ansisotropic characteristics of materials and basic selecting rules with diferent sheet metal,...

8/2/2019 Ansisotropic Characteristics of Materials and Basic Selecting Rules With Diferent Sheet Metal, Forming Processes

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Anisotropic characteristics of materials and basic selecting ruleswith different sheet metal forming processes

Weilong Hua,*, Z.R. Wangb

aTDMTroy Design & Manufacturing Inc., 12675 Berwyn, Redford, MI 48239, USA

bDivision of Metal Forming, #435, School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China

Abstract

Anisotropy is one of the important properties of sheet metals, which affects the forming results very seriously in some forming cases. The

effects of anisotropy are very sensitively dependent on the different forming processes and forming failures. In order to apply anisotropy toimprove the forming processes and obtain better forming parts, each of the forming processes should require different behavior of the

anisotropy to compatibly match it. The reasons why the fundamental characteristics of deformation depend on the anisotropy of sheet metals

are described in this paper. Then, applications of the anisotropy, which combine the anisotropic behavior with the main forming failures of the

different sheet metal processes, are discussed. Analyzing the results should provide a basic rule for production to select the anisotropic

properties for any forming process.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Anisotropy; Forming failure; Wrinkling; Thinning; Splitting; Sheet metal forming; Plastic deformation

1. Introduction

With anisotropic materials being applied in manufactur-

ing industry extensively, the anisotropic behavior of materi-als are being paid more and more attention by production.

The anisotropy of materials not only affects the formability

and the forming precision, but also has a bearing on selecting

an optimal cropping shape of the blank [1]. One of the

aspects of anisotropic behavior is the different thinning

values when measured in the plane of the sheet as opposed

to its through-thickness direction under a uniaxial-tension

state. Assuming ew and et are the principal strains in the

width direction and through-thickness direction, respec-

tively, generally, this characteristic can be expressed by a

parameter R as

R ewet

(1)

For the common stamping processes, the R-value of sheet

metals is such that when it is greater, the thinning should

be smaller, and the formability better. However, a greater

R-value does not satisfy all sheet metal forming processes

such as necking and bending processes. This means that each

of the forming processes should have individual forming

properties related to the anisotropy of the materials, and the

different strain states would cause different forming failure

[2]. Because the effects of anisotropy of materials depends

on the different strain type of forming parts, then to apply

anisotropy effectively the anisotropic behavior in severalspecial deformation areas should be known rst. Then,

according to the analysis result, the anisotropic properties

relating to the forming processes should be selected cor-

rectly.

Based on fundamental metal forming theory, anisotropic

behavior with different deformation conditions will be dis-

cussed below. This work will deal with the aspect that the

different R-values should result in different principal strain

component values in different deformation areas. Finally,

combining with some main forming process in production,

anisotropic properties will be shown as being a basic rule to

guide material properties selection. The effects of anisotropywill concerned with the stress state, strain state, thinning,

splitting, wrinkling, warping and their relations under the

different deformation conditions.

2. Stress and strain space with anisotropic sheet metals

Because of the existing anisotropic properties, many

deforming characteristics will be changed depending on

the anisotropic parameter R-value, such as the yield curve,

strain space, and thinning etc. This basic behavior will affect

the forming results very seriously. The effects have to be

Journal of Materials Processing Technology 127 (2002) 374381

* Corresponding author.

E-mail address: [email protected] (W. Hu).

0924-0136/02/$ see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 4 1 0 - 7


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known rst so as to be able to apply the anisotropic proper-

ties to enhance the forming processes and material property

options.

2.1. Stress spaces dependence on different R-values

In order to simplify analyses, suppose that anisotropicsheet metals are in planar isotropy but that this is different

from thorough-thickness. A yield criterion proposed by Hill

[3] is used to discuss the anisotropic properties of materials,

expressed by

Fs si s211 s222 2R

R 1 s11s22 22R 1R 1 s

212

1=2(2)

Ifs11, s22 and s33 are selected as the principal stresses in the

plane stress state, Eq. (2) can be rewritten as

Fs11;s22 si s211 s222 2RR 1 s11s22 1=2

;

rsa

1 Rp si; rsb

1 R1 2R

rsi (3)

where rsa and rsb are the half long-axis and half short-axis of

the stress ellipse, respectively.

