Forming limit diagrams calculated using Hill's non-quadratic yield criterion

An analytical model is presented to predict the forming limit diagram (FLD) of a sheet metal under plastic deformation. The model is an extension of previous analyses by Jones and Gillis (JG) (6) and Choi et.al. (10). The process of plastic deformation was idealized by JG into three phases based on easily observable features of a tensile test. These were: (I) homogeneous deformation up to maximum load; (II) deformation localization under constant load: (III) local necking with a precipitous drop in load. The JG analysis was originally applied to the tension test of a round bar (8), and then to the right hand side (RHS) of the FLD (6) and assumed homogeneous straining in the minor strain direction. Later Choi et.al. (9,10) extended the analysis to the left hand side (LHS) of the FLD based on the assumption of proportional straining on the LHS. Both JG and Choi et.al. described the plastic behavior of sheet metals using a generalized quadratic flow law proposed by Jones and Gillis (5).; In the current analysis, Hill's non-quadratic flow law (11) for sheet metals having in-plane isotropy is used in conjunction with the JG three stage deformation approximation. Another major difference of the present analysis is the application of Hill's velocity discontinuity analysis (17) to the final deformation stage. This leads to a kinematic condition for LHS necks that was not recognized by Choi et.al.; Calculated FLD's compare very favorably to experimentally determined results for AK steel and aluminum alloys 2036-T4, 1100-H19, 5052-H32, 3003-0 and 3004-0. The comparison is fair for the titanium alloy Ti-6Al-4V. The agreement is poor for the rate insensitive aluminum alloys 5052-0, 5052-H241, 5154-H111 and 6061-T4. In making these calculations the Hill exponent M was treated as an adjustable parameter and varied for each material to obtain best agreement with the experimental FLD's. For certain materials, the strain-rate sensitivity parameter was treated similarly.; Besides making direct comparisons with experimental results the mathematical model is used to study the theoretical effects of changing various material properties. The properties considered are the strain hardening parameter, n, the strain rate sensitivity parameter, m, and the plastic anisotropy ratio, r. The important influence of these material properties upon formability (level of the FLD) is affirmed by the calculations.