Investigation of a new methodology for the prediction of drawing force in deep drawing process with respect to dimensionless analysis
Abstract
In this research, geometric parameters were given in dimensionless form by the Buckingham pi dimensional analysis method, and a series of dimensionless groups were found for deep drawing of the round cup. To find the best group of dimensionless geometric parameters, three scales are evaluated by commercial FE software. After analyzing all effective geometric parameters, a fittest relational model of dimensionless parameters is found. St12 sheet metals were used for experimental validation, which were formed at room temperature. In addition, results and response parameters were compared in the simulation process, experimental tests, and proposed dimensionless models. By looking at the results, it very well may be inferred that geometric qualities of a large scale can be predicted with a small scale by utilizing the proposed dimensionless model. Comparison of the outcomes for dimensionless models and experimental tests shows that the proposed dimensionless models have fine precision in determining geometrical parameters and drawing force estimation. Moreover, generalizing proposed dimensionless model was applied to ensure the estimating precision of geometric values in larger scales by smaller scales.
Keywords
Dimensional analysis Geometrical parameters Dimensionless model Π-Buckingham pi theorem Deep drawingIntroduction
Process performance and better control of product quality are the main area of active research for the deep drawing process as the complex form of the sheet metal forming process. The quality of the process is still highly dependent on trial and error over large sizes which require elevated production costs. Finite element analysis (FEM) is also used which can be computationally costly and highly dependent on constitutive laws. Because of the simplification of FEM on constitutive laws and boundary conditions, it cannot cover the wide variety of physical activities across the wide range of length scales. Therefore, the experimental tests are essential for verifying FEM results. Also, it is important to select the proper process parameters to achieve flawless parts when the process changes from small to large scale. Although many studies have been done through dimensional analysis, quite a few of them contain the metal forming and articles involving both dimensional analysis and sheet metal forming especially predicting suitable geometrical parameters in different dimensional scales are still pretty limited (Davey et al. 2017; Liu and Yin 2018). It should also be noted that although the scale changes for the generalization of experimental outcomes are not a new challenge for fluid and dynamic applications but no attention has yet been paid to its application in metal forming (Li et al. 2019; Al-Tamimi et al. 2017).
Dimensional analysis plays an important role in evaluating the process on different scales. While this method depends on the complexity of the problem, the most significant benefits of this method are simplicity, no need to understand the fundamental process model, general dimensional model diagnosis, and process trend prediction in different scales (Tan 2011). In this case, Navarrete et al. (2001) demonstrated that the prediction of required force for an open die forging can be presented with less than 15% error rate by the application of dimensional analysis and Π- Buckingham pi theorem. Although the evaluation performed by dimensional analysis was very helpful in predicting force, the friction coefficient and the complexity of the geometry could limit the process of extracting information. Pawelski (1992) reported one of the few explanations of metal forming dimensional analysis. In this research, Buckingham pi theorem was used to define the effect of lubricants on cold rolling. Jamadar and Vakharia (2016) developed a new methodology based on dimensional analysis to measure the localized faults, an improvement of the computational efficiency in the nonlinear dynamic analysis for the rolling contact bearing. A good agreement between proposed theoretical models obtained by dimensional analysis with experimental data precisely shown the validity of the theoretical model. Finally, it was mentioned that the dimensional analysis tool can strongly provide great help for investigating the actual industrial scale by the laboratory studies. Ajiboye et al. (2010) investigated the determination of the friction role in cold forging. They use dimensional analysis to detect friction effects and Buckingham pi theorem for predicting its value. They found that changes in friction trend can be strongly predicted by a linear model obtained from dimensional analysis.
