Springback Analysis in Sheet Metal Forming Using Modified Ludwik Stress-Strain Relation

Abstract

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 𝑌/𝐸 or 𝜎_𝑜/𝐸 ratio, different values of Strain hardening index (𝑛), Poisson’s ratio (𝜈), and thickness on spring back ratio (𝑅_𝑜/𝑅_𝑓). The main aim of this paper is to study the effects of the thickness, 𝑌/𝐸 ratio, 𝑛 and Poisson’s ratio in spring back ratio.

1. Introduction

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 [1], Schroeder [2], Gardiner [3], Singh and Johnson [4], and others studied the springback considered bending of sheets of different shapes and depicted springback as a function of material thickness, length, and width of the sheets taken. Their studies were limited to V- and U-shaped dies for applying bending loads and they predicted the springback as a measure of change in the curvature distribution. Huth [5], Nadai [6], and Upadhyay [7] have all done a number of excellent work on the elasto-plastic torsion of bars with rectangular sections, but their work has been limited to monotonically increasing loads only. Dwivedi et al. [8, 9] analytically predicted the residual angle of twist and torque relation, and so forth, for bars of elastic strain-hardening materials with narrow rectangular sections. This work, however, has the limitation that it is valid for thin rectangular strips only. Dwivedi et al. [10, 11] dealt with the torsional springback of square-section bars of linear and nonlinear work-hardening materials. Dwivedi et al. [12] also dealt with the torsional springback of L-shaped section bars of nonlinear work-hardening materials.

An accurate analysis of springback has been made in the past on sheet bending and tube bending operations through experiment [13–18]. Torsional springback in thin tubes with nonlinear work hardening analysis was made by Choubey et al. [19–21].

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.

2. Basic Equations and Formulation

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: 𝑀𝐼=𝜎𝑦=𝐸𝑅,𝑀=1𝐸𝐼𝑅,1𝑀=𝐶∗𝑅. (1)

From this equation, the springback is related to the applied bending moment and curvature during the formation. The modified Ludwik stress-strain relation (Figure 2) is given by 𝑌𝜎=𝐸𝜀𝜀≤𝐸𝑌𝐸𝜀𝑌^𝑛𝜀⩾̇𝑌𝐸, (2) where 𝑌=𝜎_𝑜, 𝜎=𝐸𝜀 is the elastic part and 𝜎=𝜎_𝑜(𝐸𝜀/𝜎_𝑜)^𝑛 is the plastic part, and the equation gives the total stress.

3. Assumption for Derivation of Springback Behaviour

(1)Pure bending of the sheet to cylindrical surface.(2)The cross section dimensions of the sheet are such so that the width to thickness ratio is large. (3)The stress-strain characteristics of material are the same in tension and compression.(4)The natural surface is always in the centre of the sheet, and plane sections remain plane during bending.(5)The cross section dimensions of the sheet do not change significantly in bending.(6)The radius of bending is large compared to the thickness of the sheet so radial stresses are assumed to be negligible. (7)The circumferential strains are sufficiently small so that the conventional strain and the true strain are approximately equivalent. (8)The transverse strain is zero at any point in the sheet. (9)The circumferential strain for any fiber does not vary along the bent section.

4. Derivation of Springback Equation for Tresca Yield Criteria

Figure 1 shows the schematic plot of applied bending moment versus curvature during the formation of a wide sheet of metal around a portion of a cylindrical die at point A, the material yields and plastic deformation continuous until the inside surface of the material conforms to the inside surface of the material conforms to die at point B when the applied moment is released elastic springback occurs from B to C. The change in curvature due to this elastic springback is (1/𝑅_𝑜)−(1/𝑅_𝑓)

Figure 1 

Schematic representation of curvature versus applied moment during bending.

Figure 2 

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

From Figure 3, 1𝑅_𝑜−1𝑅_𝑓=𝑀_max𝜕𝑀_𝐸()./𝜕1/𝑅 (3)

Figure 3 

Direction of principal stresses and curvature during bending.

