Download 3.6 Yield Phenomena 3.6.1 Introduction

3.6 Yield Phenomena
163
in accordance with theoretical models. They explained their observations as being
a consequence of the very low stacking-fault energy, because the annihilation of
dislocations is hindered by their high degree of dissociation into partials.
HCP and BCC metals are prone to show serrations during low-temperature
deformation. Thus, the enhanced ductility of zirconium at 4.2 may be attributed
to the ability of this metal to twin readily at low temperatures (Briottet et al.). All
stress/strain curves for the 4.2 tests exhibited sudden, characteristic drops in the
load. Transverse specimens showed a much higher incidence of twinning, greater
strain hardening, more plastic deformation between serrations and fewer serrations
than longitudinal specimens. However, these researchers concluded that twinning
per se is probably not responsible for the discontinuous yielding. In BCC, serrations
in the stress/strain curves occur at high strain rates, such as are obtained by impact
and/or low temperatures. Twin bands obtained in BCC iron by high strain rates, such
as impact, are known as ‘Neumann bands’.
The morphological change of solids has been largely developed in the last
years. The localization of plastic deformation in homogeneous materials may be
associated with instabilities of stress/strain curves. This phenomenon can have
very different aspects: the Portevin-Le Chatelier effect, Lüders bands, twinning,
thermomechanical effects and avalanches of dislocations. During the past years,
more and more researchers have doubted the ideas that twinning is the main
contributor to the instability of the stress-strain curve at low temperatures and
the contribution of dislocation is emphasized. The cooperative contribution of
the various causes of instability must be appreciated. Nevertheless, this section
focuses on twinning. Twinning is associated with the coordinated deformation
of a large number of atoms, possibly leading to serrations in the deformation
curves (giving a jagged appearance). Loud clicks are heard during the formation
of twins, commonly known as ‘tin cry’. This occurs because twin formation can
be extremely rapid. The serration of the stress-strain curve is a sign of twin
formation. Many investigators reported twin formation and serrated stress–strain
curves associated with twinning. For further details on twinning, see the literature
on the crystallography of deformation.
3.6 Yield Phenomena
3.6.1 Introduction
In Sect. 1.2.2, the elastic and proportional limits were discussed in regard to the
transition from the elastic to the plastic deformation regions. In Fig. 1.7, these
transition points were indicated together with a 0.2% offset yield point. In Fig. 1.4,
the stress/strain relation under tension was shown by means of tests done to 1020
(low-carbon) steel, 1070 steel and a polymer (polyethylene), characterizing sharp
yield points and otherwise ductile materials. Most structures are designed to act in
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3 Plastic Deformation
brittle
B
B
ductile
D
C
σ
sharp yield drop-LiF
BCC iron
C
A
A
ε
Fig. 3.15 Yielding in ductile materials, brittle materials, BCC iron and LiF. The C-D zone is the
“yield elongation” region
the elastic region, since permanent change in a structural material must be avoided.
Nonetheless, the plastic region is of great importance for the fabrication and shaping
of structural materials requiring high ductility. Thus, the transition from the elastic
to the plastic regions is of great importance. Consequently, extensive research was
carried out and is available in the literature. A number of interesting phenomena
were observed while characterizing yield and information was also provided on the
behavior of materials during their transition into the plastic range. Some of the data
on yield are considered below.
3.6.2 Sharp Yield
In metals, where the elastic–plastic transition is gradual, for practical purposes, the
deviation from linearity may be taken to be the yield point (the 0.002 offset point).
The most commonly known sharp yield was first observed in low carbon content
BCC iron, also known commercially as ‘mild steel’. In this case, sharp yield is
followed by a sudden drop to a lower value, before further deformation takes place.
In Fig. 3.15 such a yield drop in low-carbon steel can be seen. Deformation in the
C-D region occurs without an increase in the stress level beyond a specific value,
known as the ‘lower yield point’. The highest stress in the elastic region is known
as the ‘upper yield point’. The C-D region is not smooth, but jagged. This kind of
yield-point drop can be detected in what is known as a ‘hard tensile machine’, which
is characterized by very small elastic distortion.