Methods
Aim and design of the study
Although dimensional analysis and the Buckingham pi theorem have been used in the fields of fluids, dynamic analysis, friction, and bulk forming, but regarding previous studies, in the case of reducing time and costs of manufacturing and simulation process of large dimensional scales, there is no attention paid to sheet metal forming so far. This study demonstrates that how the required geometrical parameters for designing and manufacturing of deep drawing process can be made in dimensionless form by Buckingham pi theorem. Moreover, measurement of drawing force and prediction of failure in metal forming processes need accurate and costly instrumentations. Therefore, it can be inferred that exceeding tensile stresses which causes fracture and failure lead to very crucial situations in the area of sheet metal forming. This condition can be successfully controlled by the selection of accurate geometrical parameters for the deep drawing process.
Considering the fact that there is no investigation on the dimensionless parameters for sheet metal forming, the main purpose of the current study is to suggest new dimensionless models for reducing the manufacturing costs by predicting drawing force as the crucial factor to evaluate the flawless quality of the deep drawing process in its original large size by small scale laboratory samples. For this reason, the similarity law and the Buckingham π theory were used as key theories that are widely used in dimensional fields. The geometric dimensional groups of the deep drawing process of round cups are assessed on different scales. Then, the best dimensionless models to predict drawing force at the moment of tearing are determined by stepwise regression method and ANOVA taking into consideration the effective geometric parameters. Also, the accuracy of the dimensionless models was investigated using experimental results; the prediction of the drawing force was presented. Finally, the precision of the proposed dimensionless models and dimensionless analysis was also investigated using the generalization technique. Afterward, the results of experimental tests and the proposed dimensionless model were compared.
Buckingham pi theorem
There are several techniques to reduce the number of dimensional variables to a smaller number of dimensionless groups. The method provided here has been suggested by Buckingham (1914) and is now called the Buckingham pi theorem. The name pi is derived from the mathematical notation π, i.e., the product of variables. The dimensionless groups found in the theorem are denoted by π_{1}, π_{2}, π_{3}, etc. The technique enables pi groups to be discovered in sequential order without the use of free exponents. The first part of the pi theorem describes what to expect in the decrease of variables:
If a physical process satisfies the dimensional homogeneity and involves “n”-dimensional variables, it can be reduced to a relation between only “m” dimensionless variables or pi groups. The Buckingham pi theorem explains that if there are “n”-dimensional variables in a problem, the dimensions or quantities that are related in a homogeneous condition can be described quietly by “m” dimensions (Buckingham 1914; Allamraju and Srikanth 2017).
The most general form of physical equations among a number of “n” physical quantities contains “n” symbols Q_{i} · · · Q_{n}, one for each kind of quantity, and also, in general, a number of ratios r′, r″, etc., so that it may be written as Eq. (1).
The second part of the theorem demonstrates how to discover pi groups at a moment. Discovering pi groups depends on finding the reduction factor “k” and selecting “k” scaling variables which do not form a pi group among themselves. Each desired pi group will be a power product of these “k” variables. Therefore, each pi group found is independent.
The number “k,” of separate independent arguments of “f,” is the maximum number of independent dimensionless products of Eq. (8) which can be made by combining the n quantities Q_{1}, Q_{2} … Q_{n} in different ways. The value of reduction factor “k” can be determined by the factor “n” as the number of arbitrary fundamental units needed as a basis for the absolute system [Q_{1}], · · [Q_{n}]. It is mentioned by (Buckingham 1914) that there is always at least one set of “m” which may be used as fundamental units, the remaining (n − m) being derived from them. So, the relationship between the “n” quantities can always be described by the reduction factor “k” and independent π terms. The reduction factor “k” equals the maximum number of variables that do not form a pi group among themselves and is always less than or equal to the number of dimensions describing the variables
Furthermore, if [Q_{1}], [Q_{2}] · · · [Q_{m}] are “m” of the “n” units which might be used as fundamental, the relations for Eq. (5) can be written as Eq. (12).
