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Simulation of sheet metal forming processes by presenting a bending-dependent inverse isogeometric methodology

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Abstract

Recently, eliminating the gap between design and formability analysis of sheet metal parts has been studied to simulate sheet metal stamping processes. In this regard, a transfer-based inverse isogeometric formulation has been proposed. This method has various advantages such as solving the governing equations in two-dimensional networks without any concern about the convergence; however, it neglects the bending effect which is a major contributor in die/punch profile radii. The present work aims to consider the bending effects by introducing a bending-dependent inverse isogeometric formulation. The developed model deals with the minimization of potential energy, deformation theory of plasticity, classical plate theory, and considering a yield criterion in stress-resultant space. In addition to all advantages of the transfer-based inverse isogeometric formulation, one major benefit of this study is that the bending effects are included with a slight increase in the computation time. This methodology allows for accurately predicting the effects of changing die/punch profile radii and initial sheet thickness on the formability of the final part by presenting a new material updating process. To assess the credibility of this approach, an experimental setup and forward FEM software have been utilized to form a rectangular box. The results acquired by the developed method and those achieved by experiment and forward FEM reveal acceptable accuracy in the presented model. Also, strains and thicknesses predicted by the developed method, membrane inverse isogeometric model, and forward FEM for nine different values of punch radius to the sheet thickness ratio have been compared. Considering forward FEM as a reference method, the average of calculated error in the presented model for prediction of thickness at the middle of punch radius zone is around half of that in the membrane model. In solving the studied problems, the presented model requires only slightly more computation time (around 2%) than the membrane inverse isogeometric model and much less computation time than forward FEM. Therefore, the presented method is a valuable inverse forming solver especially when the bending effects are significant.

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All data generated or analyzed during this study are included in this manuscript.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9567, Tehran, Iran

    Mansoor Shamloofard & Ahmad Assempour

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  1. Mansoor Shamloofard
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Correspondence to Ahmad Assempour.

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Appendix

Appendix

The initial blank thickness, geometry of the final part, and material properties of the studied sheet are considered as the inputs of this algorithm. Also, the outputs of the presented model are strains/thicknesses of the deformed elements and the initial blank’s shape and size. Comprehensive guideline about how the presented algorithm is employed in the analysis of sheet metal forming processes is presented in the following:

  • Step #1: The desired sheet metal part is drawn in the CAD software.

  • Step #2: The knot vectors and control points required to draw the studied part are determined.

  • Step #3: In the first step of the solving procedure, thicknesses and material properties of the deformed elements are considered equal to the initial blank thickness and elastic properties of the used sheet. These assumptions are updated for the next iterations, according to steps #12 and #14.

  • Step #4: The curvatures of the final part elements are calculated using Eqs. (8) and (9).

  • Step #5: The projected (the initial guess of the blank) and transfer (representative of the final part) networks are constructed, according to Section 3.1.

  • Step #6: The in-plane force vector in each element is calculated using displacements of control points between the projected and transfer networks and also element stiffness matrix in the transfer network, according to Eq. (16).

  • Step #7: The non-uniform friction force vector is computed, according to Eq. (18).

  • Step #8: The external force vector is computed by assembling the in-plane and friction forces calculated in steps #6 and #7.

  • Step #9: The element stiffness matrices in the transfer state are assembled.

  • Step #10: By satisfying the minimum potential energy in the transfer state, the governing equation is solved using the assembled stiffness matrix and external force vector calculated in steps #8 and #9, according to Eq. (14).

  • Step #11: Displacements acquired by solving the governing equation (Eq. (14)) are applied to the projected network, and then, the new blank is constructed.

  • Step #12: Strains and thicknesses of the deformed elements are calculated using the continuum relations presented in ref. [43] between the new blank and the final part.

  • Step #13: Considering curvatures and strains calculated in steps #4 and #12, the equivalent strain of each element is computed, according to Eq. (25).

  • Step #14: Using the equivalent strains calculated in step #13, the material property matrix is updated for each element, according to Eq. (27).

  • Step #15: The external forces and stiffness matrices are re-calculated using the material properties computed in step #14 and thicknesses acquired in step #12. Then, the governing equation (Eq. (14)) is resolved, and the obtained displacements are added to the blank. Finally, strains and thicknesses are re-computed, similar to step #12.

  • Step #16: If differences between the results of the two last steps fall below a predefined threshold, the problem is converged. Otherwise, the procedure is repeated up to the convergence.

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Shamloofard, M., Assempour, A. Simulation of sheet metal forming processes by presenting a bending-dependent inverse isogeometric methodology. Int J Adv Manuf Technol 112, 1389–1408 (2021). https://doi.org/10.1007/s00170-020-06348-w

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