Fig. 1 shows several changing yield curves obtained by

selecting different R-values under complex stress compo-

nents with the same effective stress si.

According to similar deformation types, the stress space

can be separated into six areas based on different deforma-

tion types including biaxial elongation (0A0B), tensile

deformation as major with one tensile and one compressivedeformation (0B0C, 0A0F), compressive deformation as

major with one tensile and one compressive deformation

(0C0D, 0F0E), and biaxial compressive deformations

(0D0E). Since the anisotropy of materials is the existence

of the location of an interface line between two areas, which

are the plane-strain lines, they will be transferred depending

on the R-values of the anisotropic materials. The angle b of a

plane-strain line 0A is given as

tgb R

1 R (4)Each of the interface lines is in the plane-strain state. When

the R-value is larger or smaller, the biaxial elongation areas

transfer to become:

R 3I; b3 45; 0A0Band0D0E areas 3 0;s11 s22 3I;

R 3 0;b3 0; 0A0Band0D0E areas 3 45;

s11 s22 3

2p

2si (5)

That is, when the R-value increases to innity, the point A

transfers to AH, the biaxial elongation area transfers to zero,and nearly no deformation occurs in these areas; and when

the R-value is decreasing to zero, the point A transfers to A HH,the biaxial elongation areas transfer to be the biggest, and the

plane-strain line overlaps with the uniaxial-tension line.

From Fig. 1 and Eq. (5), it is also known that the bigger

R-value can make the deformation resistance increase in

the biaxial elongation area (bf) and the biaxial compression

area (ce).

2.2. Strain spaces dependence on different R-values

Because the strain space is a three-dimensional problem,in order to know the characteristics of strain distribution

dependence on the anisotropic parameter R-value clearly,

the strain space with three-dimensional principal strain

components will be transformed to the two-dimensional

state with e11e22 strain space and e11e33 strain space, where

e11 and e22 are the principal strains in the sheet plane, and e33is the principal strain in the through-thickness direction.

2.2.1. Planar strain space behavior

If the deformation theory of the stress and strain is

selected to describe the strain space behavior with the

two principal strains of the sheet plane, the strain equation

can be given by

Fe11; e22 ei 1 R1 2Rp e

211

2R

1 R e11e22 e222

1=2;

rea

1 2R1 R

rei; reb 1

1 Rp ei (6)

where rea and reb are the half long-axis and half short-axis of

the strain ellipse.

This is a plane principal strain ellipse. With the same

effective strain and several different R-values, the corre-

sponding strain loci spaces are shown in Fig. 2.Fig. 1. Stress space with different R-values.

W. Hu, Z.R. Wang / Journal of Materials Processing Technology 127 (2002) 374381 375


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When the deformation is supposed to be the simple

tension state, the effective strain ei is equal to the major

strain emaj. Substituting ei emaj into Eq. (6), the minorstrain emin can be obtained as

emin R1 R emaj (7)

From Fig. 2 it can be seen that the strain space in the sheet

plane is different from the stress space. The strain space is

limited to a circle, the radius of which is 2p

ei. The angle

between the uniaxial-tension state line (0b) and e22 axis is

tg a eminemaj

R1 R (8)

From Fig. 2 and Eqs. (7) and (8), a changing rule of the

different strain areas dependence on the R-values can be

obtained as

R 3I: 0B0Dand0E0A areas 3 Plane-strain state;0A0Band0D0E areas 3 0;

R 3 0: 0B0band0E0e areas 3 Plane-strain state;0A0Band0D0E areas 3 biggest deformation state

(9)

That is, when the R-value extends to innity the deformation

areas 0b0c and 0e0f will approach to become one line, and

the biaxial tension line will approach to the center 0 of the

strain space; and when the R-value is decreasing to zero, the

deformation areas 0b0c and 0e0f transfer to 0B0D and

0A0E, and the absolute values of thickness strain become

biggest in the 0A0B and 0D0E areas.

