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On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming. (English) Zbl 1394.74024
Summary: This paper investigates the capabilities of several non-quadratic polynomial yield functions to model the plastic anisotropy of orthotropic sheet metal (plane stress). Fourth, sixth and eighth-order homogeneous polynomials are considered. For the computation of the coefficients of the fourth-order polynomial an improved set of analytic formulas is proposed. For sixth and eighth-order polynomials the identification uses optimization. Simple constraints on the optimization process are shown to lead to real-valued convex functions. A general method to extend the above plane stress criteria to full 3D stress states is also suggested. Besides their simplicity in formulation, it is found that polynomial yield functions are capable to model a wide range of anisotropic plastic properties (e.g., the Numisheet’93 mild steel, AA2008-T4, AA2090-T3). The yield functions have then been implemented into a commercial finite element code as constitutive subroutines. The deep drawing of square (Numisheet’93) and cylindrical (AA2090-T3) cups have been simulated. In both cases excellent agreement with experimental data is obtained. In particular, it is shown that non-quadratic polynomial yield functions can simulate cylindrical cups with six or eight ears. We close with a discussion on earing and further examples.

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
90C90 Applications of mathematical programming
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[1] Arminjon, M.; Bacroix, B.; Imbault, D.; Raphanel, J. L.: A fourth-order plastic potential for anisotropic metals and its analytical calculation from texture function, Acta mech. 107, 33 (1994) · Zbl 0848.73005
[2] Barlat, F.; Maeda, Y.; Chung, K.; Yanagawa, M.; Brem, J. C.; Hayashida, Y.; Lege, D. J.; Matsui, K.; Murtha, S. J.; Hattori, S.; Becker, R. C.; Makosey, S.: Yield function development for aluminum alloy sheets, J. mech. Phys. solids 45, 1727 (1997)
[3] Barlat, F.; Aretz, H.; Yoon, J. W.; Karabin, M. E.; Brem, J. C.; Dick, R. E.: Linear transformation based anisotropic yield function, Int. J. Plast. 21, 1009 (2005) · Zbl 1161.74328
[4] Barlat, F.; Yoon, J. W.; Cazacu, O.: On linear transformations of stress tensors for the description of plastic anisotropy, Int. J. Plast. 23, 876 (2007) · Zbl 1359.74014
[5] Chung, K.; Lee, S. Y.; Barlat, F.; Keum, Y. T.; Park, J. M.: Finite element simulation of sheet forming based on a planar anisotropic strain-rate potential, Int. J. Plast. 12, 93 (1996) · Zbl 0857.73073
[6] Danckert, J.: Experimental investigation of a square-cup deep drawing process, J. mat. Proc. technol. 50, 375 (1995)
[7] Dawson, P. R.; Macewen, S. R.; Wu, P. D.: Advances in sheet metal forming analyses: dealing with mechanical anisotropy from crystallographic texture, Int. mat. Rev. 28, 86 (2003)
[8] Gotoh, M.: A theory of plastic anisotropy based on a yield function of fourth order (plane stress) - part I and II, Int. J. Mech. sci. 19, 505 (1977) · Zbl 0373.73038
[9] Groemer, H.: Geometric applications of Fourier series and spherical harmonics, (1996) · Zbl 0877.52002
[10] Hill, R.: The mathematical theory of plast., (1950) · Zbl 0041.10802
[11] Hosford, W. F.: A generalized isotropic yield criterion, J. appl. Mech. trans. ASME 39, 607 (1972)
[12] Pearce, R.: Some aspects of anisotropic plasticity in sheet metals, Int. J. Mech. sci. 10, 995 (1968)
[13] Savoie, J.; Macewen, S. R.: A sixth order inverse potential function for incorporation of crystallographic texture into predictions of properties of aluminium sheet, Texture microstruct. 26 – 27, 495 (1995)
[14] Schitkowski, K.: NLPQL: a Fortran subroutine solving constrained non-linear programming problems, Ann. oper. Res. 5, 485 (1986)
[15] Shephard, G. C.: A uniqueness theorem for the Steiner point of a convex region, J. London math. Soc. 43, 439 (1968) · Zbl 0162.25801
[16] Simo, J. C.; Hughes, T. J. R.: Computational inelasticity, (1999) · Zbl 0934.74003
[17] Tug&caron, P.; Cu; Neale, K. W.: On the implementation of anisotropic yield functions into finite strain problems of sheet metal forming, Int. J. Plast. 15, 1021 (1999) · Zbl 0944.74016
[18] Wu, P. D.; Jain, M.; Savoie, J.; Macewen, S. R.; Tugcu, P.; Neale, K. W.: Evaluation of anisotropic yield functions for aluminum sheets, Int. J. Plast. 19, 121 (2003) · Zbl 1032.74530
[19] Yoon, J. W.; Barlat, F.; Chung, K.; Pourboghrat, F.; Yang, D. Y.: Earing prediction based on asymmetric nonquadratic yield function, Int. J. Plast. 16, 1105 (2000) · Zbl 0986.74015
[20] Yoon, J. W.; Barlat, F.; Dick, R. E.; Karabin, M. E.: Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function, Int. J. Plast. 22, 174 (2006) · Zbl 1148.74325
[21] Zhou, Y.; Jonas, J. J.; Savoie, J.; Makinde, A.; Macewen, S. R.: Effect of texture on earing in fcc metals: finite element simulations, Int. J. Plast. 14, 117 (1998) · Zbl 0914.73069
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