-138-
Lecture 18
Strain Hardening And Recrystallization
Strain Hardening
We have previously seen that the flow stress (the stress necessary to produce a
certain plastic strain rate) increases with increasing plastic strain as
m
σt = κ' εt
18.1
Now we realize that the plastic strain is produced by the motion of dislocations, we
should examine the microscopic basis for this strain hardening behavior. A logical
first step is to examine the dislocations in a highly strain hardened metal. We can
do that (at least in thin foils of metal) using the transmission electron microscope
(TEM).
Please see:
http://www.mrl.ucsb.edu/~edkramer/LectureVGsMat100B/99Lecture18VGs/CellSt
ructureVG.html
Figure 18.1
(Courtesy C. Cho)
A typical TEM micrograph of a heavily deformed metal (in this case Al) is
shown in Figure 18.1. One can see very dense regions of dark lines (the
dislocations) surrounding other regions which have very few dislocations. This
structure is called a cell structure and the dense dislocation regions are called cell
walls. These are tangles of immobile dislocations (if they were able to move the
dislocations would glide out of the thin foil). If we want to produce more plastic
strain we must produce fresh dislocation from dislocation sources and push these
through the tangled immobile dislocations of the cell structure. This process
requires more applied stress (more force on the dislocation) than to push a fresh
dislocation through a region free of dislocation tangles. The (immobile)
dislocation density in the cell structure increases with plastic strains. In fact below
a certain strain dense tangles are not formed. An example of this increase in
dislocation density with strain is shown in Figure 18.2.
See
http://www.mrl.ucsb.edu/~edkramer/LectureVGsMat100B/99Lecture18VGs/Disloc
ationCellsTEMVG.html
-139Figure 18.2
From: A.S. Keh in Direct Observations of Imperfections in Crystals ed
J.B. Neukirk and J.H. Wernick (1962, Metallurgical Society AIME, Interscience Pub.) p.216.
The increasing density of immobile dislocations in the cell walls as the plastic
strain increases makes these walls or tangles more effective obstacles to fresh
dislocation motion. Hence the flow stress increases resulting in the observed strain
hardening.
Although many details of the nature of the interactions between the fresh
dislocations and the tangles are obscure (for some examples, see Barrett, Nix and
Tetelman p.270 & 271) the increase in flow stress is predicted to be proportional to
the square root of overall dislocation density ρ (since fresh dislocations are present
at much lower density than the immobile dislocations, ρ ≅ ρimmobile). In fact all
theories predict
σ = σ + α Gb ρ
t
o
18.2
where G is the shear modulus, b is the magnitude of the dislocation Burger vector
and α is a constant which varies from theory to theory but which is approximately
equal to 0.2. Experimentally this relation is obeyed quite well as can be seen from
0.4
x 10
-3
Figure 18.3 which shows typical data for copper.
x
x
0.2
x
x
x
τ
G
x
x
x x
x x x
0
x
x
x
0.2
0.4
b ρ
0.6
-3
x 10
Figure 18.3 Effect of dislocation density on the flow stress of polycrystalline
copper.
-140All that is necessary to finally describe strain hardening then is to know how ρ (the
density of immobile dislocations in the cell structure) builds up with plastic strain
εp. Unfortunately this is difficult to predict since the dislocation tangles form
statistically (especially in single crystals). In polycrystals however (where
multiple slip occurs even at low strains) one finds that
ρ = ρo + κ εp
2m
18.3
If one substitutes ρ from Eq. 18.3 into Eq. 18.2 and makes the approximation that
ρo and σy are small compared to ρ and σt one arrives at the empirical equation for
strain hardening, Eq. 18.1.
A particularly interesting application of these ideas is the prediction of the strain
hardening behavior of metals with hard uncuttable particles. We learned in Lecture
13 that dislocations circumvent such particles by extruding between them, leaving a
loop of dislocation around the particle. If we push many dislocations through the
array of particles, as we need to do to achieve large plastic strains, we will rapidly
produce a very large immobile dislocation density where the dislocations are loops
around the particles. One would predict that the metal with hard particles would
have a much larger strain hardening rate (higher) than the metal without particles.
The experiments (Fig. 18.4) show exactly the predicted behavior.
