This paper deals with the springback analysis in sheet metal forming using modified Ludwik stress-strain relation. Using the deformation theory of plasticity, formulation of the problem and spring back ratio is derived using modified Ludwik stress strain relationship with Tresca and von Mises yielding criteraia. The results have been representing the effect of different value of

In sheet metal working, sheets are deformed to cylindrical and helical shapes by plastic bending with the help of a punch and die set. When such bending is properly done, the inside contour of the section matches the surface contour of the die during the forming operation. However, after the release of the applied loads, the contour assumes a different shape than that of the die because of the release of elastic stresses in the metal. This elastic distortion is commonly called springback. Springback complicates tool design in that the die must be designed to compensate for it. Consequently, it is desirable to have a method of quantitatively predicting the magnitude of springback as a function of the properties of the material and geometry involved in the forming operation.

In bending, the springback is a measure of elastic recovery of radius on removal of the applied bending moment or load after the bending section is beyond elastic limit. While designing the die set, the springback factor should be taken into account to avoid mismatch while assembling formed different sections.

Initially, springback studies are with sheet bending operations. Sachs [

An accurate analysis of springback has been made in the past on sheet bending and tube bending operations through experiment [

In the following, approximation equations are derived in an attempt to provide a quantitative method with practical utility for predicting the springback behavior in bent section of sheet metal as a function of die radius, sheet thickness, and stress-strain characteristics of the material.

In this paper, springback prediction approach using the modified Ludwik stress-strain relation was studied. The assumption of narrow beam of a wide sheet can significantly alter the magnitude of the predicted results in sheet metal bending because this ignorance of the transverse stresses are present during forming. In this paper, it is assumed that the bend section is a wide sheet of an elastic plastic strain hardening material in an effort to obtain more accurate expression for predicting the springback behavior. In this method, use of applied moment and curvature relation during the formation of a wide sheet of metal around a portion of cylindrical die was done to derive springback relationship. From the bending equation, we directly relate the bending moment and curvature during the forming as follows:

From this equation, the springback is related to the applied bending moment and curvature during the formation. The modified Ludwik stress-strain relation (Figure

Pure bending of the sheet to cylindrical surface.

The cross section dimensions of the sheet are such so that the width to thickness ratio is large.

The stress-strain characteristics of material are the same in tension and compression.

The natural surface is always in the centre of the sheet, and plane sections remain plane during bending.

The cross section dimensions of the sheet do not change significantly in bending.

The radius of bending is large compared to the thickness of the sheet so radial stresses are assumed to be negligible.

The circumferential strains are sufficiently small so that the conventional strain and the true strain are approximately equivalent.

The transverse strain is zero at any point in the sheet.

The circumferential strain for any fiber does not vary along the bent section.

Figure

Schematic representation of curvature versus applied moment during bending.

Empirical stress-strain curve for elastic plastic material in modified ludwik stress-strain relation.

From Figure

Direction of principal stresses and curvature during bending.

Figure

Before proceeding, it is desirable to attempt to characterize the stress-strain behaviour of material in a simple tension. From the modified Ludwik stress-strain relation, we get

This is the equation for springback for Modified Ludwik stress-strain relation.

In elastic region,

For combined states of stress, the relationship between principal stresses and strain for the elastic.

By Hooke’s law,

From assumptions that the radius of bending is large compared to thickness of the sheet so radial stresses can assume to be negligible, and the transverse strain is zero at any point in the sheet,

And the circumferential strain for any fiber does not vary along the bent section.

So,

Then, from putting the value of (

Substituting (

Then,

So in yield point, the circumferential strain is

Putting yield point stress value from (

Then,

From the deformation theory of plasticity,

From assumptions that the radius of bending is large compared to thickness of the sheet so radial stresses can assume to be negligible and the transverse strain is zero at any point in the sheet,

And the circumferential strain for any fiber does not vary along the bent section. So,

Put this value in (

So,

Putting the value of

From initial assumption that the circumferential strain are sufficiently small so that the conventional strain and true strain are approximately equivalent,

Putting the value of

From (

Then

From elementary plate theory [

The slope of elastic recovery in elastic unloading is

Putting this into (

This is the equation for springback ratio for modified Ludwik stress-strain relationship using maximum shear stress theory.

From maximum shear stress theory of failure (von Mises yielding criteria),

Substituting (

Now,

So, in yield point, the circumferential strain is

Putting yield point stress value from (

This is the approximate value of circumferential strain at elastic plastic interface.

Then in elastic region, the circumferential stress is

Putting value of circumferential strain, we get

for

From the deformation theory of plasticity [

From assumptions that the radius of bending is large compared to thickness of the sheet so radial stresses can assume to be negligible and the transverse strain is zero at any point in the sheet,

So,

So,

From initial assumption that the circumferential strain are sufficiently small so that the conventional strain and true strain are approximately equivalent,

From elementary plate theory [

For the sheet metal bending, the springback is calculated from derived equation (using modified Ludwik stress-strain relationship with Tresca and von Mises yield criteria).

Then,

We can see that both equations have a last term that is very negligible because the

Then we write,

Therefore, we can say that if we apply Tresca or von Mises yield criteria, then we are getting same result. This equation is dependent on the

Figure

Springback ratio with

Springback ratio with

Springback ratio with

Springback ratio with

Figures

Springback ratio with

Springback ratio with

Springback ratio with

Figures

Springback ratio with

Springback ratio with

Springback ratio with

Figures

Springback ratio with

Springback ratio with

Springback ratio with

Based on the results presented in chapter of results and discussion, following conclusions about the springback analysis in sheet metal forming using nonlinear constitutive equation, the following applies.

The theoretical analysis for the sheet metal under pure bending has been done, and it was found that the prediction of springback is quite successfull.

As expected, elastic recovery is found to be more with decreasing values of work hardening coefficient. At lower values of n, the material will approach to an elasto-ideally plastic behavior.

Springback ratio increases with increasing thickness.

Springback ratio increases with decreasing ratio of yield point stress to Young’s modulus of elasticity

Springback ratio is increasing with increasing Possion’s ratio.

Springback ratio for Tresca and von Mises yield criteria, we say that

Radius of die

Radius of final product after removing load

Maximum applied bending moment per unit width

Distance from midsection

Thickness

True stress

True strain

Modulus of elasticity

Poisson’s ratio

Strain hardening index

Empirical constant

Constant

Yield point stress

Strain at yield point

Strain in plastic region.