Probabilistic design of aluminum sheet drawing for reduced risk of wrinkling and fracture
Introduction
Deep drawing is a process for transforming flat sheet blanks into cup- or box-shaped parts without defects. These parts typically find applications in the appliance, automotive and aerospace industries. In industrial deep drawing operations, dominant defects are springback, excessive localized thinning [1] (and fracture) of the part walls, and flange and side-wall wrinkling [2]. Uncompensated springback in the part dimensions can lead to assembly problems; thinning and fracture directly relate to part performance and safety, while wrinkling is often related to part quality. Consequently, enormous design and control efforts are directed towards reducing and eliminating these defects through the proper design of the blank, the design of tooling configuration, and the selection and control of process parameters. The blank design includes optimizing blank geometries and sheet thickness [3], the tooling design involves determination of optimal punch and die radii, punch and die clearance, and draw-bead shape and location [4], [5], and finally, the process design seeks to find the best setting for process parameters such as friction (at the blank holder, die radii and punch corners), the punch speed, the blank holder force (BHF), etc. While blank and tooling designs are often done a priori and require considerable effort to change, process parameters can be designed off-line, and controlled during the process execution to compensate for process transients. Draw bead control, BHF control and flexible binder are some of the strategies used to handle process variabilities.
Even if the drawing process is optimally designed and controlled, the part during production has significant scatter in dimensions and properties that degrade its safety and quality. This process uncertainty could be due to the process inputs or uncontrolled parameters, shown in Fig. 1, such as BHF, press dynamics or friction. Majeske and Hammett [6] analyzed data from several leading automobile manufacturers, and highlighted that part variability still remains a predominant issue especially in the drawing of complex and thin-walled parts. They reported that within the same batch (same die and process setup) part-to-part geometric variation can be as high as 21%. This high variation inevitably results in high scrap rate, frequent rework, machine shut down and huge loss in production. Recently, a few researchers have investigated the effect of process variability on part quality. For example, Gantar and Kuzman [7] identified 12 most important influencing input parameters in the deep drawing process for square-shaped parts and measured their influence in the scatter in performance. They found the material properties (hardening coefficient, yield stress, flow strength and anisotropy), the initial sheet thickness and the friction coefficient to be the dominant variables influencing wrinkling and necking (thinning) behavior. Karthik et al. [8] investigated the coil-to-coil variability in sheet material properties by carrying out measurements using more than 45 coils of the same material. They found that the strain hardening coefficient “n” can have coil-to-coil variation of up to 14%. This significantly influences springback and thinning behavior during deep drawing. Finally, Cao and Kinsey [9] pointed out that during sheet bending process the variation of material strength “K” can be as high as 20%, the strain hardening coefficient 16% and the friction coefficient 65%. To give a better idea about how much the material property can vary, the results from [7] and [8] are listed together in Table 1.
There are two strategies to handle these material, tooling and process uncertainties: process design or feedback/adaptive control [9]. Extensive research has been done in exploring the deterministic effect of each design factor on the part quality, but very little on the impact of process uncertainties and their interaction with part and process design in influencing part variability [10], [11], [12], [13], [14]. The inclusion of uncertainty in the design and optimization cycle should lead to better understanding of the impact of uncertainty associated with system input on the system output. This understanding can then be applied for managing such uncertainties. Such designs that incorporate uncertainty are more popularly referred to as probabilistic design (PD) [15]. For example, Buranithi et al. [10] used the three-point-based weighted method to estimate the statistical characteristics, mean and variance, of the margins of failure for tearing and wrinkling responses. Margins to failure were determined by using the forming limit diagrams (FLD) for the selected material. Finite element model of a wheel house stamping was used for process simulations and a Latin-hyper cube experimental design used to generate a surrogate model which is optimized using sequential quadratic programming. Results from Monte Carlo Simulations (MCS) were used as a benchmark for validating their results. Klieber et al. [11] investigated uncertainty in the drawing of square box. They used interface friction and blank holding force as design parameters and the sheet FLD to estimate the safe zone “feasible sample space.” An adaptive MCS method was chosen for reliability assessment. They observed very high sensitivity of local necking with friction coefficients. They found the adaptive MCS to be computationally intensive and suitable for small- to medium-sized parts. Jansson and Nilsson [12], [13] studied process optimization. They found the FEM high-fidelity models and surrogate response surface models together with FLD diagrams suitable for uncertainty analysis and risk assessment. They used BHF, die design (unloading holes and performing punches) and size of draw beads to be the design variables. Sahai et al. [14] carried out sequential optimization (SORA) for reliability assessment of springback during two-dimensional sheet bending process. They found the computational intensity of probabilistic optimization to be the main constraint in the application of SORA to complex problems.
