Dislocation

History of a concept

The paradox of the deformation

It is clear that the plastic deformation requires an important rearrangement of the matter. However this situation seems paradoxical in metals, where the internal structure (that of a Cristal where the atoms are distributed on a three-dimensional periodic network) must be preserved in spite of the changes in external form. The most intuitive way to imagine the deformation is to consider that it proceeds by a series of elementary slips along atomic plans, with the manner of the sheets of a paper oar which slip the ones on the others. This is called a Cisaillement. The metallurgists of the beginning of the century had understood that the deformation was carried out by such “slips”. They had noticed in particular that the fragile crystals broke along certain plans by forming quite particular facets, called surfaces of Clivage, and that one observes so easily in the minerals (quartz, Diamant…).

When a crystal is deformed, one can observe under certain conditions of small steps on their surfaces. It is easy to understand that when one makes slip a plan of atoms one on the other (one speaks about a slip surface), one creates a shift which forms a walk on the surface. It is noted that in this process, the structure of the crystal remains preserved if shearing is a multiple of the period of the crystal lattice.

If one calculates the constraint necessary to shear a perfect crystal (without defects), one realizes that it would be of 1000 to 10.000 times the real constraint observed. If such a perfect crystal existed, one could suspend a car with a wire of Acier 10 mms in diameter!

The explained enigma

In the Years 1930, Orowan, Polanyi and Taylor proposed that shearing could occur by the propagation of called elementary linear defects dislocations.

Let us suppose that an elementary shearing of a interatomic distance B occurs only along part of the plan of shearing. The line which separates the part which was sheared of that which is not it, is the line of fracture. It seems here the limit of an atomic plane half which strongly distorts the close plans.

Although observed in the Liquid crystals at the beginning of the century per G. Friedel, it will have been necessary to await the Années 1950 and the invention of the Electron microscope in transmission to observe them in metals.

Model dislocations

The concept of dislocation in a “continuous” medium is well-known since work of the mathematician Volterra at the beginning of the 20th century. The construction known as “of Volterra” makes it possible to create a dislocation formally. It consists:

initially to cut a volume according to an unspecified surface being pressed on a line L ;
then to move one of the lips of cut compared to the other according to a vector $\ vec {B}$ (called vector of Burgers ); in the crystalline solids (a discontinuous medium), this vector is always a translation of the network;
finally, in a last stage, it acts to restick the two lips of the cut and to release the constraints necessary to displacement.

When this displacement is apart from the plan of cut, the sticking together requires the matter addition.

This construction leads to the formation of a linear discontinuity (purely elastic in a continuous medium) bordering surface. Dislocation thus created is defined by the geometrical position of the line L and the force necessary to the relative displacement of the two lips. It does not depend on the position of the surface of cut. In a discontinuous medium, the line " L" mark the " cœur" dislocation. In this area, the displacement of the atoms of their initial position cannot be defined by an elastic strain.

The line of fracture cannot stop inside the crystal but must either emerge on an imperfection (surface, grain boundary, another dislocation) or to be even closed again on it. Two particular cases of rectilinear dislocations are interesting: dislocation corner ( $\ vec {B}$ perpendicular to $\ vec {U}$ , unit vector of the line L) and dislocation screw ( $\ vec {B}$ parallel with $\ vec {U}$ ).

Dislocation corner

It can be visualized easily if one carries out the process of Volterra while inserting an additional atomic half-plane in the perfect structure, with the way in which one would insert a corner in a piece of wood.

This mode of adaptation is used by certain Plante S when parallel lines follow a variable form of width, such as for example the lines of grains on ears of Maïs or the lines of needles on a Cactus.

In a dislocation corner, the force is perpendicular to dislocation

Dislocation screw

Dislocation Screw car its name owing to the fact that each point on the “plan” atomic perpendicular with the line of fracture goes up of a step equal to $\ vec {B}$ with each turn of a trajectory which rolls up dislocation. The topology of the stress field around dislocation is thus that of a Hélice, or, if one makes it tower of the line while jumping of atom in atom, one assembles $\ vec {B}$ when a turn is made.

Real dislocation

In the general case, dislocation is known as mixed, where the vector of Burgers $\ vec {B}$ and the unit vector $\ vec {U}$ of the line form an unspecified angle.

The line of fracture is in a general way curves, the dislocation present in fact of the portions at mixed character.

Movement of dislocations

There exist two types of movement:

slip, corresponding to the movement of dislocation in the plan defined by its vector of Burgers and the direction of its line
rise corresponding to the movement apart from the slip surface.

Slip

This movement is known as " conservatif" because it does not require transport of matter. It is carried out gradually by the rupture and the sticking together of the atomic connections to the way in which one would make slip a double zipper. This type of movement is particularly effective to propagate the deformation, and generally occurs without another energy contribution that a low external pressure. In a coloured way, one thinks very well that it is easier to trail a carpet on the ground while making propagate a series of small bumps rather than to draw the unit from the carpet.

Rise

To move a dislocation apart from its slip surface, it is necessary to move atoms on long distances: the process is nonconservative and takes place thanks to the diffusion of the gaps or interstitial atoms in material towards the heart of dislocation. As the quantity of the gaps/interstitial and their diffusion is a process thermically active, the rise generally appears at high temperature.