Figure 3 shows a section of bent sheet and the coordinate system adopted. Then calculation of maximum applied bending moment by balancing the external and internal moments is as follows: 𝑀_max=^+𝑡/2_−𝑡/2𝜎_𝑥𝑦𝑑𝑦=2₀^𝑡/2𝜎_𝑥𝑦𝑑𝑦. (4)

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 𝑌𝜎=𝐸𝜀𝜀≤𝐸𝑌𝐸𝜀𝑌^𝑛𝜀⩾̇𝑌𝐸. (5)

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

In elastic region, 𝜎=𝐸𝜀. (6) In plastic region, where 𝑌=𝜎_𝑜, 𝜎=𝜎_𝑜𝐸𝜎_𝑜^𝑛𝜀^𝑛. (7) Let 𝜎_𝑜𝐸𝜎_𝑜^𝑛=𝐾. (8) Then 𝜎=𝐾𝜀^𝑛. (9) At yield point, the stress in elastic and plastic region is equal; so we can write 𝐸𝜀_𝑜=𝐾𝜀^𝑛_𝑜,𝜀_𝑜=𝐾𝐸^1/(1−𝑛). (10) From (10), stress in yield point is 𝜎_𝑜=𝐾𝜀^𝑛_𝑜𝐾=𝐾𝐸^{𝑛/(1−𝑛)}. (11) It will be assumed for stresses less than 𝜎_𝑜 that the stress-strain behaviour of the material in a simple tension test is elastic (6) and that yielding occurs at 𝜎_𝑜 and (7) is applicable for stresses greater than 𝜎_𝑜.

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

By Hooke’s law, 𝜀_𝑥=1𝐸𝜎_𝑥−𝜈𝜎_𝑦+𝜎_𝑧,𝜀_𝑦=1𝐸𝜎_𝑦−𝜈𝜎_𝑥+𝜎_𝑧,𝜀_𝑧=1𝐸𝜎_𝑧−𝜈𝜎_𝑦+𝜎_𝑥. (12)

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, 𝜎_𝑦=𝜀_𝑧=𝛿_𝑧=0. (13)

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

So, 𝜀_𝑥=𝛿_𝑥. (14)

Then, from putting the value of (12) in (13), 𝜀_𝑧=1𝐸𝜎_𝑧−𝜈𝜎_𝑦+𝜎_𝑥,10=𝐸𝜎_𝑧−𝜈𝜎_𝑦+𝜎_𝑥,1𝐸𝜎_𝑧−𝜈𝜎_𝑥𝜎=0,_𝑧=𝜈𝜎_𝑥. (15) From maximum shear stress theory of failure (Tresca yield criteria), 𝜎_𝑜2=𝜎_𝑥−𝜎_𝑧2,𝜎_𝑜=𝜎_𝑥−𝜈𝜎_𝑥,𝜎_𝑜=𝜎_𝑥(1−𝜈). (16)

Substituting (16) in (11), 𝜎_𝑜=𝜎_𝑥()𝐾1−𝜈=𝐾𝐸^{𝑛/(1−𝑛)},𝜎_𝑥=𝐾(𝐾/𝐸)^1/(1−𝑛)().1−𝜈 (17)

Then, 𝜀_𝑥=1𝐸𝜎_𝑥−𝜈𝜎_𝑦+𝜎_𝑧,𝜀_𝑥=1𝐸𝜎_𝑥−𝜈²𝜎_𝑥,𝜀_𝑥=1𝐸𝜎_𝑥1−𝜈². (18)

So in yield point, the circumferential strain is 𝜀_𝑥𝑜=1𝐸𝜎_𝑜1−𝜈². (19)

Putting yield point stress value from (11), we get 𝜀_𝑥𝑜=𝐾(𝐾/𝐸)^{𝑛/(1−𝑛)}𝐸(1−𝜈)1−𝜈²,𝜀_𝑥𝑜=𝐾𝐸^1/(1−𝑛)(1+𝜈). (20) This is the approximate value of circumferential strain at elastic plastic interface. Then in elastic region, the circumferential stress is 𝜎_𝑥=𝐸1−𝜈²𝜀_𝑥. (21) Here, 𝜀_𝑥=𝑦/𝑅_𝑜.

