Finite Element Modeling of the Punch Stretching of Square Plates

The paper presents an updated Lagrangian-type finite element procedure, formulated with reference to a surface embedded coordinated system. Membrane shell theory is employed, and an attempt is made to calculate the strain distribution incurred by a peripherally clamped square plate, when impressed by a rigid punch. Three different punch geometries were considered. The material is treated as a rate insentive, elastic work-hardening solid, which obeys the J₂ flow theory; both finite deformation and normal anisotropy can be considered. A linear relationship between the Jaumann rate of Cauchy stress and the Eulerian rate of Green’s strain tensor is derived. A slip-stick model was adopted for the interfacial frictional conditions. This was achieved by considering the equilibrium of a constant strain-element node in contact with the tools, and deciding whether such a node would stick or slip under Coulomb friction conditions. It is demonstrated that the punch geometry and frictional conditions exert a strong influence on the deformation mode, and hence, upon the overall strain distribution. The predictions were checked against experimental observations when stretch-forming square plates of pure aluminum, 0.5-mm thick. Contours of equal height on the deforming blanks were determined using a Moir´e fringe technique. The agreement between theory and experiment was favorable.