2.2.2. Thinning behavior of strain space

In terms of the incompressible assumption of volume

emine22 emaje11 ete33 (10)Substituting Eq. (10) into Eq. (6), an equation related to the

major strain (or any selecting principal strain in the sheet

plane) and thickness strain et is obtained as

Fe11; et ei 1 R

1 2Rp

2

1 R e211

2

1 R e11et e2t

1=2(11)

This is an elliptical equation also named as a thinning ellipse

or thickness strain space. Its half long-axis rta and half short-

axis rtb are given respectively as

rta ei 21 2R1 R3 R

R 12 4

q

264

375

1=2

;

rtb ei 21 2R1 R3 R

R 12 4

q

264

375

1=2

(12)

Fig. 3 displays the shapes of two ellipses (etemaj ellipse (T)and eminemaj ellipse (P)) with several different R-values andthe rotation angles of thinning ellipse long-axes, where

eu-min and eu-t are the planar minor strain and thickness

strain under the uniaxial-tension state.

From Fig. 3 it can be seen that when the R-value is equal

to 1, two ellipses overlap together, and following the

R-value increase the thickness strain ellipse rotates clock-

wise and the rotation angle increases, jeu-minj 4 ; jeu-tj 5 .When the R-value is decreasing, the thickness strain ellipse

rotates anti-clockwise and the rotation angle decreases,

jeu-min

j 5;

jeu-t

j 4.

Fig. 2. Strain space with different R-values in the sheet plane.

Fig. 3. Thinning behavior based on anisotropic property.

376 W. Hu, Z.R. Wang / Journal of Materials Processing Technology 127 (2002) 374381


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The rotation angle between the thinning ellipses long-

axes and the abscissa of the principal stress coordinate

system can be expressed as

f arctg 12R 1

R 12 4

q

(13)

From Eq. (13),

f > 45;R > 1; f3 90;R 3I; f 45;R 1;

f < 45;R < 1; f3 arctg

5p 1

2

!;R 3 0 (14)

3. Anisotropic characteristics associated with

different deformation areas

Relating to the basic anisotropic behavior of materials, the

deforming characteristics in the different deformation area

will be discussed further. From Figs. 1 and 2, the different

deformation areas are respectively discussed in the sectionsbelow.

3.1. Biaxial tensile stress state with planar

two elongation deformations

In this deformation area 0A0B, the two principal strains

in the sheet plane are in elongation. According to Eqs. (6)

and (11), the strain relations are

emaj

1 2Rp

1 R ei;

emin

emaj R

1 R 1 2Rp

1 Re2

ie2maj

1 !1=2

;

et

emaj 1

1 R

1 2Rp

1 Re2ie2maj

1 !1=224

35 (15)

Fig. 4 displays the strain distributions at points A and B with

the different R-values, where e22ARj; etARj; e11BRj; etBRj(j 0, 1, 30) are the principal strains e22, e11 and thethickness strains et on the points A and B with the different

R-values R 0;R 1;R 30.

From Eq. (15), the effects of the anisotropy of the material

will result in

R 4 ; e22AR1e11BR1 > e22AR30e11BR30;jetAR1etBR1j > jetAR30etBR30j; R 3I; eij 3 0;

R 5 ; e22AR1e11BR1 < e22AR0e11BR0;jetAR1etBR1j < jetAR0etBR0j; R 3 0; 0 e11 ei;0 e22 ei;

2

pei et ei; e11 ! 0;

e22 ! 0; et < 0 (16)That is, when the R-value is increasing, all strains and

thinning decreases. When the R-value is decreasing, the

two principal strains and thinning increases.