Note that N can also produce a sharp yield point followed by a sudden drop. In
the theory put forth by Cottrell and Bilby, the yield in BCC iron was explained
by the involvement of interstitial solutes, such as C or N, in the sharp yield.
The nonhomogeneous deformation in the ‘yield-elongation’ region (the C-D zone)
begins at a point of stress concentration, often at the grips, and propagates through
the specimen as bands. The grips used for holding the specimens during tensile
stress are sites of stress concentration. Beyond point D, deformation proceeds as
3.6 Yield Phenomena
165
stress increases with further straining, as shown in Fig. 3.15. The sharp-yield drop
occurs not only in low C (or N) ’Fe single crystals, but also in polycrystalline
iron, where the yield-elongation region is well developed. Cottrell has indicated
that interstitial solid solutions, such as Cd, etc., may show similar yield-point drops,
as seen in ’Fe, but less pronounced. Wain and Cottrell observed sharp yield points
also in crystals of zinc containing N. Their experiments showed not only that yield
points could be produced in zinc crystals, but also that this effect is associated with
impurities in the metal, as in the case of yield in ’Fe. Johnson and Gilman, who
studied LiF, revealed that sharp yield points also occur in this crystal. They reported
that, in order to obtain a sharp yield drop, the necessary criteria are: (a) an increase
in the number of moving dislocations and (b) a direct relation between the stress
and the velocity of the dislocations. By knowing the strain rate, given as:
"P D nvb
(3.5)
and the velocity of dislocation motion:
v D k£m
(3.6)
they were able to calculate stress-elongation curves for LiF showing sharp yield
drops. By varying the exponent, m, in Eq. 3.6 and the density of the mobile
dislocation ¡ of Eq. 3.7, the magnitude of the yield drop could be changed. In the
early stages of deformation, the density of mobile dislocations is given as:
¡ D ¡0 C C©’
(3.7)
The exponent, m, is in the range 1–100. For a certain value of ¡, increasing
m decreases the yield drop. The value of m for LiF is 16.5 and for ’Fe, 35.
Equation 3.6 is an empirical relation with k being a constant. In Eq. 3.5, n is the
number of moving dislocations/cm2 . Calculations were performed for © < 0:1,
namely in the early stages of the deformation. A dislocation-density evaluation is
done either using the etch pits technique or by electron microscopy in the range
of © < 0:1. Equation 3.7 gives ¡0 as the initial dislocation density, the constant
C D 108 /cm2 and the constant ’ is 1 ˙ 0.5.
3.6.3 Lüders Bands
When a yield drop is observed in mild steel, surface marks usually develop at
the point of stress concentration. These band-like markings are called ‘Lüders
bands’. Deformation in this range is non-uniform. Grips in the vicinity of specimen
fillets are considered be locations of stress concentration. To best see these bands,
the specimens should be reasonably well polished. Schematic Fig. 3.16 shows a
specimen in which such bands may develop. Note that these bands should not be
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3 Plastic Deformation
Fig. 3.16 Lüders bands at
45ı from the tensile axis
considered as slip lines or slip bands. The propagation of the Lüders bands continues
while the deformation is going on, until the entire specimen is covered by such
bands. In industry, especially in the automobile industry, these bands are liable
to appear, particularly on fenders, as veins called ‘stretcher strains’, that ruin the
surface finish. The appearance of Lüders bands happens when the sharp yield drop
occurs, meaning that the propagation of such bands occurs without the further need
of stress; as long as the yield-point elongation continues, these bands will spread.
Only after the yield-point elongation stops will the stress rise again. The appearance
of Lüders bands is a result of dislocations having been pinned by C (or N), and
the higher stress is required to free pinned dislocations that have been anchored
by interstitial atoms. As will be detailed below, the avoidance of the formation of
stretcher strains can be achieved by applying a small prestrain shortly before the
fabrication, but somewhat beyond the lower yield point (e.g., by skin rolling) and
only then starting the actual fabrication of a mild steel part (e.g., to be used in
the automobile industry). After reloading, no Lüders bands appear, since the sharpyield drop and the low yield-point elongation have been destroyed by the prestrain
(or prefabrication straining). Adding Ti or similar elements, which react with either
C or N or both, is another method for avoiding the appearance of stretcher strain
marks.
