Response and residual curvature of bent-stretched circular rods with applications to metal forming: Closed-form solutions for elastic-perfectly plastic and hyperbolic hardening materials

Here we develop an analytical expression for the curvature of rods of circular cross-section, when subjected to combined bending and axial force. Small deformations are assumed here so that the deflections are small compared to the cross-sectional dimensions. An enhanced material constitutive law having a linear elastic part followed by smoothly matched hyperbolic hardening is proposed. Analysis for two material models are presented which includes elastic-perfectly plastic material as well as hardening. The effects of the axial force on the response and springback are analytically assessed. The analysis is then applied to the bending of rods of circular cross-section. The analytical results are in excellent agreement with numerical calculations.