plasticity of materialsplasticity of materials

7/28/2019 plasticity of materialsplasticity of materials

http://slidepdf.com/reader/full/plasticity-of-materialsplasticity-of-materials 1/9

-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

σt= κ ' ε

t

m18.1

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. Thisstructure 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



-139-

Figure 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 predictσ

t= σ

o+ α Gb ρ

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

Figure 18.3 which shows typical data for copper.

0 0.2 0.4 0.6

0.2

0.4

b ρ

G τ

x 10-3

x 1 0 - 3

x x

x

xx

x

x

x

x

x

x

x

xxx

Figure 18.3 Effect of dislocation density on the flow stress of polycrystalline

copper.



-140-

All 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

ε p. 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 2 m

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

200

100

20 40 60 80 100 %

Shear Strain γ

ShearStress

(MN/m )2

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



-141-

strain 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 largegrained 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 dislocation-free 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



-142-

fact 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 themand 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 therecrystallized 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.



-143-

R e c r y s t a l l i z a t i o

n g r a i n s i z e , m m

Initial grain size, 0.5 mm

Initial grain size, 0.05 mm

% deformation, rolling

0.12

0.06

0

20 40 60 80

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



-144-

a single number one can use the time to 50% recrystallization, t50%.

100

50

0

R e c r y s t a

l l i z a t i o n ( % )

Time of Annealing

Time for 50%recrystallizationIncubation

period {

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.)

R e c r y s t a l l i z a t i o n ,

%

Time of Heating, min.

275°F 235°F 191°F 109°F

100

50

0

10 100 1000 10000

Figure 18.8



-145-

T e m p e r a t u r e ,

° F

39

103

183

290

24 °F

Time for 50% Recrystallization, min.

1 100 10,000 1,000,000

25 years

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 inTR with prior plastic strain as shown in Figure 18.10.



-146-

99% Al

1200

900

600

20 40 60 80%

(°F)

TR

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.

True FlowStress

Annealing Temperature

recrystallizationbegins

microstructure

is fully recrystallized

σt

original flow stress

before strain hardeningT

R

recovery

Figure 18.11

plasticity of materialsplasticity of materials

Documents