A theory has been developed to provide a solution for axisymmetrical shells in the plastic range for large deformations up to fracture. It includes the effects of strain history, nonlinear strain-hardening characteristics of materials, plastic anisotropy in the thickness direction, prestrain, through-thickness stress, and boundary tractions. It is also possible to use nonuniform initial conditions such as varying thickness and varying prestrain. A numerical solution has been developed especially suitable for stretch forming by a rigid punch and for hydraulic bulging of shells or diaphragms. It can easily be modified for the deep-drawing problem. Different instability criteria have been studied. It was found that the conventional criteria would not yield satisfactory results. A new criterion called the “strain propagation” criterion gave satisfactory results in the prediction of the onset of fracture. It could expalin the fracture taking place at increasing or decreasing pressures in the hydraulic bulging problem.

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