Finite Element Analysis of High Tensile Strength Steel Sheet by Using Complex Step Derivative Approximations


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The framework for the complex step derivative approximations (hereafter CDSA) to calculate the consistent tangent moduli is studied. The present methods is one of the most effective methods to implement any material constitutive equations to the commercial finite element codes and does not suffer from calculation conditions and errors. In order to confirm the efficiency of CDSA, we developed the user subroutine code based on the CDSA using associative J2 flow rules with general nonlinear isotropic hardening rules that is commonly and widely utilized in commercial finite element codes. In this study, the user material subroutine ‘Hypela2’ of MSC.Marc (ver.2013.0.0) was utilized. The finite element calculation result by the proposal method shows a good agreement with the corresponding result by the MSC.Marc default setting. Also we apply the Yoshida-Uemori back stress model to the CDSA and evaluate this new technique to predict the deformation behavior of high tensile strength steel sheet.



Edited by:

Yeong-Maw Hwang and Cho-Pei Jiang




T. Uemori et al., "Finite Element Analysis of High Tensile Strength Steel Sheet by Using Complex Step Derivative Approximations", Key Engineering Materials, Vol. 626, pp. 187-192, 2015

Online since:

August 2014




* - Corresponding Author

[1] M. Tanaka, H. Noguchi, M. Fujikawa, et al, Development of large strain shell elements for woven fabrics with application to clothing pressure distribution problem, CEMS, vol. 62, No. 3, 2010, pp.265-290.

[2] M. Tanaka, and M. Fujikawa, Numerical Approximation of Consistent Tangent Moduli Using Complex Step Derivative and Its Application to Finite Deformation Problems, Transactions of the JSME vol. 77-773, 2011, pp.27-38 (in Japanese).


[3] K.L. Lai and J.L. Crassidis, Extension of the first and second complex step derivative approximations, J. Comput. Appl. Math., vol. 219, 2008, pp.276-293.

[4] C. Pozrikidis, Numerical Computation in Science and Engineering, Oxford University Press, New York, NY, 1998, pp.45-47.

[5] W. Squire, G. Trapp, Using complex variables to estimate derivatives of read functions, SIAM Rev. 40, 1998, 110-112.


[6] L.I. Cervino, T.R. Bewley, On the extension of the complex step derivative technique to pseudospectral algorithms, J. Comput. Phys. 187, 2003, pp.544-549.


[7] H. Al-Mohy, A. and N.J. Highram, The complex step approximation to the frechet derivative of a matrix function, Numerical Algorithms, vol. 53, 2010, pp.133-148.


[8] M. Gotoh, A theory of plastic anisotropy based on a yield function of fourth order, Int. J. Mechanical Sciences, 19 (1979), pp.505-520.

[9] F. Yoshida and T. Uemori, Elastic-plastic behavior of steel sheets under in-plane cyclic tension-compression at large strai Int. J. Plasticity, 18, 2002, pp.661-686.


[10] F. Yoshida and T. Uemori, A model of large-strain cyclic plasticity for sheet metals describing the Bauschinger effect and workhardening stagnation, International Journal of Plasticity, 18 , 2002, pp.661-686.


[11] F. Yoshida and T. Uemori, A Model of Large-Strain Cyclic Plasticity and its Application to Springback Simulation, Key Engineering Materials, Vols. 233-236, 2003, pp.47-58.


[12] F. Yoshida, T. Uemori and S. Abe, Modeling of Large-Strain Cyclic Plasticity for Accurate Springback Simulation, Key Engineering Materials Vols. 340-341, 2007, pp.811-816.


[13] T. Uemori, T. Kuramitsu, R. Hino, T. Naka and F. Yoshida, Plastic Deformation Behavior of High Strength Steel Sheet under Non-proportional Loading and Its Modeling, Key Engineering Materials Vols. 340-341, 2007, pp.895-900.


[14] T. Uemori, S. Sumikawa, S. Tamura, H. Akagi, T. Naka and F. Yoshida: Springback simulation of high strength steel sheets calculated by Yoshida-Uemori model, Steel Research International, 81-9, 2010, pp.825-828.