thinsheets.pdf

Thin sheets-Anisotropy, formability &strain localisation

THIN SHEETS story PLASTIC ANISOTROPY, FORMABILITY and STRAIN LOCALISATION J. Gil Sevillano, TECNUN, Materials Engineering PLASTIC ANISOTROPY OF THIN SHEETS AND DEEP-DRAWABILITY A thin sheet material: when its (constant) thickness, t, is much smaller than any other of its in-plane dimensions of interest (length, width).

w,l<<t A consequence of this relative (not absolute) definition: it can most often be assumed that thin sheets deform under plane-stress conditions, the stress components acting on the sheet plane being negligible in comparison with the in-plane components in most circumstances. Let 3 be the normal to sheet plane,

0323133 ≅σ≅σ≅σ Thin sheets are most often very anisotropic because of their production history (involving some CVD or PVD deposition method or cold rolling and recrystallization processes). Because of the same reason they display either planar isotropy (properties are independent of the measuring direction on the sheet plane although their value is different if the property is measured along the direction normal to that plane) or orthotropy (three mutually perpendicular symmetry planes, one of them the plane of the sheet). As a first option to describe their plastic behaviour we can use the Hill’s (1948) orthotropic extension to the Von Mises yield criterion and the associated flow rule based on the minimum plastic work principle. The criterion is

( ) ( ) ( ) ( ) 012 212

231

223

22211

21133

23322 =−σ+σ+σ+σ−σ+σ−σ+σ−σ NMLHGF

where the axis are the principal axis of symmetry. Assuming planar isotropy,

GF = By choosing the reference system in the principal stress directions, σ : 321 ,, σσ

( ) ( ) ( )

FFH 12

212

132

32 =σ−σ+σ−σ+σ−σ

Let X and Z be respectively the (tensile) equivalent stresses in the sheet plane and along the normal to the sheet plane. Then

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HFX

+=

1 F

Z21

= 122

−

=XZ

FH

The ratio XZ is an indicator of the anisotropy of the sheet (it is 1 for isotropy). The criterion in terms of the equivalent stresses is

( ) ( ) ( ) 22

221

213

232 212 Z

XZ

=

−

σ−σ+σ−σ+σ−σ

From the criterion we can now derive the plastic strain increments for this particular reference system (directions of principal stress):

( ) ( )

−

σ−σ+σ−σ−=

λε

12222

21131

XZ

dd p

( ) ( )

−

σ−σ−σ−σ=

λε

12222

21322

XZ

dd p

( ) ( ) ([ ]21313323 2222 σ+σ−σ=σ−σ+σ−σ−=λ

ε

dd p

)

For the plane-stress condition:

( ) 22

221

22

21 212 Z

XZ

=

−

σ−σ+σ+σ

( )

−

σ−σ+σ=

λ

ε1222

2

2111

XZ

dd p

( )

−

σ−σ−σ=

λ

ε1222

2

2122

XZ

dd p

( )213 2 σ+σ−=λ

ε

dd p

The plastic yield loci in the 2-dimensional ( ) space are ellipses with their axis oriented at 45º and 135º, their eccentricity increasing as the anisotropy ratio

21 ,σσ

XZ increases.

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Measurement of the XZ ratio needs to perform tensile or compressive tests in the sheet plane and in the through-thickness direction, the last case being an impossible or quite difficult task. It is easy however to show that the through-thickness strength is equal to the balanced biaxial in-plane tensile tests (look at the criterion equation or simply consider the superposition of a hydrostatic stress of the right amount – without any plastic effect - to a through-thickness compression test). Some laboratories can do that using a hydrostatic bulging testing device (in which a sheet circle is inflated like a membrane under the pressure of a fluid) but such equipment is much less common than the ubiquitous universal tension-compression testing machine. The need for an additional test can be circumvented by an ingenious exploitation of the in-plane tensile test. Lankford (1950) realised that the contraction of the cross-section of an in-plane tensile extended specimen contained the same information provided by the XZ ratio of strengths. For instance, after an elongation in a tensile test in the direction 1: pd 1ε