3.6.4 Stain Aging
Schematic Fig. 3.17 illustrates strain aging in mild steel. The yield phenomenon indicated by a sharp yield-point drop is shown in region I. Following inhomogeneous
Tensile stress
3.6 Yield Phenomena
167
I
II
III
Tensile strain
Fig. 3.17 Strain aging. In region I, a sharp yield drop is seen with Lüders bands formation. In
region II, no yield drop is seen. The reappearance of a sharp yield drop in region III is also shown
deformation (represented by lower yield-point elongation) and after Lüders bands
have formed and propagated, and passing it by some small deformation, if one
interrupts the tensile test by unloading the specimen, one can observe an interesting
behavior having practical applications in the automotive industry. If this specimen
is reloaded immediately – or after the elapse of a short time – following unloading,
the yield-point drop does not reappear and the Lüders bands do not form. A smooth
transition from the elastic to the plastic regions is the result of the reloading, as
shown in region II of Fig. 3.17. However, if a specimen has been unloaded and is
left for a few days in the unloaded state, then the yield-point phenomenon reappears.
This time, the upper yield point is higher than the one initially observed in that
specimen, as shown in region III of Fig. 3.17. This is explained by the fact that
the interstitial atoms C or N return to the dislocations and form what is called an
‘atmosphere’ around the dislocations re-pinning them. This re-pinning requires the
time-dependent diffusion of C or N to the dislocations. This may take several days
at room temperature, but, at higher temperatures, the increased diffusion rates of
the interstitial C atoms shorten considerably the time for the yield point return. The
involvement of the element of time in yield-point return also clarifies the use of
the term ‘aging’ (strain aging), which is appropriate, since the process of straining
also involves a temperature factor. That the new yield point is higher than the one
initially observed is associated with the fact that some work hardening occurred in
region I before unloading the specimen.
The C or N atoms occupy the octahedral interstitial, distorting the unit cell
tetragonally (i.e., into a body-centered tetragonal structure) and a large volume
expansion occurs.
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3 Plastic Deformation
Fig. 3.18 The Portevin
Le Chatelier effect. The
influence of temperature
on the appearance of a
stress–strain curve under
tension
273 K
~273 – 373
~ 473 K
σ
~ >600K
ε
3.6.4.1 The Portevin Le Chatelier Effect (PLE)
Another observed phenomenon, closely related to the yield-point phenomena
associated with interstitial atoms, is that of temperature-dependent serrated curves.
Serrated stress/strain curves are similar to those observed in twinned specimens,
but their origin is different, as mentioned. In Fig. 3.18, a set of curves is shown
along with the effect of temperature on their appearance. Such behavior, although
observed initially in ’ Fe, is exhibited by several materials as they undergo plastic
deformation, e.g., Al-Cu alloys (see, for example, Liang et al.), substitutional
Al-2.5% Mg alloy (P. Barat et al.), etc. For the Portevin Le Chatelier effect
[henceforth: PLE] to occur, solute atoms must segregate at the dislocation core.
This requires sufficient mobility by diffusion of the segregated atoms. The local
site of the dislocation core is energetically favorable, since it has space available
to accommodate the solute atom which locks the dislocation, hindering its motion.
A larger force (stress) is necessary to move the dislocation, as the cloud of solute
atoms is dragged with it. At some stage, the dislocation eventually breaks away
from the atmosphere of solute atoms, resulting in reduced drag stress for dislocation
movement. Note that this process results in the formation of stretcher strains,
mentioned above, which make the surface rough, prohibiting the use of that material
in the automotive industry, unless remedied.