300
Shear
Stress
(MN/m
200
2
)
100
20
40
60
80
100 %
Shear Strain γ
Figure 18.4
Strain hardening then is the hardening of the metal by immobile dislocation
debris left as a result of plastic strain. In many pure metals it is the most important
way to harden the material. On the other hand there are many instances where
-141strain hardening is a nuisance or worse. For example, in attempting to roll a bar of
metal into a thin sheet the metal may become so hardened that cracks develop in the
outside edges of the sheet. In such situations one would like to periodically soften
the metal as the sheet is rolled thinner and thinner, i.e., one wants to remove the
strain hardening. To do that it is necessary to remove the immobile dislocation
debris. The primary way of removing the dislocation debris is by recrystallization.
Recrystallization
Not only can one remove the dislocation tangles built up in the course of strain
hardening by the process of recrystallization but one also can use it to manipulate
and control the grain size of the metal. The grain size is defined as the average
diameter of the crystallites or "grains" in the polycrystal. One usually desires a
small grain size. Small grained samples have higher yield stress σy than large
grained samples. In steels the ductile-to-brittle transition moves to lower
temperatures (the steel becomes more ductile at room temperature) as the grain
size is decreased. A large grain size leads to a large scale rumpling of the surface
("orange peel" effect) if the material is to undergo further plastic deformation
during fabrication. Only for certain high temperature creep applications does one
want a large grain size (in fact in these one would like a single crystal).
Recrystallization takes place on annealing at elevated temperatures by the
nucleation of new nearly dislocation-free grains in the dense dislocation tangles of
the cell walls. The nuclei of these new grains are very small (often <100Å
diameter) and the microscopic processes by which they form are not well
understood. What is known is that they form in the densest parts of the dislocation
tangles and that the driving force for their formation (nucleation) is the very high
(local) strain energy in the dislocation tangle. [Remember the strain energy of a
dislocation per unit length is ~Gb2 (its line tension); if there is a high local ρ, there
is a high local strain energy.] That strain energy is wiped out if a new dislocationfree grain is formed. It is known that the nuclei do not form by a melting and
refreezing as implied by the word "recrystallization", but rather they form entirely
in the solid state, for example, by a shuffling of atoms across an existing grain
boundary.
The number of nuclei formed per unit volume is directly proportional to the
number of high strain energy (very dense) dislocation tangles per unit volume. In
-142fact recrystallization will not take place at all until such dense tangles are formed.
Thus a metal given no, or only a small plastic strain, will not recrystallize. It would
not be possible to recrystallize the metal in Fig. 18.2a whereas in Fig. 18.2d where
dense tangles are present recrystallization is possible.
The fact that the number of new grain nuclei is proportional to the number of
dense dislocation tangles is what allows us to control the final grain size of the
metal after recrystallization is complete. After nucleation the new grains grow in
size by migration outwards of the grain boundary surrounding the new grain. As
the boundary moves outwards (actually by the shuffling of atoms across the grain
boundary it destroys the high strain energy dislocation debris structure and replaces
it with a low strain energy nearly dislocation-free material resulting from the grain
boundary migration). After a certain time however, new grains growing from
different nuclei eliminate all the old high dislocation density metal between them
and impinge, forming a grain boundary between two strain free grains. When this
impingement has occurred everywhere in the sample recrystallization is over.
There is no more dislocation debris to cause either nucleation of new grains or
migration of grain boundaries. (But see the process of grain growth, next lecture).
Since each new grain nucleus ends up as a larger, strain free grain in the final
recrystallized grain structure, the more nuclei we have to begin with, the smaller the
size of the final grains will be. Thus if we want a large grain size we plastically
deform to a strain where only a few very dense dislocation tangles have formed so
that only a few new grains nucleate. The final recrystallized grain size will thus be
very large. [It is even possible, but tedious, to grow a single crystal this way by
knowing there is only one dislocation tangle in the entire sample capable of
nucleating a new grain]. Conversely if one wants a very small grain size, one
deforms to very high plastic strains where the number of dense dislocation tangles
per unit volume is very high. The number of nuclei of new grains is very large and
when they finally impinge, the grain size will be very small. Fig. 18.5 shows the
recrystallized grain diameter vs. prior plastic strain for a brass. Notice that at low
prestrains there is no recrystallization and one has the original grain structure of the
metal. It is possible to produce grain sizes larger than or smaller than the original
grain size by suitable plastic prestrain.
