Extrapolation and Ce-based implicit integration of anisotropic constitutive behavior

For finite strain plasticity with both anisotropic yield functions and anisotropic hyperelasticity, we use the Kroner-Lee decomposition of the deformation gradient to obtain a differential-algebraic system (DAE) in the semi-implicit form and solve it by an implicit Richardson-extrapolated method based on intermediate substeps. The source is here the right Cauchy-Green tensor and the consistent Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first-order differential equation and a non-smooth algebraic equation. The development of a Richardson-extrapolated implicit integrator for any hyperelastic case and any yield function is the goal of this work. The integration makes use of a backward-Euler method for the flow law complemented by the solution of a yield constraint. The resulting system is solved by the Newton-Raphson method to obtain the plastic multiplier and the elastic right Cauchy-Green tensor Ce. To ensure power consistency, we make use of the elastic Mandel stress construction. Iso-error maps for three yield functions and three numerical examples are presented.