To make use of any one of the equations of Eq. (12) for finding the specific fom1 of the corresponding Eq. (12), each of the [Q]s is replaced by the known dimensional equivalent for it, in terms of whatever set of “k” fundamental units (such as mass, length, time, etc.). The resulting equation contains the “k” independent fundamental units, and since both members are of zero dimensions, the exponent of each unit must vanish. The “k” independent linear equations are obtained which suffice to determine the “k” exponents. It is still however one arbitrary choice left which is sometimes convenient to make use of it.
As in the above relations, each of them is dimensionless and the exponents a, b, c, d... m, are defined by the dimensions homogeneity. Lastly, as shown in Eq. (13), the general relation for the phenomenon can be obtained by specifying any of the terms as a function of the others (Buckingham 1914; De Rosa et al. 2016).
Dimensionless groups
Buckingham pi theorem is based on the dimensionless groups so that dimensionless groups can be created after determining the effective independent parameters on the response variable, considering the similarity law and the principle of dimensional homogeneity. According to the similarity law, the prototype and physical model for dimensional analysis should be in complete similarity which means that all relevant dimensionless parameters for the conditions of the process have the same corresponding values for the model and the prototype. Since the deep drawing process is considered as a quasi-static process, the dynamic similarity was passed up in this research. In this research, the prototype (small scale) and physical model (large scale) were drawn in the same velocity (similar strain ration). So the kinematic similarity is considered to be achievable. Therefore, it is necessary for this research to obtain geometric similarity. To achieve comparable results during scale changing, all process parameters that can affect the final process quality including geometrical parameters such as blank and punch dimensions, punch and die edge radius, drawing depth, sheet thickness, and the gap between the punch and die must be in the same scale. According to the similarity law, the same material with the same mechanical and thermal properties should be considered for the small and large scales. It is mentioned in (Tan 2011; Zare et al. 2012) that the material properties for small and large scales should be considered equal if the kinematic similarity is achievable. It means that in addition to the same ratio for velocities (kinematic similarity), the homologous particles can lie at homologous points at homologous times if the mechanical properties such as density, young modulus, and tension strength are the same for the prototype (small scale) and physical model (large scale). On the other hand, all parameters affecting the failure or damage of the material should be considered the same to investigate tearing in the original sample (De Rosa et al. 2016).
Dimensional matrix for independent and dependent parameters
Dimension | F | D | d | t | r_{p} | r_{d} | U | μ |
---|---|---|---|---|---|---|---|---|
M | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
L | 1 | 1 | 1 | 1 | 1 | 1 | − 1 | 0 |
T | − 2 | 0 | 0 | 0 | 0 | 0 | − 2 | 0 |
In accordance with Eq. (28), in the next step, the equation of dimensional equality is written as
Dimensionless groups
