Theoretical and Numerical Comparison of Limit Point Bifurcation and Maximum Force Criteria. Application to the Prediction of Diffuse Necking

ABSTRACT

A large number of criteria has been developed to predict material instabilities, but their choice is limited by the lack of existing comparison of their theoretical basis and application domains. To overcome this limitation, a theoretical and numerical comparison of two major models used to predict diffuse necking is present in this paper. Limit Point Bifurcation criterion is first introduced. An original formulation of the Maximum Force Criterion (MFC), taking into account the effects of damage and isotropic and kinematic hardenings, is then proposed. Strong connections are shown between them by comparing their theoretical basis. Numerical Forming Limit Diagrams at diffuse necking obtained with these criteria for different metallic materials are given. They illustrate the theoretical link and similar predictions are shown for both models.

A large number of criteria has been developed to predict material instabilities, but their choice is limited by the lack of existing comparison of their theoretical basis and application domains. To overcome this limitation, a theoretical and numerical comparison of two major models used to predict diffuse necking is present in this paper. Limit Point Bifurcation criterion is first introduced. An original formulation of the Maximum Force Criterion (MFC), taking into account the effects of damage and isotropic and kinematic hardenings, is then proposed. Strong connections are shown between them by comparing their theoretical basis. Numerical Forming Limit Diagrams at diffuse necking obtained with these criteria for different metallic materials are given. They illustrate the theoretical link and similar predictions are shown for both models.

KEYWORDS

Forming; Bifurcation Analysis; Maximum Force Criterion; Diffuse Necking; Forming Limit Diagram

Forming; Bifurcation Analysis; Maximum Force Criterion; Diffuse Necking; Forming Limit Diagram

Cite this paper

G. Altmeyer, "Theoretical and Numerical Comparison of Limit Point Bifurcation and Maximum Force Criteria. Application to the Prediction of Diffuse Necking,"*Modeling and Numerical Simulation of Material Science*, Vol. 3 No. 1, 2013, pp. 39-47. doi: 10.4236/mnsms.2013.31006.

G. Altmeyer, "Theoretical and Numerical Comparison of Limit Point Bifurcation and Maximum Force Criteria. Application to the Prediction of Diffuse Necking,"

References

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[2] K. Nakazima, T. Kikuma and K. Hasuka “Study on the Formability of Steel Sheets,” Yamata Technical Report, No. 264, 1968, pp. 8517-8530.

[3] G. Altmeyer, “Theoretical and Numerical Modeling of Plastic Instability Criteria. Application to the Prediction of Necking and Localization Phenomena,” Ph.D. Thesis, Arts et Métiers ParisTech, Metz, 2011.

[4] Z. Marciniak and K. Kuczyński “Limit Strains in the Processes of Stretch-Forming Sheet Metal,” International Journal of Mechanical Sciences, Vol. 9, No. 9, 1967, pp. 613-620. doi:10.1016/0020-7403(67)90066-5

[5] A. Reyes, O. S. Hopperstad, T. Berstad and O. G. Lademo, “Prediction of Necking for Two Aluminium Alloys under Non-Proportional Loading by Using an FEBased Approach,” International Journal of Material Forming, Vol. 1, No. 4, 2008, pp. 211-232. doi:10.1007/s12289-008-0384-6

[6] P. Eyckens, A. V. Bael and P. V. Houtte, “MarciniakKuczynski Type Modelling of the Effect of throughThickness Shear on the Forming Limits of Sheet Metal,” International Journal of Plasticity, Vol. 25, No. 12, 2009, pp. 2249-2268. doi:10.1016/j.ijplas.2009.02.002

[7] A. Considère, “Report on the Use of Iron and Steel in Constructions,” Annals of Ponts and Chaussées, Vol. 9, 1885, p. 574.

[8] H. W. Swift, “Plastic Instability under Plane Stress,” Journal of the Mechanics and Physics of Solids, Vol. 1, No. 1, 1952, pp. 1-18. doi:10.1016/0022-5096(52)90002-1

[9] R. Hill, “On Discontinuous Plastic States, with Special Reference to Localized Necking in Thin Sheets,” Journal of the Mechanics and Physics of Solids, Vol. 1, No. 1, 1952, pp. 19-30. doi:10.1016/0022-5096(52)90003-3

[10] P. Hora, L. Tong and J. Reissner, “A Prediction Method of Ductile Sheet Metal Failure in FE Simulation,” Proceedings of Numisheet 1996, Zaragoza, 1996, pp. 252256.

