forming limit diagrams - laur.lau.edu.lb:8443

FORMING LIMIT DIAGRAMS

By

Jimmy Issa

A THESIS

Submitted to Michigan State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2004

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UMI Number: 1424692

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ABSTRACT

FORMING LIMIT DIAGRAMS

By

Jimmy Issa

In this work, the Marciniak and Kuczynski model (MK) will be

reviewed and will be generalized to a more general approach for the

calculation of forming limit diagrams. In the new model we will

account for all directions of the imperfection. This new approach has

been used to calculate the theoretical FLDs (forming limit diagrams)

for Vonmises, Barlat Yield96 and Barlat Yield2000 for aluminum alloy

AI2008-T4 and compared with an experimental one. All the derivations

needed for the implementation of the code are derived in the appendix

for all yield functions. Finally the stretch stamping test will be modeled

using LS-DYNA. A very good agreement has been achieved between

the experimental and theoretical results.


To my family

i i i


ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Farhang Pourboghrat, who

supported me throughout my study, Dr. Cloud and Dr. Liu for being on

my committee. I would also like to thank Nader Abedrabo and the

faculty members of the department of Mechanical Engineering at

Michigan State University who taught me through my masters or

helped me with my research. Finally thanks to the mechanical

engineering department at Michigan state university for giving me

admission and financial support.

IV


TABLE OF CONTENTS

LIST OF TABLES....................................................................................................vii

LIST OF FIGURES................................................................................................viii

LIST OF ABBREVIATIONS................................................................................... x

INTRODUCTION..................................................................................................... 1

CHAPTER 1: Experimental forming lim its.......................................................3

CHAPTER 2: MK model.........................................................................................62.1 Introduction .............................................................................................62.2 Right hand side of forming limit diagrams........................................7

2.2.1 Algorithm.............................................................................................82.2.2 Vonmises........................................................................................ 112.2.3 Hill's quadratic 1948.................................................................... 142.2.4 Hill's non-quadratic 1 9 7 9 ............................................................ 16

2.3 Left hand side of forming limit diagrams..................................... 192.3.1 Algorithm.......................................................................................... 202.3.2 Thichness lim it ............................................................................... 212.3.3 Results...............................................................................................23

CHAPTER 3: General MK model.......................................................................253.1 Introduction............................................................................................253.2 Modified MK m odel................................................................................27

3.2.1 Algorithm.......................................................................................... 293.2.1.1 Region a ....................................................... 293.2 .1 .2 Region b ..................................................................................... 32

3.3 Yield functions......................................................................................... 373.4 Hardening rules.......................................................................................373.5 Material constants................................................................................. 373.6 Results and discussions........................................................................39

CHAPTER 4: Simulation of Stretch Stamping............................................. 404.1 Introduction............................................................................................404.2 Description of the m odel................................................................... 41

4.2.1 Part dimensions.............................................................................41

V


4.2.2 Element type and material model............................................. 414.2.3 Material properties.........................................................................424.2 .4 Constrained and prescribed m otion.......................................... 44

4.3 LS-DYNA.................................................................................................. 464.3.1 Swift and Voce FLDs......................................................................464.3.2 DYNA FLDs........................................................................................47

4.4 Experimental findings........................................................................... 494.5 Numerical results.................................................................................. 50

CHAPTER 5: Conclusions................................................................................... 53

Appendix: Numerical implementations..........................................................55A Vonmises................................................................................................. 55

A1 Yield functioned.............................................................................55A2 First derivative................................................................................55A3 Second derivative.......................................................................... 55

B Barlat Yld96............................................................................................. 56B1 Yield functioned.............................................................................56B2 First derivative................................................................................58B3 Second derivative.......................................................................... 59

C Barlat Yld2000 ...................................................................................... 63C l Yield functioned.............................................................................63C2 First derivative................................................................................64C3 Second derivative.......................................................................... 66

VI


LIST OF TABLES

Table 1 : Constants for voce and swift - AI2008T4............................. 38

Table 2 : Constants for Barlat Yield96 - A I2008T4............................. 38

Table 3 : Constants for Barlat Yield2000 - AI2008T4....................38

Table 4 : Properties of Steel......................................................................42

Table 5 : Properties of A A 3003-H 111.................................................... 42

Table 6 : Constants for voce equation...................................................43

Table 7 : Constants for swift equation...................................................43

Table 8 : Constants for Barlat Yield2000......................................... 43

Table 9 : FLD user............................................................................................. 47

v i i


LIST OF FIGURES

Figure 1 : Fractured specimen......................................................................... 4

Figure 2 : Failure under biaxial tension.........................................................5

Figure 3 : RFIS model..........................................................................................7

Figure 4 : Different n values........................................................................ 12

Figure 5 : Different m values....................................................................... 12

Figure 6 : Different f values.......................................................................... 13

Figure 7 : Different R values.......................................................................... 15

Figure 8 : Different R values.......................................................................... 17

Figure 9 : Different a values.......................................................................... 18

Figure 10 : LHS m odel.................................................................................. 19

Figure 11 : Thickness limit Different n values..................................21

Figure 12 : Thickness limit Different m values................................ 22

Figure 13 : Thickness limit Different f values...................................22

Figure 14 : LHS Different n values............................................................ 23

Figure 15 : LHS Different m values............................................................. 24

Figure 16 : LHS Different f values................................................................ 24

Figure 17 : Modified MK m odel......................................................................27

Figure 18 : Fids A I2008-T4-Sw ift............................................................... 38

Figure 19 : Fids AI2008-T4-Voce.................................................................. 39

Figure 20 : Model stretch stamping............................................................. 44

Figure 21 : Blank holder force.......................................................................45

v i i i


Figure 22 : Punch prescribed motion.......................................................... 45

Figure 23 : FLD AA3003-H111.......................................................................46

Figure 24 : LS-DYNA FLDs..............................................................................48

Figure 25 : Failed sheet................................................................................... 49

Figure 26 : FLD AA3003-H111 (t: 15 m s )................................................. 50

Figure 27 : FLD AA3003-H111 (t: 23 m s )................................................. 51

Figure 28 : Middle Element displacement...................................................51

Figure 29 : FLD contour at fa ilure................................................................ 52

Figure 30 : Experimental and numerical findings....................................54

IX


LIST OF ABBREVIATIONS

a - Cauchy stress

a,aYF- Yield stress from yield function

oHR - Yield stress from hardening rule

s,s - Cauchy and effective strain

e- Engineering Strain

n,k,e0 - Swift constant

A,B,C- Voce constants

/ - Imperfection parameter

p- Strain ratio

a - Stress ratio

® - Yield ratio

t - Thickness

R- Ratio of plastic strain in the width and thickness direction

R- Average of lva lues

a - E x p o n e n t to a c c o u n t fo r crystallographically textured fee and bee m etal

J - Jacob ien

0- Imperfection orientation

C1,C2,C3,C6ax,ay,az0azl - Barlat yield96 material constants

al,a2,a3,a4,a5,a6,a1,ai- Barlat yield2000 material constants

x


Introduction

Forming lim it d iagram s (FLDs) have been valuable tools for analyzing

sheet m etal form ing a fte r the introduction of the idea in 1965 . Stretching

operations of sheet m etals will norm ally lead to a developm ent of a sharp

localized neck on the surface w here the plastic flow will localize and lead to

fa ilure. Forming lim it diagram s have been first introduced by Keeler [1 -2 ]

and are still used as a failure criterion to assess the form ability o f sheet

m etals. Many tried to model the localized necking analytically a fte r the

introduction of the idea, starting w ith the pioneer w ork o f Marciniak and

Kuczynski [3 ] who first presented a m athem atical model to help understand

sheet m etal deform ation during form ing. They m odeled the im perfection as a

long groove, perpendicular to the direction of the largest stress for positive

m ajor strains (R H S) or along Hill's direction of zero-extension (th e strain

increm ent parallel to the groove inside and outside of it is equal to zero) for

th e negative m ajor strains (LH S). Using this approach theoretical FLDs have

been calculated for linear strain path using different yield criteria , von Mises,

Hill's 1948 [4 -5 -6 ] , Hill's 1979 [5 -6 ] and m any other yield functions. G raf

and Hosford [6 ] used an anisotropic yield function (Hill's 1 9 7 9 ) and

investigated the effect o f the hardening exponent n, the strain rate sensitivity

m , the im perfection f, the R-value and the yield function exponent a. F.

Barlat [7 ] calculated the FLD using a yield function based on von Mises but

with a high exponent. The effect o f nonlinear strain paths has been

1


investigated on form ing limits w here experim ental and num erical FLD have

been predicted for changing strain paths for d ifferent m ateria ls [8 -9 -1 0 -1 1 ] .

M.C. Butuc [1 2 -1 3 ] used a m ore general MK model and calculated th e FLDs

fo r von Mises, Hill's 1 94 8 , Hill's 1979 and Barlat Y ld96 for A I6061 .

