Crystal Dislocation

While most crystal dislocations are total dislocations when viewed from sufficiently
far away, it is not uncommon to find them dissociated locally into a configuration
that can be described as two parallel partial dislocations connected by a planar defect
that is called a stacking fault in the crystal.

From: Encyclopedia of Materials: Science and Technology, 2001

Related terms:

Gallium Arsenide, Chemical Mechanical Polishing, Crystal Structure, Hydrogen,

Bulk Metallic Glass, Grain Boundary, Burger Vector, Dislocation Density, Radiation
Property, Slip Plane

View all Topics

Calculation of cross-slip parameters in

f.c.c. crystals
W. Püschl, G. Schoeck, in Fundamental Aspects of Dislocation Interactions, 1993

1 Introduction
In f.c.c. crystals dislocations can split into Shockley partials connected by a stacking
fault in a {111} glide plane. In order that cross-slip can take place a screw dislocation
must be constricted locally before it can dissociate in an intersecting {111} plane.
This process can occur by thermal activation aided by an external stress. Since
cross-slip events can control the overall plastic behaviour of crystals, knowledge of
the activation energy and its stress dependence is of considerable interest.

The first calculation of the cross-slip configuration by Schoeck and Seeger [1] was
on the energy level based on the Peierls [2] model taking account of anisotropy.
Although the energy of the critical transition configuration where a metastable
equilibrium exists was calculated correctly, it was assumed on the basis of entropy
arguments that this configuration is reached by a reaction path with higher energy.
The calculations were refined afterwards by Wolf [3] who also made allowance for
the narrowing of the stacking fault due to an applied stress.
Later Escaig [4], following a suggestion of Friedel, calculated the equilibrium con-
figuration on the force level using a method first proposed by Stroh [5] based on
the line tension approximation. This rests on questionable foundations since the
governing differential equation requires small slopes y of the partial dislocations
whereas the solution gives y infinity at the nodes. Further, the interaction energy
of the partials is calculated as if they consisted of parallel straight segments whereas
in the cross-slip nucleus they are strongly curved. The uncertainty of the calculation
is aggravated by the fact that no reliable procedure is given to determine the inner
cut-off radius r0 for the elastic solution or the critical distance for recombination rc
of the partials. They are arbitrarily chosen as r0=rc=b.

More recently a treatment was given by Saada [6] along the same Unes. He based
his arguments on the assumption that the driving force for cross-slip always results
from an applied stress 2 acting in the cross-slip plane. This overlooks completely
the fact that cross-slip can occur even without such a stress when the stacking fault
ribbon is narrowed in the primary plane and/or widened in the cross-slip plane. As
Escaig [4] has emphasized this narrowing-widening (NW) effect generally produces
a larger driving force than the action of 2. Saada [6] recalculates the constriction
energy and points out that without a knowledge of the cut-off radius r0 only the
order of magnitude of the activation energy can be obtained.

Another treatment was given by Duesbery et al. [7] who first calculated the equilibri-
um configuration by the relaxation method of Bacon [8] and Foreman [9]. However,
the resulting cut-off radius r0 also remains uncertain. Duesbery et al. choose r0 = 2b.
Hence values for constriction energies for aluminium (d ≈ 2b), nickel (d ≈ 3b), and
gold (d ≈ 4b) can hardly be viewed with much confidence.

In the usual treatment in the literature in linear elasticity theory, the cut-off radius
r0 is chosen so that the elastic energy outside r0 equals the total energy (i.e. elastic
energy plus core energy) but usually some ad hoc value of r0 is chosen. The ambiguity
of this procedure can be avoided when we choose an r0 in a region where the linear
elastic solution is still very reliable and include explicitly the (atomistic) core energy
in the calculations as done for instance by Piischl et al. [10]. The uncertainty in the
second kind of cut-off radius, the approach distance rc of the partials, can be removed
by demanding that the energy of dissociation in the elastic treatment equals that of
the Peierls model. In the following we give a treatment of the energy level taking
account of anisotropy which is based on this approach avoiding the ambiguity of
the previous treatments.