Then, 𝜎_𝑥=𝐸1−𝜈²𝑦𝑅_𝑜, (22) for 0≤𝜀_𝑥≤(𝐾/𝐸)^1/(1−𝑛)(1+𝜈).

From the deformation theory of plasticity, 𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑥−𝜎_𝑦2−𝜎_𝑧2,𝛿_𝑦=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑦−𝜎_𝑥2−𝜎_𝑧2,𝛿_𝑧=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑧−𝜎_𝑥2−𝜎_𝑦2. (23)

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, 𝜎_𝑦=𝜀_𝑧=𝛿_𝑧=0.

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

Put this value in (23). We get 𝛿_𝑧=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑧−𝜎_𝑥2−𝜎_𝑦2,10=𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}𝜎_𝑧−𝜎_𝑥2. (24)

So, 0=𝜎_𝑧−𝜎_𝑥2,𝜎_𝑧=𝜎_𝑥2. (25)

Putting the value of 𝜎_𝑧 into (23), we found that 𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑥−𝜎_𝑦2−𝜎_𝑧2,𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑥4−𝜎²_𝑥2^{(1−𝑛)/2𝑛}𝜎_𝑥−𝜎_𝑥4,𝛿_𝑥=1𝐾^1/𝑛3𝜎²_𝑥4^{(1−𝑛)/2𝑛}3𝜎_𝑥4,𝛿_𝑥=1𝐾^1/𝑛34^{(1+𝑛)/2𝑛}𝜎_𝑥^1/𝑛,𝜎_𝑥=𝐾(3/4)^(1+𝑛)/2𝛿^𝑛_𝑥. (26)

From initial assumption that the circumferential strain are sufficiently small so that the conventional strain and true strain are approximately equivalent, 𝜀_𝑥=𝑦𝑅_𝑜=𝛿_𝑥. (27)

Putting the value of 𝛿_𝑥 into (26), we get 𝜎_𝑥=𝐾(3/4)^(1+𝑛)/2𝑦𝑅_𝑜^𝑛, (28) where 𝜎_𝑥 is valid for plastic region; that is, 𝐾𝐸^1/(1−𝑛)(1+𝜈)≤𝜀_𝑥≤𝑡2. (29)

From (4), we found maximum bending moment is 𝑀_max=2₀^𝑡/2𝜎_𝑥𝑦𝑑𝑦, (30) where we can split the limit into elastic or plastic part, from 0 to 𝑅_𝑜𝑒𝑥 elastic region and 𝑅_𝑜𝑒𝑥 to (𝑡/2) plastic region, where 𝑅_𝑜𝑒𝑥 is calculated by 𝑅_𝑜𝑒𝑥=𝑅_𝑜𝐾𝐸^1/(1−𝑛)(1+𝜈). (31) Now, 𝑀_max=2^{𝑅𝑜𝑒𝑥}₀𝜎_{𝑥(elastic)}𝑦𝑑𝑦+_𝑅^𝑡/2_𝑜𝑒𝑥𝜎_{𝑥(plastic)}𝑦𝑑𝑦=2^{𝑅𝑜𝑒𝑥}₀𝐸1−𝜈²𝑦²+𝑑𝑦_{𝑅𝑡/2𝑜𝑒𝑥}𝐾(3/4)^(1+𝑛)/2𝑦𝑅_𝑜^𝑛𝑦𝑑𝑦=2^{𝑅𝑜𝑒𝑥}₀𝐸1−𝜈²𝑦²+𝑑𝑦_{𝑅𝑡/2𝑜𝑒𝑥}𝐾(3/4)^(1+𝑛)/21𝑅_𝑜^𝑛𝑦^𝑛+11𝑑𝑦=2𝑅_𝑜𝐸1−𝜈²𝑦³3^{𝑅𝑜𝑒𝑥}₀𝐾+2(3/4)^(1+𝑛)/21𝑅_𝑜^𝑛𝑦^𝑛+2𝑛+2_𝑅^𝑡/2_𝑜𝑒𝑥. (32) Put the value of 𝑅_𝑜𝑒𝑥 in above equation.