In this w ork the MK model and a m ore general approach for the calculation of

form ing lim it d iagram s are explained in detail. The general approach is based

on the MK model but is m ore general since there is no constraint on the

groove direction and the lim iting strain is calculated for all possible angels

and m inim ized versus the angel o f the groove. This new approach uses the

New ton-Raphson m ethod to solve for the transverse stress and effective

strain in the groove. For this reason the second derivative of the yield

function is needed for the evaluation of the Jacobian. I calculated th e form ing

lim it d iagram for A I2 00 8 -T 4 alum inum alloy using the m odified MK model

w here von Mises, Y ield96 and Y ie ld2000 w ere used to describe the yield locus

and modeled the hardening using both Voce and Swift models. The first and

second derivatives for the th ree yield functions are evaluated in th e appendix

to m ake fu ture im plem entation easier. F. Barlat [1 4 ] calculated the FLD by

incorporating m icrostructural param eters , internal dam age (voids) and

crystallographic tex tu res in his model and it has given very accurate results.

Using a macroscopic approach, a very good agreem ent has been achieved

betw een the calculated and the experim ental FLD when the yield locus is

described w ith Y ld96 and the m aterial hardening with Sw ift model.

2


CHAPTER 1: Experimental Forming Limits

O ver the years experim ental form ing lim it d iagram s have been

determ ined [4 -1 5 ] for d ifferent linear strain paths and even for m aterial p re

strained in d ifferent directions (biaxial tension, uniaxial tension and plane

strain).

The m ost widely used technique involves printing or etching a grid of

small circles o f d iam eter d0 on the sheet m etal before deform ing it. A fter

stretching or form ing the sheet, the circles will deform into a shape which

resem bles an ellipse. The form ing process will stop when the first perceptible

neck is observed. To find the lim iting strains for d ifferent strain paths,

lubrication and specim en widths are varied. To obtain th e lim it fo r negative

m inor strains, very narrow specim ens are used and by increasing the width

to full w idth specim en, we can find the lim its up to balanced biaxial. A fter

deform ation the ellipses wholly or partially in the neck are considered failed,

while th e circles one or m ore d iam eters aw ay form the neck are considered

safe. The principal d iam eters o f the failed ellipses are m easured (m a jo r di

and m inor d2). The lim iting strain can then be represented as engineering or

true strains using the tw o form ulas w ritten below.

3


True Limiting Strains

ts, = In

Engineering Limiting Strains

s2 - In

I d,

\d0 J

fd ?\d0 j

e x =

e2 =

dj d0

d2 d0

Finally, Forming lim it d iagram s determ ined in d ifferent laboratories

m ay be som ew hat d ifferent. This is because the determ ination of the first

perceptible neck is subjective. Many techniques have been used to determ ine

the onset o f the neck; visual inspection and feeling with the fingers. Also,

th ere is no restriction placed on how fa r one should go from th e neck before

considering a circle to be safe. I t is recom m ended th a t safe readings should

be m ade a t a distance a t least equal to 1 .5 circle d iam eter from the cen ter of

th e neck. Below are tw o pictures of tw o d ifferent failed specim ens.

Fig 1: Fractured specimen

4


Fig2: Failure under biaxial tension

5


CHAPTER 2: MK Model

2.1 Introduction

The MK model is th e first m athem atical model presented by Marciniak

and Kuczynski [3 ] to help understand localized necking in sheet m etals. They

assum ed a long groove, a locally thinned region within which the strain will

concentrate and will lead to failure due to the onset of localized necking. The

origin o f this im perfection is a variation in the sheet thickness [3 ] , difference

of the m aterial strength [1 6 ] or a com bination of both. The fa ilure is

assumed to occur when the ratio of the strain increm ent w ithin the groove

and outside of it reaches a critical value. There are some constraints on the

orientation of the im perfection w here it has been taken to be perpendicular

to the axis o f the largest principal stress and strain for positive m ajor strains

(RHS of th e FLD), and in the direction of zero-extension for negative m ajor

strains (LHS of th e FLD). In this section we will explain the algorithm for the

calculation of FLDs based on the MK model. W e will investigate th e effect o f

the hardening exponent n, the strain rate sensitivity exponent m , the

im perfection p aram eter f, the R-value and the yield function exponent a,

w here the yield locus will be described by Hill's 1979 yield function. W e will

investigate yield function differences in predicting localized necking. The

corresponding algorithm s are all w ritten in M atlab.

6


2.2 Right hand side of forming limit diagrams

For positive m inor strain the groove is assum ed to be perpendicular to

th e axis o f the largest principal stress and strain. The strain increm ents

parallel to the groove w ithin and outside of it are equal. The algorithm can be

sum m arized by assum ing, fo r a known strain path , the largest principal strain

increm ent in the im perfection (region b) and then calculating th e one in the

hom ogeneous region (region a ). The failure is assumed to occur when th e ir

ratio reaches a critical value (in the calculation below the critical value is

assum ed to be equal to 1 0 ). The algorithm below is w ritten for von Mises

yield function. The changes th a t will need to be im plem ented for o ther yield

functions are described in the definition of each yield function. The M atlab

program s are w ritten for von Mises (Appendix A ), Hill's quadratic (Appendix

B), and non-quadratic yield functions (Appendix C). In the algorithm below

indices a and b re fer to the hom ogeneous and the defected region

respectively. Indices 1 and 2 re fer to the principal values. The figure below

describes well the model.

2

Fig3: RHS model

7


2.2.1: Algorithm

Under plane stress condition (<x3 = 0 ) , for a known strain ratio in region a

The stress ratio is calculated:

aa = ^ = (2pa+l)/(pa+2)°a\

The yield function is then evaluated:

°a\

Im pose a value for the strain increm ent in the groove:

d£bl = 0.001

Assume a value for the strain increm ent in region a:

d£ai

Calculate the rest o f the variables needed for region a:

d£„2 ~ Pad£a\dsa 3 = —dsa j — dsa2

dsa = dsal (l + a aPa )/O fl

n _ d-Sa• a ~ J

d £ a l

The equation below holds due to the com patib ility requirem ent:

d£b 2 d£a 2


The rest of the variables in region b are evaluated:

dsb 3 — dsb] ds,4 2

dsb - dsbx (l + abpb ) /O t

dsb

P b =

ds

ds,b 1

42

ds41

« 4 = ( 2 / ^ 4 + O O 4 + 2 )

>b ab + 1

From the force balance equation betw een region a and b the largest strain

increm ent in th e region a is recalculated. Then it is correspondingly adjusted

and the process is repeated until the force balance is satisfied. The force

balance equation is derived below:

F a i = F bl

For a unit width and thickness t, the force in the 1-direction is:

F x = <Tj4 = <7 t =

The hardening rule w ith strain rate dependence can be w ritten :

a = ksnsm

9


Using the variables defined above the force can be expressed by:

F] = kt{e + ds)" sm/o - kt{s + ds)n(/3/p)s™ / o

By applying the above force expression to regions a and b, and a fte r some

adjustm ents, the force balance can be reduced to:

/ x m /

— \ £ a i = -^ -exp(^3b - £ i a t ? b + d S b ) \ — y Pa I ^ y P>-Pb.

'b2

W here the im perfection p aram eter f and the current thickness are:

f = kbtbo/KtaOt = t0 exp s3

To solve th e force balance equation, the New ton-Raphson or the M id-point

m ethod can be used. Both m ethods w ere program m ed in M atlab, but the

New ton-Raphson proved to be faster. The sam e algorithm can be used for

any yield function provided th a t the equations th a t involve th e use of the

yield function are changed correspondingly. Below, the algorithm will be

applied to several d ifferent yield functions.

10


2.2.2 von Mises

The von Mises yield function is th e m ost basic yield function describing

the plastic behavior o f isotropic m ateria ls. An isotropic m aterial responds in

th e sam e m anner when strained or stressed in any direction. All the

derivations needed for the im plem entation of the algorithm for von Mises

yield function are shown above. In plane stress applications the yield function

is:

2<x2 = (cr, - a 2 )2 + + <j22

The calculated form ing lim it diagram s below are for von Mises yield function

w here the im perfection p aram eter f, th e hardening exponent n and the strain

rate exponent m are variables. The FLD curves shown below have been

plotted using the MK m odel. The groove does not change direction when the

deform ation occurs. So it is assumed to be perpendicular to the direction of

the largest stress and strain and does not ro tate during the straining of the

sheet. The plots shown below indicate th a t form ability increases and failure

occurs a t higher strains, with increased strain rate sensitivity and stra in -

hardening exponent, and decreased im perfection param eter.