> Read full chapter

DISLOCATIONS
J.P. HIRTH, in Physical Metallurgy (Fourth Edition), 1996

1. Elementary geometrical properties

The concept of crystal dislocations was introduced by Polanyi [1934], Orowan [1934]
and Taylor [1934], although the elastic properties of dislocations in isotropic continua
had been known since 1905 (Timpe, also Volterra [1907]). Dislocations are defects
whose motion produces plastic deformation of crystals at stresses well below the
theoretical shear strength of a perfect crystal. In fig. 1a, b and c the glide motion
of an edge dislocation is shown to cause plastic shear strain. One can imagine a
virtual process of cutting the crystal on a glide plane, shearing the cut surface by a
shear displacement vector b, the Burgers vector, and gluing the cut surfaces together,
creating the edge dislocation in fig. 1b. The dislocation bounds a slipped area and is
a line defect. It is characterized by the Burgers vector b and by a unit vector tangent
to the dislocation line at a point in question. The same dislocation could be formed
by opening a cut under normal tractions, fig. 1d, and inserting a plane of matter.

Fig. 1. (a-c) shear of a crystal under a shear stress by an amount b by passage of an

edge dislocation; (d) creation of same dislocation under normal stress once added
material is placed in the opened cut.

In order that the deformation not produce a high energy fault on the cut surface, b
is usually a perfect lattice vector as illustrated for an edge dislocation in fig. 2. The
choice of the ± sense of is arbitrary, but once chosen, the ± sense of b is fixed by
the following convention: imagine a perfect reference crystal, select a vector in it,
and construct a closed circuit in it, right-handed relative to . Then construct the
same Burgers circuit in the real crystal, as shown for example in fig. 2. The vector SF
connecting the start of the circuit to the finish is the Burgers vector of a dislocation if
it is contained within the circuit. In this operation, the circuit must not pass through
the nonlinear core region within an atomic spacing or two of the dislocation line.
Fig. 2. An edge dislocation in a simple cubic crystal. A Burgers circuit is also shown,
projected toward the viewer for clarity so that it passes through the center of atoms
not shown. The sense vector points out of the page.

For the edge dislocation,b is seen to be perpendicular to . In fig. 3 a screw dislocation

can be imagined to have been created by a cut and displacement operation, or simply
by shearing the slipped area by motion of the dislocation in from the surface of a
perfect crystal. The screw dislocation has b parallel to . It is right-handed if b points
in the same direction as as in fig. 3; left-handed otherwise. In general, fig. 4, the
slipped or displaced surface can be arbitrary, the dislocation line can be arbitrarily
curved, and the dislocation is called mixed when the angle between b and is
neither 0 nor an integer multiple of π/2.

Fig. 3. A screw dislocation in a simple cubic crystal.

Fig. 4. A mixed dislocation.

Some other properties follow directly from the above definitions. If is reversed,
the sense of b is also reversed as seen from fig. 2 since the circuit is also reversed
when is reversed. Since the dislocation line bounds a displaced area, the line
cannot end within otherwise perfect crystal but can only end at a free surface, a
grain boundary, a second-phase interface, or a dislocation node. A node is a point
where two or more dislocation lines join. Translation of a Burgers circuit along
without the circuit cutting through a dislocation core does not change the vector
SF or thus the total b (imagine such an operation for fig. 2); such circuits are called
equivalent Burgers circuits. Thus, if a dislocation denoted by its Burgers vector, b1,
splits into two dislocations b2 and b3, enclosed by equivalent circuits, fig. 5, an analog
of Kirchhoff 's law applies and b1=b2+b3 If the for dislocations 2 and 3 are reversed,
then the signs of b2 and b3 change by the earlier axiom and: Σibi = 0 for dislocations
meeting at a node if all sense vectors are selected to point toward the node.

Fig. 5. Three dislocations meeting at a node. Arrows on lines indicate sense of .

Equivalent Burgers circuits also shown.