Then =2𝐸𝑅²_𝑜(𝐾/𝐸)^3/(1−𝑛)3(1−𝜈)(1+𝜈)²𝐾()+2𝑡/2^𝑛+2(3/4)^(1+𝑛)/2𝑅^𝑛_𝑜()−𝑛+2𝐾𝑅²_𝑜(𝐾/𝐸)^{(𝑛+2)/(1−𝑛)}(1+𝜈)^𝑛+2(3/4)^(1+𝑛)/2().𝑛+2 (33)

From elementary plate theory [17] applied moment per unit width during elastic unloading, 𝑀_𝐸𝐼=𝜎_𝑥𝑦,𝑀_𝐸=2⋅𝐸⋅(𝑡/2)³31−𝜈².⋅𝑅 (34)

The slope of elastic recovery in elastic unloading is 𝜕𝑀_𝐸𝜕()=()1/𝑅2⋅𝐸⋅𝑡/2³31−𝜈². (35)

Putting this into (3), we get 1𝑅_𝑜−1𝑅_𝑓=𝑀_max𝜕𝑀_𝐸()𝑅/𝜕1/𝑅𝑅^𝑜_𝑓=1−𝑅_𝑜𝑀_max𝜕𝑀_𝐸(),𝑅/𝜕1/𝑅_𝑜𝑅_𝑓=1−𝑅_𝑜2𝑅_𝑜𝑡³𝐾𝐸^3/(1−𝑛)(1+𝜈)³+3𝐾1−𝜈²2𝑅_𝑜/𝑡^1−𝑛𝐸(3/4)^(1+𝑛)/2()−3𝑛+2(2𝑅_𝑜)/𝑡³(𝐾/𝐸)^3/(1−𝑛)(1+𝜈)^𝑛+3(1−𝜈)(3/4)^(1+𝑛)/2(),𝑅𝑛+2_𝑜𝑅_𝑓=1−3𝐾1−𝜈²𝐸(3/4)^(1+𝑛)/2(𝑛+2)2𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝐾𝐸^1/(1−𝑛)(1+𝜈)³×3(1+𝜈)^𝑛(1−𝜈)(3/4)^(1+𝑛)/2(),𝑅𝑛+2−1_𝑜𝑅_𝑓3=1−1−𝜈²(3/4)^(1+𝑛)/2()𝐸𝑛+2𝜎_𝑜^𝑛−12𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝜎_𝑜𝐸¹(1+𝜈)³×3(1+𝜈)^𝑛(1−𝜈)(3/4)^(1+𝑛)/2().𝑛+2−1 (36)

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

5. Derivation of Springback Equation for Von Mises Yielding

From maximum shear stress theory of failure (von Mises yielding criteria), 𝜎²_𝑜=12𝜎_𝑥−𝜎_𝑦²+𝜎_𝑦−𝜎_𝑧²+𝜎_𝑧−𝜎_𝑥²,𝜎²_𝑜=12𝜎²_𝑥+𝜎²_𝑧+𝜎²_𝑧−2𝜎_𝑧𝜎_𝑥+𝜎²_𝑥,𝜎_𝑜=𝜎_𝑥1+𝜈²−𝜈^1/2, (37) by using (13) and (14).

