Abstract
The present work demonstrates that a non-associated Barlat’s Yld 2000-2D plastic flow stress potential gives better correlation accuracy with the Lankford and equal biaxial coefficients of plastic anisotropy than the associated flow rule. Additionally, new generalized exact equations are presented to calculate the Lankford and equal biaxial anisotropy coefficients deduced from the Yld 2000-2D function. The investigated metals were mildly and highly anisotropic Al 2024, Al 6022, Al 2090 aluminum alloys and AISI 409 steel sheets. The non-associated Barlat’s Yld 2000-2D flow stress potential is validated by plotting on the same graph predicted r-value, normalized yield stress curves and experimental data. Newton–Raphson numerical method with a relaxation factor was employed to calculate accurately the anisotropy coefficients. Present findings for slightly and highly anisotropic aluminum alloys and AISI 409 steel revealed that Barlat’s Yld 2000-2D function can be employed for accurate characterization of metal plastic anisotropy behavior by using two independent functions: the non-associated flow stress potential and the yield stress criterion. Consequently, this procedure requires a total of 12 experimental parameters of anisotropy in calibration for accurate r-value and s-value independent curves fitting. Therefore, the proposed non-associated Barlat’s Yld 2000-12p plastic potential and yield criterion give better correlation with experimental r-value and s-value data than the associated Barlat’s Yld 2004-18p flow rule. In addition, the predicted forming limit strain curves of AISI 409 steel are in good agreement with the experimental FLC, using the non-associated Barlat’s Yld 2000-2d plastic potential, better than the associated flow potential rule.
Similar content being viewed by others
References
R.W.K. Honeycombe, The Plastic Deformation of Metals, Edward Arnold, London, 1977.
U.F. Kocks, C.N. Tomé and H.-R. Wenk, Texture and Anisotropy, Cambridge University Press, Cambridge, 1998.
H. Tresca, Mémoire sur L’écoulement des Corps Solides Soumis à de Fortes Pressions, Vol 29 C.R. Acad. Sciences, Paris, 1864.
W. Johnson, R. Sowerby and R.D. Venter, Plane-Strain Slip-line Fields for Metal Deformation Processes, Pergamon Press, Oxford, 1982.
R. von Mises, Mechanik der Festen Köper im Plastischen Deformablen Zustand, Gött. Nachr. Math. Phys. Klasse, 1913, 1913, p 582–592.
J. Lemaitre and J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.
D.C. Drucker, Relation of Experiments to Mathematical Theories of Plasticity, J. Appl. Mech Trans. ASME, 1949, 71, p A349–A357. https://doi.org/10.1115/1.4010009
O. Cazacu and F. Barlat, Generalization of Drucker’s Yield Criterion to Orthotropy, Math. Mech. Solids, 2001, 6, p 613–630. https://doi.org/10.1177/108128650100600603
N. Medeiros, L.P. Moreira, J.D. Bressan, J.F.C. Lins and J.P. Gouvea JP, Upper-Bound Sensitivity Analysis of the ECAE Process, Mater. Sci. Eng. A, 2010, 527, p 2831–2844. https://doi.org/10.1016/j.msea.2009.12.049
R. Hill, A Theory of the Yielding and Plastic Flow of Anisotropic Metals, Proc. R. Soc. Lond. A, 1948, 193, p 281–297. https://doi.org/10.1098/rspa.1948.0045