−

σ−=

λ

ε122

2

12

XZ

dd p

13 2σ−=λ

ε

dd p

and the ratio of the width to the thickness contractions is

21+

=R

XZ

−

=

ε

ε=

ε

122

3

2

1

XZ

d

dR

pdp

p

Substituting in the equation of the plane-stress criterion and operating

221

22

21 1

2 XRR

=σσ

+−σ+σ

The strain ratio ( ) is known as the anisotropy index. I do not know if made Lankford as rich as it made him famous but his index has since then

R ∞≤≤ R0R

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saved millions or even billions of dollars or euros to the steel and automotive industries all over the world! The reason for such success is that constitutes a very efficient index for qualifying the formability of a metallic sheet with respect to deep-drawing operations. From the steel-maker point of view, maximising optimises the drawability of the sheets and understanding the structural basis of the anisotropy described by gives clues for reaching such goal. From the user point of view (i.e., the car body maker, the maker of bodies of electric appliances, etc.), he can quantitatively specify the quality of the sheet he needs according to the drawing difficulty of a part or he can employ the index as a parameter for numerical die design. Before all decisions concerning such matters were based on qualitative grounds and sometimes on misunderstandings of what was really important for a successful deep-drawing operation!

R

R

R

RR

In a typical sheet drawing operation, a flat sheet blank is forced to totally or partially pass through the gorge of a die by the action of a punch (i.e., in the simplest case, the forming of a cylindrical cup from a circular blank). The flat region approaching the die must contract in the direction around the punch in order to finally accommodate to its perimeter. The region already formed by the punch (bottom and walls) is pulling through the border of the flat deforming region transmitting the force of the punch. The flat deforming region is pressed against the die by a blank-holder in order to avoid wrinkling, but the required through-thickness pressure is very small compared with the in-plane plastic stresses.

The stress state of the deforming flat region is located in the second or fourth quadrants of the locus, the free border being under pure compression, the other stress component being a traction increasing towards the die trough as each annular element is transmitting the pulling force through a smaller perimeter. The strain state in this region is approximately a plane strain without any important thickness change, . Once the material is adapted to the punch perimeter after passing through the die gorge, its stress state shifts to a biaxial tensile one, i.e., to the first quadrant of the locus. At the bottom of the punch, there is a balanced biaxial state. At the walls, one of the in-plane strains is

03 =ε pd

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impeded by the rigid punch, i.e., the strain state is also a plane-strain state there but now with either or . 02 =ε pd 01 =ε pd

03εd

0 ⇒1

2 =σσ

=R

0= ⇒ 11

2 −=σσ

σ

For a successful operation the bottom and the walls of this region must resist, without plastic flow, the force necessary for the complete forming of the flat region. Any plastic deformation there would imply an undesirable local thinning of the and, most probably, its failure. From such point of view an optimal sheet would have an anisotropy such that its resistance to plastic deformation with

or would be very high compared to its resistance to plastic deformation without thinning, . Coming back to the plastic strain increment equations:

02 =ε pd 01 =ε pd

=p

2 =ε pd 1+

βR ⇒

1211

++

=σ

RR

X wall

3ε pd ⇒ ( )12211

++

=σ

RR

X flat

( )121

1+=

σR

X

X

flat

wall

In conclusion: the higher the anisotropy index, , the better the drawability. R

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PLASTIC STRAIN HETEROGENEITIES IN THIN SHEETS. DIFFUSE AND LOCALISED NECKING It is well known that, in a tensile test along direction 1, instability occurs when

