eqn. (2) can be simplified to Eqn. (3), which can be written as The sheet material is assumed to follow the power-law work hardening as in Eqn. (5) , Eqn (7) gives the internal moment generated on the sheet. where is the distance from the neutral axis to a strip of sheet thickness , and 1 × × 1 is the tension force acting on the strip of the sheet with unit width. By substituting Eqn. (4) and Eqn. (6) into Eqn. (7) and simplifying the equation, the internal moment can be expressed as in Eqn. (8), Assuming the principal strain ε 1 ≈ /ρ [4] , where ρ is the radius of curvature and = /2 , Eqn. (8) To obtain the external moment generated by the punch on the sheet, we analyze the moment of bending force, , about point as expressed in Eqn. Based on moment equilibrium condition, equating the internal moment in Eqn. (8) to an external moment in Eqn. (12) will result in Eqn. (13) . Solving Eqn. (13) for , we obtain the bending force acting on the sheet as in Eqn. (14) . Assuming the radius of curvature is small in OSM bending and is equal to the sheet thickness, t. Eqn. (13) can be expressed as in Eqn. Eqn. (15) is the concluded analytical form of bending force required for a sheet without MD features to bend the sheet with an angle of θ using the OSM bending setup. In this work, we designed an experiment to measure the OSM bending force magnitude required to bend a sheet with MD. During the experiment, the needed force to bend the sheet from a flat state to 90° was continuously measured. The CNC machining center-DMG Mori NMV1500 shown in Fig. 4 -was used to achieve OSM bending. The CNC controlled relative motion between the punch and the sheet. Two fixtures were designed to hold the sheet and the punch. One fixture was mounted on the CNC spindle to hold the sheet. The other fixture was installed on a Kistler threecomponent dynamometer to hold the punch. The dynamometer was screwed on the CNC bed. During the bending process, the CNC spindle controlled the motion of the sheet. As a result, the sheet moved against the punch, which is the opposite of the schematic shown in Fig. 2 . The experiment was set up in this way due to the impracticality of installing the dynamometer on the spindle. However, the current experiment setup still provides the same force measurement because the relative motion between the punch and the OSM specimen does not change. To collect data from the Kistler Type 9257B Dynamometer, a Yokogawa SL-1000 data acquisition unit was used to collect three voltage signalsone for each of the orthogonal directions. The voltage readings could then be converted to force (in Newtons). The experiment was performed on three different configurations of MDs. MD-1, MD-2, and MD-3 are shown in Fig.5 . These three MD patterns were studied since they are the bases for other MD patterns. The used sheet material for the samples is aluminum alloy 6061-O temper. To fabricate the MD patterns, a micro waterjet cutting with a tolerance of ±0.025 (mm) was used. The sheet specimen dimensions are 50 (mm) x 50 (mm) in length and width, respectively. MD patterns were applied in the middle of the sheet, as shown in Fig. 5 . There were three different thicknesses for the sheet specimens. To achieve precise bending, the thickness of the sheet before the experiment was measured. The measured thicknesses were 1.548 (mm), 2.228 (mm), and 3.192 (mm), respectively. In previously published work, we identified the significant parameters that influence the OSM bending process [23] . It was found that OSM bending force is influenced by an offset distance ( ), punch placement ( ), web-to-width ratio ( / ), and sheet thickness ( ). In this current study, these parameters are taken into consideration during the OSM bending force formulation. Table 1 . lists investigated parameters, baseline value, and a full list of the investigated values for each parameter. In this experiment, only one parameter is changed at a time while keeping all others at the baseline value. Thus, 42 different cases (i.e., samples) were considered. Each case is tested with three repetitions; thus 126 experiment runs were performed. Table 2 lists the 42 cases with each associated OSM parameter, while the changed parameters are indicated by bold font. The parameters considered are the kerf-to-thickness ratio ( / ), the thickness of the sheet ( ) and the web-to-width ratio ( / ), punch placement ( ), punch radius ( ) and offset distance from the bending line ( ). These parameters are illustrated in Fig. 2 and Fig. 6 and further explained below: • The thickness of the sheet ( ): it refers to the thickness of the sheet material used. • Kerf-to-thickness ratio ( / ): it is the ratio between the kerf ( ) and the sheet thickness ( ). Kerf is the cutting width that the laser or water jet cut out of the sheet. • Web-to-width ratio ( / ): the ratio between the web ( ) and the width of the sheet ( ). web ( ) is defined as the remaining distance between two consecutive MD patterns after the removal of the material along the bend line to create MD pattern. It is the section that connects neighboring MD along the bend line. The length of the web is measured along the bend line. • Punch placement ( ): the distance between the blank holder and punch. • Punch radius ( ): the radius of punch at the region that stays in contact with the sheet. • Offset distance ( ): it is the distance between the imaginary (predetermined) bend line determined by MD and the edge of the blank holder (or support). The measurement data