Properties of dislocations

Vector of Burgers

The vector of Burgers is defined as being the vector necessary to buckle initially completed in the perfect crystal and which is open when it intertwines the line of fracture. This vector is not unspecified in a crystal but represents a translation of the network. For example in cubic aluminum centered faces, the vector of Burgers traditionally met is b= A/2, of standard |B|= 0.29 Nm. In more mathematical terms, it is about the integral of displacement on a closed circuit intertwining the line of fracture U: b= int of

Physically, the vector of Burgers represents the amplitude of the deformation transported by a dislocation. As dislocations are flexible objects, two dislocations can interact to form the third dislocation if and only if the quantity of deformation is preserved: one speaks about gravitational junction. It follows that with a node between several dislocations, the sum of the vectors of Burgers is null (analog of the law of Kirchhoff).

Elastic stress field

As an isolated dislocation is an elastic singularity, it develops a stress field with long distance, in the same way that an electron is surrounded by an electromagnetic field of infinite range. In the case of a dislocation screw, it is form: sigma m. driven B/2 pi R (it is actually about a tensor whose only nonnull components correspond to pure shearings in the radial plans parallel with U and in the horizontal plans perpendicular to a ray around dislocation)

It is seen that it is proportional to the vector of Burgers (what is similar to the load electric), with the modulus of rigidity (similar to the electric permittivity of the medium), and inversely proportional to the distance. Thus, a dislocation can be seen like one elementary quantum of deformation.

Interaction with an external pressure

As dislocations have an elastic field, they can interact with an external field.

Interaction with another dislocation

Interaction with the network

The crystal lattice being periodic, there are positions where dislocation has an elastic energy more important than others. The displacement of dislocation requires to overcome these “energy barriers”; there is thus a phenomenon similar to the Frottement. This induced force of friction is called “force of Peierls-Nabarro”.

In fact, when a metal undergoes a plastic deformation, it warms up.

Interaction with specific defects

Dislocations attract the atoms not forming part of the network (foreign atoms: alloy impurities or elements). If these foreign atoms are mobile, they migrate towards dislocations and constitute a “cloud of Cottrell”. This cloud of Cottrell obstructs the movement of dislocations, this explains why pure metals are more ductile than allied metals.

When the force of deformation (the Forced ) is sufficient to tear off dislocation with its cloud, mobility increases suddenly; this explains the setback observed sometimes on the traction diagrams (see the mechanical article Essai ).

If the atoms are mobile (sufficient temperature to allow the diffusion) and that dislocation does not move too quickly (speed of moderate deformation), the atoms can join dislocation and pin it again. One thus notes oscillations on the traction diagram, it is the “phenomenon of Portevin-Lechatelier”.

When dislocation is strongly pinned on motionless atoms, only the central part will move, she thus will curve herself. If it is curved until its branches touch, it is formed a circular dislocation which will move freely. One thus has a phenomenon of multiplication of dislocations, the “mechanism of Frank and Read”, which explains the work hardening.

Dislocation and polycrystal

Dependence of the yield stress with the crystallite size.

Interaction with precipitates

Two different cases can occur when a dislocation meets a precipitate and thus tries to shear it to pass with-through:

1.Le precipitate is sufficiently small to be able to be sheared by dislocation. However, the precipitate will exert a force of recall on the dislocation which tries to shear it while moving and this force will be all the more large as the precipitate is large. This force is in particular due to the stress field surrounding the precipitate because of its non-homogenity with the matrix. So dislocation will have more and more evil to progress and will become deformed more and more. To with difficulty dislocation will pass through the precipitate, with hardest will be the material concerned since it is precisely the difficulty of movement of dislocations within a material which is responsible for its hardness.

2.Le precipitate is too large that so that dislocation can shear it. In this case, dislocation will be closed again more and more on itself while trying to shear and circumvent the precipitate until being closed again completely, thus forming a new line of dislocation in front of the precipitate while leaving a small dislocation around the precipitate: it is what is called the mechanism of Orowan and it is one of the mechanisms of multiplication of dislocations. In this case, the hardness of the sample will decrease with the size of the precipitate.

In short, one can say that the hardness of material initially will increase with the size of the precipitates then to decrease, while passing by a maximum, a peak of hardness corresponding in a state of material says T6 state. In general, one will support the samples comprising of the precipitates slightly larger than those which one would find with the T6 state to avoid any mechanism of shearing, which would cause to reduce the size of the precipitates, and thus to reduce the hardness of material.

In the case or the sample contains several families of precipitates of different sizes, two for example, the hardness of material will depend on the relative size and the position of the peaks of hardness corresponding to each family of precipitates:

If one of the two peaks is clearly smaller than the other (i.e: one of the two curves of hardness is always lower than the other), the hardness of material will be that which is given by the highest curve. So on the other hand the two peaks are close height but correspond to various diameters of precipitates, hardness will follow the envelope of the two curves of hardness, i.e. it will take the maximum of the two values given by the curves for each diameter of precipitates. There will be then a total curve of hardness divided into 4 phases: increasing (mechanism of shearing) to the first peak, then decreasing (mechanism of Orowan) until the intersection of the two curves; again increasing to the second peak and finally second decreasing once.

Arrangement of dislocations

See too

Restoration

External bonds

Plastic deformation and dislocations
Of the vermiculations in metals, one trotts in the world of the plastic deformation

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