[11] M. Brunet and F. Morestin, “Experimental and Analytical Necking Studies of Anisotropic Sheet Metals,” Journal of Materials Processing Technology, Vol. 112, No. 2-3, 2001, pp. 214-226. doi:10.1016/S0924-0136(01)00578-7

[12] R. Hill, “On the Mechanics of Localized Necking in Anisotropic Sheet Metals,” Journal of the Mechanics and Physics of Solids, Vol. 49, No. 9, 2001, pp. 2055-2070. doi:10.1016/S0022-5096(01)00031-X

[13] P. Hora and L. Tong, “Numerical Prediction of FLC Using the Enhanced Modified Maximum Force Criterion (EMMFC),” Proceedings of FLC Zurich 2006, 2006, pp. 31-36.

[14] K. Mattiasson, M. Sigvant and M. Larson, “Methods for Forming Limit Prediction in Ductile Metal Sheets,” Proceedings of IDDRG 2006, Porto, 2006, pp. 1-9.

[15] H. Aretz, “Numerical Analysis of Diffuse and Localized Necking in Orthotropic Sheet Metals,” International Journal of Plasticity, Vol. 23, No. 5, 2007, pp. 798-840. doi:10.1016/j.ijplas.2006.07.005

[16] D. Banabic, S. Comsa, P. Jurco, S. Wagner, S. He and P. Van Houtte, “Prediction of Forming Limit Curves from Two Anisotropic Constitutive Models,” In: D. Banabic, Ed., Formability of Metallic Materials, Springer, Berlin, 2000.

[17] D. C. Drucker, “Some Implications of Work Hardening and Ideal Plasticity,” Quarterly of Applied Mathmatics, Vol. 7, 1950, pp. 411-418.

[18] R. Hill, “A General Theory of Uniqueness and Stability in Elastic-Plastic Solids,” Journal of the Mechanics and Physics of Solids, Vol. 6, No. 3, 1958, pp. 236-249. doi:10.1016/0022-5096(58)90029-2

[19] J. R. Rice, “The Localization of Plastic Deformation,” Proceedings of the 14th IUTAM Congress, Delft, 1976, pp. 207-220.

[20] J. W. Rudnicki and J. R. Rice, “Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials,” Journal of the Mechanics and Physics of Solids, Vol. 23, No. 6, 1975, pp. 371-394. doi:10.1016/0022-5096(75)90001-0

[21] S. St?ren and J. R. Rice, “Localized Necking in Thin Sheets,” Journal of the Mechanics and Physics of Solids, Vol. 23, No. 6, 1975, pp. 421-441. doi:10.1016/0022-5096(75)90004-6

[22] A. Molinari and R. J. Clifton, “Viscoplastic Strain Localization in Simple Shear Mode: Exact Results in Non-Linear Theory,” CRAS Série II, Vol. 296, 1983, pp. 1-4.

[23] G. Barbier, A. Benallal and V. Cano, “Theoretical Relation between Linear Perturbation Method and Bifurcation Analysis Method for Strain Localization Prediction,” CRAS-Series IIB-Mechanics-Physics-Chemistry-Astronomy, Vol. 326, No. 3, 1998, pp. 153-158.

[24] A. Benallal, “Perturbation and Stability of Rate-Dependent Solids,” European Journal of Mechanics-A/Solids, Vol. 19, 2000, pp. 45-60.

[25] A. Lejeune, N. Boudeau, and J. C. Gelin, “Influence of Material and Process Parameters on Bursting During Hydroforming Process,” Journal of Materials Processing Technology, Vol. 143-144, No. 1, 2003, pp. 11-17. doi:10.1016/S0924-0136(03)00295-4

[26] K. C. Valanis, “Banding and Stability in Plastic Materials,” Acta Mechanica, Vol. 79, No. 1-2, 1989, pp. 113-141. doi:10.1007/BF01181483

[27] M. K. Neilsen and H. L. Schreyer, “Bifurcations in Elastic-Plastic Materials,” International Journal of Solids and Structures, Vol. 30, No. 4, 1993, pp. 521-544. doi:10.1016/0020-7683(93)90185-A

[28] M. Habbad, “Plastic Instabilities in Anisotropic ElastoPlasticity and Large Transformations,” Ph.D. Thesis, Centrale Lyon, Lyon, 1994.

[29] M. Ben Tahar, “Contribution to Study and Simulation of Hydroforming Process,” Ph.D. Thesis, ENSMP, Nice, 2005.

