Free Shipping Free global shipping No minimum order. Moriarty Dislocations in Silicon at High Stress J. Rabier, L. Pizzagalli and J. Demenet Metadislocations M. Feuerbacher and M. Heggen Dislocations in Minerals D. Barber, H. Wenk, G. Hirth and D. Kohlstedt Dislocations in Colloidal Crystals P. Schall and F. Kink mechanisms for dislocation motion at high pressure in bcc metals Dislocation core structures identified in silicon at high stress Metadislocations, a new type of defect, identified and described Extension of dislocation concepts to complex minerals First observations of dislocations in colloidal crystals.

Powered by. You are connected as. Connect with:. Use your name:. Thank you for posting a review! We value your input. Share your review so everyone else can enjoy it too. Your review was sent successfully and is now waiting for our team to publish it. Reviews 0. Updating Results. If you wish to place a tax exempt order please contact us. The com- putation involved is non-trivial and several methods have been developed to provide computational efficiency see Bulatov and Cai in Further Reading.

The dislocation will want to change its shape if the force on a node is unbalanced. One way of allowing for this is to give the node a veloc- ity which is proportional to the force and inversely proportional to a drag coefficient, as in equation 3. Realism can be introduced by setting the drag coefficient for climb in response to a force component out of the glide plane to be much larger than that for glide, and allowing the coefficient for glide to vary with dislocation line direction between the edge and screw orientations. The effects of temperature can also be incorporated with a temperature-dependent drag coefficient and by probability-determined wait- ing times for thermally-activated processes section The other important issue is concerned with the density of nodes that define the dislocation line.

Computational efficiency requires the number of nodes in the system to be as small as possible, but accurate representation of the line shape is achieved if the nodal density is large. It is clear from Fig. Thus, the computer code has to be able to add and remove nodes in response to shape changes of disloca- tions during a simulation. Furthermore, segments on the same or different dis- locations can react to either annihilate or form segments with a different Burgers vector, and again nodes have to be added or removed.

With permission from Oxford University Press www. However, dislocation dynamics has more complexity. First, the response of a node depends on the direction of the force on it. Second, the number of atoms is constant in molecular dynamics, whereas the chang- ing number of nodes in dislocation dynamics requires that the inventory of nodes, nodal coordinates and Burgers vectors be updated after every time- step.

The choice of boundary conditions can also be problematic. Articles listed under Further Reading provide more detail. Dislocation dynamics has been applied to situations where either a few or many dislocations are involved. Interactions between two dislocations that are important in work hardening section As an example of the latter, Fig. The model has axes in , The initial dislocation arrangement shown in Fig. The large increase in dislocation density is apparent. The com- puting demands of this simulation were such that it used a parallelized code on computer clusters in which the number of processors increased from 16 to as the dislocations, and therefore number of nodes, multiplied.

The plots of stress and dislocation density versus strain for this simulation are described in section Humphreys CJ: Imaging of dislocations. Cherkaoui M, Capolungo L: Atomistic and continuum modeling of nanocrystalline materials: deforma- tion mechanisms and scale transition, , Springer.

Frederiksen SL, Jacobsen K: Density functional theory studies of screw dislocation core structures in bcc metals, Phil Mag , Guo ZX, editor: Multiscale materials modelling: fundamentals and applications, , Woodhead Publishing. Phillips R: Crystals, defects and microstructures: modeling across scales, , Cambridge University Press. Glide or conservative motion occurs when the dislocation moves in the surface which contains both its line and Burgers vector: a dislocation able to move in this way is glissile, one which cannot is sessile.

Climb or non-conservative motion occurs when the dislo- cation moves out of the glide surface, and thus normal to the Burgers vector. Glide of many dislocations results in slip, which is the most common manifesta- tion of plastic deformation in crystalline solids. It can be envisaged as sliding or successive displacement of one plane of atoms over another on so-called slip planes. Discrete blocks of crystal between two slip planes remain undis- torted as illustrated in Fig.

Further deformation occurs either by more movement on existing slip planes or by the formation of new slip planes. The slip planes and slip directions in a crystal have specific crystallographic forms. The slip planes are normally the planes with the highest density of atoms, i. Often, this direction is one in which the atoms are most closely spaced. In body-centered cubic metals the slip direction is the hi close-packed direction, but the slip plane is not well defined on a macro- scopic scale.

A slip plane and a slip direction in the plane constitute a slip system. These are readily detected if the surface is carefully polished before plastic deformation. Figure 3. Slip plane A characteristic shear stress is required for slip.

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Consider the crystal illustrated in Fig. From Hull, Proc. A, 5, Each band is made up of a large number of slip steps on closely spaced parallel slip planes. Consider the edge dislocation represented in Fig. This could be formed in a different way to that described in Chapter 1, as follows: cut a slot along AEFD in the crystal shown in Fig. Although it is emphasized that dis- locations are not formed in this way in practice, this approach demonstrates that the dislocation can be defined as the boundary between the slipped and unslipped parts of the crystal.

