Material compatibility and some understanding of the ball swaging process

For a ball swaging process, an attempt has been made to get a closed-form solution for in-plane deformation. Inner and outer rings and arm have been approximated as thick cylinders subjected to uniform internal pressure. Based upon plastic deformations, residual stresses are calculated for perfect elasto-plastic materials and values of retention torques have been obtained. When treated as thin cylinders, it is proved that for perfect elasto-plastic materials, the sufficient, but not necessary, simple inequality shown below, if satisfied, ensures that retention torque is positive: σ_y3/E₃>σ_y2/E₂ >σ_y1/E₁ where σ_yi and E_i are yield stress and Young´s Modulus of cylinder i. Cylinder 1 is the innermost cylinder. This has also been validated for thick cylinders, numerically; but not proven mathematically