The typical forming failures in this deformation area are

the thinning and splitting. Thus the bigger R-value may

restrain to occurrence of these forming problems.

3.2. Biaxial compression state with planar two

compression deformations

With this deformation area 0D0E, two planar principal

strains are negative (compression deformation) and the

thickness increases. The strain change at points D and E

with the different R-value is shown in Fig. 5, where

e22DRj; etDRj; e11ERj; etERj (j 0, 1, 30) are the principalstrains e22, e11 and the thickness strains et on the points D and

E with the different R-values R 0;R 1;R 30.The principal strains with the R-value in 0D0E area is

R 4 ; je22DR1e11ER1j > je22DR30e11ER30j;etD

R1

etE

R1

> etD

R30

etE

R30

; R

3I; eij

30;

R 5 ; je22DR1e11ER1j < je22DR0e11ER0j;etDR1etER1 < etDR0etER0; R 3 0; 0 ! e11 ! ei;0 ! e22 ! ei;

2

pei ! et ! ei;

e11 0; e22 0; et > 0 (17)This means that when the R-value is increasing, the absolute

values of two principal strains and thickening decreases.

When the R-value is decreasing, the absolute values of two

principalstrainsandthickeningincreases.Thetypicalforming

Fig. 4. Strain change with the R-value.



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failure in this deformation area is wrinkling. Because a larger

R-value will cause a larger compressive stress, selecting a

smaller R-value can slow down the occurrence of wrinkles.

3.3. Stretching as the major deformation in the planar

tensioncompression strain state

In the tensioncompression strain state (0B0C, 0A0F),

the tension as the major deformation denotes that the

thickness is reduced. Suppose emaj is the tensile principal

strain, the compression strain can be expressed as

emin R1 R emaj

1 2Rp

1 R e2i e2maj1=2 (18)

The three principal strain distributions with the R-value

changing (which is shown in Fig. 6) are given as

R

4; e11B

R1

jetB

R1

j> e11B

R30

jetB

R30

j;

je22CR1j < je22CR30j; etCR1 etCR30 0;R 3I; e22 3 e11; et 3 0;R 5 ; e11BR1jetBR1j < e11BR0jetBR0j;je22CR1j > je22CR0j; etCR1 etCR0 0;

R 3 0; et 3 e11; e22 3 0 in bi-tension stress area;e11 ! 0; e22 0; et 0 (19)

where e22CRj; etCRj; e11BRj; etBRj (j 0, 1, 30) are the abs-cissa principal strains and the thickness strains on the points

C and B with the different R-values R 0;R 1;R 30.This means that when the R-value is increasing, the values

of e11 (or e22) and thinning near to the plane-strain state

(points B or A) decreases, and the absolute values of twoprincipal strains near to the plane-strain state (points C and

F) are increasing a little. When the R-value is approaching

zero, the thinning is most serious (refer to Fig. 3).

Between the uniaxial-tension and plane-strain state

(points B and A), the main forming failure types are the

thinning and splitting. Nearer to the plane-strain state (points

C and F), the main forming failure is fracture [2].

3.4. Compression as the major deformation in the

planar tensioncompression strain state

According to Eqs. (6) and (11) and Fig. 2, the compression

as major deformation areas are 0C0D and 0E0F, and the

variation of the three principal strains with the R-value can

be given by

R 4 ; je22DR1jetDR1 > je22DR30jetDR30;je22CR1j < je22CR30j; etCR1 etCR30 0; R 3I;e22 3 e11; et 3 0;





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R 5 ; je22DR1jetDR1 < je22DR0jetDR0;je22CR1j < je22CR0j; etCR1 etCR0 0;

R 3 0; et 3 emin;emaj 3 0 in the bi-compression stress area;e11 ! 0; e22 0; et ! 0 (20)That is, when the R-value is increasing, the values ofe11 (or

e22

) and thinning near to the plane-strain state (point D or E)

decreases, and the absolute values of two principal strains

near to the plane-strain state (points C and F) are increasing a

little. When the R-value is approaching zero, the thickening

is the greatest. The strains distributions on the points C and

D with the different R-values are shown in Fig. 7, where

e22CRj; etCRj; e22DRj; etDRj (j 0, 1, 30) are the abscissaprincipal strains and the thickness strains on the points C and

D with the different R-values R 0;R 1;R 30.The main forming failure type is wrinkling between the

uniaxial-compression and plane-strain state (points D and

E), and near to the plane-strain state (points C and F), the

main forming failure is fracture [2].