Substituting (15) in (11), at yield point, 𝜎_𝑜𝐾=𝐾𝐸^{𝑛/(1−𝑛)}=𝜎_𝑜𝑥1+𝜈²−𝜈^1/2,𝜎_𝑜𝑥1+𝜈²−𝜈^1/2𝐾=𝐾𝐸^{𝑛/(1−𝑛)},𝜎_𝑥𝑜=𝐾(𝐾/𝐸)^{𝑛/(1−𝑛)}1−𝜈+𝜈^21/2. (38)

Now, 𝜀_𝑥=1𝐸𝜎_𝑥−𝜈𝜎_𝑦+𝜎_𝑧,𝜀_𝑥=1𝐸𝜎_𝑥−𝜈²𝜎_𝑥,𝜀_𝑥=1𝐸𝜎_𝑥1−𝜈². (39)

So, in yield point, the circumferential strain is 𝜀_𝑥𝑜=1𝐸𝜎_𝑜𝑥1−𝜈². (40)

Putting yield point stress value from (38) in (40), we get 𝜀_𝑥𝑜=𝐾(𝐾/𝐸)^{𝑛/(1−𝑛)}𝐸1−𝜗+𝜗^21/21−𝜈²,𝜀_𝑥𝑜=𝐾𝐸^1/(1−𝑛)1−𝜈²1−𝜈+𝜈^21/2. (41)

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

Then in elastic region, the circumferential stress is 𝜎_𝑥=𝐸1−𝜈²𝜀_𝑥. (42) Here, 𝜀_𝑥=𝑦/𝑅_𝑜.

Putting value of circumferential strain, we get 𝜎_𝑥=𝐸1−𝜈²𝑦𝑅_𝑜, (43)

for 0≤𝜀_𝑥≤(𝐾/𝐸)^1/(1−𝑛)((1−𝜈²)/(1−𝜈+𝜈²)^1/2).

From the deformation theory of plasticity [16], 𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑥−𝜎_𝑦2−𝜎_𝑧2,𝛿_𝑦=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑦−𝜎_𝑥2−𝜎_𝑧2,𝛿_𝑧=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑧−𝜎_𝑥2−𝜎_𝑦2. (44)

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, 𝜎_𝑦=𝜀_𝑧=𝛿_𝑧=0. (45) And the circumferential strain for any fiber does not vary along the bent section.

So, 𝜀_𝑥=𝛿_𝑥. (46) Put these values in (44). We get 𝛿_𝑧=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑧−𝜎_𝑥2−𝜎_𝑦2,10=𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}𝜎_𝑧−𝜎_𝑥2. (47)

So, 0=𝜎_𝑧−𝜎_𝑥2,𝜎_𝑧=𝜎_𝑥2. (48) Putting the value of 𝜎_𝑧 into (44), we get 𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑦+𝜎²_𝑧−𝜎_𝑥𝜎_𝑦−𝜎_𝑦𝜎_𝑧−𝜎_𝑧𝜎_𝑥^{(1−𝑛)/2𝑛}×𝜎_𝑥−𝜎_𝑦2−𝜎_𝑧2,𝛿_𝑥=1𝐾^1/𝑛𝜎²_𝑥+𝜎²_𝑥4−𝜎²_𝑥2^{(1−𝑛)/2𝑛}𝜎_𝑥−𝜎_𝑥4,𝛿_𝑥=1𝐾^1/𝑛3𝜎²_𝑥4^{(1−𝑛)/2𝑛}3𝜎_𝑥4,𝛿_𝑥=1𝐾^1/𝜆34^{(1+𝑛)/2𝑛}𝜎_𝑥^1/𝑛,𝜎_𝑥=𝐾(3/4)^{(1+𝑛)/𝑛}𝛿^𝑛_𝑥. (49)