> Read full chapter

Dislocation Arrays and Crystal Bound-

aries
D. Hull, D.J. Bacon, in Introduction to Dislocations (Fifth Edition), 2011

Glide of Interfacial Defects

The discussion above is concerned with crystal dislocations. Now consider bound-
aries containing more general interfacial defects (section 9.6). The glide plane of a
dislocation whose Burgers vector lies in the interface is the plane of the interface
itself. A disconnection (Fig. 9.16(b)) with h =hµ, as in Fig. 9.18(c), meets this condition
and can glide if the resolved shear stress is high enough. Glide of the step allows
one crystal to grow at the expense of the other: for example, glide of the defect in
Fig. 9.18(c) to the right would allow black atoms to transfer to sites in the white
crystal as the step passes. The simplest, yet very important, example of this occurs
in deformation twinning.

Deformation twinning on the system is common in the body-centered cubic metals,

as noted in section 6.3. The direction and magnitude of the twinning shear is
consistent with the glide of dislocations with on every successive atomic plane. This
can be understood in terms of the description of admissible interfacial defects given
in section 9.6. The defect-free interface is plotted in [110] projection in Fig. 9.24(a).
(Visualization of the atomic positions shown is assisted by comparison with Fig. 1.6.)
The twin habit plane is K1 and the twinning shear in the direction 1 reorientates
the complementary twinning plane K2 as indicated. The dichromatic pattern for the
bicrystal is plotted in Fig. 9.24(b), which is seen to be a CSL with Σ=3. The interfacial
defect with a step up (from left to right) and height d, the spacing of the {112} planes,
and the shortest Burgers vector lying parallel to the interface, is created by joining
the steps defined by and as indicated. This disconnection has and is a twinning
dislocation, as described in section 6.3, and, with the convention that positive line
sense is out of the paper (section 9.6), is a negative edge dislocation. For a step
of opposite sign, i.e. step down from left to right, the defect would be a positive
dislocation.

Figure 9.24. (a) Projection of the atomic positions in two adjacent (110) planes in a
bicrystal formed by 111 {112} twinning in a body-centered cubic metal. The twin
habit plane is K1, the direction of the twinning shear is 1 and [110] is out of the
paper. (b) Dichromatic pattern associated with (a). b=(t −tμ) is the Burgers vector
of a twinning dislocation formed by joining steps t and tµ on the two crystals, as
indicated.

Thus, the atomic displacements that give rise to the macroscopic twinning shear
occur by the glide of these defects on successive planes, as illustrated schematically
in Fig. 9.25. An experimental observation of dislocations of this type is presented
in Fig. 9.26. Figure 9.26(a) illustrates the cross-section shape of the small twin seen
in the transmission electron microscope image in (b) and an explanation for the
contrast from the individual twinning dislocations is sketched in (c).
Figure 9.25. Schematic illustration of twinning in a body-centered cubic metal.
The projection is the same as that in Fig. 9.24 and the dislocations are the same
as the one defined there. (a) Untwinned crystal in [110] projection showing the
stacking sequence A, B, … of the planes. (b) Twinning dislocations with glide to
the right on successive planes under the applied shear stress indicated to produce
a twin-orientated region.

Figure 9.26. Experimental observation of a small deformation twin in a molybde-

num–35% rhenium alloy by transmission electron microscopy. (a) Illustration of the
shape of the small twin shown in (b); the dislocations are represented by dots. (b)
Diffraction contrast produced by twin which lies at an angle of 20° to the plane of
the thin foil. (c) Diagrammatic illustration of the diffraction contrast observed in (b).
The K1 plane is (112) and the dislocations are in screw orientations. Each change in
the fringe sequence is due to a twinning dislocation.(From Hull (1962), Proc. 5th Int.
Conf. Electron Microscopy, p. B9, Academic Press.)