According to the above discussions, it is known that the

R-value depends on the different deforming areas. The

different strain state should cause different forming failures.

A basic rule may be concluded as

bf area : R 4 ;Thinning 5 ;sij 4 ; b 3 C;ce area : R 5 ;Wrinkling 5 ; sij

5 ; e 3 F;cbandef : R 4 ; cb 3 Candef3 F;

R 5 ; cb 3 DBandef3 EA (21)which is shown in Fig. 8.

4. Applications of anisotropic properties with

different forming processes

From the above analyses, it is known that the effects of the

anisotropy of materials are very different in the different

deformation areas. In order to apply these results of analysis

in production, the main forming failures with different

forming processes due to the anisotropy of materials such


Fig. 8. R-value with different deformation areas.



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as thinning, splitting, wrinkling and warping will be dis-

cussed below. Then how to improve the forming problems

based on applying the anisotropic properties of materials

also will be described simply. These studies on the applica-

tion of the anisotropy probably can become a basic rule

to guide engineers selecting the anisotropic properties of

sheet metals with different forming processes and different

forming failure types. Fig. 9 shows several sheet metal

forming processes [4] and their strain distributions of the

main deformation areas in the FLD-space [57] and planar

principal-strain space.

4.1. Deep drawing

Deep drawing is the most basic sheet metal forming

process. Principal strain distributions in the main deforma-

tion area generally include the tensioncompression strain

state and the tensiontension strain state. According to the

above analysis and Fig. 8, it is known that increasing the

R-value may improve the forming failures of thinning and

splitting. This means that a greater R-value of materials is

suited for the deep-drawing process option, but a greater

R-value also increases the deforming resistance (see Fig. 2).

If the main metal ow area is scattered about AB (the

tensiontension deformation area shown in Fig. 9), the

R-value cannot be selected to be too large. In the ten-

sioncompression area, the R-value can be selected to be

much larger. The common rule of material option for deep

drawing process is

R > 1 (22)

4.2. Bulging

The bulging process involves the biaxial stretch stress state.

Thus a greater R-value can improve the formability of part

and reduce forming failure such as thinning and splitting;

but much greater R-value also increases the deformation

resistance. The common selecting rule for the anisotropic

properties is

R > 1 (23)

4.3. Flanging

The anging process commonly includes shrink anging,

stretch anging and hole anging, shown in Fig. 9. Stretch

anging and hole anging process involve the two-tension

stress state (from the uniaxial-tension state to the plane-

strain state), and stretching is the major deformation. Themain forming failure is thinning and tearing [8]. In order to

reduce these forming problems, a greater R-value should be

selected.

With shrink anging, the main deformation strain is

compressive strain and the main forming failure is wrink-

ling. A smaller R-value can be selected to improve com-

pressive instability. Based on the two kinds of completely

different deforming properties, the material option with the

anisotropy should conform to the rule below:

R > 1 stretch flanging processes;R < 1

shrink flanging processes

(24)

4.4. Necking

Necking is a process in which compressive deformation is

the major deformation with the planar compressioncom-

pression stress state. This means that the wrinkling should be

the main forming failure. Referring to Fig. 1 it is known that

when the R-value increases the compressive stress value also

increases; the possibility of causing wrinkles is higher. By

the way, in the compressivecompressive strain area, thick-

ening deformation is easy, which also can slow down the

wrinkling occurrence. Thus the necking process should

Fig. 9. Strain distributions of the main deformation areas with some forming processes.