From initial assumption that the circumferential strain are sufficiently small so that the conventional strain and true strain are approximately equivalent, 𝜀_𝑥=𝑦𝑅_𝑜=𝛿_𝑥. (50) Putting the value of 𝛿_𝑥 into (49), we get 𝜎_𝑥=𝐾(3/4)^{(1+𝑛)/𝑛}𝑦𝑅_𝑜^𝑛, (51) where 𝜎_𝑥 is valid for plastic region; that is, 𝐾for𝐸^1/(1−𝑛)1−𝜈²1−𝜈+𝜈²≤𝜀_𝑥≤𝑡2. (52) From (4), maximum bending moment is 𝑀_max=2₀^𝑡/2𝜎_𝑥𝑦𝑑𝑦. (53) We can split the limit into elastic or plastic part from 0 to 𝑡/2 to 0 to 𝑅_𝑜𝑒𝑥 elastic region and 𝑅_𝑜𝑒𝑥 to 𝑡/2 plastic region where 𝑅_𝑜𝑒𝑥 is calculated by 𝑅_𝑜𝑒𝑥=𝐾𝐸^1/(1−𝑛)𝑅_𝑜1−𝜈²1−𝜈+𝜈^21/2,𝑀_max=2^{𝑅𝑜𝑒𝑥}₀𝜎_{𝑥(elastic)}𝑦𝑑𝑦+_𝑅^𝑡/2_𝑜𝑒𝑥𝜎_{𝑥(plastic)}𝑦𝑑𝑦=2^{𝑅𝑜𝑒𝑥}₀𝐸1−𝜈²𝑦²1𝑅_𝑜𝑑𝑦+_𝑅^𝑡/2_𝑜𝑒𝑥𝐾(3/4)^(1+𝑛)/2𝑦𝑅_𝑜^𝑛𝑦𝑑𝑦=2^{𝑅𝑜𝑒𝑥}₀𝐸1−𝜈²𝑦²1𝑅_𝑜+𝑑𝑦_{𝑅𝑡/2𝑜𝑒𝑥}𝐾(3/4)^(1+𝑛)/21𝑅_𝑜^𝑛𝑦^𝑛+11𝑑𝑦=2𝑅_𝑜𝐸1−𝜈²𝑦³3^{𝑅𝑜𝑒𝑥}₀𝐾+2(3/4)^(1+𝑛)/21𝑅_𝑜^𝑛𝑦^𝑛+2𝑛+2_𝑅^𝑡/2_𝑜𝑒𝑥. (54) Putting the value of 𝑅_𝑜𝑒𝑥, we get 𝑅_𝑜𝑒𝑥=𝐾𝐸^1/(1−𝑛)𝑅_𝑜1−𝜈²1−𝜈+𝜈^21/2. (55) Then 𝑀_max=2𝐸𝑅³_𝑜(𝐾/𝐸)^3/(1−𝑛)1−𝜈²³/1−𝜈+𝜈^23/231−𝜈²𝑅_𝑜𝐾()+2𝑡/2^𝑛+2(3/4)^(1+𝑛)/2𝑅^𝑛_𝑜()−𝑛+2𝐾𝑅_𝑜^(𝑛+2)(𝐾/𝐸)^{(𝑛+2)/(1−𝑛)}(1−𝜈²)^𝑛+2(3/4)^(1+𝑛)/2𝑅^𝑛_𝑜(𝑛+2)1−𝜈+𝜈^2(𝑛+2)/2. (56)

From elementary plate theory [17] applied moment per unit width during elastic unloading, 𝑀_𝐸=2⋅𝐸⋅(𝑡/2)³31−𝑣².⋅𝑅 (57) Now, the slope of elastic recovery in elastic unloading is 𝜕𝑀_𝐸𝜕()=()1/𝑅2⋅𝐸⋅𝑡/2³31−𝜈². (58) Putting this into (3), we get 1𝑅₀−1𝑅_𝑓=𝑀_max𝜕𝑀_𝐸(),𝑅/𝜕1/𝑅_𝑜𝑅_𝑓=1−𝑅_𝑜𝑀_max𝜕𝑀_𝐸(),𝑅/𝜕1/𝑅_𝑜𝑅_𝑓=1−3𝐾1−𝜈²𝐸(𝑛+2)(3/4)^(𝑛+1)/22𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝐾𝐸^{1/(1−𝑛)3}×3(1−𝜈²)^𝑛+3(3/4)^(𝑛+1)/2(𝑛+2)1−𝜈+𝜈^2(𝑛+2)/2−1−𝜈²³1−𝜈+𝜈^23/2,𝑅_𝑜𝑅_𝑓3=1−1−𝜈²(𝑛+2)(3/4)^(𝑛+1)/2𝜎_𝑜𝐸^1−𝑛2𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝜎_𝑜𝐸³×31−𝜈^2𝑛+3(3/4)^(𝑛+1)/2(𝑛+2)1−𝜈+𝜈^2(𝑛+2)/2−1−𝜈²³1−𝜈+𝜈^23/2. (59) This is the equation for springback ratio for modified Ludwik stress-strain relationship using von Mises yield criteria.

