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Calibration and fast evaluation algorithms for homogeneous orthotropic polynomial yield functions

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Abstract

Homogeneous polynomial functions have the potential to provide a general modeling framework for yield surfaces in metal plasticity. They incorporate as particular cases many of the previously proposed yield functions and their fitting capabilities allow for capturing a wide range of yield surface shapes. And yet, there are still two unsolved problems which turn into major obstacles when it comes to actual implementations in both academic and industrial environments: The lack of a general optimization algorithm for the calibration of their parameters and the lack of an efficient computational scheme for their value, gradient and hessian. The difficulty of the first problem is two-fold, necessitating an adequate specification of the experimental input data set and satisfaction of the convexity constraint. The second problem is specific to all high degree polynomials and is comprised of issues such as numerical stability, precision and implementation efficiency. We present practical solutions to both problems: An optimization algorithm that reduces to solving a sequence of quadratic problems and a double Horner evaluation scheme that is optimal (featuring the least number of multiplications). The resulting modeling framework can account for arbitrary input data, experimental or from crystal plasticity predictions. As illustration we show new results regarding the relationship between generalized r-values and the earing profile of deep-drawn cylindrical cups. Practicality is demonstrated by the high level of automation of the entire workflow, from material parameters calibration to finite element simulations, and supporting code (Python scripts and constitutive subroutine) made available at https://github.com/stefanSCS/PolyN.

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Data availability

All supporting code is made public at: https://github.com/stefanSCS/PolyN. Numerical data (parameters and cup height profiles) for all models and deep-drawing simulations reported here is available in the CM_Data sub-directory of the repository.

Notes

  1. A third, is a constructive approach based on CAD-like interpolation tools [30, 38].

  2. An alternative deformation ratio is \(q=r/(1+r)\) (not to be confused with our usage of q), with the advantage that its variation is limited to [0, 1], e.g., Ch. 10 in [10, 20]. However, the nature of the disturbance remains the same—the asymptote of the extended r-value being replaced by zeros of q.

  3. Regarding notation: Boldface symbols represent tensors/matrices; For a second order tensor/matrix \(\varvec{M}\) and a vector \(\varvec{a}\) the notation \(\varvec{b}=\varvec{M}:\varvec{a}\) stands for the component form \(b_i=\sum _jM_{ij}a_j\); The scalar (dot) product of two vectors is denoted by \(\varvec{a}\cdot \varvec{b}\); The tensor product of two vectors \(\varvec{a}\otimes \varvec{b}\) is defined by \(\left( \varvec{a}\otimes \varvec{b}\right) :\varvec{v}=\left( \varvec{b}\cdot \varvec{v}\right) \varvec{a}\); Under plane stress conditions, a function of the stress tensor \(f=f(\varvec{\sigma })\) is, by virtue of the symmetry of its argument, always regarded a function \(f=f(\sigma _x,\sigma _y,\sigma _{xy})\); In this condensed notation for arguments, the tensor components of the gradient are: \(\nabla f = [\partial f/\partial \sigma _x,\partial f/\partial \sigma _y, (1/2)\partial f/\partial \sigma _{xy}(1/2)\partial f/\partial \sigma _{xy}]\). The summation convention on repeated indexes is not implied; Whenever used, this is stated explicitly.

  4. This requirement can be easily relaxed to 7 data points by interpolating the uniaxial data.

  5. For any even integer \(N\ge 2\), there exists a unique set of coefficients denoted by \(\varvec{a}=M\) that reduces PolyN to the Von Mises yield function. These are found by reducing to identity the equality \(P_N(\varvec{\sigma }) = \left( \sigma _x^2-\sigma _x\sigma _y+\sigma _y^2+3\sigma _{xy}^2\right) ^{(N/2)}\), with \(P_N\) defined in Eq. (8).

  6. For all simulations reported in this subsection we used the corresponding friction and hardening data in Tables-10,32 of [12]. Also, with reference to Fig. 6 and Eq. (33): \(\Phi _D=62.4\), \(\Phi _P=60.0\), \(R_N=5.0\), \(R_S=10.0\), \(\Phi _B=107.5\) \(t_B=1.0\). Finally, we recall that for simulations with classical shell elements the thickness of the blank has no relevance (besides the integration points used for calculating the response to bending).

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Acknowledgements

The Max-Planck-Institut für Eisenforschung GmbH is gratefully acknowledged for providing the computational resources.

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Authors and Affiliations

  1. Technical University of Cluj-Napoca, Cluj-Napoca, 400641, Romania

    Stefan C. Soare

  2. Dept. Mater. Eng., KU Leuven, Kasteelpark Arenberg 44, Leuven, 3001, Belgium

    Martin Diehl

  3. Dept. Comp. Sci., KU Leuven, Celestijnenlaan 200 A, Leuven, 3001, Belgium

    Martin Diehl

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Soare, S.C., Diehl, M. Calibration and fast evaluation algorithms for homogeneous orthotropic polynomial yield functions. Comput Mech (2023). https://doi.org/10.1007/s00466-023-02408-6

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