An Analytical Study On Springback Of Sheet Metal Cold Bend-forming Based On Deformation History

Springback is a key defect existing in sheet metal cold bend-forming,which greatly affects the product geometrical accuracy.According to the classical springback theory,springback is defined as the elastic deformation of metals after unloading due to the action of internal stresses.In an early attempt,Gardiner proposed an analytical solution for the springback of elastic and ideal-plastic narrow plate under pure bending based on the classical springback theory.Then Crandall expanded on the work by Gardiner from narrow plate bend-forming to wide plate bend-forming.Considering that the analysis method can quickly determine the springback and reflect the associated mechanism,law and major influence factors,several studies further expanded on Gardiner’s research to provide improved solutions of springback.However,the fore-mentioned studies did not account for mechanical analysis considering deformation history in their improved solutions,and the fore-mentioned studies also did not make a deep investigation about the mechanics principle of sheet metal cold bend-forming.In this study,the Fyh plane-stress state meets that the principal direction 1 and the principal direction 2 of plane-stress state are not opposite.The study also pointed out that the stress state of the narrow plate pure bend-forming,wide plate pure bend-forming and spherical surface bend-forming are three typical cases of Fyh plane-stress state:1)σ₂≡0;2)σ₁>σ₂;3)σ₁≡σ₂.Using Tresca yield criterion,the study applied Hook’s law in plastic regions,the two principal stress-strain relationship of Fyh plane-stress state was obtained and the simplified tangential stress-strain relationship of the narrow plate under pure bending,wide plate under pure bending and sheet metal under spherical surface bending was presented,respectively.In the meantime,the present study divides the deformation history of sheet metal bend-forming into three stages,namely a loading stage,a free unloading stage,and a non-free unloading stage.Based on small deformation assumption,an elastoplastic analysis method was established to solve the mechanical modals of the narrow plate under pure bending,wide plate under pure bending and sheet metal under spherical surface bending in which the deformation history is cosidered.Based on mechanical analysis considering deformation history,the improved solution of the study for springback prediction expanded on Gardiner’s research from ideal-elastoplastic materials to bilinear work-hardening materials,and the application of springback prediction of narrow plate under pure bending indicates that the numerical predictions with respect to the improved solution agreed better with the test results than those obtained using the solution proposed by Gardiner;the improved solution of the study for springback prediction expanded on Crandall’s research from ideal-elastoplastic materials to bilinear work-hardening materials,and the application of springback prediction of wide plate under pure bending indicates that the numerical predictions with respect to the improved solution agreed better with the test results than those obtained using the solution proposed by Crandall;the study also presented an analytical solution for the springback of bilinear kinematic hardening elastoplastic sheet metal under spherical surface bending in which the mathematical expression is similar to Gardiner’s solution,the application of springback prediction of sheet metal under spherical surface bending by square head adjustable die indicates that the analytical solution of the study constitutes an approximate reference expression for the cold bending springback computation of sheet metals under spherical surface bending.With respect to common metal materials,the springback of the narrow plate under pure bending,wide plate under pure bending and sheet metal under spherical surface bending occurs in the elastic stage of reverse loading,and this is an elastic recovery which is in agreement with the classical springback theory.The study chose ideal-elastoplastic Euler-Bernoulli monosymmetric beams in pure bend-forming as the research subject,the evaluation of elastic/elastoplastic springback for various monosymmetric beams was presented to show the existence of elastoplastic stage of reverse loading,and the springback of stiffened monosymmetric plates occurs in the elastic/elastoplastic stage of reverse loading.