1 | π_{1} = F/(D^{2}.U) | π_{2} = t/D | π_{3} = d/D | π_{4} = R/D | π_{5} = r/D | F/(D^{2}.U) = f(t/D,d/D,R/D, r/D,μ) |
2 | π_{1} = F/(d^{2}.U) | π_{2} = t/d | π_{3} = D/d | π_{4} = R/d | π_{5} = r/d | F/(d^{2}.U) = f(t/d,D/d,R/d, r/d,μ) |
3 | π_{1} = F/(t^{2}.U) | π_{2} = D/t | π_{3} = d/t | π_{4} = R/t | π_{5} = r/t | F/(t^{2}.U) = f(D/t,d/t,R/t, r/t,μ) |
4 | π_{1} = F/(R^{2}.U) | π_{2} = D/R | π_{3} = d/R | π_{4} = t/R | π_{5} = r/R | F/(R^{2}.U) = f(D/R,d/R,t/R, r/R,μ) |
5 | π_{1} = F/(r^{2}.U) | π_{2} = D/r | π_{3} = d/r | π_{4} = t/r | π_{5} = R/r | F/(R^{2}.U) = f(D/R,d/R,t/R, R/r,μ) |
Simulation
Properties of St12
Strength in tension (MPa) | 350 |
Modulus of elasticity (GPa) | 200 |
Yield strength (MPa) | 210 |
Density (kg/m3) | 7800 |
Input and output parameters of the simulation
No. | d (mm) | D (mm) | t (mm) | R (mm) | r (mm) | μ | F (kN) |
---|---|---|---|---|---|---|---|
1 | 40 | 80 | 0.1 | 6.5 | 5 | 0.1 | 4.625 |
2 | 120 | 240 | 0.3 | 19.5 | 15 | 0.1 | 42 |
3 | 200 | 400 | 0.5 | 32.5 | 25 | 0.1 | 112 |
4 | 40 | 80 | 0.1 | 6.5 | 5 | 0.05 | 3.95 |
5 | 120 | 240 | 0.3 | 19.5 | 15 | 0.05 | 35.5 |
6 | 200 | 400 | 0.5 | 32.5 | 25 | 0.05 | 98.7 |
7 | 40 | 80 | 0.1 | 6.5 | 5 | 0.2 | 6.24 |
8 | 120 | 240 | 0.3 | 19.5 | 15 | 0.2 | 56 |
9 | 200 | 400 | 0.5 | 32.5 | 25 | 0.2 | 156 |
10 | 33.3 | 66.6 | 0.08 | 5.4 | 4.2 | 0.1 | 4.105 |
11 | 50 | 100 | 0.12 | 8.1 | 6.3 | 0.1 | 9.235 |
12 | 75 | 150 | 0.18 | 12.15 | 9.5 | 0.1 | 37 |
13 | 33.3 | 66.6 | 0.08 | 5.4 | 4.2 | 0.05 | 3.435 |
14 | 50 | 100 | 0.12 | 8.1 | 6.3 | 0.05 | 7.728 |
15 | 75 | 150 | 0.18 | 12.15 | 9.5 | 0.05 | 31 |
16 | 33.3 | 66.6 | 0.08 | 5.4 | 4.2 | 0.2 | 5.72 |
17 | 50 | 100 | 0.12 | 8.1 | 6.3 | 0.2 | 12.87 |
18 | 75 | 150 | 0.18 | 12.15 | 9.5 | 0.2 | 51.48 |
19 | 25.6 | 51.2 | 0.06 | 4.15 | 3.23 | 0.1 | 27.745 |
20 | 38.4 | 76.8 | 0.09 | 6.22 | 4.85 | 0.1 | 80.183 |
21 | 57.6 | 115.2 | 0.13 | 9.34 | 7.27 | 0.1 | 202.261 |
22 | 25.6 | 51.2 | 0.06 | 4.15 | 3.23 | 0.05 | 25.382 |
23 | 38.4 | 76.8 | 0.09 | 6.22 | 4.85 | 0.05 | 73.354 |
24 | 57.6 | 115.2 | 0.13 | 9.34 | 7.27 | 0.05 | 534.75 |
25 | 25.6 | 51.2 | 0.06 | 4.15 | 3.23 | 0.2 | 33.07 |
26 | 38.4 | 76.8 | 0.09 | 6.22 | 4.85 | 0.2 | 95.572 |
27 | 57.6 | 115.2 | 0.13 | 9.34 | 7.27 | 0.2 | 241.08 |
Experimental setup
Experimental design
Process factors and their levels
d | D | t | R | μ | |
---|---|---|---|---|---|
1 | 30 | 54 | 0.5 | 3 | 0.2 |
2 | 60 | 108 | 0.5 | 6 | 0.18 |
3 | 120 | 216 | 0.5 | 12 | 0.15 |
4 | 30 | 54 | 1 | 3 | 0.2 |
5 | 60 | 108 | 1 | 6 | 0.18 |
6 | 120 | 216 | 1 | 12 | 0.15 |
7 | 30 | 54 | 2 | 3 | 0.2 |
8 | 60 | 108 | 2 | 6 | 0.18 |
9 | 120 | 216 | 2 | 12 | 0.15 |
Dimensionless model
Dimensional combinations with the best correlation
Combination | R^{2} | Regression equation | |
---|---|---|---|
1 | π_{1} = f(π_{2},π_{3}) | 0.9 | 0.0011 × π_{2}^{0.011} × π_{3}^{0.8} |
2 | π_{1} = f(π_{2},π_{4}) | 0.9 | 2.126 × π_{2}^{1.366} × π_{4}^{2.795} |
3 | π_{1} = f(π_{2},π_{5}) | 0.86 | 2.109 × π_{2}^{1.358} × π_{5}^{2.776} |
4 | π_{1} = f(π_{2},π_{6}) | 0.89 | 0.63 × π_{2}^{1.01} × π_{6}^{0.26} |
However, it should be noted that despite the high correlation coefficient for combinations 1 and 2, the friction coefficient as an important parameter for predicting drawing force which is considered as dimensionless parameter π_{6} has not any role in predicting drawing force. For this reason, to specify a precision and significant model by considering effective dimensionless parameters, the stepwise regression method was used.
