# Theoretical Determination of Forming Limit Diagram for Steel

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## Transcript Of Theoretical Determination of Forming Limit Diagram for Steel

Theoretical Determination of Forming Limit Diagram for Steel ,Brass and Aluminum alloy sheets

Waleed J. Ali

Anas O. Edrees

Mechanical Engineering Department, University of Mosul

Abstract :
Sheet metal forming is defined as the ability of metal to deform plastically (deformation by Stretching or drawing) or changing the shape of the sheet into a new desirable shape with out necking or crack . To control the operation of sheet metal forming with out failure. A diagram is used in which the range accepted , failure and critical deformation range are shown . This diagram is known as the Forming limit diagram. It is considered as one of the important tool to determine the formability of sheet metals. Every sheet metal has its own forming limit diagram which determines its formability, strain limit and the forming regions. These diagrams can be assessed using theoretical and experimental approaches, In this paper, the FLD is determined using different yield criteria Hill1948, Hosford1979 and modified Hosford 1985. It is shown that the determination of forming limit curve using the modified Hosford 1985 criterion with the (M-K) analysis , gave the best results compared with the other used criteria .Using this criterion gave the closest forming limit curve to that obtained experimentally, but with different criterion index for different alloy . The value of the index (a=6) gave the best results for brass, while (a=8) gave the best results for aluminum alloy and mild steel.

‫اﻟﺘﻌﯿﯿﻦ اﻟﻨﻈﺮي ﻟﻤﻨﺤﻨﻲ ﺣﺪ اﻟﺘﺸﻜﯿﻞ ﻟﺼﻔﺎﺋﺢ اﻟﺼﻠﺐ واﻟﺒﺮاص واﻷﻟﻤﻨﯿﻮم‬

‫اﻧﺲ ﻋﺒﯿﺪ إدرﯾﺲ‬

‫د.وﻟﯿﺪ ﺟﻼل ﻋﻠﻲ‬

‫ﻗﺴﻢ اﻟﮭﻨﺪﺳﺔ اﻟﻤﯿﻜﺎﻧﯿﻜﯿﺔ- ﺟﺎﻣﻌﺔ اﻟﻤﻮﺻﻞ‬

‫اﻟﺨﻼﺻﺔ‬

‫(اﻟﺘﺸﻜﯿﻞ ﺑﺎﻟﻤﻂ أو اﻟﺴﺤﺐ)‬ ‫أو ﻛﺴﺮ ، وﻟﻜﻲ ﻧﺴﯿﻄﺮ ﻋﻠﻰ ﻋﻤﻠﯿﺔ ﺗﺸﻜﯿﻞ اﻟﺼﻔﺎﺋﺢ دون ﺣﺪوث ﻓﺸﻞ ﯾﺘﻢ اﺳﺘﺨﺪام ﻣﺨﻄﻂ ﯾﺒﯿﻦ ﻓﯿﮫ‬
،

‫ﻣﺨﻄﻂ ﺣﺪ اﻟﺘﺸﻜﯿﻞ ﻣﻦ اﻷدوات واﻟﻮﺳﺎﺋﻞ اﻟﻤﮭﻤﺔ ﻓﻲ ﺗﺤﺪﯾﺪ ﻗﺎﺑﻠﯿﺔ ﺗﺸﻜﯿﻞ اﻟﺼﻔﺎﺋﺢ اﻟﻤ . ﻟﻜﻞ‬

. ‫ و‬(Hosford1985)، (Hosford1979)، (Hill1948)‫ﻟﻠﺨﻀﻮع‬
.‫ﻋﻤﻠﯿﺎ‬

(M-K)

(Hosford 1985)

(a=6)

، ‫ا ﻟﺨﻀﻮع ﻟﻜﻞ ﻣﻦ ا ﻟﻤﻌﺎدن اﻟﻤ ﺴﺘﺨﺪﻣﺔ‬

. ‫ ﻛﺎن اﻷﻓﻀﻞ ﺑﺎﻟﻨﺴﺒﺔ ﻟﻸﻟﻤﻨﯿﻮم واﻟﺼﻠﺐ‬(a=8) ‫ﻟﻠﺒﺮاص ، ﺑﯿﻨﻤﺎ ﻋﻨﺪ اﺳﺘﺨﺪام اﻷس‬

Notation

1,2 Principle stresses ,3Received ١ July. 2005

Accepted 7 June 2006

ε1 ε2 ε3 Principle strains

m

Strain rate sensitivity

n

Strain hardening exponent



effective stress

έ

effective strain

ε˙

strain rate



effective strain rate

ratio of minor strain to major strain

ta

thickness of the sheet

tb

Thickness of groove

α

Principle stress ratio

ƒ

Imperfection factor

a

Yield criterion index

K

Strength coefficient

Normal plastic anisotropic ratio

R1,R0 Plastic anisotropic ratio with rolling direction

R2,R90
φ β M-K

Plastic anisotropic ratio transverse to rolling direction ratio of principle stress to effective stress Ratio of effective strain to principle strain Marciniak-Kuczynski analysis