Many interstitial and substitutional alloys exhibit repeated yield-stress drops
under tension, as influenced by strain rate and temperature. At sufficiently high
temperatures and at a specific strain rate, these serrations gradually disappear, due to
the relatively higher diffusion of the solute atoms, preventing the recapture of their
atmosphere. The effect of temperature is also shown in Fig. 3.18.
Not only mild steels used in the automotive industry show Lüders bands as
surface markings (see Fig. 3.19) or serrations resulting from the PLE effect, but
other materials do as well. Al-Mg alloy sheets, potentially useful for automotive
3.6 Yield Phenomena
169
Fig. 3.19 Lüders markings on soft steel: (a) a tensile specimen; (b) a stamped sheet part (From
Chadwick and Hooper 1953; B. B. Hundy 1953)
applications, may also show yield-point related phenomena, as seen in Fig. 3.20. In
(a) the surface markings are illustrated and in (b) serrated curves are illustrated.
3.6.5 The Cottrell-Bilby Theory
Cottrell and Bilby did a theoretical study on yield-point phenomena (discussed in
Sects. 3.6.2, 3.6.3, and 3.6.4). Their original concepts, regarding the dependence of
the yield drop on impurities, discontinuous yielding and the strain aging associated
with BCC iron, are based on dislocation locking by interstitial solutes (impurities).
Various BCC metals and other structures may also show similar occurrences, but the
first and most obvious encounter with yield phenomena occurred in ’ Fe. Cottrell
and Bilby invented the concept of ‘atmosphere’, known ever since as ‘Cottrell
atmosphere’, in order to explain how dislocations are anchored by interstitials,
specifically C and N. These interstitial atoms settle down under the dislocation line
to form an atmosphere. These atoms somewhat distort the lattice by forming a stress
field around themselves and their own vicinity. This distortion may be reduced or
relaxed once they settle down beneath the dislocation line. The preferential location
of the C (N) atom below the dislocation is controlled by diffusion, which are timeand/or temperature dependent. Usually, after fabricating an iron billet (or some other
shape), time goes by before it is used or tested. This elapsed time is sufficient for the
C to diffuse into a dislocation and become mutually anchored there. Once an atom
or atoms have diffused into a dislocation core, they will stay there. The dislocation
has become pinned down and now, in order to tear such a dislocation away from its
atmosphere, a somewhat higher force is required. This extra force is responsible for
the upper yield point. After unpinning a dislocation, it is free to move at a lower
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3 Plastic Deformation
a
x15
x7
b
a)r=5-20%
1
b)r=30-70%
a)r=5-10%
2
b)r=15-70%
F
a)r=5-20%
3
b)r=30-70%
-3 -1
2 ε2=6.7-10 s
-2 -1
3 ε3=6.7-10 s
1KN
1 ε1=6.7-10-4s-1
1mm
Δl
Fig. 3.20 Al-Mg alloy: (a) the surface appearance of a fender made of Al-Mg sheets; (b) load
extension curves of AlMg6.5Mn sheets at different strain rates (Romahanji et al. 2004. Courtesy
of Dr. Marija Korać, Technical Editor, metalurgija.Org., rs)
stress, producing the lower yield point. Once a dislocation is free to move, the
material can continue to deform plastically in the usual way, i.e., the stress increase
will produce some strain in accordance with the work-hardening concept.
Leaving a sample to age at room temperature for a few hours enables the
carbon atoms to re-diffuse into the dislocation cores, resulting in the return of the
upper yield point, as strain aging (region III in Fig. 3.17). Contrary to this, if a
test specimen is pulled by tension immediately following unloading or within a
short time afterwards, the yield point does not reoccur (region II, Fig. 3.17). The
automotive industry is aware of this phenomenon and construction parts are shaped
shortly after some prestrain is applied. Furthermore, as mentioned earlier, elements
such as Ti, may be added to react with the interstitial atoms, such as N, in order to
prevent the formation of Lüders bands. Cottrell atmospheres lead to the formation of
Lüders bands, where the serration is believed to be the result of repeated yielding.
Dislocations are repeatedly released from and recaptured by anchoring interstitial
atoms.