Predicting the Complete Forming Limit Curve of Steel in Sheet Metal Forming using Artificial Neural Networks

Forming Limit Curves (FLCs) are crucial for predicting the formability of materials in the sheet metal forming industry and preventing defects. Traditionally, FLCs are determined through the Nakajima test and the Marciniak test, which assess the material’s response under various strain paths until the onset of local necking. However, these methods are costly, time – consuming, and sensitive to factors such as friction. To address these challenges, alternative methods have been developed, including theoretical models based on tensile test data and empirical approaches. This study explores the modeling of FLCs using Artificial Neural Networks (ANNs) with the aim of improving prediction accuracy and efficiency. The input data for the ANN model are derived from tensile tests and cover parameters such as yield strength, ultimate tensile strength, uniform elongation, total elongation, normal anisotropy coefficient, and strain hardening exponent. The ANN model is trained to predict FLC₀ and the complete FLC, and its output results are compared with the experimentally measured FLC from the Nakajima test and empirical formulas from the literature. The results demonstrate the great potential of ANN technology in enhancing the reliability and efficiency of FLC prediction.

1. Introduction

Forming Limit Diagrams (FLDs) and Forming Limit Curves (FLCs) are fundamental tools in the sheet metal forming industry, playing a crucial role in predicting the formability of materials during the manufacturing process. Although closely related, they serve different purposes and have significant differences that are essential to understand. An FLD is a graphical representation that depicts the strain states experienced by sheet metal under various forming conditions. It shows the critical strain levels beyond which the material may develop local thinning, leading to defects such as necking or tearing. The FLD provides a comprehensive view of the material’s behavior under different biaxial strain conditions and is valuable for assessing the overall formability of the material. In contrast, the FLC is a specific curve within the FLD that represents the boundary between safe and unsafe forming conditions. The FLC indicates the maximum strain that the material can withstand under different strain paths before failure (the onset of local necking). While the FLD offers a macroscopic perspective on formability, the FLC provides a precise threshold for safe forming operations, enabling manufacturers to optimize processes and prevent defects. By referring to the FLC, engineers can design forming operations that stay within the safe range, avoiding conditions that could lead to defects such as necking or tearing, thus ensuring that the produced sheet metal parts are both structurally sound and aesthetically acceptable.
 
Figure 1 shows a schematic diagram of an FLD. The FLC is divided into two sub – branches, the “right branch” and the “left branch”, which converge at a common point called FLC₀. These branches represent different strain conditions experienced by the sheet metal during forming and are crucial for understanding the material’s behavior under various scenarios. The right branch corresponds to positive minor strain, where the material is stretched in both principal directions. This situation is common in biaxial stretching processes such as stretch forming and deep drawing, where the metal is drawn into a die. The right branch represents the maximum strain the material can withstand before necking or failure occurs during stretching. Conversely, the left branch corresponds to negative minor strain, where the material is compressed in one direction and stretched in the other, which is typical in drawing operations. This branch shows the forming limits under such conditions, where one axis is in compression and the other in tension. Understanding these branches is essential for designing forming processes. The right branch helps prevent tearing during stretching, while the left branch is key to avoiding wrinkling or excessive thinning during drawing[1,2]. By precisely determining these limits, engineers can optimize processes, ensuring material integrity and preventing defects. In the context of the FLC, FLC₀ refers to the forming limit under plane strain conditions, which is a critical aspect in understanding material formability[3,4,5]. FLC₀ is typically located near the vertical axis of the FLC plot and represents a situation where one principal strain is zero, and the other principal strain reaches its maximum value. Besides the Nakajima and Marciniak tests, FLC₀ can also be determined through the notched tensile test[6], providing an alternative method for accurately assessing this important forming limit.

