A Study of Ductile Damage and Failure of Pure Copper – Part II: Analysis of the Deep Drawing Process of a Cylindrical Shell
The analysis of the stress-strain state and strain induced damage of a cylindrical shell made of copper during the process of deep drawing is presented. The stresses on the contact surface of operating tools (punch and die) are assigned implicitly, which leads to the mixed boundary-value problem. The results are obtained on the basis of the solution of the constitutive differential equations presenting plane plastic flow in curvilinear characteristic coordinates. The material functions required for the analysis of deep drawing were obtained by experimental studies of ductile damage and failure of pure copper (cf., Zapara et al., 2011). The process of deformation with discontinuities of the tangential velocities at the plastic zone boundaries is discussed. An estimate of the local strains and damage in the material is given both for the plastic zone and for its boundaries. The distributions of strains and of the damage within the wall of a finished part are determined. These distributions strongly affect the strength properties of a shell. The modeling of ductile damage in a material during deep drawing is based on experimental results and considerably extends them for a wider range of stress triaxialities. It is shown that the use of a drawing die with a cone angle of 12°...13° leads to a noticeable shift of the stress triaxialities into a range of negative values as compared to deep drawing with a die of larger angle ( 15°...18° ). The modeling reveals a smoother increase and decrease in damage of the finished part in case of the smaller cone angle as well as the absence of void coalescence. This fact is very important when manufacturing such products at high operating speeds. The obtained results in combination with similar ones from the literature can be applied to the analysis of metal forming processes with dominant tensile deformation (e.g., drawing, deep drawing, stripdrawing, ironing).