011111 =σ+σ= dAdAdF ⇒ 11

1 σ=εσdd ≡ σ=

εσdd

From that moment on a diffuse neck begins to develop in the sample because the strain hardening is not able to compensate anymore the weakening due to the thinning of the cross section. In a cylindrical sample of a ductile enough material the initial neck region affects a length of the order of the sample diameter where the plastic deformation is progressively concentrated until cup-and-cone rupture occurs. The fracture separates two pointed truncated conical shaped ends. In a long specimen the extent of the necked zone depends on the strain hardening ability of the material. Beyond the force maximum the necking monotonously progresses until fracture or – in ideally ductile, very “clean” materials – until exhaustion of the cross section in the neck, a limit situation termed rupture. In thin sheet strips pulled in tension something different occurs. As in the general case, when the above mechanical instability condition is met, a diffuse plastic strain concentration begins to develop somewhere in a relatively large region of the strip length. However, the final failure does not occur by a monotonous development of such diffuse necking. After some mildly heterogeneous elongation under decreasing load a very localised neck suddenly appears in the region of the diffuse neck in the form of a narrow band on the sheet plane, oriented approximately 55º away from the tensile direction. The cross section suffers there an intense thinning that leads to failure terminating in two knife-edges.

Thin tin-plate steel samples after tensile testing. Initial width: 20mm Initial thickness: 0.3 mm The localised necking develops inside the diffuse necking region. Sometimes two localised bands develop.

3 2

The failure mode and the near 55º angle of inclination of the band of strain localisation is common to all thin sheet materials independently of their natur

e

6


(crystalline or not) and, in the case of crystalline materials, it is only influenced by their anisotropy.

The explanation for such distinctly oriented localised neck specific to thin sheets is found by an examination of the plastic strain state induced by a tensile stress in an isotropic sheet. A localised neck where the thickness decreases along a line on the sheet plane can only occur in an in-plane direction whose length is invariant to the tensile elongation, i.e., in a direction 2’ deforming by plane strain

0'2'2 =ε pd when the sheet strip is pulled along direction 1. looking at the stress - plastic strain increment equations, it is easy to see that such condition is met for

'2'2'1'1 2σ=σ For a strip pulled along direction 1 by a tensile stress such condition is met for a line along 2’ inclined 54.7º from direction 1.

11σ

Notice that the material lines (i.e., lines inscribed in the sheet surface) other than the principal directions 1, 2, 3 are rotating during the tensile tests. However the material line on the sheet plane oriented at 54.7º is at any moment length invariant (but rotating). When the instability condition is met for the plane strain deforming line, localisation occurs. This happens after the force maximum condition determining the diffuse necking development, because the section across the 54.7º line evolves more slowly than the cross section perpendicular to the pulling direction 1. Across the 54.7º line, the vector tension is

'1

1

AF . For a

7


tensile elongation increment the intrinsic and geometrical hardening or weakening terms across the section are respectively and . The instability condition is

1εd

'1A

11 +

11'1 σdA '111dAσ

11 ε11

'1 = d− dA

εσdd

0'111'1 ≤σσ dAdA

Assuming that the deformation in the diffuse necking region remains near the pure tensile stress state, i.e., σ=σ , ε= , and because of the plane-strain taking place in the 54.7º orientation,

2/2/11'1 ε=ε dA Consequently, the localised necking condition inside the diffuse necking region is reached for:

2σ

=

(in contrast with the diffuse necking instability taking place earlier, when σ=εσ dd ).

Plastic anisotropy (or a yield criterion different from the Von Mises one here assumed) induces deviations of the localised necks from the 54.7º angle.

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REFERENCES

• Backofen W. A., “Deformation Processing”. Addison-Wesley, Reading, Mass., USA, 1972.

• Hosford, W. F., “The Mechanics of Crystals and Textured Polycrystals”, Oxford University Press, Oxford, UK, 1993.

• Marciniak, Z., Duncan, J. L., Hu, S. J., “Mechanics of Sheet Metal Forming”, Butterworth-Heinemann, Oxford, UK, 2002.

Problems

1. Try to generalize the deep-drawing analysis to the more general case of orthotropy, where there is planar anisotropy besides the normal anisotropy (i.e., R is dependent on the orientation of the tensile tests in the sheet plane).

2. In case of planar anisotropy, what will be the result of drawing a circular disk with the intention of forming a cylindrical cup?

3. Imagine the necking in tension of a thin walled tube. 4. How will be the necking of a Tresca thin sheet material? 5. Try to obtain the diffuse and localised necking conditions for thin sheets

under arbitrary plane-stress states. Last version: December 2003.

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