The distortion due to the dislocations in Figs 1. These defects are Volterra dislocations, named after the Italian mathematician who first considered such distortions. More general forms arising from variable displacements are possible in principle, but are not treated here. Only a relatively small applied stress is required to move the dislocation along the plane ABCD of the crystal in the way demonstrated in Fig.

This can be understood from the following argument. Near the disloca- tion line itself, some atom spacings are far from the ideal values, and small relative changes in position of only a few atoms are required for the disloca- tion to move. For example, a small shift of atom 1 relative to atoms 2 and 3 in Fig.

Two neighboring atoms say 1 and 3 on sites adjacent across the slip plane are displaced relative to each other by the Burgers vector when the dislocation glides past. Thus, the slip direction see Fig. The glide of one dislocation across the slip plane to the surface of the crystal pro- duces a surface step equal to the Burgers vector. Each surface step produced by a slip band in Fig. The plastic shear strain in the slip direction result- ing from dislocation movement is derived in section 3.

This is uniquely defined as the plane which contains both the line and the Burgers vector of the dislocation. LMNO, and a slip step is formed. M However, the line of the screw dislocation and the Burgers vector do not define a unique plane L and the glide of the dislocation is not restricted to a specific plane. This can be dem- onstrated further by considering a plan view of b the atoms above and below a slip plane contain- ing a screw dislocation Fig.

Movement of the screw dislocation produces a displacement b parallel to the dislocation line. The direction in which a dislocation glides under stress can be determined by physical rea- Axis of screw soning. Consider material under an applied dislocation shear stress Fig. From the description of section 3.

It is seen from Figs 3. A right-handed screw glides towards of diagram, filled circles the front in order to extend the surface step in the required manner Fig. These screw. In the examples illustrated in Figs 3. However, dislocations are generally bent and irregular, particularly after plastic deformation, as can be seen in the electron micrographs presented throughout this book.

A more general shape of a dislocation is shown in Fig. The boundary separating the slipped and unslipped regions of the crystal is curved, i. The remainder of the dislocation M has a mixed edge and screw character. The Burgers vector b of a mixed dislocation, XY in Fig.

A screw dislocation at S is free to glide in either of these planes. Cross slip produces a non-planar slip surface. Double cross slip is shown in d. This process, known as cross slip, is illustrated in Fig. Double cross slip is illustrated in Fig. Slip often wanders from one plane to surface of a single crystal another producing wavy slip lines on prepolished surfaces.

An example of of 3. A direct method of measuring dislocation velocity was developed by Johnston and Gilman using etch pits to reveal the position of dislocations at different stages of deformation, as illustrated in Fig. A crystal containing freshly intro- duced dislocations, usually produced by lightly deforming the surface, is subjected to a constant stress pulse for a given time. From the positions of the dislocations before and after the stress pulse, the distance each disloca- tion has moved, and hence the average dislocation velocity, can be deter- mined.

By repeating the experiment for different times and stress levels the velocity can be determined as a function of stress as shown in Fig. The dislocation velocity was measured over 12 orders of magnitude and was a very sensitive function of the resolved shear stress. From Johnston and Gilman, J. After Stein and Low, J. It must be emphasized that equation 3. The velocity of edge and screw components was measured independently and in the low velocity range edge dislocations moved 50 times faster than screw dislocations. There is a critical stress, which represents the onset of plastic deformation, required to start the dislocations moving.

The effect of tem- perature on dislocation velocity is illustrated in Fig. The curves are of the Cu, Zn same form as equation 3. For a given material, the Ge velocity of transverse shear-wave propagation is 10—7 the limiting velocity for uniform dislocation motion. However, damping forces increasingly KBr Nb 10—10 oppose motion when the velocity increases above LiF b Fe-Si about 10 2 3 of this limit, and thus n in equation 10—11 3.

Studies on 10—2 10—1 10 face-centered cubic and hexagonal close-packed Applied shear stress, MNm—2 crystals have shown that at the critical resolved FIGURE 3. This University Press. When m 5 1, equation 3. In the velocity range where this applies B0. These are less numerous at low temperature and so B decreases. Effects due to phonon damping and the limitation of shear wave velocity have been studied by molecular dynamics computer simulation of dislocation glide section 2. Additional effects such as thermoelastic dissipation and radiative emission of phonons can be important in other materials.

From Khater and Bacon, Acta Mater. B 70, , Copyright by the American Physical Society. Arrows indicate sense of vacancy motion. In b the dislocation is centered on the row of atoms A normal to the plane of the diagram. If the vacancies in the crystal diffuse to the dislocation at A the dislocation will climb in a positive sense as in a.

If vacancies are generated at the dislocation line and then diffuse away the dislocation will climb in the negative sense as in c.

