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

• Published in: Computational mechanics 2021-08, Vol.68 (2), p.391-404
• Main Authors: Areias, P., Rosa, P. A. R., Rabczuk, T.
• Format: Article
• Language: English
• Subjects: Algorithms; Anisotropy; Approximation; Classical and Continuum Physics; Comparative analysis; Computational Science and Engineering; Deep drawing; Deformation effects; Elastic deformation; Engineering; Flux density; Laws, regulations and rules; Mathematical analysis; Original Paper; Plastic properties; Power series; Strain; Tensors; Theoretical and Applied Mechanics
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-021-02038-w

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 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.