An efficient numerical integration system for stiff unified constitutive equations for metal forming applications

Unified constitutive equations have been developed in recent years to predict viscoplastic flow and microstructural evolution of metal alloys for metal forming applications. These equations can be implemented into commercial FE code, such as ABAQUS and PAMSTAMP, to predict mechanical and physical properties of materials in a wide range of metal forming processes. These equations are normally stiff and need significant computer CPU time to solve. In this research, a series of numerical analyses are performed to investigate the difficulties within MATLAB of solving these stiff unified constitutive equations. A metric is introduced to allow evaluation of the numerical stiffness to assess the most appropriate numerical integration method. This metric is based on the ratio of maximum to minimum eigenvalue. This metric allows for an appropriate numerical method to be chosen giving more effective modelling of deformation and plasticity processes. Based on the theoretical work described above, a user-friendly system, based on MATLAB, is then developed for numerically integrating these types of stiff constitutive equations. This is particularly useful for metal forming engineers and researchers who need an effective computational tool to determine constitutive properties well based on numerical integration theories.