11


Major

Strain

Major

Stra

in

von Mises - different n values0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0 .0001

0.90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Minor Strain

Fig4: Different n values

von Mises - different m values0.9

0.8,<-0.01 <- 0.005 0.0010.7 <- 0.0005

110.00010.6

0.5

0.4

0.3

0.2

0 .9 9

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Minor Strain

Fig5: Different m values

12


Major

Stra

in

von Mises - different f values0.8

0.9980.7

0.9960.6

0.994

0.9920.99

0.4

0.3

0.2

0.0001

0.2 0.3 0.4 0.60.5 0.7 0.8Minor Strain

Fig6: Different f values

13


2.2.3: Hill's Quadratic 1948

The Hill's quadratic yield function was one of the first yield functions to

describe the plastic anisotropy of m etals. Hill has form ulated a quantita tive

tre a tm en t o f plastic anisotropy w ithout regard to its crystallographic origin. A

new p aram eter R has been used which is th e ratio o f plastic strain in the

width and thickness direction for a tensile specim en cut from a sheet. The R-

values usually vary with the tes t direction and it is com m on to characterize a

m aterial by an average R-value w ritten below.

— = R0 + 2 R45 + Rgo 4

W here th e indices represent the angles (w ith th e rolling direction) to which

the tensile specimen has been cut from the sheet. For R equal to 1 the

quadratic hill's yield function will simply be reduced to von Mises. For plane

stress loading th e yield function sim plify to the equation below:

2 2 2£X — (J. + ------------- (JA (7-,1 2 1+R 1 2

The only tw o expressions in the algorithm th a t should be changed in order to

get the form ing lim it diagram s using Hill's 1948 are the stress ratio and the

yield ratio. The stress ratio is a function of th e strain ratio and the yield ratio

is a function of the stress ratio. Those tw o functions are w ritten below.

14


Quadratic Hill's - different R values

0.9 0.75

0.7

ifl444-1

w 0.5MOfl 0 4 a

0.3

0.2

0.00010 .9 9 8

0.3 0.4 0.6 0.7 0.8 0.90 0.1 0.2 0.5Minor Strain

Fig7: Different R values

The figure above shows th a t the form ing lim it is very sensitive to the value of

R and it decrease with increasing R. The form ing lim it for this case is

calculated w ith th e largest strain assum ed to occur in the RD direction.

15


2.2.4: Hill's Non-Quadratic 1979

Hill's quadratic yield function overestim ated the effect o f the R-value on

the shape of the yield loci. A new yield function has been developed to

overcom e this weakness w here a higher exponent "a" has been used to

account for crystallographically tex tu red fee and bcc m etals. W ith a equal to

2 this criterion simplifies to Hill's quadratic. H ow ever, crystallographic

calculations are much b e tte r approxim ated by a being equal to 6 for bcc and

8 for fee m etals. For plane stress loading this criterion simplifies to:

The yield ratio fo r this yield function can be expressed explicitly as a function

of the stress ratio but th e stress ratio cannot be expressed explicitly as a

function of the strain ratio. A nonlinear equation should be solved in order to

first get the stress ratio and then evaluate the yield ratio. The Newton

Raphson m ethod or the M idpoint m ethod can be used to solve this equation.

Below th e non-linear equation and the yield ratio are w ritten :

o a(1 + R) = cr“ + <t“ + - cr2)

p (1 + i?(l - a )" '1) - a a~' + R{1 - a )“_1 = 0

16


0.5Hill's 1979 - different R value

O.l

0.05

0

0.45

0.4

0.35

.3 03it

W 0.25

o0.2

■n

I

0.15

0.05 0.1 0.15 0.2 0.25 0.3M in o r S t r a in

Fig8: Different R values

0.35

n = 0 .2 m = 0.0001 f = 0 .9 9 8 a = 8

_L_

0.4 0.45 0.5

The plot above proves th a t the form ing lim it diagram is insensitive to the

level o f R, w here it was very dependant when the quadratic hill's yield

function was used. This independence of the FLD of the R-value gave the

non-quadratic function a g reat im portance because it m atched w ith the

experim ental FLD w here it has been shown to not depend on R values.

17


Hill's 1979 - different a values0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0 . 0001 0 .9 9 8

m0.05

00.050 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Minor StrainFig9: Different a values

As m entioned before the exponent a, connected to the crystal structure , is

equal to 6 or 8 fo r bcc and fee alloys respectively. For the special case w here

R is equal to 1 and a equal to 2 , Hill's 1 979 will simply reduce to von Mises.

The plot above shows a dependence of th e FLD on the value of a.

18


2.3: Left hand side of forming limit diagrams

For the negative m inor strain regim e the groove direction is assumed to

line up with Hill's direction of zero-extension such th a t the plane strain

condition is satisfied. The existence of a critical thickness strain criterion for

localized necking of sheets in the negative m inor strain regim e has been

dem onstrated. The effective lim it strain in region a increases as the strain

ratio becomes m ore negative and tends towards uniaxial tension. The limit

thickness strain is a constant regardless of the imposed strain ratio , thus

indicating the existence of a critical thickness strain for the onset o f localized

necking. The lim iting thickness strain is dependant only on n, m and f. I t is

independent o f the plastic anisotropy, the yield function and the plasticity

theories (flow or deform ation th eo ry ).

2

FiglO: LHS model

19


2 .3 .1 : Algorithm

The algorithm is sim ple here because it involves the solution of a

nonlinear equation w ith the thickness strain. A thickness strain increm ent is

assum ed in the im perfection and from the nonlinear equation the thickness

strain in the hom ogeneous region is calculated until th e thickness strain in

the homogeneous region reaches a lim iting value. This equation is solved

using e ither Newton Raphson m ethod or the Mid-Point m ethod. A fter lengthy

m athem atical m anipulation w here the force balance equation coupled with

the condition of Hill's zero extension have been used, the nonlinear equation

simplifies to the equation below. The full derivation of this equation is in


[1 7 ].

20

2.3.2: Thickness limit

The lim iting thickness strain in th e hom ogeneous region is independent

from the strain ratio. For each value of n, m and f the lim iting strain in region

a is constant. The variation of the lim iting strain with n, m and f is shown in

the plots below. By increasing the values of n and m, th e lim iting thickness

strain increases by keeping a constant ratio with the thickness strain in the

im perfection. By increasing the f values, the limiting thickness strain will

increase w ith a small increase in its ratio with the lim iting strain in the

im perfection.

Thickness limit - different n values

<- 0.280.25

ao■H -■e- 0.24

0.220.2

ajU4JCOCOCO(U£V•H

I0.05

0.00010 .9 9 8

0.05 0.10 0.15 0.2 0.25 0.3 0.35- Thickness strain in region b

Fig 11: Thickness lim it Different n values

21


Thickness limit - different m values

0.25nj

Iu 0.005

•H0.0001c

« 0.15 u4JU1toto<uIu

0.05

0 .9 9 8

0 0.1 0.2 0.3 0.4 0.5- T h ic k n e s s s t r a i n i n r e g io n b

Fig 12: Thickness lim it Different m values

Thickness limit - different f values0.2

0.180.998

0.9960.9940.9920.99

0.16nj

I 0.14 U)Q)Ua 0.12-rd

c'id 0.1Vd4Jto[n 0.08toa)G^ 0.06 ■H£1 0.04

0.02 0.0001

0.050 0.1 0.15 0.2 0.25- T h ic k n e s s s t r a i n i n r e g io n b

Fig 13: Thickness lim it Different f values

22


2.3.3: Results

The plots below show th e effect of th e hardening exponent n, the strain

rate sensitivity p aram eter m and the im perfect param eter f. The FLD for the

LHS increases with th e increase in the values of n, m, and decrease with the

increasing value of f. This is reasonable because an increase in the

im perfection p aram eter m eans a deeper defect, which will lead to an earlier

failure.

LHS - different n values0.5

0.28

0.45 0.26

0.240.4

C■H0.2

>4■° 0.3§,

0.25

0.20.00010 . 9 9 8

m

- 0.1- 0.2 -0.15 -0.05 0-0.25M in o r S t r a in

Fig 14: LHS Different n values

23


LHS - different m values0.5

0.010.45

0.0050.4

fi* 0.35 uJJuju

0.3

0.0010.0005

0.0001

0.25

0.2

0 . 9 9 8

-0.25 - 0.2 -0.15 0.1 -0.05 0M in o r S t r a in

F ig l5 : LHS Different m values

LHS - different f values0.35

0.998

0.996 - K .

0.994 - j 0.992

0.3

0.99

0.15

0.0001

- 0.2 -0.18 -0.16 -0.14 -0.12 ■0.1 -0.08 -0.06 -0.04 -0.02 0M in o r S t r a in

Fig 16: LHS Different f values

24


CHAPTER 3: General MK Model

3.1: Introduction

In this chapter, we will explain in detail a general approach for the

calculation of form ing lim it diagram s. This present model is an extension of a

previous analysis by Marciniak and Kaczynski (M K -m o d e l), w here it is

generalized to account for all possible directions of the im perfection. The

lim iting strain is calculated for all possible angels and m inim ized versus the

angel o f the groove. W e will use this approach to com pare d ifferent yield

functions starting w ith the m ost basic one for isotropic m aterials von Mises

and tw o anisotropic yield functions Barlat Y ld96 and Barlat Y ld2000 proven to

describe well the plastic behavior o f alum inum sheet m etals. Two w ork-

hardening laws, swift and voce, will be used to model the effective stress-

strain behavior o f the m ateria l. This new approach uses the New ton-Raphson

m ethod to solve for the transverse stress and effective strain in the groove.