S. Kobayashi, S. Oh and T. Altan, Metal Forming and the Finite-Element Method, Oxford University Press, Oxford, 1989.
J. Woodthorpe and R. Pearce, The Anomalous Behaviour of Aluminium Sheet Under Balanced Biaxial Tension, Int. J. Mech. Sci., 1970, 12, p 341–347. https://doi.org/10.1016/0020-7403(70)90087-1
A.N. Bramley and P.B. Mellor, Plastic Anisotropy of Titanium and Zinc Sheet-I Macroscopic Approach, Int. J. Mech. Sci., 1968, 10, p 211–219. https://doi.org/10.1177/1056789509103482
Y. Kurosaki, M. Tokiwa and K. Murai, Studies on Anisotropic Yield Criteria and Press Formability of Metal Sheets (Investigation into Bassani-Type Criteria), Bull. Jpn. Soc. Mech. Eng. (JSME), 1986, 29(255), p 3202–3208. https://doi.org/10.1299/jsme1958.29.3202
R. Hill, Theoretical Plasticity of Textured Aggregates, Math. Proc. Cam. Philos. Soc., 1979, 85, p 179–191. https://doi.org/10.1017/S0305004100055596
A. Parmar and P.B. Mellor, Plastic Expansion of a Circular Hole in Sheet Metal Subjected to Biaxial Tensile Stress, Int. J. Mech. Sci., 1978, 20, p 707–720. https://doi.org/10.1016/0020-7403(78)90057-7
A.V. Hershey, The Plasticity of an Isotropic Aggregate of Anisotropic Face Centred Cubic Crystals, J. Appl. Mech. Trans. ASME, 1954, 21, p 241–249. https://doi.org/10.1115/1.3443401
W.F. Hosford, A Generalized Isotropic Yield Criterion, J. Appl. Mech. Trans. ASME, 1972, 39, p 607–609. https://doi.org/10.1115/1.3422732
R.W. Logan and W.F. Hosford, Upper-Bound Anisotropic Yield Locus Calculations Assuming <111> Pencil Glide, Int. J. Mech. Sci., 1980, 22, p 419–430. https://doi.org/10.1016/0020-7403(80)90011-9
C.S. Viana, J.S. Kallend and G.J. Davies, The Use of Texture Data to Predict the Yield Locus of Metal Sheets, Int. J. Mech. Sci., 1979, 21, p 355–371. https://doi.org/10.1016/0020-7403(79)90016-X
J.L. Bassani, Yield Characterization of Metals with Transversely Isotropic Plastic Properties, Int. J. Mech. Sci., 1977, 19, p 651–660. https://doi.org/10.1016/0020-7403(77)90070-4
Y. Kurosaki, M. Matsumoto and M. Kobayashi, Studies on Anisotropic Yield Characteristic and Press Formability of Metal Sheets (Investigation into Pure Stretch-Forming), JSME International Journal, Serie III, 1988, 31(4), p 789–795. https://doi.org/10.1299/jsmec1988.31.789
M.G. Stout and P.S. Follansbee, Strain Rate Sensitivity, Strain Hardening, and Yield Behavior of 304L Stainless Steel, J. Eng. Mater. Technol. Trans. ASME, 1986, 108(4), p 344–353. https://doi.org/10.1115/1.3225893
M. Gotoh, A Theory of Plastic Anisotropy Based on a Yield Function of Fourth Order (Plane Stress State)—I, Int. J. Mech. Sci., 1977, 19, p 505–512. https://doi.org/10.1016/0020-7403(77)90043-1
P.B. Mellor and A. Parmar, Plasticity Analysis of Sheet Metal Forming, Mechanics of Sheet Metal Forming. D.P. Koistinen, N.-M. Wang Ed., Plenum Press, New York, 1978, p 53–74
M. Gotoh, A theory of Plastic Anisotropy Based on a Yield Function of Fourth Order (Plane Stress State)-II, Int. J. Mech. Sci., 1977, 19, p 513–520. https://doi.org/10.1016/0020-7403(77)90044-3
G. Ferron, R. Makkouk and J. Morreale, A Parametric Description of Orthotropic Plasticity in Metal Sheets, Int. J. Plast., 1994, 10, p 431–449. https://doi.org/10.1016/0749-6419(94)90008-6
F. Barlat and J. Lian, Plastic Behavior and Stretchability of Sheet Metals. Part I a Yield Function for Orthotropic Sheets Under Plane Stress Conditions, Int. J. Plast, 1989, 5, p 51–66. https://doi.org/10.1016/0749-6419(89)90019-3
F. Barlat and O. Richmond, Prediction of Tricomponent Plane Stress Yield Surfaces and Associated Flow and Failure Behavior of Strongly Textured F.C.C. Polycrystalline Sheets, Mat. Sci. Eng., 1987, 95, p 15–29. https://doi.org/10.1016/0025-5416(87)90494-0
F. Barlat, D.J. Lege and J.C. Brem, A Six-component Yield Function for Anisotropic Materials, Int. J. Plast., 1991, 7, p 693–712. https://doi.org/10.1016/0749-6419(91)90052-Z
F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker and S. Makosey, Yield Function Development for Aluminium Alloy Sheets, J. Mech. Phys. Solids, 1997, 45, p 1727–1763. https://doi.org/10.1016/S0022-5096(97)00034-3
J.W. Yoon, F. Barlat, K. Chung, F. Pourboghrat and D.Y. Yang, Earing Predictions Based on Asymmetric Nonquadratic Yield Function, Int. J. Plast., 2000, 16, p 1075–1104.
