Implementation of a Robust Algorithm for Prediction of Forming Limit Diagrams

Abstract

In this article, a robust algorithm for prediction of forming limit diagrams (FLD) has been presented. The presented model is based on the ''Marciniak and Kuczynski'' (M-K) theory. Solution to the system of equations has been obtained by applying the NewtonÕs method. Since the NewtonÕs method usually has nonconverging problem, a particular backtracking algorithm has been developed and applied. In this algorithm, a technique for step length selection in the frame of gradient descent method has been implemented. Also for the convergence criterion the so-called ''Armijo'' condition has been used. For verification of the results, BBC2000 yield function and Swift hardening law for AK steel metal have been used. To obtain the necking angle, the effect of groove orientation on the left-and right-hand sides of FLD has been considered. Finally, the predicted FLD has been compared with the published experimental results.

Experimental analysis and theoretical predictions of the limit strains of a hot-dip galvanized interstitial-free steel sheet

“…Given that the main governing equations of the M-K problem are defined by two equilibrium forces and one deformation compatibility condition, given by Equations 29-31, an additional relationship is derived from the equivalent principle of plastic-work in the zone b, see Equation 8, so as to obtain a system of four nonlinear equations as proposed by Ganjiani and Assempour 39 :…”

“…Moreover, a backtracking algorithm is adopted to assure the convergence of the full Newton's method which strongly depends on the initial guess 39,40 . The localized necking criterion is defined when the ratio between the effective plastic-strain increments in the two zones reaches the condition of an unstable flow, which is given by ε , determined as a function of the geometrical imperfection orientation Ψ, which, in turn, is varied between 0 and 90 degrees.…”

Advances in anisotropy and formability

“…Vegter et al have implemented their own yield criterion [282][283][284] in the MK model. Ganjiani and Assempour [100][101][102] have improved the analytical approach for determination of FLC considering the effects of yield functions (Hosford [138], Karafillis-Boyce [159] and BBC2000 [18]). The Teodosiu hardening model [270] associated with different yield criteria has been implemented by Butuc et al [65] and Haddag et al [111] in a MK theory for studying the influence of the loading path change on the limit strains.…”

Determination of Forming - Limit Diagrams Considering Various Models for Steel Sheets

“…Because of the complexity of the experimental determination of the FLD, a number of theoretical calculating models have been set up on the basis of the classical or modified Swift and Hill instability criteria [3,4]. In recent years, the knowledge and principles of damage mechanics, plastic mechanics of porous materials, and microscopic materials science combined with the finite-element method (FEM) have also been introduced into the theoretical predictions of the FLD [5,6,7]. These results have significantly enriched and improved the understanding and application of the FLD.…”