Issue 45

G. Giuliano et alii, Frattura ed Integrità Strutturale, 45 (2018) 164-172; DOI: 10.3221/IGF-ESIS.45.14 169 In the sheet peripheral area, the presence of a blank holder was simulated by imposing suitable constraint conditions. In particular, such constraint conditions prevented the sheet from sliding on the die. The contact conditions between bodies required the sheet to be defined as a deformable body while the die and the punch were defined as rigid bodies. In order to take account of the friction conditions between the punch and sheet and between the sheet and the die, the modified Coulomb friction model was chosen, whose relation between the tangential force f t and the normal force f n can be defined as: 2 r t n sv v f f arctg R                (2) where v r is the relative sliding speed and R sv is the relative sliding speed at which the friction force tends to vanish. The coefficient of friction μ ranged from 0 to 0.2. The material behavior in the plastic field was described by the power law as Eqn. (1). In the adopted FEM code, to associate a formability limit curve to the deformable material was possible for shell elements, while to adopt a user-defined subroutine by introducing the most suitable FLC was necessary for axisymmetric elements. The introduction of the FLC allowed defining a formability limit parameter (FLP). The FLC, derived from Hill's local necking theory and Swift's diffuse necking theory, was dependent only on the material constant n present in the constitutive Eqn. (1). In particular, defining as β the ratio between the principal strains (ε max and ε min ) measured in the sheet plane: min max     (3) the formability limit parameter was determined through the following ratio: max min ( ) FLP FLC    (4) where by FLC (ε min ) an analytical description of the formability limit curve as a function of the principal strain ε min is intended. In the case where β ≤0 it can be obtained: min ( ) 1 n FLC     (5) while for β >0: 2 min 2 1 ( ) 2 (1 )(2 2 ) FLC n             (6) Fig. 9 represents the FLC curve for the material under examination. The use of the FLP parameter allows monitoring, during a plastic deformation process, the position and the moment in which the instability condition occurs (FLP = 1). N UMERICAL - EXPERIMENTAL COMPARISON he finite element analysis allows to represent the FLP parameter distribution in the sheet as well as to define the trend of the force-displacement curve of the punch. Fig. 10 shows the reaching of the instability condition (FLP = 1) at the pole of the sheet, both in the case of three-dimensional and two-dimensional modeling, in perfect lubrication conditions (μ = 0). It should be noted that the simulation results are independent of the considered element type. Moreover, from Fig. 11 it is possible to compare the FEM results regarding the force-displacement curve of the punch in the case of μ=0. T