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Appendix D
Ductile Materials Notes
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The plastic behaviour of ductile metals
Recap on terms:
Elastic deformation is a change of size or shape which is reversed when the deforming stress
is removed.
If an object undergoes plastic deformation it doesn’t return to its original size or shape when
the stress is removed, but suffers permanent set.
The key to understanding the behaviour of ductile metals at stresses above the elastic limit
lies in the almost perfectly regular nature of their structure. They are crystalline metals; that is
they possess long-range order, consisting of metal ions arranged in flat planes. Each ion is
surrounded by the same number of ions as all the others, usually 12, in an arrangement
known as close packing.
A two dimensional representation would be:
In this diagram, the black circles represent ions of the lattice, and each is surrounded by 6
neighbours in the plane, and this is just part of one of the planes of ions, with other planes
situated above and below it. The diagram below shows a (grey) plane of ions below.
I’m not going to attempt adding a layer above!
In these diagrams, the size of the ions should not be taken too seriously. It would be better to
think of them as touching one another, but the diagrams are more difficult to cope with. The
ions are bonded into the lattice by bonds between them. [If you’ve covered PH2 already, you
may know that the “glue” that holds them in place comes from free conduction electrons
which move around in the lattice, but that is an issue for another day!]
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In the rest of this article, we are going to be thinking more about the bonds between the atoms
than about the atoms themselves, so it will be better to represent the planes of atoms like this:
We are not going to be concerned with the precise atomic arrangement, so we’ll stick to a
square array. The diagrams are easier to draw and to interpret. There are also horizontal
bonds between the ions, but these have been left off to make the diagram simpler.
Polycrystalline materials
This is really a note for the purists. Strictly, we should refer to metals as polycrystalline,
rather than crystalline. They consist of an interlocking mass of individual crystals. The
orientations of the lattice planes in neighbouring crystals are independent of one another.
Plastic ‘slip’
Sometimes textbooks describe plastic strain in terms of crystal planes sliding over one
another.
force
Plane x
Plane y
force
According to this idea, when a force is applied, every atom in plane x has to break bonds with
atoms in plane y and then form new ones with different atoms in plane y. This process does
not happen however because is would requires stresses many times greater than obtained in
practice. Real metals are not as strong as this model suggests. Edge dislocations can account
for this difference.
Edge dislocations
No one puts these crystals together. They grow spontaneously as the molten metal cools
down when it is smelted. Sometimes mistakes happen as the metal ions join the crystal. A
frequent mistake – one every million atomic planes or so – is that half a plane of atoms is
missed out. This is known as an edge dislocation [see next page].
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X
Edge dislocations are the key to plastic deformation. The secret is in the bonds around the ion
X. These bonds are all under strain, so they are places of weakness. Suppose forces are
applied to this region as in the diagram below:
F
X
Plane x
Plane
y
F
If the forces, F, are small, the horizontal bonds are stretched reversibly [if the force is
removed, they contract again] and the material behaves elastically.
If the forces become larger, the already strained bonds below and to the right of X [shown in
red] are stretched even more and at some point, the Yield Point of the material, they snap
making the dislocation migrate to the right:
F
X
F
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If the same force that causes this movement continues to be applied, the dislocation carries on
moving, through the snapping and reformation of bonds, until it reaches the edge of the
crystal.
In this way, the dislocation has moved from left to right through the crystal. It appears as if
plane x and plane y have slipped over each other. However, it has been achieved more easily
since only one (short) line of bonds, rather than a whole plane, has been broken at a time.
The crystal is now more elongated in the horizontal direction. If the forces are removed, the
dislocation does not move back as there is no force to make it do so, and so the deformation
is permanent.
In the plastic region of deformation, as the bonds are not being stretched, the material does
not increase in volume as it stretches. It contracts laterally. This “necking” can be seen
dramatically if the break point in a ductile fracture is examined. Signs of flow, rather like a
fluid, can be seen around the break point.
In the diagram, the crystal does not look dramatically longer than it did before, but imagine
the effect of a large number of edge dislocations at the top of the crystal migrating to the right
and a similar number of edge dislocation at the bottom moving to the left. The crystal is more
elongated (in the horizontal direction).
And finally [not for examination]
Of course, this isn’t the whole story. With a small amount of increased stress, new edge
dislocations can also be created; especially in locations where there is an atom missing from
the lattice [it happens!] or an impurity atom has a different size from the others. The extra
edge dislocations can produce more and more plastic strain as they migrate. For a fuller
treatment see Gordon.
Reference: THE NEW SCIENCE OF STRONG MATERIALS
Why You Don't Fall Through the Floor
by J. E. Gordon
London, 1976.
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GCE Physics – Teacher Guidance
4 December 2007