Analytical model for predicting sheet springback
Department of Metallurgical Engineering and Materials Science INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY
As an important manufacturing method, bending has been widely used in modern industries to produce stamping parts such as frames, channels, braces, brackets and other structural parts. The understanding and development of bending mechanics are aimed at achieving two kinds of information which are very important for industrial production. One is to predict springback for dies design and compensation in order to obtain high dimension accuracy of bending parts. The other is to determine the limit bending ratio Ri/t0 for a given sheet thickness and material properties. Different methods such as analytical method, semi-analytical method and finite element method (FEM) have been applied to analyze the bending process. FEM is a time-consuming method and also is very sensitive to numerical parameters such as element type and size, algorithms, contact definition and convergence criterion for solution, etc. Analytical method is a time-saving method and has been widely used for predicting springback of bending parts. But most of these researches ignored the effects of contact pressure, transverse stress, neutral surface shifting and thickness thinning on sheet springback of bending parts. Previous research ignored the effect of neutral surface shift and contact pressure between the sheet and die, they were limited to the springback problem of bending ratio Ri/t0 ≥5. For the bending ratio Ri/t0≥5, the effects of contact pressure, transverse stress, neutral surface shifting on sheet springback of bending are significant, with the analysis models which ignore them being inaccurate as the bending ratio decreases. An analytical model to predict the sheet springback of V-bending has been studied. This model is based on Hill’s yielding criterion and plane strain condition, and takes into account contact pressure, transverse stress, neutral surface shifting and sheet thickness thinning. The effects of contact pressure, the length of bending arm between the punch and die, neutral surface shifting and sheet thickness thinning on sheet springback were studied.
2) Analysis of Sheet Bending
Fig.1 The scheme of sheet bending
The round corner of sheet V-bending can be considered as bending under the actions of contact pressure q and bending moment M as shown in Fig.1. The following assumptions are applied: (1) The sheet is wide enough relative to its thickness. Therefore the strain in the width direction is zero; (2) Straight lines perpendicular to the neutral surface remain straight during the V-bending process; (3) Volume conservation is kept during V-bending process; (4) Bauschinger effect is neglected and only elastic deformation occurs during the unloading process. Sheet thinning and neutral surface shifting after V-bending
According to Assumptions (1) and (3), the area of sheet cross-section remains constant during V-bending process, that is:
where L0, t0 are initial sheet length and thickness, respectively; θ is the bending angle; Ri, Ro are the radii of concave and convex surface.
Since the neutral layer length remains constant during V-bending, the radius of neutral surface Rn can be defined as:
Substitution of Eq.(2) into Eq.(1) yields:
where t is the sheet thickness after V-bending; Rm is the radius of middle surface and can be written as: Rm = (R i + Ro ) / 2.
Assuming the distribution of tangential strain through thickness is:
where εθ is the tangential strain; r is the radius of the studied bending layer; c is half thickness of elastic region. Substitution of Eq.(3) into Eq.(4) yields:
It can be seen from Eq.(5) that the tangential strain in the plastic region has two parts. One is bending...