6. Results and Discussion

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). 𝑅_𝑜𝑅_𝑓3=1−1−𝜈²(3/4)^(1+𝑛)/2()𝐸𝑛+2𝜎_𝑜^𝑛−12𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝜎_𝑜𝐸¹(1+𝜈)³×3(1+𝜈)^𝑛(1−𝜈)(3/4)^(1+𝑛)/2(),𝑅𝑛+2−1_𝑜𝑅_𝑓3=1−1−𝜈²(𝑛+2)(3/4)^(𝑛+1)/2𝜎_𝑜𝐸^1−𝑛2𝑅_𝑜𝑡^1−𝑛+2𝑅_𝑜𝑡𝜎_𝑜𝐸³×3(1−𝜈²)^𝑛+3(3/4)^(𝑛+1)/2(𝑛+2)1−𝜈+𝜈^2(𝑛+2)/2−1−𝜈²³1−𝜈+𝜈^23/2. (60) Here, from (8), 𝐸𝐸𝐾=𝜎_𝑜^𝑛−1. (61)

Then, 𝐾/𝐸=(𝜎_𝑜/𝐸)^1−𝑛.

We can see that both equations have a last term that is very negligible because the 𝜎_𝑜/𝐸  or  𝑌/𝐸=5.5×10⁻⁴ and its three times will be 10⁻¹²; therefore, we have to neglect the last term of both equations.

Then we write, 𝑅_𝑜𝑅_𝑓3=1−1−𝜈²(3/4)^(1+𝑛)/2()𝐸𝑛+2𝑌^𝑛−12𝑅_𝑜𝑡^1−𝑛,𝑅_𝑜𝑅_𝑓3=1−1−𝜈²(𝑛+2)(3/4)^(𝑛+1)/2𝑌𝐸^1−𝑛2𝑅_𝑜𝑡^1−𝑛. (62)(63) Here, we can see that both (62) and (63) are the same if we can apply Tresca or von Mises, getting same result but last term is different. The values of last term are much less; therefore, we neglect this term.

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 𝑅_𝑜/𝑡, 𝜎_𝑜/𝐸, strain hardening coefficient 𝑛, and 𝜈 is Possion’s ratio.

Figure 4 (𝑛=0.1), Figure 5 (𝑛=0.2), Figure 6 (𝑛=0.3), and Figure 7 (𝑛=0.4) show the variation in springback ratio along with change in 𝑅_𝑜/𝑡 with different 𝜎_𝑜/𝐸 ratio. The higher value of 𝑛 (strain hardening coefficient) material to closer to elasto-ideal plastic material, and thickness is one mm, and 𝜎_𝑜/𝐸 value is 5.5×10⁻⁴,1.522×10⁻³, and 2.4×10⁻³ (for different material c10100 copper, 1100al, and 1065steel). From the figure, we come to know that as the with increasing ratio of 𝑅_𝑜/𝑡 is decrease the springback ratio and with decreasing 𝜎_𝑜/𝐸 springback ratio is increases.

Figure 4 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜎𝑜/𝐸 or 𝑌/𝐸 at 𝑛=0.1, 𝜈=0.33.

Figure 5 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜎𝑜/𝐸 or 𝑌/𝐸 at 𝑛=0.2, 𝜈=0.33.

