Prediction of Forming Limit Diagram with Damage Analysis

Based on the theory of damage mechanics, an orthotropic damage model for the prediction of forming limit diagram (FLD) is developed. The conventional method of FLD used to predict localized necking adopts two fundamentally different approaches. Under biaxial loading, the Hill's plasticity method is often chosen when α (= ε₂/ε₁) < 0. On the other hand, the M-K method is adopted for the prediction of localized necking when α > 0 or the biaxial stretching of sheet metal is pronounced. The M-K method however suffers from the arbitrary selection of the imperfection size, thus resulting in inconsistent predictions.

The orthotropic damage model developed for predicting the FLD is based on the anisotropic damage model recently proposed by Chow et al (1993). The model is extended to take into account, during the sheet forming process, orthotropic plasticity and damage. The orthotropic FLD model consists of the constitutive equations of elasticity and plasticity coupled with damage. With the aid of the damage dissipative potential, a damage criterion is proposed together with the damage energy release rate. This is followed by the development of the damage evolution equations. With the incremental theory of plasticity coupled with damage, the proposed approach can then be used to predict not only the forming limit diagram but also the fracture limit diagram under either proportional or nonproportional loading. In view of two distinct physical phenomena governing the cases when α < or α > 0, a set of instability criteria is proposed to characterize diffuse necking, localized necking and final rupture. The localized necking for α < 0 is postulated to take place when the calculated damage perpendicular to necking direction reaches the critical damage of the material. The critical damage is measured from the uniaxial tensile test and considered an intrinsic material property. The condition of the localized necking for α > 0 is postulated to have been satisfied when the localized damage perpendicular to the necking direction reaches the critical value of deviator damage. The orthotropic damage model is employed to predict the FLD of VDIF steel. The predicted results agree well with those determined experimentally.