ANOVA results for the dimensionless model by stepwise regression
Source | Sum of squares | Degree of freedom | Mean square | F value | P value | % contribution |
---|---|---|---|---|---|---|
Model | 0.956 | 10 | 0.0956 | 7.468 | 0.008 | |
π_{2} | 0.412 | 2 | 0.206 | 16.093 | 0.0102 | 32.07 |
π_{3} | 0.312 | 2 | 0.156 | 12.187 | 0.0145 | 24.28 |
π_{4} | 0.227 | 2 | 0.113 | 8.828 | 0.0175 | 17.61 |
π_{5} | 0.036 | 2 | 0.018 | 1.406 | 0.346 | 2.81 |
π_{6} | 0.246 | 2 | 0.123 | 9.6093 | 0.0135 | 19.15 |
Lack of fit | 0.066 | 5 | 0.0132 | 1.031 | 0.495 | 2.05 |
Results and discussion
Confirmation run
Analysis of dimensionless parameters
Generalization
Process factors for conducting verification runs
Scale | \( \frac{t}{D} \) | |||
---|---|---|---|---|
1:1 | 0.001 | 0.006 | 0.01 | 0.02 |
2:1 | 0.004 | 0.01 | 0.025 | 0.04 |
4:1 | 0.01 | 0.03 | 0.05 | 0.07 |
Conclusion
- 1-
Similarity conditions and Buckingham pi theorem have shown that various dimensionless groups, each containing different dimensionless parameters, can be used to predict deep drawing force.
- 2-
The correlation coefficients of dimensionless models for each group which were obtained by simulation showed that there is no adequate confidence for all the specified dimensionless on drawing force prediction.
- 3-
It was determined that dimensionless parameters with more than 90% correlation coefficient can be valid for predicting drawing force. The results of ANOVA and the validation experiments confirm that the proposed dimensionless model shows good accuracy with an average error of less than 9% in predicting drawing force for round cups.
- 4-
It was shown from the ANOVA results that dimensionless parameters t/D, d/D, R/D, and friction coefficient are majorly significant. It was also shown that dimensionless ratio t/D is the most dominant dimensionless parameters for estimating the drawing force.
- 5-
The results of the dimensionless analysis and proposed dimensionless models have an excellent capability for generalizing which was verified by simulation. Therefore, it can be said that geometric values in larger scales can be estimated with good precision by smaller scales for the same material.
Finally, it is worth to mention that the dimensional analysis tool applied in the present study has given a general outline of drawing force estimation for correlating the laboratory studies on the actual industrial scale.
Notes
Acknowledgments
Not applicable.
Authors’ contributions
SH took on most of the research work, including the theoretical research and modeling, proposal and establishment of the new method and simulation work, and paper writing of the manuscript. ME and MS put forward a great variety of valuable suggestions on some key theory points and assisted with the theory researching and method validity so that the research work can be carried out smoothly. All authors read and approved the final manuscript.
Funding
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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