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Technical Brief

Algebraic Convexity Conditions for Gotoh's Nonquadratic Yield Function

[+] Author and Article Information
Wei Tong

Professor
Mem. ASME
Department of Mechanical Engineering,
Lyle School of Engineering,
Southern Methodist University,
Dallas, TX 75275-0337
e-mail: wtong@smu.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 12, 2017; final manuscript received March 31, 2018; published online May 8, 2018. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 85(7), 074501 (May 08, 2018) (7 pages) Paper No: JAM-17-1636; doi: 10.1115/1.4039880 History: Received November 12, 2017; Revised March 31, 2018

A necessary and sufficient condition in terms of explicit algebraic inequalities on its five on-axis material constants and a similarly formulated sufficient condition on its entire set of nine material constants are given for the first time to guarantee a calibrated Gotoh's fourth-order yield function to be convex. When considering the Gotoh's yield function to model a sheet metal with planar isotropy, a single algebraic inequality has also been obtained on the admissible upper and lower bound values of the ratio of uniaxial tensile yield stress over equal-biaxial tensile yield stress at a given plastic thinning ratio. The convexity domain of yield stress ratio and plastic thinning ratio defined by these two bounds may be used to quickly assess the applicability of Gotoh's yield function for a particular sheet metal. The algebraic convexity conditions presented in this study for Gotoh's nonquadratic yield function complement the convexity certification based on a fully numerical minimization algorithm and should facilitate its wider acceptance in modeling sheet metal anisotropic plasticity.

