Numerical procedures for predicting localization in sheet metals using crystal plasticity

In order to obtain reliable solutions for the initiation of localized instability in sheet metal forming, numerical procedures are developed for predicting forming limits using crystal plasticity. The Marciniak–Kuczynski (M–K) method is employed to calculate the initiation of the instability with rate-dependent elastic-crystal plasticity for cubic metals. The Nelder–Mead (N–M) direct search and Newton–Raphson (N–R) methods are used to solve the highly nonlinear equilibrium and associated compatibility equations under the M–K scheme. The N–M method eliminates the need of the first-order derivatives of the constitutive equations that may not be readily available for complex material models. The numerical results show that the calculation time using the N–M method is comparable to that using the N–R for the highly nonlinear M–K equations if the simplex size parameters are properly optimized.