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The importance of FLCs in the sheet metal forming industry cannot be overstated. These tools are essential for ensuring the successful manufacture of high – quality components, as they enable engineers to predict and address common forming issues before they occur. FLCs help engineers understand the deformation limits of materials under various forming conditions. By accurately identifying these forming limits, manufacturers can design forming processes that operate within a safe zone, thereby preventing defects such as wrinkling, tearing, and necking[1,2]. Wrinkling can lead to poor product appearance and structural damage, while tearing and necking are more severe, often resulting in complete component[2] failure. Incorporating FLCs into the forming process aids in optimizing several key manufacturing aspects. For example, by understanding the material’s strain limits, engineers can fine – tune forming parameters such as die geometry, lubrication conditions, and strain paths to achieve the desired shape without exceeding the material’s formability limits. This optimization not only improves product quality by reducing defects but also decreases material waste. In mass production, material waste is a significant concern, and even minor inefficiencies can lead to substantial cost increases. Moreover, by preventing defects and reducing waste, FLCs contribute to enhanced overall production efficiency. They make the manufacturing process more predictable and reliable, reducing the reliance on expensive trial – and – error approaches and minimizing downtime due to defects. In an industry where speed, cost, and quality are paramount, this efficiency is crucial for maintaining competitiveness.
 
In experiments, determining the FLC requires controlled tests to assess the behavior of sheet metal before the onset of local necking. The Nakajima test and the Marciniak test[1,7,8] are the most commonly used methods. In the Nakajima test, specimens of varying widths are stretched using a hemispherical punch, and the strain is monitored until necking occurs. In the Marciniak test, a similar deformation operation is carried out using a flat – bottom punch. Both tests measure the strain at the onset of necking. Multiple specimens are tested under different strain conditions, and the resulting data are then used to construct the FLC, which shows the maximum strain tolerance of the material under various loading conditions.
 
The concept of FLCs was first introduced in 1963[9] and has since become a widely used tool for predicting the risk of failure in sheet metal forming processes. Despite their widespread application, the experimental determination of FLCs still poses challenges, as it is costly, time – consuming, and the results are variable, as the test results depend heavily on the mechanical properties of the specimens. Additionally, friction during the experimental testing process can significantly affect the position of the necking point, the deformation behavior, and the strain path, further complicating the accuracy of FLC measurements[1,10]. To improve the quality and efficiency of FLC testing, various procedures have been developed. For example, a new technique using hydraulic bulging of double specimens was introduced to minimize the influence of friction[10,11] and a testing method was proposed to determine the entire left – hand side of the FLC through a frictionless tensile test[6]. In recent years, significant efforts have been made to develop more cost – effective, efficient, and accurate methods for determining FLCs. One research direction focuses on theoretical models, such as the Hill–Swift[12,13] model and the Marciniak–Kuczyński (MK)[7] model, which are used to estimate the necking limit strain of sheet metal. These models typically calculate the forming limit under plane stress conditions but often neglect the effect of non – constant normal stress in the thickness direction, which is crucial for predicting the FLC of medium – thick plate materials.

 

Another approach is empirical methods based on simple, low – cost experiments. These methods utilize the basic mechanical properties obtained from tensile tests to predict FLCs. For example, Keeler and Brazier[14] developed an analytical equation to predict FLC₀ from the strain hardening exponent and sheet thickness; Raghavan[15] proposed a similar equation incorporating total elongation and thickness; Paul[16,17] introduced a non – linear regression equation considering sheet thickness, strain hardening exponent, total elongation, and ultimate tensile strength; Cayssials[18,19] extended this equation by incorporating strain rate sensitivity and anisotropy to predict the FLC of ultra – high – strength steels, known as the Arcelor model; the Tata Steel model developed by Abspoel[20] calculates four key strain points using total elongation, the Lankford coefficient, and sheet thickness; Gerlach[21] provided equations for determining three characteristic points of the FLC based on sheet thickness, total elongation, and ultimate tensile strength.

 

The literature review shows that FLCs can be effectively modeled using the results of tensile tests, and non – linear relationships are often included in empirical formulas. Many studies have successfully used Artificial Neural Networks (ANNs) to achieve this goal[16,22,23,24]. In this study, we build on this approach and explore the relationship between tensile test results and FLCs using ANN technology, but with a new approach different from previous studies.

 

The structure of this paper is as follows: Section 2 introduces the analytical models for predicting FLC₀ and outlines the equations required for their calculations; Section 3 is divided into two parts, first introducing the calculation of FLC₀ using the selected ANN model, and then calculating the right – hand side of the FLC through the ANN model; Section 4 models the complete FLC based on tensile properties, and compares the predicted values with the experimentally measured values to evaluate the accuracy of the estimates; finally, Section 5 summarizes the research results and draws conclusions.