8/8

follow the rule below to select the anisotropic properties of

materials.

R < 1 the larger the wrinkles; the smaller theR-value(25)

4.5. Flaring

The main deformation type of the aring process is

stretching as the major deformation in the tensilecompres-

sive deformation areas (from uniaxial-tension to the plane-

strain state). The main forming failures are thinning and

splitting. The common rule for selecting the anisotropic

properties of materials is

R > 1 (26)

4.6. Bending

Bending deformation includes stretching as the majordeformation with the outside ber through-thickness of

the bending sheet and compression as the major deformation

with the inside ber of the bending sheet both in the tension

compressive deformation areas. For wide plate bending, the

warping problem in the width direction of sheet plate often

causes forming failure. In order to reduce this forming

defect, the deformation in the width direction should be

more difcult than the deformation through the thickness.

Thus the material properties relating to the anisotropic

behavior should obey the rule below:

R < 1 (27)

5. Conclusions

The stress and strain spaces relating to the anisotropic

property of the materials have been described, in particular

discussing the thickness strain space. Two strain loci ellipses

change depending on the R-values: when R 1 two strainellipses overlap together; when R > 1 the thickness strainellipse rotates clockwise; when R < 1 the thickness strainellipse rotates anti-clockwise. The strain space is divided

into six different deformation areas with different stress

states, including: biaxial tension with planar two elonga-

tion deformations; biaxial compression with planar two

compressive deformations; stretching as major deforma-

tion; the compression as the major deformation in planar

tensilecompressive deformations. Three principal strain

distributions with the R-value change are discussed. Some

forming failures, such as thinning, splitting, wrinkling and

warping of a wide bending plate, are discussed based on

anisotropic characteristics. Relating to several main forming

processes in production, some basic rules for selecting the

anisotropic properties of materials are proposed, summarized

as follows:

(a) Deep drawing process: The R-value should be selected

to be more than 1. The thinning and splitting problems

are more serious when the R-value is greater.

(b) Bulging process: The R-value should be selected to be

greater.

(c) Flanging process: With stretch flanging and hole

flanging, the R-value should be selected to be greater.

In shrink flanging, the R-value should be selected to be

smaller.

(d) Necking process: The R-value should be selected to be

less than 1 and greater wrinkles occur for a smaller

R-value.(e) Flaring process: The R-value should be selected to be

more than 1 and the thinning problem is more serious

when the R-value is greater.

(f) Bending process: With the warping phenomenon, the

R-value should be selected to be smaller.

For other forming processes, everything can depend on

the strain-changing rule relating to the different R-values to

select the anisotropic properties of materials.

References

[1] W. Hu, J. He, Anisotropic behaviors and its application on sheet metal

stamping processes, in: Proceedings of the Sixth International LS-

DYNA Users Conference Simulation, Detroit, MI, USA, April 911,

2000.

[2] W.F. Hosford, J.L. Duncan, Sheet metal forming: a review, JOM 51

(11) (1999) 3944.

[3] R. Hill, A theory of the yielding and plastic flow of anisotropic metals,

Proc. R. Soc. London, Ser. A 193 (1948) 281.

[4] Z.R. Wang, Fundamental of Plastic Working Mechanics, National

Defence Press, Beijing, 1989.

[5] S.P. Keeler, W.A.B. Backofen, Plastic instability and fracture in sheet

stretched over rigid punches, Trans. ASM. 56 (1) (1963) 2548.

[6] S.P. Keeler, Determination of forming limits in automotive stampings,

Sheet Met. Ind. 9 (1965) 683691.

[7] G.M. Goodwim, Application of strain analysis to sheet metal formingproblems in the press shop, SAE Paper 680093, 1968.

[8] K.N. Shah, M. Li, E. Chu, Effects of part and tool design on surface

distortion after down flanging of aluminum panels, NUMISHEET'99,

Besancon, France, September 1317, 1999.


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