A constitutive model for spring-back prediction in which the change of Young’s modulus with plastic

A constitutive model for spring-back prediction in which the change of Young’s modulus with plastic deformation is considered

S.L. Zang

School of Mechanical Engineering, Xi’an Jiaotong University, No. 28, Xianning Road, Xi’an, Shaanxi, China

In order to improve the prediction capability of spring-back in the computational analysis of sheet metal forming processes, a stress–strain constitutive formulation of non-linear combined hardening rule has been proposed in this paper according to non-linear kinematic hardening theory of Lemaitre and Chaboche and Hill’s 1948 anisotropic yielding function. Traditionally, Young’s modulus is considered as a constant in engineering application and numerical simulation. In fact, it decreases with plastic deformation. So the effect of the change of Young’s modulus with plastic strain on spring-back is considered in the constitutive model. The algorithm of stress update is elastic prediction, plastic correcting and radial returning. Numerical results and experimental results show that the proposed constitutive model significantly improves the prediction accuracy of spring-back.

In recent years, the high strength steels and aluminum alloys, which have higher ratios of yield strength to elastic odulus, are increasingly used for sheet metal parts in automotive industry to reduce mass [1]. Hence, spring-back is one of the most important problem faced in sheet metal forming processes. Introducing computational methods based on the finite element method in the design stage is one way to reduce the try-error correct, especially utilizing proper mechanical properties of sheets [2].

Lems [3] revealed that Young’s modulus decreases with plastic deformation. Due to the change of Young’s modulus with plastic deformation, actual spring-back is larger than that computed with a constant Young’s modulus. However, the most commercial FEM codes do not provide non-linear elastic and plastic constitutive model. To improve the spring-back prediction capability, the relationship of plastic deformation and Young’s modulus should be considered in numerical simulation. A variety of theoretical and experimental researches on the phenomenon of the change of Young’s modulus with plastic deformation and elasto-plastic constitutive model have been investigated. Vin [4] gave a simple mathematical model describing the relationship between plastic deformation and Young’s modulus based on experiment results.

Morestin [5,6] formulated an elasto-plastic model using kinematic hardening model for spring-back analysis in   sheet metal forming. In his calculations the change of the Young’s modulus versus plastic strain was taken into account. Yan [7] addressed the effect of non-linear elastic unloading on spring-back prediction of V-shape bend.

Accurate spring-back prediction based on FEM also strongly depends on the hardening rule [1,8,9]. Isotropic hardening (IH) may not be so effective when the material points experience cyclic loads; for example, bending–unbending on the die shoulder and reverse bending–unbending at the punch [10]. The linear kinematic hardening proposed by Prager [11] and Ziegler [12] underestimates the spring-back because of the assumption of yield stress rface without changing its shape   size. The Chaboche [13,14] model has gained some popularity because a recall term is introduced to realize the smooth elastic–plastic transition behavior in recent years. The combined isotropic-kinematic hardening law is considered suitable in predicting the spring-back [2,10].

The objective of the present study was to formulate an elasto-plastic constitutive equation based on the Chaboche non-linear kinematic (NLK) hardening rule and Hill’s 1948 anisotropic yielding function. The Young’s modulus has a significant influence on sheet spring-back. Usually, Young’s modulus is considered as a constant in engineering   plication and numerical simulation, but it decreases with plastic deformation. So the change of Young’s modulus with plastic strain is considered in the suggested constitutive model. A backward-Euler method with r-residual algorithm is utilized to   the stress increment. The constitutive equation for isotropic-kinematic hardening behaviors was implemented into the ABAQUS codes using the implicit user subroutine UMAT.   The influences of hardening rules on the sheet spring-back are also discussed.