For this reason, the second derivative of the yield function is needed for the

evaluation of the Jacobian. The first and second derivatives of the th ree yield

functions are evaluated in the appendix for im plem entation purposes.

Forming lim it d iagram s will be calculated for alum inum alloy A I2 00 8 -T 4 and

will be com pared w ith experim ental data . Barlat calculated the FLD for

A I2 00 8 -T 4 by incorporating micro structural param eters, internal dam age

(voids) and crystallographic tex tu res in his model and showed very accurate

results. All com puted FLDs assum e th a t the largest strain is in the y-direction

25


(TD d irection) in order to be on the safe side. A very good agreem ent

betw een the calculated and th e experim ental FLD was obtained, when the

Y ld2000 and the voce hardening law w ere used. M.C. Butuc [1 2 -1 3 ] already

developed FLD for the A I6061 alum inum alloy based on von Mises, Hill's

1948 , Hill's 1979 and Barlat Y ld96yield function.

26


3.2: Modified M-K model

This model uses the sam e assumption proposed by Marciniak and

Kuczynski [3 ] who assumed a long groove, a locally thinned region within

which the strain will concentrate and will lead to failure due to the onset of

localized necking. The origin of this im perfection is a variation in the sheet

thickness [3 ] , d ifference of the m aterial strength [1 6 ], or a com bination of

both. The fa ilure is assum ed to occur when the ratio of the strain increm ent

within th e groove and outside of it reaches a critical value. As we mentioned

earlier th ere is som e constraints on the orientation of the im perfection. The

modified M -K model will consider all possible orientations for th e groove and

consider the fa ilure to occur when the ratio o f the total strain increm ent

inside and outside of the im perfection reach the value of 10.

2

Fig 17: Modified MK model

27


The model above consist o f tw o regions, th e homogeneous region "a" w here

the axis 1, 2 and 3 represent th e principal directions of stress and strain and

the groove region "b" w here the axis n, t and 3 are the norm al to the

direction of the groove, the longitudinal direction, and the thickness

direction. The M -K im perfection p aram eter is defined as the initial thickness

ratio o f th e defected and the hom ogeneous area .The deform ation within the

groove occurs a t a fas ter rate than the one in the hom ogeneous region and

the plastic flow will localize in the im perfection area leading to eventual

failure. Under deform ation , the strains parallel to the im perfection direction

within and outside of the groove are equal and forces perpendicular to the

groove in both regions are the sam e. Those two constraints are called

com patibility and equilibrium conditions. The groove is assum ed to rotate

while the deform ation occurs. Let e and e0 be the current and the initial sheet

thickness. The value of the im perfection param eter, the com patib ility and the

equilibrium conditions can then be w ritten as

Im perfection p aram eter: f -ebjea

Com patibility condition: ds“ = dsb

Equilibrium condition : <7amea = & bnrieb , cr“e° = <jbnteb

28


3.2.1 Algorithm

Given the stress state in region "a", the plastic strain is evaluated in

region "b" for a known orientation of the groove. In th e figure above we will

assum e the x -, y - and z-axis to be the orthotropic axes of the m aterial

(rolling, transverse and norm al) th a t line up with axes 1, 2 and 3

respectively. For one increm ent we will explain clearly how to calculate the

plastic strain in region "b". First, assum e a small increm ent o f the effective

strain in region "a" and then follow the steps below. The details below show

how this algorithm should be used in order to calculate th e FLD for

proportional loading, by assuming th a t no shear stress exists in region "a".

This algorithm could be used to find th e critical strain for an unknown strain

path w here the strain s tate will be an input from an FEM analysis.

3 .2 .1 .1 Region "a"

• Strain ratio

Usually in the calculation of form ing lim it diagram s for linear strain path the

strain ratio in region "a" will be an input.

29


• Stress ratio

The stress ratio is then evaluated by assum ing th a t it is a variable equal to

th e stress in the y -d irection , if the stress in th e x-direction is set equal to 1

and shear stress equal to zero.

For most yield functions, to get the stress ratio one will need to solve the

following nonlinear equation:

The Newton Raphson or M id-Point m ethod could be used for this purpose. For

the Mid-Point m ethod the initial interval w here the solution is located can be

taken as [-1 ; 2 ].

• Yield ratio

The yield ratio is the ratio of the yield function to the stress in the x -

direction. The yield ratio becomes the yield stress by assuming th e stress in

the x-direction to be equal to 1, the stress in the y-direction to be equal to

the stress ratio , and the shear stress to be equal to zero.

<7O = <j yf For cr = a

0cr

XX

30


• Stress state

Having assum ed an increm ent for the effective strain , the yield stress can be

evaluated using th e hardening rule (sw ift or voce). Coupled with the yield

ratio, th e stress in the x-d irection can be calculated, knowing the stress in

the y-d irection , and by setting the shear stress equal to zero (no shear stress

in region "a").

o HR - hardening^a + dsa)

= ° hr/®

e°yy =

<,=0

• Strain state

The strain state is calculated from the Flow Rule knowing th e stress state in

the orthotropic referential fram e.

del ds„ 8(Tyf“ dal

e l = e l + d e l

d£l = dz 2^ILa*;,

dsa = dea' 2 dal

C = < y+ de;y

YF £° = £ a + d£ a**xy xy xyxy

31


• Output needed

To solve for the effective strain increm ent in region "b" four values are

needed; the strain increm ent parallel to the groove, the strain in the

thickness direction, th e stress norm al to th e groove and the shear stress in

the groove coordinate axis. Those four values are calculated as following:

3 .2 .1 .2 Region "b"

• Nonlinear Equations

For this region to com plete our study we have to com pute the effective strain

increm ent and com pare it with th e one in the hom ogeneous region. Two

nonlinear equations in the effective strain increm ent and the stress in the

longitudinal direction m ust be satisfied. The first state th a t the yield stress

com puted from th e yield function m ust equal the one com puted form the

hardening rule. The second is the com patibility equation. The tw o equations

are w ritten below.

del = dsaxx sin2 6 + dsayy cos2 6 - dsaxy sin29

Gm = c°s 2 0 + o ayy sin2 9 + a £ sin 29

a “t = ° \^ (cos2 0 - s in 2 9) + (ayy - axx ) sin 9 cos 9

32


To solve these equations the New ton-Raphson m ethod is used. This m ethod

involves the com putation of the Jacobian needed in the iterative evaluation of

the solution. The solution is reached by iteration using the form ula below:

W here J is th e Jacobian of the nonlinear equations. The Jacobian is the

m atrix o f the derivatives of G i and G2 w ith respect to the effective strain

increm ent and the stress in the longitudinal direction. For this case the

Jacobian is sym m etric and its com ponents are described below:

' dGx oG] ", _ ddeb dcrb

dG2 dG2

ddsb d a b _

The first com ponent o f the Jacobian is stra ightforw ard and is obtained by

deriving e ither hardening rules (S w ift or Voce)


dGx _ nK(sQ +6Jj + d s b Swi f t hardening

ddsb C x B exp[- c(sb + deb)]Voce hardening

33

As we m entioned above th e Jacobian is sym m etric and the non-diagonal

com ponent involve the first derivative of the yield function, which is derived

in the Appendix.

8G, 8G-,

8 a t t d d s-xb

s i n 2 6d a bF d a b F d a b F

c o s 2 98 a b 8 a b

X X y y xy- s i n # c o s #

The derivative o f G2 with respect to a tt involves the second derivative of the

yield function, which is derived in the appendix.