Recrystallization grain si ze, mm
-1430.12
Initial grain size, 0.5 mm
0.06
Initial grain size, 0.05 mm
0
20
40
60
80
% deformation, rolling
Figure 18.5
After S. Channon and H. Walker, Trans. Am. Soc. Metals, 45, 200 (1953).
Kinetics of Recrystallization
From optical microscopy one can determine the percentage of the material that
is recrystallized (that exists as new, rather than old, high dislocation density,
grains.) Fig. 18.6 shows micrographs as various percentages of recrystallization.
Figure 18.6
From: Nes and Ryum, Acta. Met. 23 979 (1975)
The fraction recrystallized increases as a function of time at a given T as shown
in Fig. 18.7. The process may take as much as 2 decades of time from start
(nucleation) to finish (impingement) and the result is a characteristic sigmoidal (S
shaped) curve. If one wanted to characterize the kinetics of the recrystallization by
Recrystallization (%)
-144a single number one can use the time to 50% recrystallization, t50%.
100
50
Time for 50%
recrystallization
0
{
Incubation
period
Time of Annealing
Figure 18.7
If we anneal at different temperatures we will find that the sigmoidal curve
shifts to shorter times at higher temperatures as shown in Fig. 18.8. If we plot log
t50% from Fig. 18.7 vs. 1/T where T is the absolute temperature we find that the
data are described by a straight line (Fig. 18.9). One can use that straight line
relationship to extrapolate to conditions where experiments are impractical. For
example one can use Figure 18.8 to predict that very pure copper will be 50%
recrystallized at 24° F (–3° C) in 25 years. Processes or mechanisms which give
kinetic laws where log rate or log time varies as 1/T are said to be thermally
activated. (Diffusion is another such process.)
Recrystallization, %
275°F
235°F
191°F
109°F
100
50
0
10
100
1000
Time of Heating, min.
Figure 18.8
10000
-145-
Temperature, °F
39
24 °F
103
25 years
183
290
1
100
10,000
1,000,000
Time for 50% Recrystallization, mi n.
Figure 18.9 Isothermal recrystallization of 99.999% pure copper. (After Decker and
Harker.)
One should mention that if one looks in handbooks (such as the Metals
Handbook) one will find for various metals something called the "recrystallization
temperature". Clearly from the example above there is no such thing since if we
wait long enough we can get recrystallization at almost any annealing temperature.
However the Metals Handbook is designed for practical engineers and practical
engineers do not like to anneal anything for much more than one hour. The
recrystallization temperature TR is defined as the temperature at which T50%, the
time for 50% recrystallization, is one hour. For the example of Cu used above, TR
is about 240° F (116° C).
A number of other material variables affect the recrystallization kinetics and
thus the recrystallization temperature. Small amounts of trace impurities can
drastically increase the TR since they slow down outward growth of the grain
boundary surrounding the new dislocation-free grain. For example TR for
99.9999% Al (less than 1 part per million impurities) is –50° C whereas
commercially pure Al (99% pure) has a TR of 250° C ! The amount of prior
plastic strain also affects the TR (can you think why?) giving rise to a decrease in
TR with prior plastic strain as shown in Figure 18.10.
-1461200
T
R
(°F)
900
600
99% Al
20
40
60
80%
Plastic Strain
Figure 18.10 Effects of the Amount of Cold Work on Kinetics - the greater
the amount of cold work, the lower the TR on the lower t50% at a given T.
Just as strain hardening can produce large increases in the flow stress,
recrystallization annuls these increases. Starting with a highly strain hardened
metal (well-developed dislocation cell structure) the decreases in flow stress (or
hardness) as a function of annealing temperature are shown schematically in Figure
18.11. (Specimens are given a one hour anneal at each temperature before testing).
There is a small decrease in flow stress below temperatures where recrystallization
begins. This decrease is due to processes of dislocation rearrangement and a small
amount of dislocation annihilation in the cell walls, processes which are called
recovery processes. The major decrease in flow stress, to approximately its value
before plastic deformation, occurs as a result of recrystallization.
recovery
recrystallization
begins
True Flow
Stress
σt
original flow stress
before strain hardening
microstructure
is fully recrystallized
T
R
Annealing Temperature
Figure 18.11