An approach for prediction of coil specific forming limit curves

1. An approach for prediction of coil specific forming limit curves Timo Manninen, Mikko Palosaari Outokumpu, Process R&D, Tornio, Finland Jari LarkiolaCentre of Advanced Steel Research University of Oulu, Oulu, Finland ESSC & DUPLEX 2019 Wien, 30th of September 2019

2. Contents • Introduction • Predicting coil specific forming limit curves • Application example: cold rolled stabilized ferritic grades produced in Outokumpu Tornio Works

3. Forming Limit Curve (FLC) • One of the most important tools in the forming engineer’s tool box • Used for predicting material failure in sheet metal forming processes • Defines much a specific material can deformed without localized necking and incipient failure • Largely used with FE-simulations for designing press tools • Commonly determined in the lab according to ISO 12004-2

4. Application

5. Experimental determination acc. ISO 12004-2 Tools for the Nakajima test

6. Cross-section method in ISO 12004-2 defines an objective mathematical criterion for limit strains Section lines ●= measured strain x = point in fitting window ● =limit strain value Fitted inverse parabola Crack Strain distribution just before crack initiation Major and minor principal strain along one section line. The resulting data point.

7. Prediction of forming limit curves • Statistical methods: FLC estimated based on strip properties • Pioneering work by Keeler and Brazier for mild steel (1977). • Practically all models developed for carbon steel. • Theoretical models • Bifurcation theory, geometrical imperfection theory or damage mechanics. • Pioneering works by Hill (1950’s) and Marciniak and Kuczynski (1967). • Challenge: Description of plastic flow in different regions of principal strain space • Small changes in hardening or yield surface shape may have a strong influence on the FLC. Keeler and Brazier equation

8. Prediction of coil specific FLCs • Measurement time-consuming => can measure one coil per 1000 coils produced. • Differences exist between coils produced by different process routes and process practices. • Influencing factors: • Steel family (austenitic, ferritic, duplex). • Physical metallurgy. • Product thickness. • Total amount of cold-rolling reduction and the number of number of cold rolling steps. • Line annealing practices (heat cycle). • Grainsize. • Currently we do not have enough knowledge on how these factors affect the shape yield surface shape to use the theoretical calculation methods. Fundamental difference => need separate models

9. Our approach One empirical model can describe the FLC for a group of steels if and only if the steels are sufficiently similar in terms of their similar physical metallurgy and mechanical properties. • Narrow case => Success. • No problem in having separate models for different steels and process routes. • The algorithm: • Select sufficiently narrow target group of steels. Do not be overly ambitious. • Gather set of test materials to capture the variation of the most important process parameters in their respective process windows • Measure FLCs and mechanical properties of sampled test materials. • Derive predictive equations for describing the FLC in terms of steel properties. First apply Tikhonov regularized least squares for reduction of parameter space. Then use regression analysis to solve the unknown model parameters in terms of the strip properties and process parameters.

10. Illustrative example case • Grade 1.4509, 1.4521 and 1.4622 ferritic stainless steels with 2B surface condition. • Representative sample of 12 coils • All coils produced with process route “A”. • Cold-rolling reduction from 60% to 90%. • Thickness from 0.6 mm to 2.0 mm. • Grainsize from 5.9 to 10.0 in the ASTM scale. • Hand picked test materials with independent variation of the three parameters. • Tensile test results and r-values also used as independent variables in the regression analysis. Windows for important process variables

11. Test materials

12. Parametrized function used to present FLC Parameter a0given by the continuity requirement Five unknown parameters to be estimated by regularized least squares and regression analysis. J. Gerlach, L. Kessler, A. Köhler, The forming limit curve as a measure of formability - Is an increase of testing necessary for robustness simulations? Proceedings of the 50th IDDRG Anniversary Conference, Graz, Austria, 2010: pp. 479–488.

13. Experimental uncertainly in the measurement data • Points have normal distribution with respect to their center of mass. • Experimental uncertainty in major principal strain 0.025 at the confidence level of 95%.

14. Application of Tikhonov regularized least squares method for estimation of model parameters • Ordinary least squares estimation results in over-fitted solution for full set of parameters (p=60). • Tikhonov regularization can be used to remove redundant parameters. • In the regularized least squares method an additional term is included in the target function to give preference to solutions with desired properties. • Morozov discrepancy principle: Desired accuracy level is achieved when the standard error of estimation (SEest) equals to the measurement error in data. • In our case: • Give preference to solutions where one or several model parameters have same value for all test materials. • Increase regularization in iterative manner until SEest ≈ 0.025

15. Results of regularized least squares analysis Optimal Estimated values of parameters common for all test materials

16. The outcome of the least squares analyses • Effective reduction of parameter space. • Root Mean Square Error (RMSE) is below measurement error in the data for all materials • Five parameters per test material (p=60) is serious over-fitting. • We need only 16 parameters to describe this data with good accuracy. RMSE for all test materials obtained with the optimal level of regularization

17. Regression analysis to predict the remaining parameters • The remaining parameter b0 can be modelled as a linear function of strip thickness t. RMSE of prediction obtained with the linear regression model for b0.

18. Two examples of achieved accuracy One 1.4509 coil with the thickness of 2.0 mm and having ASTM grainsize of 5.9. Three 1.4509 coils with the thickness of 0.6 mm and different ASTM grainsizes from 8.0 to 10.0.

19. Final touch – correction for the biaxial pre-strain • Small equibiaxial strain produced in the beginning of Nakajima test decreases the forming limits on the right hand side of the diagram and shifts the lowest point to the right from the plane strain axis • Guidelines for adjusting FLC given in the draftstandard ISO/DIS 12004-2:2012. • Adjustment applied directly to the parametrized FLC: FLC before and after the adjustment. Solid line = before. dashed line = after.

20. The final result

21. Discussion • According to Hill’s model for diffuse and localized necking the critical strain equals to Where the critical subtangent Z depends on the shape of the yield surface. For stainless steels the work hardening exponent is approximately: • How come we can predict the forming limits without tensile test results? • Work hardening exponent very similar to all materials. • Strong collinearity between thickness are several independent parameters. • Influence of grain size smaller than the experimental uncertainty in FLC data => disregarded.

22. Summary • An approach was proposed for predicting coil specific FLCs in the steel industry. • Original features: • Application to carefully selected, narrow target group. • Gathering test materials capturing the complete variation of process parameters. • Application of Tikhonov regularized least squares for eliminating redundant parameters. • Thanks to narrow target group and reduced of parameter space, simple regression model suffices for predicting the remaining model parameters. • The method was applied to a group of ferritic stainless steels produced by process route “A”. • Predictive equations were derived for describing the FLC for the target group. • Achieved accuracy of prediction is on the same level as measurement error in the data.

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