Plastic Deformation of a Circular Diaphragm Under Pressure

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Abstract
In this report a numerical solution is given of a set of equations consisting essentially of three plasticity laws, two strain-displacement laws, and two equilibrium laws which describe the action of a clamped, thin, circular diaphragm as it yields plastically when pressure is applied to one side. The stresses, strains, thickness variation, and deflections for any thin circular diaphragm of a given material may be computed by the numerical integration of the equilibrium conditions, the geometric conditions relating displacements and strains, and the stress-strain laws. The solution may be reduced to the solution of a second-order differential equation with the radial distance r as independent variable. The solution depends upon an experimentally determined function, τ (γ), which describes the stress-strain properties of the material, and upon three parameters, the pressure p, the original thickness h0, and the radius a of the clamping ring. It is found that for a given material, a family of curves with pa/h0 as a parameter serves to predict the solution for any thin circular diaphragm of the same material. This analysis has been carried out for a particular function τ(γ) based upon results of a tensile test made on a specimen of medium steel. Graphs of theoretically and experimentally determined values of deflection, radial and circumferential strains, radial and circumferential stresses, and thickness corresponding to various pressures are presented which apply to all diaphragms made of the same steel as this specimen.

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