8G2

8 a b= - d s b \ s i n 2 <9 c o s 2 # - s i n 20

8 2a bF

8°xx 8 2a bYF

1 8 2 a2—b

YF

2 8 < j 8 ctxy x

2—b YFd a

2 —b8 l a YF

2—b1 d a YF

2 dander„

2—b YF8 a

d°XXd°Xy2—b8 a YF

daYY8cx*y1 82a bF

2 d a Xy

s i n 2 0

c o s 2 0

s i n 9 c o s 9

• System Solution

To solve th e system of the tw o nonlinear equations we will use the N ew ton-

Raphson m ethod. The initial values are assumed to be the one of the

hom ogeneous region. Using the equilibrium condition we are able to com pute

the norm al stress to the groove and the shear stress in the groove. W e

assum e the initial im perfection p a ram ete r to be the current fo r the first

iteration so we can proceed with the calculations and provided th a t the

variation of the im perfection p aram eter is too small a t the beginning of the

34


deform ation. Using the rotation rule below and th e flow rule the stresses and

strains in th e groove in the orthotropic referentia l fram e are determ ined and

finally we calculate the strain increm ent in the groove in the longitudinal

direction using th e rotation form ula:

dea, f f a„IN IT IA L

VALUES

1i 1<3Itc

i

. 1 1 0 _ J

a a rV n n ^ n t J E Q U IL IB R IU M *

= < , / /

'crnn

<jb =XX

< R O T A T IO N * < =

_<T»«. ^ =

° bxx_ b

y y FLOW R U LE ^ £ y y

xy _ S x y

crb - <7bnt(cos2 0 - sin2 0) - (cr* - crbm) sin# cos#

R O T A T IO N * dstt = dsxx sin # + dzyy cos 6 - dsxy sin 20

35


Now having all the values needed to solve the nonlinear equation, th e strain

increm ent in the groove and the stress in the longitudinal direction of the

groove are com puted by iterating until a convergence value is reached. A fter

th a t the current im perfection p aram eter and th e angle o f the band are

updated. The process continue by assuming an effective strain increm ent in

region a and calculating the one in the groove until the ratio of these two

approaches a critical value corresponding to local instability. I t is usually

assumed th a t fa ilure occurs when the strain increm ent in the hom ogeneous

region reaches 10 percent or less than the one in the groove.

Im perfection param eter: f = f0 exp(f* - )

/ 1 + dsaGroove orientation : tan(# + d6)= tan# —

l + ds-„

The study above is done assuming the m a jo r strain to occur along the rolling

direction (x -a x is ), but to obtain the full FLD we have to assum e the case

w here th e m ajor strain occur along the transverse direction w here it will give

the lowest values for the strain and will be used as the FLD for m ateria l. To

calculate the FLD w here the m ajor strain occurs along the (y -a x is ), the only

thing to do is to switch the x and y m aterial coefficient and use the sam e

algorithm . For Vonm ises both FLDs are the sam e due to the sym m etry o f the

yield function w ith th e rolling and transverse directions, but for Barlat Y ield96

and Barlat 2 00 0 the full FLD is not sym m etric w ith the first bisector.

36


3.3: Yield Functions

In this w ork th ree yield functions are used to calculate the theoretical

FLDs for A I2 00 8 -T 4 , von Mises, Barlat Y ield96 and Barlat Y ie ld2000 . A

description of the yield functions and the derivation needed for the

im plem entation of the codes are presented in the appendix.

3.4: Hardening rules

The coefficients o f an effective stress-strain curve th a t defines the strain

hardening of a m aterial is found by fitting the experim ental data obtained

from a uni-axia l tensile tes t to one of the following equations:.

Sw ift ct = K (s0 + e )”

Voce a - A - B exp(-CTr)

W here a a n d s a r e the effective stress and effective strain respectively, and

the rest o f the variables are constants th a t are obtained from theby fitting

the above equations of to experim ental results from a tensile test.

3.5: Material constants

The m aterial constants, needed for the im plem entation of the code for

alum inum alloy A I2 0 0 8 -T 4 , describes for both tw o hardening rules and yield

functions are presented below. The im perfection p aram eter is assum ed to be

0 .9 9 6 .

37


Swift Voce£ o 0.002

n 0.1882

K 456.82 Mpa

A 447 Mpa

B 248 Mpa

C 4.3 Mpa

Table 1: Constants for voce and swift equation - AI2008T4

Yield96

a Ci c2 c3 c6 a* Cty <*/0 azi

8 1.1227 0.9154 1.0137 1.0578 1.55 1.75 1.0 0.6

Table 2: Constants for Barlat Yield96 - AI2008T4

Yield2000

a ai a2 a3 a4 a5 a6 a7 a8

8 0.91777 1.14937 1.17360 1.14556 1.05262 1.19593 1.01494 1.06188

Table 3: Constants for Barlat Yield2000 - AI2008T4

0.6 -

1COoQ>

/

------ vonMises

(r —•— Yld96

—*■—Yld2000

-------------------------,-------------------- 0 -O E^erimental

-0.4 -0.2 0 0.2 0.4 0.6

RD Strain

F ig l8 : Fids AI2008-T4-Swift

38


.sasaCZ3

-0.4 -0.2 0 0.2 0.4 0.6

RD Strain

Fig 19: Fids AI2008-T4-Voce

3.6: Results and discussions

The experim ental and the com puted FLDs for von Mises, Barlat Y ld96 and

Barlat Y ld2000 yield function for A I2 00 8 -T 4 using Sw ift hardening law are

shown in F ig -18 . I t can be seen th a t von Mises over predicted the

experim entally m easured failure points due to the anisotropy of the m etal

and Y ld96 and Y I2000 have given good result with Y ld96 being more

conservative. In F ig -18 the FLDs are calculated using Sw ift hardening rule

w here it is shown th a t failure points are well predicted when Y ield96

describes the yield locus. For A I2 00 8 -T 4 it is best to use Y ield96 to describe

the yield locus and Sw ift equation to model the hardening of th e m ateria l to

be on the safe side.

39

0.6 -

0.3 -

------ von Mises

— Yld96

— Yld2000

O Experimental----------------------- 1--------------------9 - . ' u ----- -


CHAPTER 4: Simulation of Stretch Stamping

4.1: Introduction

In m any sheet m etal form ing sim ulations a form ing lim it d iagram is used

to predict fa ilure. In sheet m etal stam ping sim ulations, the m inor and m ajor

strains for all e lem ents are plotted against each other and com pared with

with the FLD curve to check w hether or not strains a t any e lem ent exceed

the lim iting curve. Usually th e FLD is generated using both Voce and Swift

hardening rule, and the m ost conservative one is used to check for the

possibility of the developm ent of a localized neck in the sheet. In this chapter

stretch stam ping of a sheet m etal w ith a hemispherical punch is sim ulated

using LS-DYNA and the m ajor and m inor strains o f all the elem ents are

tracked through the sim ulation to see how and when they exceed its the

FLDIimit. W hen one e lem ent reaches the lim iting curve the sim ulation is

stopped and the lim iting punch depth is recorded. Although strain paths in

this type of sim ulations are not linear, FLDs are still used to check for

necking and has been shown to provide qualitatively accurate and failure

validprediction. The pure-stretch stam ping tes t o f a A I2 00 8 -T 4 sheet using a

hem ispherical punch has been done experim entally and initial fa ilure

occurredwas recorded for a t a punch depth of 27m m at room tem p eratu re .

The finite e lem en t sim ulation result w ith LS-DYNA alsos m atched the

experim ental results, w here it was showed predicted th a t the first tim e the

strains in a first e lem en t exceeded reached the FLD, occurred when the

punch depth was a t 27m m .

40


4.2: Description of the model

In pure-stretch stam ping, a sheet is clam ped betw een a die and the

blank holder, and a punch is used to deform the sheet. A full model has been

used in this sim ulation to capture th e anisotropic behavior o f th e A A 3003-

H l l l sheet. The dimensions of the part, the m aterial properties, and all the

data needed to model this experim ent is described below.

4.2.1: Part dimensions

The punch d iam eter is 4 in. The d iam eter o f the die and the blank holder

is 7 .5 in. The d iam eter o f th e sheet is 3 .5 4 in. The height o f the die is 1 .81

in, and 0 .4 in for the blank holder. The radius of the surface of the die and

blank holder parallel to the sheet is 2 .3 in. The sheet is assum ed to have a

uniform thickness of 1m m . The initial distance betw een the die and the sheet

was taken to be zero, and th e one betw een the blank holder and the sheet

was taken to be 0 .5 m m to prevent any initial penetration . The initial

distance betw een the punch and the sheet was assumed to be 3 m m . The

radial distance betw een the punch and the blank holder was assum ed to be

0 .7 m m .

4.2.2: Element type and material models

For all parts the "Belytschko-Tsay" (Shell e lem ent num ber 2 in LS-DYNA)

was used, using 5 integration points for the sheet and 2 for the rest of the

41


parts. The punch and the blank holder w ere m odeled as rigid m aterials

"Steel" (m ateria l num ber 33 in LS-DYNA) and constrained to m ove only

norm al to the initial plane of the sheet. The die was modeled as rigid "S teel"

but constrained to rem ain fixed during the form ing process. The sheet was

alum inum A A 3003 -H 111 and modeled to follow BARLAT-YLD2000 (user

m aterial option has been used) and assum ed to follow Voce and Sw ift

hardening rule.

4.2.3: Material properties

The m aterial constants fo r the alum inum and steel are shown below.