 

2. Review of Analytical Models for FLC₀ Prediction

2.1 Keeler – Brazier (KB) Model

Keeler and Brazier[14] developed a standardized FLC shape to distinguish between safe and unsafe regions for major – minor strain points, divided into the right – hand side and the left – hand side. They established the relationship between FLC₀ and the strain hardening exponent (n) and sheet thickness (t), indicating that FLC₀ increases with the increase of these two parameters. However, this relationship is only valid when the thickness does not exceed 3.1 mm. 

1

 

For the left – hand side of the curve:

2

are the major and minor strains, respectively.

 

For the right – hand side of the curve:

3

 

2.2 Paul Model

Paul’s[16] non – linear regression model uses a non – linear regression approach to calculate FLC₀ based on selected variables, such as ultimate tensile strength (σUTS), total strain (ϵt), Lankford coefficient (r), strain hardening exponent (n), and sheet thickness (t). 

4

According to Paul’s fitting correlation, p and FLC₀ show an exponential decay relationship. Therefore, in the reference literature, p is expressed as:

5

where p is the material constant

 

For the left – hand side of the curve:

6

For the right – hand side of the curve:

7

 

3. Review of ANN Models for Complete FLC Prediction

3.1 Overview of the ANN Model for FLC₀ Prediction

An Artificial Neural Network (ANN) is a machine – learning model designed to mimic the structure and function of the human brain. It consists of interconnected layers of neurons (nodes) that can learn from data, process information, and recognize patterns. A typical structure includes an input layer, one or more hidden layers, and an output layer. These layers enable the network to adjust based on feedback, thus performing tasks such as classification, prediction, and optimization.
 
Figure 2 shows a schematic diagram of the ANN model for FLC₀ prediction. The list of data sources used in this study is shown in Table 1. For FLC₀ prediction, the input layer of the ANN model contains seven nodes, each representing a different material parameter: thickness, yield strength, ultimate tensile strength, normal anisotropy coefficient, uniform elongation, total elongation, and strain hardening exponent. The data are standardized using a Min – Max normalizer to ensure that all features have the same weight during training, and no bias is introduced at this stage. 

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The network contains two hidden layers, which help in learning complex relationships in the data. The first hidden layer has two neurons, and the second hidden layer has one neuron, both of which use the ReLU (Rectified Linear Unit) activation function. ReLU is a widely used function that introduces non – linearity into the model, enabling it to handle more complex data. The output layer consists of one node, which uses a linear activation function to handle continuous outputs and produces the predicted FLC₀ value.
 
The network is trained using the ADAM (Adaptive Moment Estimation) optimizer. ADAM is favored for its ability to dynamically adjust the learning rate, accelerate convergence, and improve performance. The dataset is divided into 80% for training and 20% for testing to ensure that the model generalizes well to unseen data. During training, a batch size of 4 is selected to balance speed and accuracy. The model training process is shown in Figure 3. [Insert Figure 3 here: Model training process of ANN]
 
To avoid overfitting, the model uses an early – stopping strategy. If the validation loss does not improve for five consecutive epochs, the training is stopped. This prevents unnecessary iterations and saves computational resources. After training, the model is evaluated on the test data and saved for future use.
 
The accuracy of the model is evaluated using the Mean Squared Error (MSE) as the loss function. MSE measures the average squared difference between the predicted and actual values, reflecting the prediction accuracy of the model. MSE can be calculated by formula (8):

8

 where n is the number of data points, \(FLC_{0e}\) is the experimentally measured FLC₀ value, and FLC0p is the predicted FLC₀ value. MSE is chosen because of its smooth, differentiable gradient, which is crucial for optimization. Additionally, MSE penalizes larger prediction errors more heavily, ensuring that the model focuses on minimizing significant deviations, thus achieving more accurate predictions.
 

3.2 Overview of the ANN Model for the Exponent p in FLC Prediction

The ANN model for predicting the exponent p on the right – hand side of the FLC has a structure similar to that of the model for FLC₀ prediction. The data sources for this study are shown in Table 2. The number of input nodes, hidden layers, neurons, activation functions, optimizers, and the training process are the same as those of the previous model. This model also takes FLC₀ as an additional input parameter. The training process of the model follows the same steps outlined in Figure 3. 
 