However, at higher temperatures an edge dislocation can move out of its slip plane by a process called climb. Consider the diagram of an edge dislocation in Fig. If the row of atoms A normal to the plane of the diagram is removed, the dislocation line moves up one atom spacing out of its original slip plane; this is called positive climb.

Positive s climb can occur by either diffusion of vacancies to line A or the formation of a self-interstitial atom at A and its diffusion away. Negative climb can occur either by an interstitial atom diffusing to A or the formation of a vacancy at A and its diffusion away. The area of the More generally, if a small segment l of line under- shaded element is jl 3 sj.

The glide plane of the element is by defini- tion perpendicular to b 3 l, and so dV is zero when either s is perpendicular to b 3 l or b 3 l 5 0, Jogs which means the element is pure screw. In both Dislocation cases s lies in the glide plane and this is the condi- tion for glide conservative motion discussed pre- viously. The mass transport involved A pair of jogs on an edge occurs by diffusion and therefore climb requires thermal activation.

It has been implied above that a complete row of atoms is removed simulta- neously, whereas in practice individual vacancies or small clusters of vacancies diffuse to the dislocation. The effect of this is illustrated in Fig. Both positive and negative climb proceeds by the nucle- ation and motion of jogs. Conversely, jogs are sources and sinks for vacancies. Jogs are steps on the dislocation which move it from one atomic slip plane to another. Steps which displace it on the same slip plane are called kinks. The two are distinguished in Fig.

Jogs and kinks are short elements of dislocation with the same Burgers vector as the line on which they lie, and the usual rules apply for their conservative and non-conservative movement. Thus, the kink, having the same slip plane as the dislocation line, does not impede glide of the line. The jog on a screw dis- location Fig. This impedes glide of the screw and results in point defect production a b during slip Chapter 7.

Since Eif is much Burgers vector b larger than Evf in metals section 1. The jogs described have a height of one lat- tice spacing and a characteristic energy EjB1 eV 0. The climb of dislocations by jog formation and migration is analogous to crystal growth by surface step transport. There are two possible mechanisms. In one, a pre-existing single jog as in Fig.

In the other, thermal jogs are nucleated on an otherwise straight line and the jog migration energy is Ed 1 Ej. In most situations the first process dominates. The effective activation energy in some circum- stances can be less than one half the value of Ed measured in the bulk, for the crystal is distorted at atom sites close to the dislocation line itself and Ed is smaller there. Several factors are involved, as discussed in section 4. Pure screw dislocations have no extra half-plane and in principle cannot climb.

However, a small edge component or a jog on a screw dislocation can provide a site for the start of climb. Two examples will serve to illustrate the climb process in both edge and predominantly screw dislocations. If a dislocation forms a loop in a plane and b lies in that plane Fig. When the Burgers vector is not in the plane of the loop, the glide surface defined by the dislocation line and its Burgers vector is a cylindrical surface Fig. The dislocation is called a prismatic dislocation. It follows that the dislo- cation can only move conservatively, i. They can be formed in the following way.

The Glide plane and glide supersaturation excess concentration of vacancies resulting from either cylinder for dislocation rapid quenching from a high temperature see section 1. If the disc is large enough, it is energetically favorable for it to collapse to produce a disloca- Glide cylinder tion loop Fig. The Burgers or prism vector of the loop is normal to the plane of the loop, so that an edge b b dislocation has been formed.

In the presence of an excess concen- tration of vacancies the loop Glide plane will expand by positive climb. In b the vacancies have collected on a close-packed plane and in c the disc has collapsed to form an edge dislocation loop. The dislocation loops were formed in a thin sheet of aluminum by quenching. The sheet was thinned to about nm and examined by trans- mission electron microscopy. Electron transmission In a similar fashion, platelets of self-interstitial atoms can form dislocations photographic sequence in irradiated materials Fig.

Thus, in , and min, Fig. From Tartour and Washburn, Phil. A mechanism for the formation of helical dislocations is 18, , The dislocation AB, in Fig. Motion of the dislocation in the plane ABA0 corresponds to glide since this plane contains the line and the Burgers vector. Motion at right angles to this plane corresponds to climb. The configuration of the dislocation after a certain amount of climb is shown in Fig.

The dislocation now lies on the surface of a cylin- der whose axis is parallel to the Burgers vector i. Further climb displaces each Spiral or helical dislocation part of the dislocation in a direction normal to the surface of the cylinder. The radius of the spiral is smallest technique. After Bontinck at the nodes, since, for a given number of vacancies, the angle of rotation is and Amelinckx, Phil.