Twinning is an important mode of deformation for many crystalline solids, particu-

larly if the number of independent slip systems associated with the glide of crystal
dislocations is restricted (section 10.9). In all cases, the b and h characteristics of the
twinning dislocations can be analyzed using treatments similar to those presented
here. (Further details can be found in references at the end of the chapter.) The glide
motion of twinning dislocations can occur under low resolved shear stress in many
materials and is therefore relatively easy. Note, however, that if the crystal structure
has more than one atom per lattice site, then the simple shear associated with the
passage of such dislocations may restore the lattice in the twinned orientation but
not all the atoms. Shuffles of these atoms will be required. This is illustrated by the
twin in an hexagonal-close-packed metal, which has two atoms per lattice site. The
structure of the twinning dislocation found by computer simulation (section 2.4) is
shown in Fig. 9.27(a). The positive edge dislocation defined by t −tµ results in a step
down (from left to right) of height 2d, where d is the spacing of the lattice planes.

This structure has been verified experimentally, as demonstrated by the high resolu-
tion transmission electron microscopy image in Fig. 9.27(b). The black dots indicate
the match between the positions of atoms near the interface in this image and those
in Fig. 9.27(a). It can be seen from Fig. 9.27(a) that the atoms in the two atomic
planes labelled S that traverse the step have to shuffle as the dislocation glides along
the boundary, because atoms such as 1 and 2 are closer than 2 and 3 on the left
whereas 2 and 3 are closer than 1 and 2 on the right. The shuffles are short and
easily achieved in this particular case, and so the core of the dislocation spreads
along the interface. Computer simulation shows that this twinning dislocation
moves easily. For boundaries where complex shuffles are necessary, the glide of
twinning dislocations can require relatively high stress and the assistance of elevated
temperature.
Figure 9.27. (a) Atomic structure obtained by computer simulation of the structure
of a twinning dislocation in a twin boundary in titanium, an hexagonal-close-packed
metal. Unit cells are shown in outline and the position of the boundary is indicated
by a dashed line. The twinning dislocation, defined by the lattice vectors t and tμ,
has a very small Burgers vector, but requires shuffling of atoms in the layers labelled
S. (After Bacon and Serra (1994), Twinning in Advanced Materials, eds. M. H. Yoo and
M. Wuttig, p. 83. The Minerals, Metals and Materials Society (TMS).) (b) Experimental
HRTEM image of a boundary in titanium containing a twinning dislocation. The
dashed lines show the location of the interface and the dots indicate the positions
of some atoms near the interface.(From Braisaz, Nouet, Serra, Komninou, Kehagias
and Karakostas, Phil Mag. Letters 74, 331 (1996), with permission from Taylor and
Francis Ltd (http://www.tandf.co.uk/journals).)

> Read full chapter

Dislocations
V. Vitek, in Encyclopedia of Condensed Matter Physics, 2005

Dislocation Cores
It has already been mentioned above that every crystal dislocation possesses a core
region in which the linear elasticity does not apply and the structure and properties
of the core can only be fully understood when the atomic structure is adequately
accounted for. When a dislocation glides, its core undergoes changes that are the
source of an intrinsic lattice friction. This friction is periodic with the period of the
crystallographic direction in which the dislocation moves. The applied stress needed
to overcome this friction at 0 K temperature is called the Peierls stress and the
corresponding periodic energy barrier is called the Peierls barrier.

In general, the dislocation core structure can be described in terms of the relative
displacements of atoms in the core region. These displacements are usually not
distributed isotropically but are confined to certain crystallographic planes. Two
distinct types of dislocation cores have been found, depending on the mode of the
distribution of significant atomic displacements. When the core displacements
are confined to a single crystallographic plane, the core is planar (a more complex
planar core, called zonal, may be spread into several parallel crystallographic planes
of the same type). Dislocations with such cores usually glide easily in the planes of
the core spreading and their Peierls stress is commonly low. In contrast, if the core
displacements spread into several nonparallel planes of the zone of the dislocation
line, the core is nonplanar, extended spatially. The glide planes of dislocations with
such cores are often not defined uniquely, their Peierls stress is high, and their glide
well below the melting temperature is enabled by thermal activation over the Peierls
barriers.