Figure 6 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜎𝑜/𝐸 or 𝑌/𝐸 at 𝑛=0.2, 𝜈=0.33.

Figure 7 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜎𝑜/𝐸 or 𝑌/𝐸 at 𝑛=0.4, 𝜈=0.33.

Figures 8, 9, and 10 show the variation in springback ratio along with 𝑅_𝑜/𝑡 with the different value of 𝑛 with fixed value of 𝜎_𝑜/𝐸=5.5×10⁻⁴, 𝜎_𝑜/𝐸=1.522×10⁻³, and 𝜎_𝑜/𝐸=2.4×10⁻³, here we take the values of 𝑛=0.1,0.2,0.3, and 0.4. From the figure, we come to know that with increasing value of 𝑛, springback ratio is decreasing.

Figure 8 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝑛 at 𝑌/𝐸=5.5×10−4, 𝜈=0.33.

Figure 9 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝑛 at 𝑌/𝐸=1.5×10−3, 𝜈=0.33.

Figure 10 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝑛 at 𝑌/𝐸=2.4×10−3, 𝜈=0.33.

Figures 11, 12, and 13 show the variation in springback along with 𝑅_𝑜/𝑡 with different value of Possion’s ratio 𝜈=0.25, 𝜈=0.35, and 𝜈=0.45 with fixed 𝜎_𝑜/𝐸=5.5×10⁻⁴, 𝜎_𝑜/𝐸=1.522×10⁻³, 𝜎_𝑜/𝐸=2.4×10⁻³, and 𝑛=0.3. From the figure, we come to know that with increasing value of 𝜈, springback ratio is increasing.

Figure 11 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜈 at 𝑌/𝐸=5.5×10−4.

Figure 12 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜈 at 𝑌/𝐸=1.5×10−3.

Figure 13 

Springback ratio with 𝑅𝑜/𝑡 with different values of 𝜈 at 𝑌/𝐸=2.4×10−3.

Figures 14, 15, and 16 show the variation in springback along with 𝑡 with different value of 𝑛=0.1, 0.2, 0.3, and 0.4 and where 𝜎_𝑜/𝐸=5.5×10⁻⁴, 𝜎_𝑜/𝐸=1.522×10⁻³, 𝜎_𝑜/𝐸=2.4×10⁻³, 𝜈=0.33, and 𝑅_𝑜=40 is fixed. From the figure, we come to know that the springback ratio rapidly increases in range of 0 to 2 mm thickness, then the increase in springback ratio is stable.

Figure 14 

Springback ratio with 𝑡 for different values of 𝑛 at 𝑌/𝐸=5.5×10−4.

Figure 15 

Springback ratio with 𝑡 for different values of 𝑛 at 𝑌/𝐸=1.5×10−3.

Figure 16 

Springback ratio with 𝑡 for different values of 𝑛 at 𝑌/𝐸=2.4×10−3.

7. Conclusions

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.(i)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.(ii)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. (iii)Springback ratio increases with increasing thickness.(iv)Springback ratio increases with decreasing ratio of yield point stress to Young’s modulus of elasticity (v)Springback ratio is increasing with increasing Possion’s ratio.(vi)Springback ratio for Tresca and von Mises yield criteria, we say that 𝑅_𝑜/𝑡 is less than 20, there will be slightly change in that is negligible and for high value of 𝑅_𝑜/𝑡 or more than 20, differences in springback ratio is increases.

Nomenclature

𝑅𝑜:	Radius of die
𝑅𝑓:	Radius of final product after removing load
𝑀max:	Maximum applied bending moment per unit width
𝑦:	Distance from midsection
𝑡:	Thickness
𝜎:	True stress
𝜀:	True strain
𝐸:	Modulus of elasticity
𝜈:	Poisson’s ratio
𝑛:	Strain hardening index
𝛼:	Empirical constant
𝐾:	Constant
𝑌,𝜎𝑜:	Yield point stress
𝜀𝑜:	Strain at yield point
𝛿:	Strain in plastic region.