If A0 B is small compared with AA0 , combination of prismatic 2, 94, The Burgers vector of the helical dislocations in Figs 3. The helix consists essentially of a screw disloca- tion parallel to the axis of the helix and a set of prismatic loops. This interpretation is made more apparent by considering the interaction between a screw dislocation and a dislocation loop with the same Burgers vector. A0 is the projection of A on to plane PN. AB is now curved and lies on a cylinder whose axis is parallel to b. The dislocation can glide on this cylinder. The area F is proportional to the amount of material added or lost in climb.

The projection of this dislocation on to PN is the double spiral shown in the diagram. The sense of the b resulting helical turn is different in a and b. It b plus b equals would be reversed for a left-handed screw. However, if a prismatic loop Fig. An example is seen in Fig.

Loop translation occurs by the transfer of vacancies a around the loop by pipe diffusion, the vacancies producing positive climb at one side of the loop and negative climb at the other side. The process is called conservative climb. The relation between climb motion of a dislocation loop, with a Burgers vector normal plastic strain and the applied stress is more com- to the plane of the loop, due to its interaction with an edge plicated and depends on factors such as tempera- dislocation.

From Kroupa and Price, Phil Mag. Dislocation i has moved a distance xi as shown. It is based on the fact that when a dislocation moves, two atoms on sites adjacent across the plane of motion are displaced relative to each other by the Burgers vector b section 3.

The relationship for slip is derived first. Consider a crystal of volume hld con- taining for simplicity straight edge dislocations Fig. Under a high enough applied shear stress acting on the slip plane in the direction of b, as shown, the dislocations will glide, positive ones to the right, negative ones to the left. The top surface of the sample is therefore displaced plastically by D relative to the bottom surface as demonstrated in Fig.

If a disloca- tion moves completely across the slip plane through the distance d, it contri- butes b to the total displacement D. The same relationships hold for screw and mixed dislocations. Climb under an external tensile load is shown schematically in Fig. When an edge dislocation climbs, an extra plane of thickness b is inserted into, or removed from, the crystal in the area over which the line moves. The same relations also hold for climb of mixed dislocations, except that b is then the magnitude of the edge component of the Burgers vector.

Imura T: Dynamic studies of plastic deformation by means of high voltage electron microscopy, Electron microscopy and strength of materials, , University of California Press, p. Vreeland T Jr. The dislocation is therefore a source of internal stress in the crystal. For example, consider the edge dislocation in Fig. The region above the slip plane contains the extra half-plane forced between the normal lattice planes, and is in compression: the region below is in tension.

The stresses and strains in the bulk of the crystal are sufficiently small for conventional elasticity theory to be applied to obtain them. This approach only ceases to be valid at positions very close to the center of the dislocation. Although most crystalline solids are elastically anisotropic, i. This still results in a good approximation in most cases. From a knowledge of the elastic field, the energy of the dislocation, the force it exerts on other dislocations, its energy of interaction with point defects, and other important characteristics, can be obtained.

The elastic field produced by a dislocation is not affected by the application of stress from external sources: the total stress on an element within the body is the superposition of the internal and external stresses. In linear elasticity, the nine components of strain are 63 Introduction to Dislocations. Partial differentials are used because in general each displacement component is a function of position x, y, z.

The three strains defined in 4. They repre- sent the fractional change in length of elements parallel to the x, y and z axes respectively, e. The six components defined in 4. This is demonstrated by exy in Fig. By rotating, but not deforming, the element as in Fig. The simple shear strain often used in engineering practice is 2exy, as indicated.

In elasticity theory, an element of volume experiences forces via stresses applied to its surface by the surrounding material. Stress is the force per unit area of surface. A complete description of the stresses acting therefore requires not only specification of the magnitude and direction of the force but also of the orientation of the surface, for as the orientation changes so, in general, does the force.

Consequently, nine components must be defined to specify the state of stress. They are shown with reference to an elemental cube aligned with the x, y, z axes in Fig. For a face with outward normal in the 2j direction, i. As explained in section 3. The effective pressure acting on a volume element is Components of stress in therefore cylindrical polar 1 coordinates. The stresses are still defined as above, and are shown in Fig.

The notation is easier to follow if the second subscript j is consid- ered as referring to the face of the element having a constant value of the coordinate j. The strain energy per unit volume is one-half the product of stress times strain for each compo- nent. Consider the screw dis- location AB shown in Fig. A radial slit LMNO was cut in the cylinder parallel to the z-axis and the cut surfaces displaced rigidly with respect to each other by the distance b, the magnitude of the Burgers vector of the screw dislocation, in the z-direction. The elastic field in the dislocated cylinder can be found by direct inspection.

For a dislocation of opposite sign, i. Sketch of a wire containing an axial screw dislocation b without and c with branch wires. The end of the main wire is free to rotate and contains the surface step that mediates growth, From Zhu, Peng, Marshall, Barnett, Nix and Cui, Nature Nanotechnology 3, , Reprinted with permission from Macmillan Publishers Ltd: copyright Growth of the main wire was mediated by the sur- face step due to an axial screw dislocation see section 2. Solids cannot withstand infinite stresses, and for this reason the cylinder in Fig.