F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi and E. Chu, Plane Stress Yield Function for Aluminum Alloy Sheets—Part 1: Theory, Int. J. Plast., 2003, 19, p 1297–1319. https://doi.org/10.1016/S0749-6419(02)00019-0
H. Aretz, A Non-quadratic Plane Stress Yield Function for Orthotropic Sheet Metals, J. Mater. Process. Technol., 2005, 168, p 1–9. https://doi.org/10.1016/j.jmatprotec.2004.10.008
F. Bron and J. Besson, A Yield Function for Anisotropic Materials Application to Aluminum Alloys, Int. J. Plast., 2004, 20, p 937–963. https://doi.org/10.1016/j.ijplas.2003.06.001
F. Barlat, H. Aretz, J.W. Yoon, M.E. Karabin, J.C. Brem and R.E. Dick, Linear Transfomation-Based Anisotropic Yield Functions, Int. J. Plast., 2005, 21, p 1009–1039. https://doi.org/10.1016/j.ijplas.2004.06.004
H. Aretz and F. Barlat, Unconditionally Convex Yield Functions for Sheet Metal Forming Based on Linear Stress Deviator Transformation, Key Eng. Mater., 2012, 504–506, p 667–672. https://doi.org/10.4028/www.scientific.net/KEM.504-506.667
F. Barlat, J.J. Gracio, M.-G. Lee, E.F. Rauch and G. Vincze, An Alternative to Kinematic Hardening in Classical Plasticity, Int. J. Plast., 2011, 27, p 1309–1327. https://doi.org/10.1016/j.ijplas.2011.03.003
D. Banabic, H. Aretz, D.S. Comsa and L. Paraianu, An Improved Analytical Description of Orthotropy in Metallic Sheets, Int. J. Plast., 2005, 21, p 493–512. https://doi.org/10.1016/j.ijplas.2004.04.003
F. Barlat, W.Y. Jeong and O. Cazacu, On Linear Transformations of Stress Tensors for the Description of Plastic Anisotropy, Int. J. Plast., 2007, 23, p 876–896. https://doi.org/10.1016/j.ijplas.2006.10.001
O. Cazacu, B. Plunkett and F. Barlat, Orthotropic Yield Criterion for Hexagonal Close Packed Metals, Int. J. Plast., 2006, 22, p 1171–1194. https://doi.org/10.1016/j.ijplas.2005.06.001
E.W. Kelley and W.F. Hosford, Deformation Characteristics of Textured Magnesium, Trans. TMS-AIME, 1968, 242, p 654–660.
B. Plunkett, O. Cazacu and F. Barlat, Orthotropic Yield Criteria for Description of the Anisotropy in Tension and Compression of Sheet Metals, Int. J. Plasticity, 2008, 24, p 847–866. https://doi.org/10.1016/j.ijplas.2007.07.013
Y. Tozawa, Plastic Deformation Behaviour Under Conditions of Combined Stress, Mechanics of Sheet Metal Forming. D.P. Koistinen, N.-M. Wang Ed., Plenum Press, New York, 1978, p 81–109
A.R. Ragab and Ch. Saleh, Evaluation of Constitutive Models for Voided Solids, Int. J. Plast., 1999, 15, p 1041–1065. https://doi.org/10.1016/S0749-6419(99)00024-8
V. Tvergaard and A. Needleman, Analysis of the Cup-Cone Fracture in a Round Tensile Bar, Acta Metall., 1984, 32, p 157–169. https://doi.org/10.1016/0001-6160(84)90213-X
A.L. Gurson, Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I - Yield Criteria and Flow Rules for Porous Ductile Media. J. Eng. Mater. Technol, Trans. ASME, 1977, 99, p 2–15. https://doi.org/10.1115/1.3443401
J. Zhang, L. Ma and Z.X. Zhang, Elastoplastic Damage Model for Concrete Under Triaxial Compression and Reversed Cyclic Loading, Strength Mater., 2018, 50(5), p 724–734. https://doi.org/10.1007/s11223-018-0017-3