[30] S. Boumaiza, “Instabilities Prediction during Tubes Hydroforming,” Ph.D. Thesis, Insa-Lyon, Lyon, 2005.

[31] J. Lemaitre and J. L. Chaboche, “Mechanics of Solid Materials,” Cambridge University, Cambridge, 1990.

[32] B. Haddag, “Contribution to Sheet Metal Forming: Application to Springback and Localization,” Ph.D. Thesis, Arts et Métiers ParisTech, Metz, 2007.

[33] J. Lemaitre, “Continuous Damage Mechanics Model for Ductile Fracture,” Journal of Engineering Materials and Technology, Transactions of the ASME, Vol. 107, No. 1, 1985, pp. 83-89. doi:10.1115/1.3225775

[34] G. Altmeyer, F. Abed-Meraim and T. Balan, “Comparison of Forming Limit Diagrams Predicted with Different Localization Criteria,” Steel Research International, Vol. 79, No. 1, 2008, pp. 24-31.

[1] S. P. Keeler and W. G. Brazier, “Relationship between Laboratory Material Characterization and Press-Shop Formability,” Microalloying, Vol. 75, 1975, pp. 517-530.

[2] K. Nakazima, T. Kikuma and K. Hasuka “Study on the Formability of Steel Sheets,” Yamata Technical Report, No. 264, 1968, pp. 8517-8530.

[3] G. Altmeyer, “Theoretical and Numerical Modeling of Plastic Instability Criteria. Application to the Prediction of Necking and Localization Phenomena,” Ph.D. Thesis, Arts et Métiers ParisTech, Metz, 2011.

[4] Z. Marciniak and K. Kuczyński “Limit Strains in the Processes of Stretch-Forming Sheet Metal,” International Journal of Mechanical Sciences, Vol. 9, No. 9, 1967, pp. 613-620. doi:10.1016/0020-7403(67)90066-5

[5] A. Reyes, O. S. Hopperstad, T. Berstad and O. G. Lademo, “Prediction of Necking for Two Aluminium Alloys under Non-Proportional Loading by Using an FEBased Approach,” International Journal of Material Forming, Vol. 1, No. 4, 2008, pp. 211-232. doi:10.1007/s12289-008-0384-6

[6] P. Eyckens, A. V. Bael and P. V. Houtte, “MarciniakKuczynski Type Modelling of the Effect of throughThickness Shear on the Forming Limits of Sheet Metal,” International Journal of Plasticity, Vol. 25, No. 12, 2009, pp. 2249-2268. doi:10.1016/j.ijplas.2009.02.002

[7] A. Considère, “Report on the Use of Iron and Steel in Constructions,” Annals of Ponts and Chaussées, Vol. 9, 1885, p. 574.

[8] H. W. Swift, “Plastic Instability under Plane Stress,” Journal of the Mechanics and Physics of Solids, Vol. 1, No. 1, 1952, pp. 1-18. doi:10.1016/0022-5096(52)90002-1

[9] R. Hill, “On Discontinuous Plastic States, with Special Reference to Localized Necking in Thin Sheets,” Journal of the Mechanics and Physics of Solids, Vol. 1, No. 1, 1952, pp. 19-30. doi:10.1016/0022-5096(52)90003-3

[10] P. Hora, L. Tong and J. Reissner, “A Prediction Method of Ductile Sheet Metal Failure in FE Simulation,” Proceedings of Numisheet 1996, Zaragoza, 1996, pp. 252256.

[11] M. Brunet and F. Morestin, “Experimental and Analytical Necking Studies of Anisotropic Sheet Metals,” Journal of Materials Processing Technology, Vol. 112, No. 2-3, 2001, pp. 214-226. doi:10.1016/S0924-0136(01)00578-7

[12] R. Hill, “On the Mechanics of Localized Necking in Anisotropic Sheet Metals,” Journal of the Mechanics and Physics of Solids, Vol. 49, No. 9, 2001, pp. 2055-2070. doi:10.1016/S0022-5096(01)00031-X

[13] P. Hora and L. Tong, “Numerical Prediction of FLC Using the Enhanced Modified Maximum Force Criterion (EMMFC),” Proceedings of FLC Zurich 2006, 2006, pp. 31-36.

[14] K. Mattiasson, M. Sigvant and M. Larson, “Methods for Forming Limit Prediction in Ductile Metal Sheets,” Proceedings of IDDRG 2006, Porto, 2006, pp. 1-9.