Real crys- tals are not hollow, of course, and so as the center of a dislocation in a crys- tal is approached, elasticity theory ceases to be valid and a non-linear, atomistic model must be used see section The region within which the linear-elastic solution breaks down is called the core of the dislocation. From equation 4. A rea- sonable value for the dislocation core radius r0 therefore lies in the range b to 4b, i.

Edge Dislocation The stress field is more complex than that of a screw but can be represented in an isotropic cylinder in a similar way. Considering the edge dislocation in Fig. Derivation of the field components is beyond the scope of the present treatment, however. The largest nor- edge dislocation in a. The effective pressure equation 4. These observations are implied qualitatively by the type of distortion illustrated in Figs 1. As in the case of the screw, the signs of the components are reversed for a dislocation of opposite sign, i.

Again the elastic solution has an inverse dependence on distance from the line axis and breaks down when x and y tend to zero. It is valid only outside a core of radius r0. The elastic field produced by a mixed dislocation Fig. The two sets are independent of each other in isotropic elasticity. The extra energy is the strain energy. This is a simple calculation for the screw dislocation, because from the symmetry the appropriate vol- ume element is a cylindrical shell of radius r and thickness dr.

From equa- tion 4. The above approach is much more complicated for other dislocations having less symmetric fields. The factor 12 enters because the stresses build up from zero to the final values given by equations 4. In crystals containing many disloca- 0. The energy per dislocation is thereby reduced The strain energy within a and an appropriate value of R is approximately half the average spacing of cylinder of radius R that the dislocations arranged at random.

However, the estimates that have been made suggest that the The data was obtained by core energy will be of the order of 1 eV for each atom plane threaded by the computer simulation for a dislocation, and is thus only a small fraction of the elastic energy. However, model of iron. Courtesy Yu. The validity of elasticity theory for treating dislocation energy outside a core region has been demonstrated by atomic scale computer simulation section 2.

Figure 4. Etotal is the strain energy within a cylinder of radius R with the dislocation along its axis. The energy varies logarithmically with R, as pre- dicted by equation 4. The core energy is about 7 eV nm It was mentioned in the preceding section that the elastic field of a mixed dis- location see Fig. Consider the two dislocations in Fig.

Allow them to combine to form a new dislocation with Burgers vector b3 as indicated. In this argument the assumption is made that there is no additional interaction energy involved, i. If this is not so, the reactions are still favorable and unfavorable, but the energy changes are smaller than implied above. The load producing the applied stress therefore does work on the crystal when a dislocation moves, and so the dislocation responds to the stress as though it experiences a force equal to the work done divided by the distance it moves.

The force defined in this way is a virtual, rather than real, force, but the force concept is useful for treating the mechanics of dislo- cation behavior. The glide force is considered in this section and the climb force in section 4. When an element dl of the disloca- b tion line of Burgers vector b moves forward a distance ds the crystal planes above and below the slip plane will be displaced relative to each other by b. The positive sense of the force is given by the physical reasoning of section 3.

In addition to the force due to an externally applied stress, a dislocation has a line tension which is analogous to the surface tension of a soap bubble or a liquid. This arises because, as outlined in the previous section, the strain energy of a dislocation is proportional to its length and an increase in length results in an increase in energy. The line tension has units of energy per unit length. From the approximation used in equation 4.

The line tension will produce forces tending to straighten the line and so reduce the total energy. The direction of the net force is perpendicular to the dislocation and towards the center of curvature. The line will only remain curved if there is a shear stress which produces a force on the dislocation line in the opposite sense. Equation 4. In all other cases, the line experiences a torque tending to rotate it towards the screw orientation where its energy per unit length is lower. Thus, for a line bowing under a uniform stress, the radius of curvature at any point is still given by equation 4.

For most calculations, however, equation 4. Consider two parallel edge dislocations lying in the same slip plane. When the dislocations are separated by a large Slip plane distance the total elastic energy per unit length of the dislocations in both situations will be, from a equation 4. Thus the dislocations will tend to repel each other to c B reduce their total elastic energy. When disloca- tions of opposite sign Fig.

Thus dislocations of dislocations with parallel Burgers vectors lying in opposite sign will attract each other to reduce their total elastic energy. The parallel slip planes. Similar effects occur when the two dislocations do slip plane, and c unlike not lie in the same slip plane Fig. Consider two dis- locations lying parallel to the z-axis in Fig. The total energy of the system consists of the self- energy of dislocation I plus the self-energy of dislocation II plus the elastic interaction energy between I and II.