of the OSM bending force is plotted against the bending angle. The results are shown in Fig. 7-Fig. 12 . The force magnitude represents an averaged force from three repetitions of the same experiment case listed in Table 2 . The individual OSM bending force curves are compared to evaluate the repeatability of the experiment before the average is calculated from the three repetitions. Good repeatability, as shown in Fig. 13 has been observed for all cases. In Fig. 13 , the three repetitions for one case are labeled with the full name of the case in addition to letters A, B, and C. The comparison among the replications A, B, and C indicates the excellent repeatability of the experimental measurement. The experiment confirmed that the magnitude of OSM bending force is affected by the sheet thickness ( ), web-towidth ( / ) ratio, offset distance ( ), and punch placement ( ) [23] . The remaining two parameters-punch radius ( ) and kerf-to-thickness ratio ( / )-have shown little effect on OSM bending force. Hence, in the final formulation of OSM bending force, kerf-to-thickness ratio ( / ) and punch radius ( ) are excluded. Further, Eqn. (15)-the analytical bending force for a sheet without MD-already includes the impact of these three parameters. These parameters are the punch placement ( ), offset distance ( ), and the sheet thickness ( ). The mechanics of the bending operation suggest that the bending angle theta (θ), the web-to-width ratio ( / ), and the MD type have an influence on the magnitude of OSM bending force. Therefore, these parameters must be incorporated in the final form of the OSM bending force prediction model. Consequently, the final OSM bending force, , is a function of as listed in Eqn. (15) , bending angle theta (θ), web-towith ratio ( / ), and MD type as listed in Eqn. (16) , To find the explicit form of OSM bending force in Eqn. (16) , the function form is shown in Eqn. (17) is proposed. In Eqn. (17), θ is bending angle, is the effective width of the sheet along the bending line after applying MD. is equal to , the width of the sheet, multiplying web-to-widthratio, / . Except for MD type, each independent variable in Eqn. (16) is correlated to OSM bending force, , through an exponential function. MD type is considered as a categorical variable in the regression process, and it represents the shape factor of each MD. Constants 1 , 2 , 3 , 4 are the coefficients that need to be determined through regression. To obtain the constants in Eqn. (17) and validate Eqn. (17) , the OSM bending force measured data is divided into two sets. One set is made of 34 cases and used to find the constants in Eqn. (17) . The remaining eight cases constitute the validation set. The experiment cases are numbered from Table 2 . These eight-cases in the validation set are selected in such a way that they represent the variance of MD type and all six different parameters considered in the experiment. In the regression process, 50 equal-distanced data points-consisting of bending angle and force magnitude-were extracted from each experiment case measurement Thus, the extracted data points represent the full range of the OSM bending process from 0° to 90°. The regression process is carried out in MATLAB using the built-in regression function "fitlm". The nonlinear Eqn. (17) is linearized by taking the natural logarithm, as shown in Eqn. (18), In addition, the type of MD is given as a categorical variable in "fitlm" function. The regression yielded the following coefficients, 1 = 0.6, 2 = 0.93817, 3 = 0.4392, 4 = 1.1368 for MD-1. The coefficient 1 is a shape factor that changes as the MD type changes. Thus, the value of 1 for MD-2 and MD-3 are identified to be 0.591 and 0.107, respectively. Therefore, the regression provided three equations representing each MD, respectively. The final form of the OSM bending force equation can be written as presented in Eqn. (19) . Eqn. (19) provided a calculation for each MD type. It should be noted that only coefficient 1 is different when MD type alters, which shows that 1 is a shape factor that represents the topology of the MD. The underlying reason why the three MD has a different value for 1 can be explained by the topology of each respective MD pattern and the amount of material removed by the MD. The topology of MD-2 differs from MD-1 only at the end of MD-2 where there is an enlarged opening. Due to this opening, MD-2 has more material removed around the bending line, which leads to slightly less material resisting bending. Hence, the bending force needed for bending MD-2 is marginally less than that of MD-1. As for MD-3, the curved cuts that span away from the bending line towards the two sides of the sheet resulted in more material removal around the bending line Name compared to both MD-1 and MD-2. In addition, the topology of MD-3 pattern has changed the transmission path of external force from one half of the specimen to the other. Hence, the bending force magnitude is much less compared to MD-1 and MD-2. Eqn. (19) indicates that the coefficient 1 is the only factor that can change the magnitude of OSM bending force, provided that the material type, the effective length, and the bending angle remain the same. This implies that only coefficient 1 needs to be identified for Eqn. (19) to be extended to other MDs configurations. The identification of 1 for other MDs using effective mathematical, statistical, or numerical methods is in our future research plan. The adequacy of the proposed model