Steel

Young's modulus 210 GPA

Poisson ratio 0 .3

Density 7 85 0 k g /m 3Table 4: Properties of Steel

A A 3003 -H 111

Young's modulus 70 GPA

Poisson ratio 0.33

Density 2710 kg/m 3Table 5: Properties of AA3003-H111


Voce

A 1 4 4 . 0 1 5 M P A

B 7 6 . 2 1 7 M P A

C 1 3 . 0 5 8

Table 6: Constants for voce equation

S w i f t

5 . 7 E - 4

n 0 . 2 1 5 7

k 1 9 9 . 8 2 M P a

Table 7: Constants fo r sw ift equation

Barlat YLD 2000

a 8

L m 0 . 6 2 0 3 2 5 6 2 4

L112 - 0 . 3 1 0 1 6 2 8 1 2

L m - 0 . 3 6 5 9 6 5 1 0 6

Ll22 0 . 7 3 1 9 3 0 2 1 3

Ll66 1 . 0 6 9 5 3 3 1 3 1

L211 0 . 6 8 1 6 5 3 1 1 4

L212 - 0 . 3 3 1 2 9 0 1 7 1

L221 - 0 . 3 5 0 7 9 8 4 9 8

L222 0 . 6 9 9 9 8 0 0 7 5

L266 1 . 0 6 0 9 6 9 3 8

Table 8: Constants for Barlat Yield2000

See Appendix C for the definition of th e param eters above.


4.2.4: Constraints and prescribed motion

The die was assum ed to follow a prescribed motion (d isp lacem ent), with

the corresponding curve shown below. A force was applied on the blank

holder to clam p the sheet to the die. This force started from zero and

increased until it reaches 4 5 0 KN a fte r nine seconds. A plot th a t describes

this motion is shown below. A picture of the full model is also shown below.

AA3003-H111, RT

Fig20: Model stretch stamping

44


The plots show the prescribed motion for the punch and the blank holding

force.

Blank holder - force cuve600

400

200

25 30 3510 15 200 5Time ms

Fig21: Blank holder force

Punch - Prescribed motion

50

40tseo£atortacns

10

10 15 20 25 30 350 5Tim e ms

Fig22: Punch prescribed motion

4 5


4.3: LS-DYNA

A fter calculating th e theoretical FLDs of the m aterial using th e program

described in th e previous chapter fo r both voce and swift hardening rules, the

more conservative curve was used to check for failure.

4.3.1: Swift and Voce FLDs

For this case the FLD curve based on voce hardening rule turned out to

give the m ore conservative FLD. Both FLDs w ere calculated w ith the larger

principal strain assumed to occur in the TD direction to be on the safe side.

The m aterial coefficients for swift and voce hardening rules are w ritten in the

previous paragraph. Both voce and swift FLDs are plotted below.

0.25 -

Swift Voce

0.3-0.3 - 0.1 0.1

RD Strain

Fig23: FLD AA3003-H111

46


4.3.2: LS-DYNA FLDs

In LS-DYNA, to open a FLD curve, th e points should be copied to a tx t file

starting w ith "fld_user" and followed by double the num ber o f points. For

A A 3003 -H 111 the input FLD file is w ritten below. In LS-DYNA the curve is

read and plotted. The FLD curve used in this sim ulation is plotted below.

Usually m any curves are generated from one data input file. The m ost useful

ones are the original FLD (Cracks curve) and a m ore conservative (Risk of

Cracks curve) curve to keep one on the safe side. Materials are considered to

be safe if they do not cross the crack curve. But it is b e tte r to be on the safer

side w here one should not cross the risk o f crack curve.

fid user360.178426 -0.089210.168248 -0.06730.159279 -0.047780.150106 -0.030020.141445 -0.014140.132489 0.0000020.137678 0.0110160.147395 0.0235850.15628 0.037510.1647 0.0527130.172611 0.0690680.180267 0.0865520.187824 0.1052360.195239 0.1249630.202804 0.1460840.210229 0.1682620.217641 0.1915880.221089 0.212251

Table9: FLD user

47


Maj

or E

ngin

eerin

g St

rain

(%

)LS-DYNA FLD VOCE25

20

15

5 0 5 10 15 20

Minor Engineering Strain (%)Fig24: LS-DYNA FLDs

4 8


4.4: Experimental Findings

The pure-stretch stam ping tes t has been perform ed in the lab a t room

tem p eratu re . The picture below shows the fractured sheet a t the depth of 27

m m .

Fig25: Failed sheet

49


4.5: Numerical results

The com puter sim ulation was stopped when the strains in an e lem ent

crossed the form ing lim it d iagram . The plots below show th e strain

distribution o f all e lem ents a fte r 10 ms and a t the failure tim e (2 3 m s). The

strain paths range from plane strain for the e lem ent on th e sides of the

sheets to biaxial tension for the elem ents in the middle of th e sheet. The

displacem ent of the middle e lem ent is plotted too.

25 LS-DYNA FLD (L15 ms)

20

c2 15-COU)coc 10a>cLDi—o‘<5*

£-5 0 10 15 20

Minor Engineering Strain (%) Fig26: FLD AA3003-H111 (t: 15 ms)

5 0


Dis

plac

emen

t-mm

M

aJ°r

Eng

inee

ring

Stra

in

(%)

LS-DYNA FLD (L23 ms)25-:--------------- -

-5 0 5 10 15 20

Minor Engineering Strain (%)Fig27: FLD AA3003-H111 (t: 23 ms)

Middle Element

-10

-20

-25

-3050 10 15 20

Time - msFig28: Middle Element Displacement

5 1


The fa ilure o f th e sheet has been found to take place in the cracks region

in the picture below. In this region th e strains in some of the e lem ents have

crossed the form ing lim it diagram and th e sheet was considered to have

failed due to the onset o f localized necking. A comparison of the actual and

predicted failed sheets indicates th a t a good qualitative match exists betw een

experim ental and num erical results.

Formability keyContours of Formability: Mid. Surface

Cracks

Risk of cracks

Severethinning

Good

Inadequatestretch

Wrinklingtendency

W rinkles

r xFig29: FLD contour at failure

5 2


CHAPTER 5: Conclusions

The experim ental and num erical results w ere very close. First the punch

depth a t fa ilure was 27 m m for the experim ent and 29 m m for th e num erical

sim ulation, which corresponds to a discrepancy of only 7 .5 % . More

im portantly , the fa ilure occurred in the sam e location in both experim ental

and num erical sim ulation. The results below show th a t form ing limits are a

very useful tool to check for th e failure of stam ped sheet m etals in num erical

sim ulations. Experim ental and num erical findings are com pared in the picture

below. The discrepancy betw een experim ental and num erical findings can be

a cause of som e accuracy issues or even because the strain paths in the

sheet are not linear for all e lem ents. For elem ents on the boundary o f the

deform ed sheet the deform ation could be assumed to be linear, how ever it

becomes non-linear as one m oves tow ard the center o f th e sheet. For the

pure-stretch stam ping it has been shown [1 0 ,1 1 ] th a t the FLD lowers, in

comparison with linear path deform ations. This m ight explain the difference

betw een the actual and predicted punch depth a t the tim e of fa ilure. And

finally in all the num erical sim ulations we have assumed fa ilure occurs when

the strain increm ent in the im perfection region is ten tim es th e value of the

strain in the hom ogeneous region, and by assuming an im perfection

p aram eter. All these assum ptions could be partially responsible fo r the

discrepancy betw een the experim ental and num erical fa ilure punch height.

5 3


Fig30: Experimental and numerical findings

54


Appendix A - von Mises

I t is the yield function th a t describes th e yield locus of isotropic m ateria l.

A 1 - Y ie ld fu n c t io n e d

= = ^ + K +3a™ - ax*0'xy xx yy

W here cr is the yield stress and the variables are the Cauchy stresses in the

x-d irection , y direction and the shear stress respectively.

A 2 - F irst d e r iv a tiv e

2 - S a _ atda„ da.

W here n refers to xx, yy and xy.

d<j>da „

~ G y y

dtj>da y y

- ^ = 6av5 ( 7 .xy

A 3 - S eco n d d e r iv a tiv e

d2a

d°nd(Tp &

1 d2d> d a _ daY <g>-

2 d a d a , da, da .

W here n and p refer to xx, yy and xy.

d2(j)

deld2<f>

= 2

d24>

d^xxdayy dGyydaxx= -1

d2(f>da

= 2y y

d2</> d2<t>da daxx xy da daxy xx

d2(f>

da= 6

xy

d20 d2(j>d° yydaxy da xyd a yy

= 0

5 5


Appendix B - Barlat Yld96

Many yield functions have been developed to describe m ore accurately

th e plastic anisotropy of sheet m etals. Barlat Y ld96 [1 8 ] has proven to

describe well the plastic behavior o f a lum inum alloys. Although its convexity

cannot be proven and has com plex derivatives, which m ake it a difficult yield

function to im p lem e n ta tio n it describes the behavior o f a lum inum alloys very

accurately.