A planar dislocation core can be regarded as a continuous distribution of dislocations

in the plane of the core spreading. The reason for the core extension is the decrease
of the dislocation energy when it is redistributed into dislocations with smaller
Burgers vectors. Such approach to the analysis of the dislocation core is the basis
of the Peierls–Nabarro model. If the coordinate system in the plane of the core
spreading is chosen such that the axes x1 and x2 are parallel and perpendicular to
the dislocation line, respectively, the corresponding density of continuous distri-
bution of dislocations has two components: , where is the -component of the
displacement vector u in the x1, x2 plane. Owing to the dislocation distribution,
the displacement u increases gradually in the direction x2 from zero to the Burgers
vector so that, where is the corresponding component of the Burgers vector. In the
continuum approximation, the elastic energy of such dislocation distribution can be
expressed as the interaction energy of the dislocation distribution with itself:

where K are constants depending on the elastic moduli and orientation of the
dislocation line. On the atomic scale, the displacement across the plane of the core
spreading causes a disregistry that leads to an energy increase. The displacement
u produces locally a generalized stacking fault, and in the local approximation the
energy associated with the disregistry can be approximated as

where is the energy of the -surface for the displacement u. The continuous distri-
bution of dislocations describing the core structure is then found by the functional
minimization of the total energy with respect to the displacement u. When the
displacement vector is always parallel to the Burgers vector, the Euler equation
corresponding to the condition leads to the well-known Peierls equation

where u is the displacement in the direction of the Burgers vector and K is a constant
depending on the elastic moduli and orientation of the dislocation line; in the
isotropic elasticity K is given earlier. In the original Peierls–Nabarro model, ∂ /∂u
was taken as a sinusoidal function and the Peierls equation then has an analytical
solution

where 2 measures the width of the core (for , the disregistry is greater than half its
maximum value) and it is proportional to the separation of the crystal planes into
which the core spreads. The Peierls stress evaluated in the framework of this model
is
where μ is the shear modulus and a constant of the order of 1. This stress is
very small even when the core is very narrow; for example, if , . This is a general
characteristic of the planar cores and this is the reason why in f.c.c. metals, in which
dislocations possess planar cores, the Peierls stress is practically negligible.

The nonplanar cores can be divided into two classes: crosslip and climb cores. In the
former case, the core displacements lie in the planes of core spreading, whereas in
the latter case, they possess components perpendicular to the planes of spreading.
Climb cores are less common and are usually formed at high temperatures by a climb
process. The best-known example of the crosslip core is the core of the 1/2 1 1 1
screw dislocations in body-centered cubic (b.c.c.) metals that is spread into three
planes intersecting along the 1 1 1 direction. Figure 3 shows two alternate
structures of such core. The core shown in Figure 3a is spread asymmetrically into
the,, and planes that belong to the [1 1 1] zone and is not invariant with respect to the
diad and another energetically equivalent configuration, related by this symmetry,
operation exists. The core in Figure 3b is invariant with respect to the diad. These
core structures were found by atomistic calculations employing (1) central-force
many-body potentials and (2) a tight binding and/or density-functional theory based
approach, respectively. Atomistic calculations employing appropriate descriptions
of atomic interactions are the principal tool for investigations of such cores since
direct observations are often outside the limits of experimental techniques. The
example presented here shows that the structure of such cores is not determined
solely by the crystal structure but may vary from material to material even when
they crystallize in the same structure. The Peierls stress of dislocations with these
cores is commonly several orders of magnitude higher than that of dislocations with
planar cores. Furthermore, the movement of the dislocations is frequently affected
not just by the shear stress parallel to the Burgers vector in the slip plane but by
other stress components. Thus, the deformation behavior of materials with non-
planar dislocation cores may be very complex, often displaying unusual orientation
dependencies and breakdown of the Schmid law. The nonplanar dislocation cores are
the more common the more complex is the crystal structure, and thus these cores
are more prevalent than planar cores. In this respect, f.c.c. materials (and also some
hexagonal close-packed (h.c.p.) materials with basal slip) in which the dislocations
possess planar cores and, consequently, the Peierls stress is very low, are a special
case rather than a prototype for more complex structures.
Figure 3. Two alternate core structures of the 1/2 [1 1 1] screw dislocation depicted
using differential displacement maps. The atomic arrangement is shown in the
projection perpendicular to the direction of the dislocation line ([1 1 1]) and circles
represent atoms within one period. The [1 1 1] (screw) component of the relative
displacement of the neighboring atoms produced by the dislocation is represented
by an arrow between them. The length of the arrows is proportional to the magnitude
of these components. The arrows, which indicate out-of-plane displacements, are
always drawn along the line connecting neighboring atoms and their length is
normalized such that it is equal to the separation of these atoms in the projection
when the magnitude of their relative displacement is equal to |1/6 [1 1 1]|.