1.Introduction Forming processes are among the most important metal working operations. The industrial process of sheet-metal forming is strongly dependent on numerous interactive variables such as material behavior, lubrication, forming equipment, etc.
Forming limit Diagram is a representation of the critical combination of the two principal surface strains major and minor above which localized necking instability is observed. Forming limit curve (FLC) provides excellent guidelines for adjusting material, tooling and lubrication conditions. Also it is strongly dependent on material parameters, The idea of forming limit diagrams was first introduced by Keeler [1], when he observed that the maximum local elongation was not enough to determine the possible straining rate of a sheet .He established that the plotting of the principal strains at fracture ε1, and ε2 on two axes of a same diagram gave a curve : the forming limit curve. This curve, first restricted on the area ε2>0, was made complete for ε2< 0 by Goodwin's works [2].This curve is interesting because it divides the plane into two zones .The success area under the forming limit curve and the fail area above it, for a deep drawing operation .The criteria to reject the drawn parts is now the onset of localized necking.
Hill [3] was the first who proposed a general criterion for localized necking in thin sheets under plane stress states. His analysis predicts localized plastic deformation in the negative minor strain region. Marciniak and Kuckzinsky (M–K) [4] have proposed the first realistic mathematical model for theoretical determination of FLDs that suppose an infinite sheet metal containing a region of local imperfection where heterogeneous plastic flow develops and localizes. Hutchinson and Neale

[5] extended M–K theory using a J2 deformation theory. Therefore, the left and right hand sides of the forming limit diagram can be calculated by M–K analysis.
Sheet metals exhibit a highly anisotropic material behavior by cold rolling. It is therefore of major importance to extend the plastic instability analysis to anisotropic materials. Constitutive relations for the plastic yielding and deformation of anisotropic metals at a macroscopic level were proposed long ago by Hill 1948[6] This theory was the simplest conceivable one for anisotropic materials, however, inevitable limitations of its range of validity have eventually became apparent. The original MK analysis [4] was based on Hill's 1948 yield criterion [6]. However , it can be seen from the comparison with experiments and predicted results of Painter and Pearce 1974[7] that this analysis overestimates the limit strains towards the equibiaxial strain region, and underestimates the limit strains towards the plane strain region, particularly for materials with R values less than unity such as aluminum or brass . In addition, the calculated limit strains for the right hand side of the FLDs are very sensitive to the material anisotropy, a phenomenon that has not been observed in experiment .Sowerby and Duncan 1971[8] argued that the difference between these two stress states depends on the yield criterion and the shape of the corresponding yield locus. The effect of R on the FLDs depends on how the R-value affects the yield locus shape. Using (Hill's 1948) yield criterion, the stress ratios for positive strain ratios depend strongly on the value of R.
Hill's 1979 [9] yield criterion, taken with the assumption of the principle of equivalence of plastic work, was proposed to account for the so-called "anomalous behavior" of aluminum. This yield criterion has undergone application. One line of attack is represented by the work of Parmar and Mellor 1978[10].
Hosford 1979 [11] developed an extension of Hill's 1948 yield criterion , which is also found to be a special case of Hill's 1979 yield criterion. This criterion has been used by Graf and Hosford 1990[12] for sheet metals with normal anisotropy. Later, Padwal and Chaturvedi 1992[13] also used Hosford's 1985[14] planar anisotropy yield criterion to analyze the insatiable behavior of strain localization. They found that the effect of planar anisotropy is negligible while the predictions are

strongly dependent on exponent "a" an exponent in yield criterion . Predictions with a=5,6 or 8 match the experimental results much better than the predictions that were obtained from Hill's yield criterion.

Friedman and Pan 2000 [15] studied the effect of different yield criteria of (Hill1948),(Hill1979)and(Hosford 1979) on the right hand side of the forming limit diagram .

Dariani and Azodi 2003[16] showed the agreement between theoretical and experimental results by changing the index of Hill1979 yield criterion for right and left hand of FLD.

Banabic2004 [17] determined the FLD of Aluminum alloy (Al-2008) using new yield criterion (BBC2000)[18] , showed the best agreement between theoretical and experiment result of right hand side of FLD.

In this paper the FLDs of different sheet metals: Steel, Brass and Aluminum alloy are obtained theoretically using, the MarciniakKuczynski Theory, the following yield criteria :Hill1948 , Hosford 1979 and modified Hosford 1985 [14] , These FLDs were compared with the experimentally obtained FLDs of the same sheets metals to obtain the best agreement between the calculated FLDs and the experimental FLDs.
2.Theoretical Analysis
The geometry of neck formation and the element of sheet undergoing plastic deformation are shown in Fig.1. Following the MK analysis , based on a simplified model with assumed pre-existing thickness imperfection in the form of a groove perpendicular to the principal strain directions Fig.1, The sheet is composed of the nominal area and weak groove area, which are denoted by `a' and `b', respectively. The initial imperfection factor of the groove, ƒ0, is defined as the thickness ratio ƒ0=(tbo/tao); where `t' denotes the thickness and subscript `0' denotes the initial state. A biaxial stress state is imposed on the nominal area and causes the development of strain increments in both the nominal (a) and the weak area (b).