4. Results and Discussion

4.1 Calculation of FLC₀ Values

The input and output data sources used in this study are detailed in Table 1. Initially, the FLC₀ values were calculated using the analytical models developed by Keeler – Brazier (KB) and Paul, according to formulas (1) and (4), respectively. For the Keeler – Brazier model, the required basic inputs include the strain hardening exponent (n) and sheet thickness (t); while the Paul model requires a broader range of input parameters, including ultimate tensile strength (σUTS), total strain (ϵt), Lankford coefficient (r), strain hardening exponent (n), and sheet thickness (t).
 
Figure 4(a) shows the prediction capabilities of the Keeler – Brazier model and the Paul model for FLC₀, and Figure 4(b) shows the prediction capabilities of the ANN model for FLC₀. It is evident from the figures that the ANN model has the highest prediction accuracy for FLC₀, followed by the Paul model, while the Keeler – Brazier model has the lowest prediction ability among the three. Figure 5 shows the training and validation loss curves as a function of epochs, which exhibit a clear downward trend and approach zero. [Insert Figure 4 here: Comparison of experimentally measured and predicted FLC₀ by different models, (a) KB model and Paul model, (b) ANN model] [Insert Figure 5 here: Training and validation loss curves as a function of epochs]
 

4.2 Calculation of p Values

The input and output data sources for calculating the p values are summarized in Table 2. In the Keeler – Brazier (KB) model, p is usually taken as a fixed value of 0.5. In contrast, in the Paul analytical model, p is determined by formula (5)
 

References:

 