Additional complex features of the dislocation core structures arise in covalent

crystals where the breaking and/or readjustment of the bonds in the core region may
be responsible for a high lattice friction stress, and in ionic solids where the cores
can be charged which then strongly affects the dislocation mobility. Such dislocation
cores affect not only the plastic behavior but also electronic and/or optical properties
of covalently bonded semiconductors and ionically bonded ceramic materials.

> Read full chapter

Protein Dynamics as Reported by NMR

Zoltán Gáspári, András Perczel, in Annual Reports on NMR Spectroscopy, 2010

4.5 X-ray diffraction

Real crystals do not fulfill the perfect repetition of the unit cell in three dimensions.
While at larger scale crystal dislocations and twinning deteriorate periodicity, at the
atomic level static and dynamic disorder take effect. Dynamic disorder is caused by
the thermal motion of atoms around their equilibrium positions, which can be larger
scale motion as well. Co-existence of different atoms at the same spots of different
unit cells (e.g. molecules adopting different conformations) causes static disorder.
Classical crystallographic methods cannot discriminate between static and dynamic
disorder because the resulting electron density maps are averages of molecular
conformations over the volume of the crystal and over the whole period of the
diffraction data collection. Disorder can be described at different levels depending on
the quality of the experimental data (i.e. resolution) using the atomic displacement
parameter (ADP) (U or B-factor: B = 8π2U).26 Assuming the thermal motion of the
atom is spherically symmetric, U is isotropic, and it is the square mean shift of
the atom with respect to its average position during vibration (). Anisotropic ADPs
are better description of atomic scale disorder in the crystal, typically used with
atomic resolution data. Macromolecular crystals, even though they usually diffract
weaker, allow only isotropic description of ADPs, or even averaged values only. An
intermediate and effective description of anisotropy in macromolecular crystals is
to model isotropic ADPs and anisotropic motion (translation, libration and screw
motion27) for more rigid domains within the same macromolecule.

As a result of static disorder, often alternate positions of atoms can be resolved

in electron density maps. In macromolecular crystals, some of the flexible regions
possess variable conformers and as a consequence none of the individual ones can
be detected in electron density maps. Thus, dynamics between several energetically
similar states and/or larger amplitude makes to vanish more mobile sequential
subunits. In practice, more than two or three conformer makes detection and/or
assignment impossible. In other words lower than 25–33% of relative population of
conformers is unseen by diffraction methods.

Using modern synchrotron X-ray sources from the late 1990s, it has been possible to
study complex dynamic behaviour of proteins using Laue techniques. The structures
of the intermediates and transient states of enzymes involved in their catalytic cycles
can be identified and the respective time scale 100 ps to 100 ms can be assigned. The
analysis of the bacterial photoreceptor photoactive yellow protein revealed that by
taking snapshots at every 150 ps throughout the photocycle of 10 ns to 100 ms, five
distinct intermediate structures could be identified and used to establish a reason-
able chemical kinetic mechanism.28 Consequently the chemical mechanism of the
process can be formulated.29 Recent advances made possible sampling picosecond
dynamics of the protein by 100–150 ps time resolution of measurement.30–33 New
frontiers include 5D time and temperature dependent studies34 as well as developing
new techniques to further shorten time scale of the experiments.35 Time-resolved
Laue studies require photoreaction-initiated systems for triggering the reaction
uniformly in the whole crystal volume rapidly with respect to the process under
study to explore dynamics of the sub-ns–ms range in these systems. For systems
presenting a significantly slower motion of s–h time scales initiation is not restricted
to fs photo reactions, but it can be carried out by for example substrate diffusion.
Intermediates of these systems can be freeze-trapped, so there is no need for using
Laue techniques.36,37