The yield criterion proposed recently by modified Hosford was used in
the calculation [14] in the plane stress state , this criterion is obtained as
follows : R2  1 a  R1 2 a  R1R2  1   2 a  R2 R1  1 a .......... .........1

1

    

 R

1
R

 1

R2  1 a

R1  2 a

R1R2  1

  2 a

 

a

..........

......

2

2 1

1

 

R2R1 1

 1 
 a .....................................3

 R2  R1a  R1R21a 

The behavior of material can be represented in the form of Power law
   K nm .......... .......... .......... .......... .......... .......... .......... ........ 4

The ratio of the principal stress and strain are define as follows:

  2 1

,    2  d 2 ...........................................................5
1 d1

The associated flow rule is expressed by

d  d   .................................................................................6

ij

 ij

Thus, the yield criterion can be written as follows:

            d1

d2

d3

R2 1 a1 R1R2 1 2 a1 R1 2 a1 R1R2 1 2 a1 R2 1 a1 R1 2 a1

d ..........................7 R2R1 1a1

and

  d   d   a 1 R  R R 1   a 1 .......... .......... .......... .......... .......... ... 8 
1 R2 R1  1 2 1 2

    d   d   a 1 R  a 1  R R 1  

2 R 2 R1  1 1

12

a 1 .......... .......... .......... ......... 9 

from eq.(5)and(7)

  d 2  R1 a1  R1R2 1 a1 .........................................................10

d1

R2  R1R2 1   a1

using condition of constant volume in plastic deformation
d1  d 2  d3  0................................................................................................11

from eq.(8)and(9),(11)
  d    d   a 1 R  R  a 1 .......... .......... .......... .......... .......... ....... 12 
3 R2 R1  1 2 1
then, by applying the principle of equivalence of plastic work
 d    1d 1   2 d 2 .......... .......... .......... .......... .......... .......... ....... 13 
 d     1 d  1 1   .......... .......... .......... .......... .......... .......... ....... 14    d     1   .......... .......... .......... .......... .......... .......... ... 15 
d1 the compatibility condition is given by
d 2a  d 2b ..........................................................................................................16
from Marciniak-Kuczynski analysis
f  t b .......... .......... .......... .......... .......... .......... .......... .......... .1 7 
ta
f  fo exp 3b   3a .......... .......... .......... .......... .......... .......... .......18
the equilibrium condition requires that the applied load remains constant along the specimen ; therefore
F1a  F1b .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .....19 
 1a t a   1btb .......... .......... .......... .......... .......... .......... .......... ........ 20 
from eq.(17)
1a  f1b...............................................................................21
from eq.(3)
a a  fb b..............................................................................22

from eq.(4)
 a  a  d  a n a m  f b  b  d  b n b m .......... .......... ...... 23 
from eq.(18)
 a  a  d  a n  a m  f o exp  3 b   3 a  b  b  d  b n  b m ........  24 
from eq.(10),(15) and (16)

 

d 

n 

a

m 

f

exp



    d  n  b m...25

aa

a   a 

o

3b

3a b b

b  b 

Fig.(1) M-K analysis model[4]

Substitute equations(10),(12),(15) in equilibrium equation (25), an equation can be found and solved numerically. Imposing a loading path (ρa), a finite increment of strain is also imposed in region (a), and by numerical computation is performed by using computer program (Fortran power Station) to determine the limit strain of a strain path in the FLD , and the limit strain is determined when [(dεb/dεa)> 10] in the range of strain ratios from (-0.5 to 1.0).
3. Experimental Procedure

In the experimental study, mild Steel, brass and aluminum alloy sheets their chemical composition are shown in table (1),(2)&(3) were used .
Table(1) chemical analysis of Aluminum alloy Material Sn% Ni% Ti% Cr% Zn% Mg% Mn% Cu% Fe% Si% Al% Aluminum 0.001 0.0006 0.016 0.009 0.027 0.01 0.015 0.15 0.58 0.38 Rem.
alloy

Table(2) chemical analysis of Mild Steel

Mat C M N C S P Si M C F

eria u o i r % % % n % e

l% % % %

%

%

Mil 0. 0. 0 0 0. 0. 0. 0. 0 R

d 0 00 . . 01 00 02 3 . e

Ste 4 7 0 0 1 4 2 5 2 m

el

34

1.

Table(3) chemical analysis of α Brass Material Mn% Pb% Fe% Zn% Cu% α Brass 0.001 0.001 0.19 26.2 Rem.
The FLDs of the sheets are determined using stretch forming tests with a hemispherical punch of (50mm) diameter and Die [19] with blank holder as shown in Fig.2.Using two sets of specimens with constant length (100mm) and having various widths with radius in one set for negative minor strain Fig.3.By changing the sheet width , major and minor strains

were measured following varied deformation paths. Circular grids of (2mm) diameter were initially printed on the surface of the specimens for the purpose of strain measurements. For each specimen the strain were directly measured from deformed grids.
Fig.(2) Stretch forming test
CriterionFldsLimit DiagramHillLimit Curve