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References

Hill, R. , 1950, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, UK.
Hill, R. , 1990, “Constitutive Modeling of Orthotropic Plasticity in Sheet Metals,” J. Mech. Phys. Solids, 38(3), pp. 403–417. [CrossRef]
Hill, R. , 1948, “A Theory of the Yielding and Plastic Flow of Anisotropic Metals,” Proc. R. Soc. London, A193(1033), pp. 281–297. [CrossRef]
Bourne, L. , and Hill, R. , 1950, “On the Correlation of the Directional Properties of Rolled Sheets in Tension and Cupping Tests,” Philos. Mag., 41(318), pp. 49–53. [CrossRef]
Pearce, R. , 1968, “Some Aspects of Anisotropic Plasticity in Sheet Metals,” Int. J. Mech. Sci., 10(12), pp. 995–1004. [CrossRef]
Gotoh, M. , 1977, “A Theory of Plastic Anisotropy Based on a Yield Function of Fourth Order (Plane Stress State) I & II,” Int. J. Mech. Sci., 19(9), pp. 505–520. [CrossRef]
Hill, R. , 1979, “Theoretical Plasticity of Textured Aggregates,” Math. Proc. Cambridge Philos. Soc., 85(01), pp. 179–191. [CrossRef]
Tamura, S. , Sumikawa, S. , Hamasaki, H. , Uemori, T. , and Yoshida, F. , 2010, “Elasto-Plasticity Behavior of Type 5000 and 6000 Aluminum Alloy Sheets and Its Constitutive Modeling,” AIP Conf. Proc., 1252, p. 630.
Kuwabaraa, T. , Hashimoto, K. , Iizuka, E. , and Yoon, J. W. , 2011, “Effect of Anisotropic Yield Functions on the Accuracy of Hole Expansion Simulations,” J. Mater. Process. Technol., 211(3), pp. 475–481. [CrossRef]
Kitayama, K. , Kobayashi, T. , Uemori, T. , and Yoshida, F. , 2012, “Elasto-Plasticity Behavior of IF Steel Sheet With Planar Anisotropy and Its Macro-Meso Modeling,” ISIJ Int., 52(4), pp. 735–742. [CrossRef]
Tong, W. , 2016, “Application of Gotoh's Orthotropic Yield Function for Modeling Advanced High-Strength Steel Sheets,” ASME J. Manuf. Sci. Eng., 138(9), p. 094502. [CrossRef]
Gotoh, M. , Iwata, N. , and Matsui, M. , 1995, “Finite-Element Simulation of Deformation and Breakage in Sheet Metal Forming,” JSME Int. J. Ser. A, 38(2), pp. 281–288.
Hu, W. , 2007, “Constitutive Modeling of Orthotropic Sheet Metals by Presenting Hardening-Induced Anisotropy,” Int. J. Plast., 23(4), pp. 620–639. [CrossRef]
Soare, S. , Yoon, J. W. , and Cazacu, O. , 2008, “On the Use of Homogeneous Polynomials to Develop Anisotropic Yield Functions With Applications to Sheet Forming,” Int. J. Plast., 24(6), pp. 915–944. [CrossRef]
Tong, W. , and Alharbi, M. , 2017, “Comparative Evaluation of Non-Associated Quadratic and Associated Quartic Plasticity Models for Orthotropic Sheet Metals,” Int. J. Solids Struct., 128, pp. 133–148. [CrossRef]
Hershey, A. , 1954, “The Plasticity of an Isotropic Aggregate of Anisotropic Face Centred Cubic Crystals,” ASME J. Appl. Mech., 21(9), pp. 241–249.
Hill, R. , 1958, “A General Theory of Uniqueness and Stability in Elastic-Plastic Solids,” J. Mech. Phys. Solids, 6(3), pp. 236–249. [CrossRef]
Drucker, D. C. , 1959, “A Definition of Stable Inelastic Material,” ASME J. Appl. Mech., 26(3), pp. 101–106.
Yang, W. , 1980, “A Useful Theorem for Constructing Convex Yield Function,” ASME J. Appl. Mech., 47(2), pp. 301–303. [CrossRef]
Hosford, W. F. , 1985, “Comments on Anisotropic Yield Criteria,” Int. J. Mech. Sci., 27(7–8), pp. 423–427. [CrossRef]
Lubliner, J. , 1990, Plasticity Theory, Macmillan, New York.
Maugin, G. A. , 1992, The Thermomechanics of Plasticity and Fracture, Cambridge University Press, Cambridge, UK. [CrossRef]
Barlat, F. , Yoon, J. W. , and Cazacu, O. , 2007, “On Linear Transformations of Stress Tensors for the Description of Plastic Anisotropy,” Int. J. Plast., 23(5), pp. 876–896. [CrossRef]
Yoshida, F. , Hamasaki, H. , and Uemori, T. , 2013, “A User-Friendly 3D Yield Function to Describe Anisotropy of Steel Sheets,” Int. J. Plasticity, 45, pp. 119–139. [CrossRef]
Tong, W. , 2016, “Generalized Fourth-Order Hill's 1979 Yield Function for Modeling Sheet Metals in Plane Stress,” Acta Mech., 227(10), pp. 2719–2733. [CrossRef]
Tong, W. , 2016, “On the Parameter Identification of Polynomial Anisotropic Yield Functions,” ASME J. Manuf. Sci. Eng., 138(7), p. 071002. [CrossRef]
Tong, W. , 2018, “An Improved Method of Determining Gotoh's Nine Material Constants for a Sheet Metal With Only Seven or Less Experimental Inputs,” Int. J. Mech. Sci, 140, pp. 394–406. [CrossRef]
Budianski, B. , 1984, “Anisotropic Plasticity of Plane Isotropic Sheets,” Mechanics of Material Behaviour (Studies in Applied Mechanics, Vol. 6), Elsevier, Amsterdam, The Netherlands, pp. 15–29. [CrossRef]
Rodin, G. J. , and Parks, D. M. , 1986, “On Consistency Relations in Nonlinear Fracture Mechanics,” ASME J. Appl. Mech., 53(4), pp. 834–838. [CrossRef]
Bigoni, D. , and Piccolroaz, A. , 2004, “Yield Criteria for Quasibrittle and Frictional Materials,” Int. J. Solids Struct., 41(11–12), pp. 2855–2878. [CrossRef]
Helton, J. W. , and Nie, J. , 2010, “Semidefinite Representation of Convex Sets,” Math. Program., 122(1), pp. 21–64. [CrossRef]
Boyd, S. , and Vandenberghe, L. , 2004, Convex Optimization, Cambridge University Press, Cambridge, UK. [CrossRef]
Ahmadi, A. , and Parrilo, P. , 2013, “A Complete Characterization of the Gap Between Convexity and SOS-Convexity,” SIAM J. Optim., 23(2), pp. 811–833. [CrossRef]
Powers, V. , and Wormann, T. , 1998, “An Algorithm for Sums of Squares of Real Polynomials,” J. Pure Appl. Algebra, 127(1), pp. 99–104. [CrossRef]
Jiang, B. , Li, Z. , and Zhang, S. , 2017, “On Cones of Nonnegative Quartic Forms,” Found. Comput. Math., 17(1), pp. 161–197. [CrossRef]
Prussing, J. , 1986, “The Principal Minor Test for Semidefinite Matrices,” J. Guid., 9(1), pp. 121–122. [CrossRef]
Hirvensalo, M. , 2005, “A Method for Computing the Characteristic Polynomial and Determining Semidefiniteness,” Turku Center for Computer Science, Turku, Finland, Technical Report No. TUCS TR727.
Nie, J. , 2010, “Positive Semidefinite Matrices (Theorem 4),” University of California, San Diego, CA, accessed Oct. 18, 2017, http://www.math.ucsd.edu/~njw/Teaching/Math271C/Lecture_03.pdf
Horn, R. A. , and Johnson, C. , 2012, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, UK. [CrossRef]
Lay, D. C. , 2003, Linear Algebra and Its Applications, 3rd ed., Addison Wesley, Boston, MA.
Tong, W. , 2018, “Calibration of a Complete Homogeneous Polynomial Yield Function of Six Degrees for Modeling Orthotropic Steel Sheets,” Acta Mechanica, epub.
Ahmadi, A. , Olshevsky, A. , Parrilo, P. , and Tsitsiklis, J. , 2013, “NP-Hardness of Deciding Convexity of Quartic Polynomials and Related Problems,” Math. Program., Ser. A, 137(1–2), pp. 453–476. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

A flow chart for the efficient convexity certification andadjustment of a calibrated yield fucntion based on a cascade of necessary, sufficient, and necessary and sufficient conditions

Grahic Jump Location
Fig. 2

Domains of admissible yield stress and plastic strain ratios for planarly isotropic Gotoh's yield function

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