T.B. Stoughton, A Non-associated Flow Rule for Sheet Metal Forming, Int. J. Plast., 2002, 18, p 687–714. https://doi.org/10.1016/S0749-6419(01)00053-5
J. Lian, F. Shen, X. Jia, D.C. Ahn, D.C. Chae, S. Münstermann and W. Bleck, An Evolving Non-associated Hill48 Plasticity Model Accounting for Anisotropic Hardening and r-value Evolution and Its Application to Forming Limit Prediction, Int. J. Solids Struct., 2018, 151, p 20–44. https://doi.org/10.1016/j.ijsolstr.2017.04.007
M. Safaei, J.W. Yoon and W. De Waele, Study on the Definition of Equivalent Plastic Strain Under Non-associated Flow Rule for Finite Element Formulation, Int. J. Plast., 2014, 58, p 219–238. https://doi.org/10.1016/j.ijplas.2013.09.010
T. Park and K. Chung, Non-Associated Flow Rule with Symmetric Stiffness Modulus for Isotropic-Kinematic Hardening and its Application for Earing in Circular Cup Drawing, Int. J. Solids Struct., 2012, 49, p 3582–3593. https://doi.org/10.1016/j.ijsolstr.2012.02.015
J.D. Bressan, S. Bruschi and A. Ghiotti, Prediction of Limit Strains in hot Forming of Aluminium Alloy Sheets, Int. J. Mech. Sci., 2016, 115–116, p 702–710. https://doi.org/10.1016/j.ijmecsci.2016.07.040
J.D. Bressan and J.A. Williams, The Use of a Shear Instability Criterion To Predict Local Necking in Sheet Metal Deformation, Int. J. Mech. Sci., 1983, 25(3), p 155–168. https://doi.org/10.1016/0020-7403(83)90089-9
J.D. Bressan and F. Barlat, A Shear Fracture Criterion to Predict Limit Strains in Sheet Metal Forming, Int. J. Mater. Form., 2010, 3(Suppl 1), p 235–238.
L. Xu, F. Barlat and D.C. Ahn, Constitutive Modelling of Ferritic Stainless Steel, Int. J. Mater. Form., 2010, 3, p 135–145. https://doi.org/10.1007/s12289-009-0430-z
S. Panich, F. Barlat, V. Uthaisangsuk, S. Suranuntchai and S. Jirathearanat, Experimental and Theoretical Formability Analysis Using Strain and Stress Based Forming Limit Diagram for Advanced High Strength Steels, Mater. Des., 2013, 51, p 756–766. https://doi.org/10.1016/j.matdes.2013.04.080
D.C. Ahn, J.W. Yoon and K.Y. Kim, Modeling of Anisotropic Plastic Behavior of Ferritic Stainless Steel Sheet, Int. J. Mech. Sci., 2009, 51, p 718–725. https://doi.org/10.1016/j.ijmecsci.2009.08.003
F. Barlat, Constitutive Descriptions for Metal Forming Simulations. In: J. M. A. Cesar de Sa and A. D. Santos (eds) NUMIFORM 07, Materials Processing and Design: Modeling, Simulation and Applications, AIP, 2007, p 3–23. https://doi.org/10.1063/1.2740786
J.W. Yoon, F. Barlat, R.E. Dick and M.E. Karabin, Prediction of Six or Eight Ears in a Drawn Cup Based on a New Anisotropic Yield Function, Int. J. Plast., 2006, 22, p 174–193. https://doi.org/10.1016/j.ijplas.2005.03.013
Acknowledgments
The authors would like to gratefully acknowledge University of Santa Catarina State–UDESC and Instituto Tecnológico de Aeronáutica—ITA, CAPES of Brazil and CNPq through Grant No. 301069/2019-0 for their support.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bressan, J.D., Donadon, M.V. An Improved Anisotropic Non-associated Plastic Potential Based on Barlat’s Yld 2000-2D Yield Stress Criterion. J. of Materi Eng and Perform 32, 9221–9243 (2023). https://doi.org/10.1007/s11665-023-07799-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11665-023-07799-4