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Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring

Abstract

For finite-strain plasticity with anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner-Lee decomposition of the deformation gradient combined with a yield function written in terms of the Mandel stress. The source is here the right Cauchy-Green tensor provided by a FE discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential, which is compared with a classical backward-Euler method. The exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source, consistent with the approximation. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function. The constitutive system is solved by a damped Newton-Raphson algorithm for the plastic multiplier and the elastic right Cauchy-Green tensor \(\varvec{C}_{e}\). To ensure power-consistency, we make use of the elastic Mandel stress construction. Two numerical examples exhibit the comparative effectiveness of the Algorithm for very large elastic and plastic deformations. The elasto-plastic pinched cylinder makes use of as few as 2 steps for the total radius displacement of 300 mm and only 25 steps are required for the cup drawing problem.

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Acknowledgements

The authors acknowledge the support of FCT, through IDMEC, under LAETA, project UIDB/50022/2020. The first author would like to thank the help from Professor Leonel Fernandes concerning ODE integrators.

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Correspondence to P. Areias.

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Areias, P., Rosa, P.A.R. & Rabczuk, T. Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring. Comput Mech 68, 391–404 (2021). https://doi.org/10.1007/s00466-021-02038-w

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Keywords

  • Hyperelasto-plasticity
  • Anisotropy
  • Matrix exponential
  • Power series
  • Scaling and squaring
  • Mandel stress