is judged based on the regression statistics in Table 3 and the analysis of variances (ANOVA) presented in Table 4 . R-squared and Adjusted R-Squared, which is desired to be close to 1, are both equal to 0.983. This implies the model can explain almost all of the response variable variation. The pValue for each parameter is under the 0-5% significance level standard, which indicates that the response variables selected are statistically significant and it is meaningful to include them in the proposed model. In the next section, the purposed model is validated using the validation set of experimental measurements. The proposed model presented in Eqn. (19) is validated against the validation set of the force measurement that consists of eight experiment cases. These eight cases were not used in the regression analysis. Similar to the regression process, 50 data points of force-angle pairs are extracted from each of these eight cases for comparison. The regression model in Eqn. (19) predicted OSM bending force values at the derived 50 angles by substituting the coefficients 1 , 2 , 3 , 4 . The prediction results were plotted and compared to the experimental measurements in Fig. 14 . The results show that force prediction is in good agreement with experimental measurements. The prediction model has good accuracy in small-angle bending. As the bending angle increases, the predictions deviated from measurements and failed to follow measurement data closely. However, the deviation is relatively small, except for case No. 32, which is shown in Fig. 14 ( ) . Usually, the repeatability of an experiment is suspected of causing the inaccuracy. Hence, repeatability of case No. 32 is examined as shown in Fig. 13 . As mentioned earlier, the three replications showed overlapping results. Hence, repeatability is not a probable cause. Further analysis showed that the force prediction deviation from experimental measurement can be categorized into two types. The first type is the prediction model cannot track experiment data closely, for example, the range 30-50° in Fig. 14 (a) . The second type, prediction model deviates when changing a particular parameter, thickness change in Fig. 14 (g) for example. The reason for the first type is that the proposed regression model in Eqn. (17) has a sine function that takes the bending angle θ as an independent variable. Hence, it appears that the sine function dictated the shape of the prediction model. Consequently, fluctuations in bending force, the range 30-50° in Fig. 14 (a) , are not captured closely. The discrepancy for the second type, shown in Fig. 14 (g) for example, is suspected to be caused by two factors. (a): the term in Eqn. (19) is function of square of OSM specimen thickness, refer to Eqn. (15) . Thus, it may have a higher weight in OSM bending force compared to other terms in Eqn. (19) . This might have resulted in a drastic change when changing thickness. (b): the number of values for the parameters in the experimental data set is three for all parameters except the kerf-to-thickness ( / ) ratio as tabulated in Table 2 . For example, the sheet thickness has only three different values, which are 1.548 mm, 2.288 mm, and 3.192 mm. Only six specimens have thickness values other than 1.548 mm in the whole study. Thus, the prediction model might not have enough variation to describe the effect of this parameter. As a result, the prediction of the regression model is noticeably different when predicting OSM bending force for a specimen having a different thickness. Hence, it needs further study to find out the cause and improve the prediction accuracy. A force prediction model is presented for OSM bending and validated through comprehensive experiments. In addition to the MD type and bending angle, there are six different parameters considered in this study. These six parameters are associated with bending configuration and MD design. The experiment confirmed the bending force is affected by sheet thickness ( ), web-to-width ratio (w/b), punch placement ( ), offset distance ( ). It was found that OSM bending force is not affected by kerf-to-thickness ( / ) ratio, and radius of punch, ( ). Unlike the other parameters, the MD type is considered a categorical variable in regression models. Experiment data were divided into two sets, of which one is used to obtain the regression coefficients in the prediction model; the other is used to validate the model. The prediction model predicts the bending force for most of the cases well. This model can be used as a design tool for OSM bending. The input for the prediction model includes MD type, the effective length along the bending line, the geometric configuration of the OSM bending setup, sheet thickness, and material properties such as the strength coefficient, and hardening exponent. The prediction model serves as an estimation tool for OSM bending based on the inputs mentioned above, which can be easily obtained. The prediction model reveals that 1 is a shape factor representing different MD topologies. This model can be extended to other MD patterns if the shape factor 1 is identified. Hence, future improvements of the model will address ways to identify 1 for different MD types and other perforated patterns by mathematical, statistical, and experimental methods. Future work extends to reducing the gap between prediction and experimental measurement to capture fluctuations in bending force magnitude. 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