Forming limits and constitutive modeling of steel tube for automotive parts

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The present study aims to clarify the nature of the path-dependence of forming limit stresses from a view point of the phenomenological plasticity analysis.In this analysis we use a non-normality flow rule proposed by Kuroda and Tvergaard (2001). This flow rule assumes a non-linear dependence of the plastic strain rate D^P on the total strain rate D, where D is decomposed asD=D^e +D^P=D^e+^^・ΦN^P (1)where e and p indicate the elastic and plastic part of strain. N^P defines the direction of D^P, ^^・Φ is a non-negative overstress function for rate-dependent cases. State variables of the material are assumed to be Cauchy stress tensor, σ, and the orthonormal vectors n, defined along the axes of orthotropy, and equivalent strain ^^-ε. These variable are calculated using following equations:^^。σ=^^・σ-ω・σ+σ・ω=C^<+e>:D^e=C:D-^^・ΦC^e : N^P (2)^^・n_i=ω・n_i (3)^^-ε=∫^^<・->εdt, ^^<・->ε=√<2/3>^^・Φ (4)where C^e is the fourth order elastic moduli tensor. With a superposed o denoting an objective rat … More e with respect to the spin ω and the superposed dot denoting a material time derivative, the elasticity relation is assumed to be given by Hooke's law.The limits to formability are simulated using the M-K model both for proportional loading and two types of combined loadings : one type includes unloading between the first and second loadings while the other type does not include unloading. The strain paths for each combined loading are almost identical, while the resulting stress paths are significantly different. The effects of changing stress/strain paths on the FLC and FLSC are discussed in detail by observing the strain localization process for each case. A basic question is: Is the forming limit stress always path-independent, irrespective of stress/strain paths? If so, why? If not, what makes the forming limit stress path-dependent?The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. Less