Probabilistic process design for improved part quality is the focus of this investigation with magnitude of side-wall thinning and wrinkling selected as the design criteria. A probabilistic formulation is presented, with a weighted approach to multi-objective optimization. The weighted linear sum approach is more relevant for industrial design optimization as it provides an intuitive feel for risk as the design approaches the probabilistic constraints (or bounds). Drawing process model is created by first representing the process in the FEM framework and then converting the high-fidelity FEM model to the RSM meta model. Process uncertainty is interrogated by MCS using a normal distribution of input and process parameters. This process is shown in Fig. 2. The drawing of a Hishida part geometry, used as a benchmark in the NUMISHEET conferences, is used to validate this approach. Details of the approach and selected results are presented in this paper.
Section snippets
General approach to probabilistic design in forming
The concept of PD was first introduced and developed in the reliability calculations of structures. A limit state function, basically a failure criterion with a deterministic model connecting the system input variables to the output variables, segments the multi-dimensional space spanned by the random variables into failure and safe domains. Given the joint probability density function of the random variables, the probability of failure can be calculated by the portion of points in the failure
Quality index to measure risk
There are several ways to incorporate risk in process simulation and design. Traditionally in single criteria design optimization, this is done by assessing the probability of failure and incorporating the impact of this failure (risk). The latter is often subjective and depends on the intended application of the product and an estimation of impact of failure. In sheet-forming analysis, the product risk is often associated with the quality of the product such as the magnitude of thinning and
Process model and problem formulation
The geometric representation of sheet and tooling for the drawing of Hishida part in the FEM framework is shown in Fig. 3, the drawn Hishida Part and the measurements on the sidewall are included in Fig. 4. The aluminum car body sheet alloy Ecodal-608-T4 (AA6181A) with 1 mm thickness is used for this study. The Krupkowsky law for work hardening (σy=K(ε0+εp)n) is used for modeling the material behavior. The drawing process is simulated using the finite element method-based commercial software
Surrogate RSM model
For the seven design variables, a Box-Behnken DOE design is selected [23]. This design allows efficient estimation of the first- and second-order coefficients of a quadratic response. Also because Box-Behnken has fewer design points, it is less expensive to run than other design methods such as central composite designs under the same number of factors. After a series of finite element simulations, the data on the magnitude of SW wrinkling and maximum thinning are collected and fitted by the
Probabilistic assessment for risk
The RSMs built in Section 5 are deterministic in nature. Given a design vector, the prediction is one certain value. This is unrealistic since many variables contained in the model, like material constants, friction coefficients, geometric dimensions, etc., are known to have a certain degree of scatter around their nominal values. The variation of system input variables should be incorporated into the process model.
To incorporate uncertainties, it is assumed that the vector of random variables Ψ
Optimal design approach
In Section 4, the criteria of defining wrinkling and fracture the two major forms of defects in the manufacturing of Hishida parts were given. The optimal design objective, therefore, is to find the right combination of BHF, Lub1 and Lub2 so as to minimize the weighted sum of the two defects rates or in other words maximize the QI. There are two approaches to achieve this objective.
The traditional way formulates the problem as
Effect of variation on the optimum design
In the previous PD, it was assumed that the variations of friction coefficient, strain hardening coefficient and strength coefficient are 10%, 13% and 6%. The real spread during manufacturing may be much larger. Consequently, the mean values and type of probability density function for the random variables were kept constant, while the percent of variation (and accordingly the standard deviations) was increased at different levels from 2% to 30%. Using MCS, the QI at each level can be
Conclusion
In this paper, the deep drawing process of the Hishida part is analyzed. Given the blank shape and tooling, a PD is successfully conducted to find the optimal combination of BHF and friction coefficient under the presence of variation of material properties. This design optimization uses an RSM model created using the FEM-based physical model and MCS-based technique to assess the effect of parameter uncertainty. A weighted sum approach is used to derive a scalar measure of efficacy, QI, for the
Acknowledgments
The authors acknowledge the financial support from the Manufacturing Research Group (MRG) at the Ohio State University, and the software support from the ESI Inc. in providing PAM-STAMP 2G software. Dr. Sheng Z.Q., Post-doctoral Research Associate at MRG, supported the PAM-STAMP based on the DOE for the Hishida part.
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