The interaction energy Eint is the work done in displacing the faces of the cut which creates II in the presence of the stress field of I. The displacements across the cut are bx, by, bz, the compo- nents of the Burgers vector b of II. Dislocations shown as edges for simplicity. The signs of the right-hand side of these equations arise because if the displace- ments of b are taken to occur on the face of a cut with outward normal in the positive y and x directions, respectively, they are in the direction of posi- tive x, y, z for the first case x-axis cut and negative x, y, z for the second y-axis cut as shown explicitly in Figs 4.

The interaction force on II is obtained simply by differentiation of these expressions, i. For the two parallel edge dislocations with parallel Burgers vectors shown in Fig. The forces are reversed if II is a negative edge i. Equal and opposite forces act on I. Fx is the force in the glide direction and Fy the force perpendicular to the glide plane. Substituting from equation 4. The full curve A is for like dislocations and the broken curve B for unlike dislocations. Since an edge dislocation can move by slip only in the plane contained by the dislocation line and its Burgers vector, the component of force which is most important in determining the behavior of the dislocations in Fig.

Fx is plotted against x, expressed in units of y, in Fig. It is zero when x 5 0, 6y, 6N, but of these, the positions of stable equilibrium are seen to be x 5 0, 6N for edges of the same sign and 6y if they have the opposite sign. It follows that an array of edge dislocations of the same sign is most stable when the dislocations lie vertically above one another as in Fig. This is the arrangement of dislocations in a small angle pure tilt y boundary described in Chapter 9. The radial and tangential components of force on the other are screw dislocations.

Fr is repulsive for screws of the same sign and attractive for screws of opposite sign. It is readily shown from either equations 4. As in the case of glide forces, climb forces can arise from external and Mechanical force F internal sources of stress. Line tension can also produce climb forces, but f f in this case the force acts to reduce the line length in the extra half-plane: shrinkage of pris- Chemical force f matic loops as in Fig.

As a result crystal. For negative climb involving vacancy emission F , 0 the sign of the chemical potential is changed so that c. The latter is used in the analysis for a dislocation climb source in section 8. The nature of these forces is illustrated schematically in Fig. The image dislocations are in space a distance d from the surface. The dislocation is attracted towards a free surface because the material is effectively more compliant there and the dislocation energy is lower: conversely, it is repelled by a rigid surface layer.

To treat this mathe- matically, extra terms must be added to the infinite-body stress components given in section 4. When evaluated at the dislocation line, as in equations 4. The analysis for infinite, straight dislocation lines parallel to the surface is relatively straightforward. Consider screw and edge dislocations parallel to, and distance d from, a sur- face x 5 0 Fig. Consideration of equation 4. The required solution for the stress in the body x. For the edge dislocation Fig. The image forces decrease slowly with increasing d and are capable of remov- ing dislocations from near-surface regions.

They are important, for example, in specimens for transmission electron microscopy section 2. It should be noted that a second dislocation near the surface would experience a force due to its own image and the surface terms in the field of the first. The inter- action of dipoles, loops and curved dislocations with surfaces is therefore complicated, and only given approximately by images.

Lardner RW: Mathematical theory of dislocations and fracture, , Univ. Mura T: Micromechanics of defects in solids, , Martinus Nijhoff. Teodosiu C: Elastic models of crystal defects, , Springer. The pure metals are soft, with critical resolved shear stress values for single crystals 0.

## Crystallographic defect

They are ductile but can be hardened considerably by plastic deformation and alloying. The deformation behavior is closely related to the atomic structure of the core of dislocations, which is more complex than that described in Chapters 1 and 3. The shortest lattice vectors, and therefore the most likely Burgers vectors for dislocations in the face-centered cubic structure, are of the type 12hi and hi.

Since the energy of a dislocation is proportional to the square of the magnitude of its Burgers vector b2 section 4. Thus, hi dislocations are much less favored energetically and, in fact, are only rarely observed.

## Dislocations in amorphous metals

Since 12hi is a translation vector for the lat- tice, glide of a dislocation with this Burgers vector leaves behind a perfect crystal and the dislocation is a perfect dislocation. Figure 5. The planes per- pendicular to b are illustrated and have a two-fold stacking sequence ABAB. Movement of this unit dislocation by glide retains continuity of the A planes and the B planes across the glide plane, except at the dislocation core where the extra half-planes terminate.

If separated, each would 85 Introduction to Dislocations. Thus, when a stacking fault Glide plane ends inside a crystal, the boundary in the plane of the fault, separating the faulted region from the perfect region of the crystal, is a partial dislo- B A B A B A B A B A Burgers vector cation. The formation of a Shockley partial edge Unit edge dislocation dislocation is illustrated in Fig. The diagram represents cubic crystal.

The close-packed planes lie at right , Dislocations and Mechanical Properties of angles to the plane of the diagram. At the right of the diagram the Crystals, p. This has produced a stacking fault and a partial dislocation. Only rarely does glide occur on other planes. A A Since slip involves the sliding of close-packed C b2 b3 planes of atoms over each other, a simple experi- ment can be made to see how this can occur. B One layer is represented by the full circles, A, the A A second identical layer rests in the sites marked B and the third takes the positions C.