B1 - Yield functioned

(/) = 2aa =a,\S2-S3\a + a 2\S3 - S . f + a 3|S, -S2\°

W here S i, S2 and S3 are the eigen values of a stress tensor m odified by a

linear operator to obtain anisotropic properties, cr is the m easure o f the yield

surface size, the coefficients ai, a2 and a3 are three functions of the angle (3

defined below and the exponent a, connected to the crystal structure , is

equal to 6 or 8 fo r BCC and FCC alloys respectively. For plane stress the

eight m ateria l coefficients needed to describe the anisotropy are

C i , C 2 , C 3 , C 6a x, ocy, clm clzX

5 6


The stress tensor modified by a linear transformation is:

$ x x(C 2 + C3)/3 - C J 3 O ' a XX

Syy - - C J 3 ( C .+ C , ) / 3 0 CTyy

S». 0 0 C6_ a*y_

The eigen values of the modified stress tensor are:

1 2

s _ S XX + S yy

2 2

s3 = - i s l + s t )

S a + S „+ Sr

xx + yy + Sr

The coefficients ak are found using the transform ation rule:

ax = ax cos2 f3 + ay sin2 f3 a2 = ax sin2 f3 + ay cos2 /3 « 3 = az0 cos2 2/3 + azX sin2 2/3

W here (3 is the angle betw een the x direction (rolling direction) and the

direction associated with principal stress S i and is found easily by:

2/3 = tan 12Sxy

s „ - s yy

51


B2 - First derivative

at _ at „ as, „ as, at v a«, v ap asr■ n- ■" i_ X x ... x ■don dSi dS d<7n da , dp dS d a n

2 a a<x _ a tScr dcr

W here n and p denote xx, yy and xy and i fo r 1, 2 and 3.

d<j>dS,

d(j)dS7

= -a a 2|53 -S ,|a 1sign(S3 — -Sj)+ aor3 |*S, x sign(Sl - 5 2)

= aor1|<S,2 - 5 ,3|fl l sign(S2 - 5 3)-ao:3|51 - 5 2|a 1 sign(Sl - S2)

= —<2<aTj|*S,2 — 531° 1 sign{S2 - 5 3)+<m2|>S3 -5,1° 1sign(S3 - 5 , )

I f A is not equal to zero the derivative below can be calculated:

A = (S - S ) 2 + 4 5V xx yy '

dS, _ 1 /1 +

V

c _ c >juc _yy ds2 1

a1

\as„ ~ 2 Va , as„ ~ 2

35, 1 (1 -

V

3S2 1 /1 +

V2 Va ^ 2

2SV dS2

V A 8S>y Va

$xx ~

Va

aSj

as3

as3a T

as,

ydsxx+ dsxxy

' as. ds2 dzT + ~dS~V yy yy J

' ds, as2 Ava ^ 3 5 ^ ,

= - i

= - i

= o

58


dSdcr„

W here L is the linear transform ation m atrix used to find the modified stress

tensor.

(C2 + C 3) / 3 - C 3 / 3 0

- C 3 / 3 (C, + C3)/3 0

d(j> da1 d</>

da2 d(j> da ,

= i ^ - s . r

= i s , - s , r

- i s . - s , r

Sa,

~s^

S a 2

~d$da2~dp

= (ay - ax ) s i n 2/3

= i a x ~ tty ) s i n 2(3

■ = 2(azX - az0 ) s i n 4/3

dp_

dfi

dp_dSm

xyA

xy

A

S„-S„

B3 - Second derivative

a y a y . .a s , . . .a s , | e ;s, . .a s , | a y a a , ..ap j s ,da,da, dSfia, a s , CCT, as, as 0<r da, da,daq d/3 3Sp da.

I B<t>.. ay,- BP .. as, | a,* . ag, a2/? „as.0a , dpda dS da, da , a/? 35 3(7 0ct,

3 2<t r l- f l

v 2a So-, S c , S c r S c rn p J

Indices n, p and q denote for xx, yy and xy and i for 1, 2 and 3.

59


W here:

d2<j> _ d 24 dSt dSp d2</> dccj _ dj3 „ dSp8Si8aq 8S2 8Sp d<jq dSpaj dp dSp daq

d2S _ d2Si dSn

dSpdcrq dSpdSn daq

a y a y .. dSi .. dSrda jdaq dapS j dSp dcrq

d2a i = 82a i ^ dp „ 8Sp

dpdaq dp 2 8Sp dcrq

d2p _ d2p 8Sn dS pd(Tq 8Sp8Sn 8crq

• The second derivative o f function cp w ith respect to Si is a sym m etric

second order tensor (3 -3 ) . The e lem ents of this tensor are w ritten below.

dS2 V " J| ' Z|

^ = a(a- ifa, |S2 - S31°"2 + a 3\S, - S21°"2) = -a(a-1 )a2 \S3 - S, \a~28S

i = a(a- 4a, |S2 - S31fl“2 + a 2 |S3 - 5, f 2) -£*=- = -£*=- = - « ( « - !>», \S2 ~ S3

d2(j) d2<t>8Sl8S2 8S2dSl

d2(j> d2<f>dSxdS3 ds3ds,

d2<t> d2(/> a-2

8S2 v A 11 2 31 21 3 11 ' dS2dS3 8S38S2

60


The second derivative of the stress vector Si with respect to the stress

tensor Sn is a third order tensor (3 -3 -3 ) . The non-zero elements of this

matrix are derived below.

d2s]dSi

d2sxdSBS

d2Sx

d S xxd S xy

S SdS

d2SxdSyydSxy

d2Sx

K= 2

" 4.S’2 ^

d % dS2„

d2S2dSJSxy

d2S2 'dS„8S„ =

d2S,

d2S2

K= -2

dSy>dS*y

f f 4S2 ^

w V a

ds dSxy xx dSxydSyy

d2s3 ( d2sx d2s2 "dStdSj ~ + dSfdSj^

- ^ = 0 dSfiSj

61


• The second derivative of function cp with respect to aj and Sj are second

order tensor (3 -3 ). The elements of this tensor are written below.

_ n d2f _ d2<f> _ Jc c 0a ac o 30 — 3 3C — 2 3 s l S n i ^ 2oapS^ daldS2 da i8Si

d't -o 9,2 * lc c l - • ,o os- u = -------------= a\S-t - S. sign(S, - S,)da2dS2 da2dS2 da2dS1

gy - fl a y a y , c e r . .

Sa3dS3 da2dS\ da2dS2 ^ 1 1 2

• The second derivative of function aj w ith respect to (3 is a vector o f th ree

elem ents derived below.

d2a.

dp

d2a

dp 2

d2 a

= 2(ay - a x)cos2fi

= 2 (ax - a y) cos 2/?

= 8(a„ -orz0)cos4/?dp2

• The second derivative o f the angle (3 w ith respect to Sp is a second order

tensor (3 -3 ) .

S2f i _ e ‘ f i = 2 sv (sa - s „ )

asi as; a 2

d2p _ d2p

d S xx d S yy d S y y d S xx A2

d2p d2p - ( s xx- S X)' f + 4Sxy

dSa dS9 dS„dSv A2

d2p _ d2p - ( s^ - sJ + as ,

dSv dS„ dSxydSyy

d2pK A2

62


Appendix C - Barlat Yld2000

This is a new yield function th a t has been proposed by F. Barlat [1 9 ] to

overcom e the weakness of y ie ld96 w here it is proven to be convex and its

m athem atical form ulation is less com plicated. This yield function was

developed for a lum inum alloys and has been shown to describe well the

anisotropic behavior o f alum inum sheets.

C l - yield functioned

^ = 2oa =\Xl- X 2\“ + |27, + Y 2|" + |272 +7,1°

W here X i, X2, Y i and Y2 are the eigenvalues of a stress tensor X modified by

a linear operator L to obtain anisotropic properties, the exponent a,

connected to the crystal structure, is equal to 6 or 8 for BCC and FCC alloys

respectively, a is the m easure of the yield surface size. For plane stress the

eight m ateria l coefficients needed to describe the anisotropy and the linear

operator L are w ritten below.

A n ■ 2 / 3 0 0 " A n ' - 2 2 8 - 2 0" a 3

A 12 - 1 / 3 0 0 a x An 2_ 1

9

1 - 4 - 4 4 0 a A

A 21 = 0 - 1 / 3 0 a 2 A 21 4 - 4 - 4 1 0 a,

A 22 0 2 / 3 0 a 7 A 22 - 2 8 2 - 2 0 «6

11 0 0 1 _^266_ 0 0 0 0 9 _ a s _

63


The linear operator L is found by a linear transformation form the anisotropy

coefficients. The modified stress tensor X can be found by a linear

transformation with L:

X = La

< J „

x rvXX

xy= L yy

1

* y

The eigenvalues of th e modified stress tensor X are easily found:

X = ? * * + X yy

* 2 = '

2+ . ■x.

X xx + X yy■ X .xy

Y =■

2 =

Y + Yxx yy

2

Y + Yx x y y

+ .

Y + Yx x y y + r

r Y + Y Vxx yy + Y „xy

C 2 - F irst d e r iv a tiv e

d ( j> d</> 8 X i d X p

8 o „ d X , 8 X p 8 o n

2 ( 7- a - 1 S C T d(/>

d o - , d o •

W here i denote indices for 1, 2 and n, p for xx, yy, xy.