In conclusion, common diffraction techniques are typically unable to refer to sub-

units and sequences of significant dynamics, but are usually able to identify
the dominant conformers/conformations. More specific approaches can report on
events (e.g. series of reaction(s) and allosteric conformational changes) happening in
the crystal state at slower (s–h) or even faster (ms–μs) time scale of motion. However,
even the latter approaches are “blind” to highly flexible regions of proteins structures
with many conformation possibilities appearing at ms–ns time scale of motion.

> Read full chapter

Tempering of martensite in carbon

steels
G. Krauss, in Phase Transformations in Steels: Diffusionless Transformations High
Strength Steels Modelling and Advanced Analytical Techniques, 2012

5.5.5 The third stage of tempering: matrix considerations

As described above, the matrix structure of as-quenched martensite is highly unsta-
ble, not only because of carbon supersaturation and unstable retained austenite, but
also because of high intra-crystal dislocation density and large areas of interfaces
bounding the fine martensite crystals of lath martensite. The latter characteristics of
a martensitic microstructure create the driving forces for recovery, recrystallization
and grain growth of the martensitic microstructure during tempering. The kinetics
and mechanisms of the matrix microstructural changes are highly dependent on
alloying, and in general are significantly slowed by alloying effects.

Changes in the matrix of as-quenched martensite have been systematically studied

in an iron-0.2% carbon alloy. The parent austenite grains are subdivided by transfor-
mation to martensite into blocks or packets of roughly parallel martensitic crystals,
many of which are very fine and not resolvable in the light microscope. There is a very
high martensite lath boundary area per unit volume of martensitic microstructure,
on the order of 65,000 cm−1 (Apple et al., 1974; Swarr and Krauss, 1976). Some of
these boundaries are low angle where adjacent crystals of martensite have the same
crystallographic orientation, and some are high angle where adjacent crystals have
different orientations. On tempering into the third stage, the lath boundary area
drops significantly. For example, in the iron-0.2% carbon alloy, after tempering for
only one minute at 400°C, the martensite boundary area per unit volume decreases
to 40,500 cm−1. similar drastic decreases in boundary area were measured over a
range of tempering temperatures, and have been shown to be largely a result of the
disappearance of low angle boundaries (Caron and Krauss, 1972).

Figure 5.10 shows the microstructure of an iron-0.2% carbon alloy after tempering
at 700°C for 2 hours. spheroidized cementite particles can be resolved in the light
microscope, and coarse blocks of structure, differentiated by etching differences, still
in the parallel packet alignment of lath martensite, characterize the microstructure.
The fine martensitic crystals within the blocks have been eliminated, a change in
microstructure attributed to recovery mechanisms associated with the low angle
dislocation boundaries that separated parallel martensite crystals with the same
orientation. With increased tempering, the large angle parallel boundaries rearrange
to form equiaxed ferritic grains to minimize grain boundary energy. Thus the
equilibrium microstructure of spheroidized carbides and equiaxed ferrite grains in
highly tempered high purity iron-carbon alloys and low-alloy steels may be a result
of a sequence of recovery and grain growth mechanisms (Caron and Krauss, 1972;
Hobbs et al., 1972). Recovery mechanisms, involved not only in the elimination of the
low angle martensite interfaces but also in the reduction of intracrystal dislocation
densities, apparently lower strain energy to below the critical amount necessary for
recrystallization.

5.10. Light micrograph, nital etch of microstructure of lath martensite in an

iron-0.2% carbon alloy after tempering at 700°C for 2h. The packet structure of lath
martensite with parallel crystals is still present, but the remnant martensitic crystals
are significantly coarsened (Caron and Krauss, 1972).