Thus, the B plane will slide over the A plane in a zig-zag motion. This simple hard-sphere description has been confirmed by computer simu- lation section 2. The locations F correspond to the intrinsic stacking fault section 1. It can be seen that these translations follow low energy paths. In terms of glide of a perfect dislocation with Burgers vector b1 5 12hi, this demonstration suggests that it will be energetically more favorable for the B atoms to move to B via the C positions, i.

This implies that the dislocation glides as two partial dislocations, one immedi- ately after the other. The first has Burgers vector b2 and the second Burgers vector b3, each of which has the form 16hi. It is necessary in dislocation reactions to ensure that the total Burgers vector is unchanged, as explained in section 1.

The right-hand side of the first of reactions 5. The same result is demonstrated diagrammatically by the vector triangles in Fig. The force may be calculated from the separate forces between their screw components equation 4. The labels ABAB. Since b2 and b3 are the Burgers vectors of Shockley partial dislocations, it follows that if they separate there will be a ribbon of stacking fault between them. This is the intrinsic fault discussed above and is equivalent to four layers of close-packed hexagonal stacking in a face-centered cubic crystal.

An equilibrium separation will be established when the repulsive and attractive forces balance. The configuration is called an extended dislocation. Note that the width, d, is inversely proportional to the stacking fault energy. Except for the size of the partial separation d, the dissociation of a perfect dislocation is independent of its character edge, screw or mixed.

During glide under stress, a dissociated dislocation moves as a pair of partials bounding the fault ribbon, the leading partial creating the fault and the trailing one removing it: the total slip vector is b1 5 12hi. Experimental observa- tions of extended dislocations in thin foils have con- firmed that this geometry is correct. The stacking fault ribbon between two partials appears as a parallel fringe pattern. The individual partials are 0. The corresponding width of the C stacking-fault ribbon given by equation 5. The widths predicted by equa- tions 5. For comparison with the b Stacking faults sketch in Fig.

Shockley partial cannot cross slip. Although extended screw Reviews, 6, , A constriction is illustrated in Fig. Energy is required to form a constriction, since the dislocation is in its lowest energy state when dissociated, and this occurs more readily in metals with a high stacking fault energy such as aluminum. It follows that cross slip will be most difficult in metals with a low stacking fault energy and this produces significant effects on the deformation behavior. The dislocation has dissociated into two Shockley partials at the positions shown. Atom displacements either into or out of the plane of the paper are indicated by smaller or larger circles, respectively.

Constriction is also assisted by a stress the Escaig stress acting on the edge component of the two partials so as to push them together. The sequence of events envisaged during the cross-slip process is illustrated in Fig. The Burgers vector of a Shockley partial is denoted by bp, that of the perfect screw by bc.

## Starting and stopping dislocations | Nature Materials

The screw has dissociated in the cross-slip plane at c. A constriction is likely to form at a region in the crystal, such as a barrier provided by a non-glissile dislocation or impenetrable particle, where the applied stress tends to push the partials together. The new extended dislocation is free to glide in the cross-slip plane and has transferred totally to this plane by stage d. The corners of the tetrahedron Fig. The Burgers vectors of dislocations are specified by their two end points on the tetrahedron. Thus, the Burgers vectors 12hi of the perfect disloca- tions are defined both in magnitude and direction by the edges of the tetra- hedron and are AB, BC, etc.

The dissociation of a 12hi dislocation described by relation 5. For the Burgers circuit construction used here section 1. For an observer inside the tetrahedron, this order is reversed. The projection and directions are the same as Fig. The latter is illustrated in Fig. See Fig. Geometrically this intrin- sic fault is identical to the intrinsic fault produced by dissociation of a per- fect dislocation section 5. Such a dislocation is said to be sessile, unlike the glissile Shockley partial.

However, it can move by climb. A closed dislocation loop of a Frank partial dislocation can be produced by the collapse of a platelet of vacancies as illustrated in Figs 1. By convention this is called a negative Frank dislocation.

A positive Frank dislocation may be formed by the precipita- tion of a close-packed platelet of interstitial atoms Figs 1. Both positive and negative Frank loops contain stacking faults. Diffraction fringes due to stacking faults section 2. An example is given in Fig. A, contain stacking faults, as seen by fringe contrast, and are Frank sessile dislocations.

In some cases no stacking fault contrast is observed. Considering the negative Frank sessile dislocation in Fig. This 16hi displacement corresponds to the glide of a Shockley partial dislocation across the fault. The Shockley partial may have one of three 16hi type vectors lying in the fault plane. It is envisaged that this partial dislocation forms inside the loop and then spreads across the loop removing the fault; at the outside it will react with the Frank partial dislocation to produce a perfect dislocation.

