HERK integration of finite-strain fully anisotropic plasticity models

Abstract

For finite strain plasticity, we use the multiplicative decomposition of the deformation gradient to obtain a differential-algebraic system (DAE) in the semi-explicit form and solve it by a half-explicit algorithm. The terminology HERK is synonym of Half-Explicit Runge-Kutta method for DAE. The source is here the right Cauchy-Green tensor and an exact 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 half-explicit constitutive integrator is the content of this work. The integration makes use of an explicit Runge-Kutta method for the flow law complemented by the yield constraint. The flow law is a first-order differential equation and the yield constraint (including the loading/unloading conditions) is seen as the invariant of system. A half-explicit method is adopted to ensure satisfaction of the invariant. The resulting scalar equation is solved by the Newton-Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. Two complete numerical examples are presented.