[15] H. Aretz, “Numerical Analysis of Diffuse and Localized Necking in Orthotropic Sheet Metals,” International Journal of Plasticity, Vol. 23, No. 5, 2007, pp. 798-840. doi:10.1016/j.ijplas.2006.07.005

[16] D. Banabic, S. Comsa, P. Jurco, S. Wagner, S. He and P. Van Houtte, “Prediction of Forming Limit Curves from Two Anisotropic Constitutive Models,” In: D. Banabic, Ed., Formability of Metallic Materials, Springer, Berlin, 2000.

[17] D. C. Drucker, “Some Implications of Work Hardening and Ideal Plasticity,” Quarterly of Applied Mathmatics, Vol. 7, 1950, pp. 411-418.

[18] R. Hill, “A General Theory of Uniqueness and Stability in Elastic-Plastic Solids,” Journal of the Mechanics and Physics of Solids, Vol. 6, No. 3, 1958, pp. 236-249. doi:10.1016/0022-5096(58)90029-2

[19] J. R. Rice, “The Localization of Plastic Deformation,” Proceedings of the 14th IUTAM Congress, Delft, 1976, pp. 207-220.

[20] J. W. Rudnicki and J. R. Rice, “Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials,” Journal of the Mechanics and Physics of Solids, Vol. 23, No. 6, 1975, pp. 371-394. doi:10.1016/0022-5096(75)90001-0

[21] S. St?ren and J. R. Rice, “Localized Necking in Thin Sheets,” Journal of the Mechanics and Physics of Solids, Vol. 23, No. 6, 1975, pp. 421-441. doi:10.1016/0022-5096(75)90004-6

[22] A. Molinari and R. J. Clifton, “Viscoplastic Strain Localization in Simple Shear Mode: Exact Results in Non-Linear Theory,” CRAS Série II, Vol. 296, 1983, pp. 1-4.

[23] G. Barbier, A. Benallal and V. Cano, “Theoretical Relation between Linear Perturbation Method and Bifurcation Analysis Method for Strain Localization Prediction,” CRAS-Series IIB-Mechanics-Physics-Chemistry-Astronomy, Vol. 326, No. 3, 1998, pp. 153-158.

[24] A. Benallal, “Perturbation and Stability of Rate-Dependent Solids,” European Journal of Mechanics-A/Solids, Vol. 19, 2000, pp. 45-60.

[25] A. Lejeune, N. Boudeau, and J. C. Gelin, “Influence of Material and Process Parameters on Bursting During Hydroforming Process,” Journal of Materials Processing Technology, Vol. 143-144, No. 1, 2003, pp. 11-17. doi:10.1016/S0924-0136(03)00295-4

[26] K. C. Valanis, “Banding and Stability in Plastic Materials,” Acta Mechanica, Vol. 79, No. 1-2, 1989, pp. 113-141. doi:10.1007/BF01181483

[27] M. K. Neilsen and H. L. Schreyer, “Bifurcations in Elastic-Plastic Materials,” International Journal of Solids and Structures, Vol. 30, No. 4, 1993, pp. 521-544. doi:10.1016/0020-7683(93)90185-A

[28] M. Habbad, “Plastic Instabilities in Anisotropic ElastoPlasticity and Large Transformations,” Ph.D. Thesis, Centrale Lyon, Lyon, 1994.

[29] M. Ben Tahar, “Contribution to Study and Simulation of Hydroforming Process,” Ph.D. Thesis, ENSMP, Nice, 2005.

[30] S. Boumaiza, “Instabilities Prediction during Tubes Hydroforming,” Ph.D. Thesis, Insa-Lyon, Lyon, 2005.

[31] J. Lemaitre and J. L. Chaboche, “Mechanics of Solid Materials,” Cambridge University, Cambridge, 1990.

[32] B. Haddag, “Contribution to Sheet Metal Forming: Application to Springback and Localization,” Ph.D. Thesis, Arts et Métiers ParisTech, Metz, 2007.

[33] J. Lemaitre, “Continuous Damage Mechanics Model for Ductile Fracture,” Journal of Engineering Materials and Technology, Transactions of the ASME, Vol. 107, No. 1, 1985, pp. 83-89. doi:10.1115/1.3225775

[34] G. Altmeyer, F. Abed-Meraim and T. Balan, “Comparison of Forming Limit Diagrams Predicted with Different Localization Criteria,” Steel Research International, Vol. 79, No. 1, 2008, pp. 24-31.