Atomic structure through vacancy and interstitial Frank loops on the plane of a face- centered-cubic metal, and b , d the perfect loops formed by the unfaulting reactions 5. For an interstitial loop, two Shockley partials are required to remove the extrinsic fault. With reference to Fig. Unfaulting reactions 5. The essential problem is whether or not the prevailing con- ditions in the Frank loop result in the nucleation of a Shockley partial dislo- cation and its spread across the stacking fault. A necessary condition is that the energy of the Frank loop with its associated stacking fault is greater than the energy of the perfect dislocation loop, i.

Taking a 5 0. If a loop grows by the absorption of point defects, it will become increasingly less stable, but if the Shockley nucleation energy is independent of r, the equilibrium unfaulted state may not be achieved. The probability of nucleation is increased by increasing temperature and the presence of external and internal sources of stress. The removal of the fault in Fig. Thus, there is no hard-and-fast rule governing loop unfaulting.

Concentric loops of Frank partial dislocation contain- ing alternating rings of intrinsic, extrinsic and perfect stacking have been observed under certain conditions. Several barriers have been proposed for the face-centered cubic metals. They form by contact reaction between dislocations of different slip systems.

If the two dislocations have the same Burgers vector but opposite line sense when they meet along the line of intersection of their glide planes, the reacting segments annihilate, as illus- trated by the collinear interaction in Fig. In other cases, favorable reac- tions create new dislocations that can act as barriers to other dislocations of the slip systems involved. One is the Lomer lock. It can be formed in the fol- lowing way. Their Burgers vector and positive line sense are indicated by arrows.

The two 12hifg dislocations in a can react to form a 2hifg dislocation as in b. The glide plane of the Lomer dislocation is and so it is sessile, i. Another reaction can occur in which b2 is unchanged but the two reacting dislocations attract to form a stable segment if their initial orientations are suitable. The product vector DB 1 AC is of , In most metals, each dislocation in Fig.

There are three possible 16hi vectors plus their reverses on each plane Fig. The perfect dislocations in Fig. By analogy with carpet on a stair, it is called a stair-rod dislocation. For some reactions with different resultant vectors, the angle between the stacking faults is obtuse. The dislocation is sessile.

### Table of Contents

The dislocations that result from the reactions in Figs 5. However, the arrangement of three par- tials in Fig. Stair-rod dislocations described above are the reaction product of disloca- tions from different slip systems. A stair-rod partial is created at the bend. Its total Burgers vector is BD, which lies in both planes. The angle between the two planes may be acute, as in a , or obtuse, as in b. The arrows on the partial dislocations show the positive line sense used to define the Burgers vectors according to the tet- rahedron drawn on each figure.

Following the rule stated at the end of sec- tion 5. The Burgers vectors of the Shockley partials change with the change of plane, and so the rule that Burgers vector be conserved section 1. These have Miller indices of the type 16hi and 13hi, respectively. According to the discussion in section 5. The Frank partial may dissociate into a low-energy stair-rod dislocation and a 0. From Cottrell, 3 18 6 Phil. With reference to the Thompson tetra- hedron notation Fig.

Taking account of dislocation line sense, it is found that the par- tetrahedron. Arrows show the positive line sense used to define the Burgers vectors, which are denoted by directions on the Thompson tetrahedron. In Miller index notation the reactions are of the type 5. As a result of the transformation to a stacking fault tetrahedron, the vacancy content of the original Frank loop becomes distributed equally over the four faces of the tetrahedron. The shape of tetrahedra observed in thin foils by transmission microscopy depends on the orientation of the tetrahedra with respect to the plane of the foil, as seen in Fig.

The complex contrast patterns inside the faults arise from overlapping stacking faults in different faces of the tetrahedron. The increase in energy due to the formation of stacking faults places a limit on the size of the fault that can be formed. If the fault energy is relatively high, the Frank loop may be stable or it may only partly dissociate, thereby form- ing a truncated tetrahedron as in Fig. The stacking fault tetrahedra observed in irradiated metals Fig.

- Starting and stopping dislocations;
- Bestselling Series;
- References.
- Dislocations Solids?

The core region of a cascade is disordered with a high concentration of vacancies and molecular dynamics computer simula- tion shows that a tetrahedron can form as the atoms reorganize to a more stable arrangement. Stacking fault tetrahedra can also result from plastic deformation due to the cross slip of a segment of jogged screw dislocation, as illustrated in Fig.

Pairs of letters denote Burgers vectors in the notation of Fig. In general, reducing the crystal symmetry, chang- ing the nature of the interatomic bonding and increasing the number of atom species in the lattice make dislocation behavior more complex. Nevertheless, many of the features of the preceding chapter carry over to other structures, as will be seen in the following. The two other major metal- lic structures are discussed first, and then some important compounds and non-metallic cases are considered. As explained in section 1.