Abstract
A fully implicit stress integration algorithm is developed for the distortional hardening model, namely the e−HAH model, capable of simulating cross−hardening/softening under orthogonal loading path changes. The implicit algorithm solves a complete set of residuals as nonlinear functions of stress, a microstructure deviator, and plastic state variables of the constitutive model, and provides a consistent tangent modulus. The number of residuals is set to be 20 or 14 for the continuum or shell elements, respectively. Comprehensive comparison programs are presented regarding the predictive accuracy and stability with different numerical algorithms, strain increments, material properties, and loading conditions. The flow stress and r−value evolutions under reverse/cross−loading conditions prove that the algorithm is robust and accurate, even with large strain increments. By contrast, the cutting−plane method and partially implicit Euler backward method, which are characterized by a reduced number of residuals, result in unstable responses under abrupt loading path changes. Finally, the algorithm is implemented into the finite element modeling of large−size, S−rail forming and the springback for two automotive steel sheets, which is often solved by a hybrid dynamic explicit–implicit scheme. The fully implicit algorithm performs well for the whole simulation with the solely static implicit scheme.
Highlights
Automotive, non-ferrous alloys and advanced high-strength steels have been extensively investigated by many researchers owing to their lightweight and excellent mechanical properties
The e-homogeneous yield function based anisotropic hardening (HAH) model features a larger number of plastic state variables than the conventional isotropic hardening model
In comparison with the Cutting-Plane Method (CPM) algorithm used by Lee et al [55], whereby the effect of the dependent variables is included in the yield surface gradient, the present CPM scheme introduces the linearization of the yield condition in Equation (20) as follows:
Summary
Automotive, non-ferrous alloys and advanced high-strength steels have been extensively investigated by many researchers owing to their lightweight and excellent mechanical properties. The series of isotropic and kinematic hardening models was well implemented into finite element (FE) simulations for sheet metals, especially when metals exhibit the Bauschinger effect, transient behavior, and permanent softening under reversed loading paths [22,23,24,25]. Some of the distortional hardening models were developed to express the yield surface evolution by using the anisotropic coefficients as a function of the plastic work or equivalent plastic strain [33,34,35,36] These models do not take into account the loading path change effect. Note that the CPPM or Euler backward scheme presented in previous studies adopted only the consistency condition and flow rule for the residuals This may lead to non-physical evolutions of the distortional hardening state variables when there is an abrupt change in the loading path. A large-scale industrial automotive forming and springback simulation is provided as a benchmark problem to assess the validity of the proposed, fully implicit numerical algorithm
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