# Production Optimization of Premium Food Can with Distortion Printing under Waving Requirement
## 3 Material Parameter Identification
### Parameter Identification via Thickness Distribution  
This section demonstrates the use of the inverse analysis to identify the material parameter, specifically the normal anisotropy ratio (\(\overline{R}\)). The thickness distribution, as shown in Figure 13, serves as the primary criterion. At each angle relative to the rolling direction, the thickness is measured at nine specified locations. While the initial sheet thickness is assumed to be constant, variations occur in certain areas due to the forming process. The thickness distribution is examined through cross-sectional analysis. Specimens are prepared by molding them with resin and sectioning them in three directions (RD, DD, and TD). The molded specimens are ground and polished to reveal the thickness profiles, as shown in Figure 14. A digital microscope (Olympus, Model DSX500-MSU, resolution 0.01 \(\upmu\)m) with a maximum magnification of 250\(\times\), along with image analysis software, is used to measure the thickness values at nine different locations per direction. For each location, three measurements are taken, and the average value is used to represent the thickness at that location. Digital image correlation (DIC) is the second technique used to obtain thickness distribution. DIC evaluates the surface strain levels of sheet metal parts after forming. Prior to forming, laminated TFS blanks undergo grid marking using offset printing with a dot pattern (dot diameter of 1.5 mm and spacing of 2.0 mm), as shown in Figure 15.  
Figure 12: Samples after tensile tests for three directions from three labs. Figure 13: Thickness measurement locations on the sidewall. After forming, the laminated TFS sheet is recorded from various viewing angles using the handheld ARGUS system, enabling the measurement of principal strains. The results provide strain distributions on the parts, including major and minor strains. ARAMIS sensors measure statically loaded specimens and parts using a contact-free, material-independent method based on the principle of DIC. Thickness is determined by the major and minor strains, assuming the constant volume deformation at RD, DD, and TD at nine different locations per direction. The thickness is calculated using the relationship \(t=t_{0}\cdot\exp(\epsilon_{t})\), where \(t\) is thickness, \(t_{0}\) is base steel thickness of 0.16 mm, major strain is \(\epsilon_{1}\), and minor strain is \(\epsilon_{2}\) (\(\epsilon_{t}=-(\epsilon_{1}+\epsilon_{2})\)), as shown in Figure 16.  
The thickness distributions at nine locations on the can wall, obtained from ten samples using both microscopy and DIC, are compared and presented in Figure 17. The results from these two methods exhibit negligible differences and align with the trends reported in [23]. The thickness measurements at the nine locations are further analyzed using the sum of squared error (SSE), also known as the residual sum of squares, which represents the difference between the experimental values and FEA values. A lower SSE indicates a better agreement with the experimental data. Figure 18 shows SSE thickness in relation to \(\overline{R}\) (independent variables ranging from 0.3 to 0.9). It identifies an optimal \(\overline{R}\) value at 0.5 in FEA for a die radius of 1.5 mm and COF of 0.03 as the process parameters. Figure 14: Thickness measurement: (**a**) molded specimens; (**b**) 250\(\times\) image; (**c**) digital microscope. Figure 15: DIC measurement: (**a**) setup; (**b**) grid configuration; (**c**) flat sheet; (**d**) drawing; (**e**) redrawing. Figure 16: Strain distribution from DIC: (**a**) major strain after drawing; (**b**) minor strain after drawing; (**c**) major strain after redrawing; (**d**) minor strain after redrawing. Figure 17: Comparison of microscope and DIC measurement. Figure 15.0 SSE average thickness as a function of \(\overline{R}\).