Metals and alloys:  Relationship between elastic deformation and dislocation motion

This relationship had already been hypothesised in the 1930s, when dislocations had only been hypothesised (transmission electron microscopes did not exist). Later, the observation of physical dislocations in the 1950s confirmed this hypothesis.

A dislocation is an additional half-plane within a perfect crystal, whose crystalline planes could only slide over each other by breaking a large number of strong bonds. In the case of real defective crystals (with dislocations), a much smaller number of atomic bonds must be broken to make the additional half-plane slide. This is the reason why plastic deformation is favoured within a defective real crystal compared to a perfect crystal. The theoretical strength of a deformation-free perfect crystal is extremely greater than the strength of a real crystal.

 This has an impact from a production-technological point of view: what we call defects (dislocations) do not have a negative connotation, in fact the presence of dislocations is an advantage. Take metals, for example: many metalworking techniques involve both cold and hot deformation. A rolling process involves thinning the thickness of a metal sheet by passing it between two rollers: this thinning is possible precisely because of the motion of the dislocations and the metal undergoes permanent deformation. With a metal consisting only of perfect crystals without dislocations, rolling would be impossible, since the force required to slide the atomic planes over each other would be too great. 

Dislocations can move along the sliding plane and eventually result in a kind of "step" on the outer surface of the crystal. If many dislocations are moved, however, the effect is a macroscopic and visible deformation.

Analogy between the motion of a caterpillar and dislocation

An analogy can be made between the motion of dislocations and the motion of a caterpillar or of a fold along a carpet: the caterpillar has many legs but moves only a couple at a time (its hump can be assimilated to the half-plane in excess that is moving), therefore the motion of the hump propagates along the whole length of the caterpillar itself, as does the motion of the dislocation from left to right inside the crystal (the latter is deformed by an atomic step). 

The same happens in the case of the carpet: to make it move you can create a hump (dislocation) and make it move along the carpet: in this way the carpet itself will move.

Since the dislocation is an additional half-plane, within the crystal there will be compressive stresses in the portion of the crystal containing the dislocation (the atoms are more compressed), while in the portion of the crystal below the dislocation there will be tensile stress fields (the atoms are displaced). 

These tensional fields can interact with each other, as if they were endowed with a sign: compressive tensional states are endowed with a + sign, while tensile tensional states are endowed with a - sign. Dislocations endowed with the same sign (equal tensional fields) tend to repel each other; conversely, opposite tensional fields tend to attract each other.

In the figure above, there are two dislocations that share a common sliding plane: there are two additional half-planes on the same side, thus compressive and tensile stress fields on the same side. The two dislocations tend to repel each other. 

In the figure below, however, there are two dislocations with the same sliding plane but opposite signs (in the left dislocation the additional half-plane will be above the sliding plane, while in the right dislocation it will be below the sliding plane). Thus there are opposite tensional fields that attract each other: the dislocations cancel each other out. Therefore, when dislocations of opposite sign meet, they will tend to cancel each other out, disappearing.

 Multiplication of dislocations

When stress is applied to a material, dislocations in the material begin to move. What happens when these dislocations encounter obstacles as they move? Obstacles can be other dislocations, inclusions or grain boundaries. In this case the dislocations multiply: the original dislocation has to get around the obstacle, giving rise to two dislocations. If these in turn meet an obstacle, they will each give rise to two more dislocations (1 to 4 dislocations). Therefore, during plastic deformation, as the dislocations move, their number increases significantly. The more the dislocations increase, the more difficult the movement becomes, since they start to interact with each other and counteract each other (dislocations are obstacles to each other). It will therefore be necessary to provide the material with a higher stress in order to allow the dislocations to move; the more the material deforms, the more it will be necessary to increase the external force in order for the material to continue to deform. 

The motion of the dislocations is favoured along those crystallographic directions in which the atoms are closest to each other, called compact directions.

Figure a shows a face-centred cubic (FCC) system in which the plane with the highest atomic compaction is the diagonal plane, so dislocations will move more easily along this plane. 

There may be more than one sliding system (with high atomic compaction) depending on the structure of the crystal lattice. In the case of body-centred cubic (BCC) or face-centred cubic (FCC) systems, there will be many sliding systems, since there are many planes of high atomic compaction. Conversely, in systems such as compact hexagonal I will have few sliding systems. 

Metallic materials are typically polycrystalline: each of the crystals has its own orientation of the lattice planes, which is different from that of the adjacent crystals. The dislocations can move within the single crystal, but when they reach the frontier of the crystal itself (i.e. the grain boundary) they should "jump" into the adjacent grain: this is possible if the adjacent grains are favourably oriented, so the misalignment between the grains must be small. If this is not the case, I can force this "jump" by increasing the external load: deformation cannot occur if the dislocations cannot move from one crystal grain to the adjacent ones.