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).
t << l, w
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,
σ 33 ≅ σ 31 ≅ σ 32 ≅ 0
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
(
)
2
2
+ Nσ12
−1 = 0
F (σ 22 − σ 33 )2 + G (σ 33 − σ11 )2 + H (σ11 − σ 22 )2 + 2 Lσ 223 + Mσ 31
where the axis are the principal axis of symmetry.
Assuming planar isotropy,
F =G
By choosing the reference system in the principal stress directions, σ1 , σ 2 , σ 3 :
(σ 2 − σ 3 )2 + (σ 3 − σ1 )2 + H (σ1 − σ 2 )2 =
F
1
F
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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Thin sheets-Anisotropy, formability &strain localisation
X =
1
Z=
F+H
2
H
Z
= 2  − 1
F
X
1
2F
The ratio Z X is an indicator of the anisotropy of the sheet (it is 1 for isotropy).
The criterion in terms of the equivalent stresses is


− 1 = 2Z 2
  X 

(σ 2 − σ 3 )2 + (σ 3 − σ1 )2 + (σ1 − σ 2 )2 2 Z 
2
From the criterion we can now derive the plastic strain increments for this
particular reference system (directions of principal stress):
  Z 2 
= −2(σ 3 − σ1 ) + 2(σ1 − σ 2 )2  − 1
dλ
  X 

  Z 2 
dε 2p
= 2(σ 2 − σ 3 ) − 2(σ1 − σ 2 )2  − 1
dλ
  X 

dε1p
dε 3p
dλ
= −2(σ 2 − σ 3 ) + 2(σ 3 − σ1 ) = 2[2σ 3 − (σ1 + σ 2 )]
For the plane-stress condition:
  Z 2 
σ12 + σ 22 + (σ1 − σ 2 )2 2  − 1 = 2Z 2
X

  
  Z 2 
= 2σ1 + 2(σ1 − σ 2 )2  − 1
dλ
  X 

  Z 2 
dε 2p
= 2σ 2 − 2(σ1 − σ 2 )2  − 1
dλ
  X 

dε1p
dε 3p
dλ
= −2(σ1 + σ 2 )
The plastic yield loci in the 2-dimensional ( σ1 , σ 2 ) space are ellipses with their
axis oriented at 45º and 135º, their eccentricity increasing as the anisotropy
ratio Z X increases.
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Thin sheets-Anisotropy, formability &strain localisation
Measurement of the Z X 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 throughthickness 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 Z X ratio of strengths. For
instance, after an elongation dε1p in a tensile test in the direction 1:
  Z 2 
= −2σ1 2  − 1
dλ

  X 
dε 2p
dε 3p
dλ
= −2σ1
and the ratio of the width to the thickness contractions is
R=
dε 2p
dε 3p dε p
1
  Z 2 
= 2  − 1

  X 
Z
=
X
R +1
2
Substituting in the equation of the plane-stress criterion and operating
 2R 
2
σ12 + σ 22 − 
σ 1 σ 2 = X
 R +1
The strain ratio R ( 0 ≤ R ≤ ∞ ) is known as the anisotropy index. I do not know if
R made Lankford as rich as it made him famous but his index has since then
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Thin sheets-Anisotropy, formability &strain localisation
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 R 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 R optimises the
drawability of the sheets and understanding the structural basis of the
anisotropy described by R 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 R index as a
parameter for numerical die design. Before R 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!
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, dε 3p = 0 . 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
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Thin sheets-Anisotropy, formability &strain localisation
impeded by the rigid punch, i.e., the strain state is also a plane-strain state
there but now with either dε 2p = 0 or dε1p = 0 .
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
dε 2p = 0 or dε1p = 0 would be very high compared to its resistance to plastic
deformation without thinning, dε 3p = 0 . Coming back to the plastic strain
increment equations:
dε 2p = 0 ⇒
dε 3p = 0 ⇒
β=
σ2
R
⇒
=
σ1 R + 1
σ2
= −1 ⇒
σ1
σ1 X
wall
σ1 X
flat
σ1
X
R +1
=
2R + 1
wall
σ1
X
=
flat
R +1
2(2 R + 1)
= 2(R + 1)
In conclusion: the higher the anisotropy index, R , the better the drawability.
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Thin sheets-Anisotropy, formability &strain localisation
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
dF1 = σ1 dA1 + A1 dσ1 = 0
⇒
dσ 1
= σ1
dε 1
dσ
=σ
dε
≡
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 nature
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Thin sheets-Anisotropy, formability &strain localisation
(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
dε 2p'2' = 0
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
σ1'1' = 2σ 2'2'
For a strip pulled along direction 1 by a tensile stress σ11 such condition is met
for a line along 2’ inclined 54.7º from direction 1.
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
F1
. For a
A1'
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Thin sheets-Anisotropy, formability &strain localisation
tensile elongation increment dε1 the intrinsic and geometrical hardening or
weakening terms across the section A1' are respectively A1' dσ11 and σ11 dA1' . The
instability condition is
A1' dσ11 + σ11 dA1' ≤ 0
Assuming that the deformation in the diffuse necking region remains near the
pure tensile stress state, i.e., σ11 = σ , ε11 = ε , and because of the plane-strain
taking place in the 54.7º orientation,
− dA1' A1' = dε11 / 2 = dε / 2
Consequently, the localised necking condition inside the diffuse necking region
is reached for:
dσ σ
=
dε 2
(in contrast with the diffuse necking instability taking place earlier, when
dσ dε = σ ).
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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Thin sheets-Anisotropy, formability &strain localisation
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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