Fracture limit prediction for sheet metal forming by damage mechanics approach

Abstract

A number of investigations have been made for understanding the metal forming process and predicting the forming limit of sheet metal. The conventional method of analyzing the deformation is based on the plasticity theory and ignores the degradation of materials which is caused by the formation of micro-defects. In fact, the micro-defects play an important role in limit strains of materials. Therefore, there is a need to introduce damage modelling technique in metal forming prediction. Previously, experimental results showed that the damage caused by uni-axial tensile loading was not isotropic, even though the material itself was originally isotropic. Therefore, it is necessary to develop an anisotropic elasto-plastic damage model for fracture limit prediction in metal forming. The objective of this research project is to develop a new method in analyzing and predicting the fracture limit of sheet metal with consideration of gradual deterioration of material. The approach used in this study takes into account the effects of initiation, coalescence and growth of these micro-defects on a material element until its final rupture. The degradation and its effects on the material have been described by introducing the field variable ψ called 'continuity'. This tensorial variable characterized the progressive deterioration of mechanical properties. For a representative volume element (RVE) of a damaged elasto-plastic material, the strain tensor could be decomposed into the elastic and plastic components under large deformation. A second order continuity tensor ψ was then proposed to characterize the anisotropic elasto-plastic damage state and its evolution process within a RVE of ductile metal material. The second order continuity tensor could be determined from the effective elastic stiffness matrix and satisfied the requirement of symmetry for derivation of the effective stress tensor, the effective elastic strain tensor and the effective elastic stiffness tensor. The corresponding anisotropic damage constitutive relations were formulated to model the damage-failure process for sheet metal. Using the load-and-unload test, the effective stiffness matrix has been calculated under different loading conditions and hence the values of the continuity tensor were then determined. The current formulation does not require the assumption that the principal coordinate system of damage must coincide with that of the material, therefore it can be used to analyze more general problems. The proposed formulation has then been applied to aluminium alloy 2024T3 specimens which were damaged under uni-axial large strain. The experimental results showed that the effects of damage caused the effective stiffness of the aluminium alloy to decrease. A damage-based criterion has been derived on the basis of the proposed formulation and was a non-linear function of the equivalent plastic strain. The criterion takes into the account of the triaxiality of stress, the elasto-plastic properties of orthotropic material, and the anisotropy of damage. Applying the damage criterion to sheet metal forming, the damage mechanics theory was extended to predict the fracture strains under different loading conditions. Experimental verification has been done by Erichsen tests and uni-axial tensile tests. The fracture limit curve (FLC), i.e. the fracture strains of A12024T3 sheet, were then determined by the damage-based fracture criterion. The microscopic examination has shown that coarse intermetallic particles exist in the aluminium alloy sheet. Breaking of these particles under high strain can open up micro-voids that initiate damage. The criterion has also been proven to be applicable for steel sheets such as HS-3 and A-K steel. The results showed that for the whole range of the strain ratio, the predicted fracture strains were in agreement with the experimental ones. To conclude, a second order continuity tensor coupled with a damage-based criterion has been proposed in predicting the fracture limit strains. The proposed formulation has also been verified under various stretching conditions by different materials. The results showed that the predicted results were acceptable and reliable. In this research, it also showed that the damage mechanics theory could be extended to analyze and predict the fracture limit of sheet metals.