    • [1]S.K.,Controlling factors of forming limit curve: a review,Adv. Ind. Manuf. Eng., 2 (2021), Article 100033
    • [2]S.K.,Theoretical analysis of strain- and stress-based forming limit diagrams,J. StrainAnal., 48 (3) (2013), pp. 177-188
    • [3]R. Narayanasamy, N.L. Parthasarathi, C. Sathiya Narayanan, T. Venugopal, H.T.,A study on fracture behaviour of three different high strength low alloy steel sheets during formation with different strain ratios,Mater. Des., 29 (2008), pp. 1868-1885
    • [4]S. Panich, F. Barlat, V. Uthaisangsuk, S. Suranuntchai, S.,Experimental and theoretical formability analysis using strain and stress based forming limit diagram for advanced high strength steels,Mater. Des., 51 (2013), pp. 756-766
    • [5]S.K.,Path independent limiting criteria in sheet metal forming,J. Manuf. Process., 20 (2015), pp. 291-303
    • [6]S.K. Paul, S. Roy, S. Sivaprasad, S.,Forming limit diagram generation from in-plane uniaxial and notch tensile test with local strain measurement through digital image correlation,Phys. Mesomech., 22 (2019), pp. 340-344
    • [7]Z. Marciniak, K.,Limit strains in the processes of stretch-forming sheet metal,Int. J. Mech. Sci., 9 (1967), pp. 609-620
    • [8]K. Nakazima, T. Kikuuma, K.,Study on the formability of steel sheets,Yawata Tech. Rep., 264 (1968), pp. 141-154
    • [9]S.P. Keeler, W.A. Backhofen,Plastic instability and fracture in sheet stretched over rigid punches,ASM Trans. Q., 48 (1964)56-25.
    • [10]D. Banabic, L. Lazarescu, L. Paraianu, I. Ciobanu, Nicodim, D.S. Comsa,Development of a new procedure for the experimental determination of the forming limit curves,Ann. Cirp., 62 (2013), pp. 255-258
    • [11]Y.M. Hwang, Y.K. Lin, J.C. Chuang,Forming limit diagrams of tubular materials by bulge tests,J. Mater. Process. Technol. (2009)
    • [12]H.W. Swift,Plastic insolubility under plane stress,J. Mech. Phys. Solid (1952), pp. 1-18
    • [13]R. Hill,On discontinuous plastic states with special reference to localized necking in thin sheets,J. Mech. Phys. Solid (1952), pp. 19-30
    • [14]S.P. Keeler, W.G.Brazier: Relationship between laboratory materialcharacterization and press shop formability, In Micro alloying, An International symposium on high strength, Low Alloy Steel Proceedings, Washington, DC, USA, (1977) 517–528.
    • [15]K.S. Raghavan, R.C. Van Kuren, H. Darlington, Recent progress in the devel opment of forming limit curves for automotive sheet steel. SAE Technical Paper (1992), 920437.
    • [16]S.K. Paul,Prediction of complete forming limit diagram from tensile properties of various steel sheets by a nonlinear regression based approach,J. Manuf. Process., 23 (2016), pp. 192-200
    • [17]S.K. Paul, M. Ganapathy, R.K. Verma,Prediction of entire forming limit diagram from simple tensile material properties,J. Strain Anal. Eng., 48 (2013), pp. 386-394
    • [18]Cayssials, F.: A new method for predicting FLC. In Proceedings of the 20th IDDRG Congress, Brussels, Belgium (1998) 443–454.
    • [19]F. Cayssials, X. Lemoine, 2005, Predictive model for FLC (arcelor model) upgraded to UHSS steels. In Proceedings of the 24th IDDRG Conference, Besançon, France (2005) 17.1–17.8.
    • [20]M. Abspoel, M.E. Scholting, J.M.M. Droog,A new method for predicting forming limit curves from mechanical properties,J. Mater. Process Technol., 213 (2013), pp. 759-769
    • [21]J. Gerlach, L. Kessler, A. Kohler, The forming limit curve as a measure of formability is an increase of testing necessary for robustness simulations. In Proceedings of the IDDRG 50th Anniversary Conference, Graz, Austria, 31 (2010) 479–488.
    • [22]N. Kotkunde, A.D. Deole, A.K. Gupta,Prediction of forming limit diagram for Ti-6Al-4V alloy using artificial neural network,Proc. Mater. Sci., 6 (2014), pp. 341-346
    • [23]S. Kannadasan, A. Senthil Kumar, C. Pandivelan, C. Sathiya Narayanan,Modelling the forming limit diagram for aluminium alloy sheets using ANN and ANFIS,Appl. Math. Inf. Sci., 11 (5) (2017), pp. 1435-1442
    • [24]I. Czinege, D. Harangozo,Application of artificial neural networks for characterisation of formability properties of sheet metals,Int. J. Lightweight Mater. Manuf., 7 (2024), pp. 37-44
    • [25]D.Ravi Kumar,Formability analysis of extra-deep drawing steel,J. Mater. Process Technol., 130 (131) (2002), pp. 31-41
    • [26]W. Bleck, Z. Deng, K. Papamantellos, C.O. Gusek,A comparative study of the forming-limit diagram models for sheet steels,J. Mater. Process Technol., 83 (1998), pp. 223-230
    • [27]K. Chung, C. Lee, H. Kim,Forming limit criterion for ductile anisotropic sheets as a material property and its deformation path insensitivity, Part II: boundary value problems,Int J. Plast., 58 (2014), pp. 35-65
    • [28]S. Panich, F. Barlat, V. Uthaisangsuk, S. Suranuntchai, S. Jirathearanat,Experimen tal and theoretical formability analysis using strain and stress based forming limit diagram for advanced high strength steels,Mater. Des., 51 (2013), pp. 756-766
    • [29]F. Shi, Strain hardening and forming limits of automotive steels, SAE Transactions, vol 104,950700 (1995) 571-577.
    • [30]A.A. Konieczny,On formability assessment of the automotive dual phase steels SAE2001-01-3075,Proceedings of SAE 2001 World,Congress, SAE (2001)
    • [31]K. Chung, K. Ahn, D.H. Yoo, K.H. Chung, M.H. Seo, S.H. Park,Formability of TWIP (twinning induced plasticity) automotive sheets,Int. J. Plast., 27 (2011), pp. 52-81
    • [32]W.J. Chen, H.W. Song, S.F. Chen, Y. Xu, S.Y. Deng, Z. Cai, X.H. Pei, S.H. Zhang, 2024, A New Phenomenological Model to Predict Forming Limit Curves from Tensile Properties for Hot-Rolled Steel Sheets, Metals (2024) 14-168. ϵt

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