64


d(f)

dx= a\Xx - X 2 p ' sign(Xx - X 2)

d(/>~dX\

= -a\Xx - Z 2 p ’ s ign(xx - X 2)

= 2a\2Yx + 72 p 1 sign(2Yx +Y2) + a\2Y2 + Yx\“~’ sign(2Y2 + y ,)

= 2a|2y2 + Yx p 1 sign(2Y2 +YX)+ a\2Yx + Y2\a~' «gw(27, + F2)

I f Ax and AY are not equal to zero the derivative below can be calculated:

A ^ = ( X „ - X w ) 2 + 4 Z xy A r =(X]a-Y„)2+4Y12xy

dXx _ 1

dX,

1

dXyy 2

dXx 2X

1 —X „ - X

V a

yy x J

xy

dx-, X

dX2 _ 1 Y X * - X yyXX V

/a y , 1

1 +7yy V

dX2

&

X ^ - X yy"\/A;

V a 7

8YX 1SY„ 2

ay, i ay,„ - 2

y - y** yy

V a7

a/ A y

ay2 iay„ 2

ay2 _ i

y - yI « yy

eyw 2i +

a/ a x

a/ A y

ay, 2 rv

3 ^ V a 7 ay^ ^ a 7

The first derivative o f X with the stress tensor is equal to L.

2 £ .a<r„

65


C3 - Second derivative

a V _ 3 V ..ax, ax, | ,, a*jr, . .ay .d a nd a B dX-do dX d a 8Xi dX d a dan p i p q n i q p n

d2a

d°nd°p■ = a 1 a

1 d2<j>2 a d a nd a p

-(a-l) d(T ^ 8(J gra-1da„ da„

W here:

a V a V dxn■ X — X -

dXid a p dXtdXj dXn da p

d X; d2X cLT

dXqdXn da p

• The second derivative o f function cp w ith respect to Xi is a sym m etric

second order tensor (4 -4 ) . The non- zero elem ents are w ritten below.

dx2 dx2 A 2\a- 2

d24> d2(j)dXxdX2 dX2dXt

d2<f>

= - a { a - l ] X x - X ,i a - 2

dY2

d2

4127, +Y2r 2 + |272 + 7 , r )

dY2

d2

£- = a ( a - i j f \ 2Y2 + 7, f 2 + |27, + 72p 2)

a VdYxdY2 dY2dYx

2 a(a - \ j \ 2Yx + Y2 p 2 + 1272 + Yx p 2)

66


• The second derivative of the stress vector X, with respect to Xn is a third

order tensor (4 -6 -6 ) . The non-zero elements are written below.

d2X l

d K

d2X ,

d2X , 82X , d2X , 1

ax; = 2 w

a2x, a2x, d2X,dX^dX^ dX^dX^ d X J X ^

= -2

/ / y v / a

W

a2x,axa2x2

a x T

a2x2a x ^ a x ^

f f 4x 2 N

vV yJ x J

d2X 0

/A,

S2X ,

a x « a x w a x ^ a x ^

a2x2ax;

fr

W

a2x, a2x, a2x0a x ^ a x ^ a x ^ a x „ a * v a * w

= 2

( x ~ - x j

yj^x

(Xjcx - xw ) x xy

W

y\

a2x,ax = - 2

" 4X 2 ^V ^ 7 -

xy

a2x,dY2XX

a2x,a x ^

/A,

a2x, a2x, a2x, 1ax axXX _yy axwax„ ax; 2 <JXy

w

a2x, a2x, a2xax ax ax ax ax axyy xy xy xx xy yy

= - 2

(X - X )2V xx xx /

~JXy

(x - X )xV xx yy 1 xy

w yj r

a2x,ax. = 2

xy w

a2x2ax2

a2x,y/ y

a2x, a2x,a x .a x yy ax ax ax2yy x t yy

r r

W

a2x, a2x, a2x, a2x,

d Y xxd Y xy WyydY" 8YvdY„ dYJ Y„

(x - X )\ xx yy 1

-\[Xy

(x - X )xV xx -1 yy ! xy

W yj y

a2x,ax. = -2

xy w yj y

67


References

[1 ] S. P. Keeler and W . A. Backofen, "Plastic Instability and Fracture in Sheets Stretched Over Rigid Punches" Trans ASM 56, pp. 2 5 -4 8 . 1963

[2 ] S. P. Keeler and W . A. Backofen, "Determination of Forming Limits inAutomotive Stampings" Sheet Metal Indust. 42, pp. 6 8 3 -6 9 1 . 1965

[3 ] Z. Marciniak, and K. Kuczynski , "Limit Strains in the Process of Stretch-Forming Sheet Metal" In t. J. Mech. Sci. 9, pp. 6 0 9 -6 2 0 . 1967

[4 ] W illiam F. Hosford and Robert M. Caddell, "Metal forming: mechanics and metallurgy" Englewood Cliffs, N.J. : Prentice Hall, c l9 9 3 . IS B N - 0 1 3 5 8 8 5 2 6 4

[5 ] A. G raf, WF. Hosford, "Calculations of Forming Limit Diagrams",Metall Trans A 21 (1 ): 8 7 -9 4 Jan 1990 .

[6 ] A. G raf, WF. Hosford, "The Effect of R-value on Calculated Forming Limit Curves", Forming lim it d iagram s: Concepts, M ethods, andapplications: 1 5 3 -1 6 3 TM S 1989 IS B N -0 8 7 3 3 9 0 9 8 9

[7 ] F. Barlat "Crystallographic Texture, Anisotropic Yield Surfaces and Forming Limits of sheet metals" M ater. Sci. Eng., 91, pp. 5 5 -7 2 . 1987

[8 ] A. Barata da Rocha, F. Barlat, J. M. Jalinier "Prediction of the Forming Limit Diagrams of Anisotropic Sheets in Linear and Non-Linear Loading" M ater Sci Eng 68 (2 ) : 1 5 1 -1 6 4 1985

[9 ] A. G raf, W . HOSFORD "Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4 Metall Trans A 24 (1 1 ): 2 5 0 3 -2 5 1 2 Nov 1993

[1 0 ] A. G raf, W . HOSFORD "Calculations of Forming Limit diagrams for Changing strain Paths" Metall Tans A 24 (1 1 ): 2 4 9 7 -2 5 0 1 Nov 1993

[1 1 ] A. G raf, W . HOSFORD "The Influence of Strain-Path Changes on Forming Limit Diagrams of A l 6111 T-4" In t J Mech Sci 36 (1 0 ): 8 9 7 - 910 Oct 1994

[1 2 ] M.C. Butuc, A. Barata da Rocha, J.J. Gracio and J. Ferrereira D uarte , "A more general model for forming limit diagrams prediction", J M ater Process Tech 125: 2 1 3 -2 1 8 Sp. Iss. S I Sep 9 2 00 2 .

[1 3 ] M.C. Butuc, J.J. Gracio and A. Barata da Rocha, "A theoretical study on forming limit diagrams prediction", J M ater Process Tech 142 (3 ) : 7 1 4 -7 2 4 Dec 10 200 3 .

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[1 4 ] F. Barlat "Forming Limit Diagrams - Predictions Based on Some Microstructural Aspects of Materials" Forming lim it d iagram s: Concepts, M ethods, and applications: 2 7 5 -3 0 1 TM S 1989 IS B N -0 8 7 3 3 9 0 9 8 9

[1 5 ] M.M. Moshksar, S. M ansorzadeh "Determination of the Forming Limit Diagram for Al 3105 Sheet", J M ater Process Tech 141 (1 ) : 138 - 142 Oct 1 2 003

[1 6 ] Z. Marciniak, K. Kuczynski and T. Pokora "Influence of Plastic Properties of a Material on Forming Limit Diagram for Sheet-Metal in Tension " In t. J. Mech. Sci. 15 (1 0 ) , pp. 7 8 0 -7 8 9 . 1973

[1 7 ] K. S. Chan "Marciniak-Kuczynski Approach To Calculating forming Limit Diagrams" Forming lim it d iagram s: Concepts, Methods, andapplications: 7 3 -1 1 0 TMS 1989 IS B N -0 8 7 3 3 9 0 9 8 9

[1 8 ] F. Barlat, Y. M aeda, K. Chung, M. Yanagaw a, J. C. Brem , Y. Hayashida,D. J. Lege, K. Matsui, S. J. M urtha, S. H attori, R. C. Becker and S. Makosey "Yield Function Development for Aluminum Alloy Sheets" J. Mech. Phys. Solids 45, pp. 1 7 2 7 -1 7 6 3 1997 .

[1 9 ] F. Barlat, J. C. Brem , J.W . Yoon, K. Chung, R.E. Dick, D. J. Lege, F. Pourboghrat, S .-H . Choi, E. Chu "Plane Stress Yield Function for Aluminum Alloy Sheets—part 1: Theory" In t. J. o f Plast. 19, pp. 1 2 9 7 - 1319 . 2 00 3 .

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