In contrast to the above development of highly tempered microstructures, recrys-

tallization by the nucleation and growth of new grains within a tempered lath
martensitic microstructure has also been documented (Galibois and Dube, 1964,
1967; Tua et al., 1992). Unrecrystallized lath martensitic microstructures around
recrystallized grains are characterized by high dislocation densities, and recrystal-
lized grains grow through retained distributions of spheroidized cementite particles.
Thus sufficient strain energy to drive recrystallization is available, perhaps as a
result of alloying-element suppression of dislocation substructure recovery in the
martensite. The latter explanation has been proposed for the role that cobalt, a
weak carbide-forming element, plays in the achievement of super-hard high-speed
tool steels during secondary hardening (Speich, 1990). Recrystallization of lath
martensite has also been documented in ultra-low-carbon steels (Tsuchiyama et al.,
2001, 2010). In these steels recystallization occurs by nucleation at prior austenite
grain boundaries, creating bulges that grow into the martensitic matrix, as effectively
shown by electron back scattered diffraction (EBSD) analysis.

> Read full chapter

Coatings for buildings

J.A. Graystone, in Paint and Surface Coatings (Second Edition), 1999

9.8.1 Characteristics of iron and steel

Pure metals have an underlying crystal structure which is substantially modified
by impurities or deliberate inclusions used to form an alloy. Alloys will normally
show a mixture of grains comprising different crystal structures or phases which
have a marked effect on mechanical properties. Controlling the movement of crystal
dislocations is a powerful method of altering strength and toughness. The most
common way to achieve this is by inclusion of carbon which is extremely efficient
in controlling dislocation movement within the iron crystal, but other elements
such as silica and manganese have specific advantages. Manufacturing processes
will also alter both physical and chemical properties; established techniques include
quenching, tempering, and work hardening. Among the most commonly encoun-
tered ferrous substrates in general building are mild steel, cast iron, and wrought
iron. Mild steel normally contains between 0.2% and 0.8% carbon, whereas cast
iron contains 4% (i.e. 20% by volume). Wrought iron has been worked so that the
morphology is changed and the carbon is present as glossy inclusions.

Mild steel will normally be covered with millscale which, over a period of time,
loosens and may fall away. It can contribute to corrosion and presents an unsound
basis for coating. Techniques for removal and preparation are well documented
[168]. Cast iron has a more adherent scale with some protective value; it corrodes at
a similar rate to mild steel, though the residue is less obviously coloured. Wrought
iron is generally similar to mild steel, though the corrosion rate may differ. Iron and
steel will frequently be coated with less corrodible non-ferrous metals such as zinc
or aluminium, which have other characteristics as described below.

> Read full chapter

Advances in Crystals and Elastic Meta-

materials, Part 1
Javier Segurado, ... Javier LLorca, in Advances in Applied Mechanics, 2018

Abstract
This paper reviews the current state of the art in the simulation of the mechanical
behavior of polycrystalline materials by means of computational homogenization.
The key ingredients of this modeling strategy are presented in detail starting with
the parameters needed to describe polycrystalline microstructures and the digital
representation of such microstructures in a suitable format to perform computation-
al homogenization. The different crystal plasticity frameworks that can describe the
physical mechanisms of deformation in single crystals (dislocation slip and twinning)
at the microscopic level are presented next. This is followed by the description of
computational homogenization methods based on mean-field approximations by
means of the viscoplastic self-consistent approach, or on the full-field simulation
of the mechanical response of a representative polycrystalline volume element by
means of the finite element method or the fast Fourier transform-based method.
Multiscale frameworks based on the combination of mean-field homogenization
and the finite element method are presented next to model the plastic deformation
of polycrystalline specimens of arbitrary geometry under complex mechanical load-
ing. Examples of application to predict the strength, fatigue life, damage, and texture
evolution under different conditions are presented to illustrate the capabilities of the
different models. Finally, current challenges and future research directions in this
field are summarized.

> Read full chapter

Mechanical Properties: Plastic Behav-

ior
G. Gottstein, ... G.V.S.S. Prasad, in Encyclopedia of Condensed Matter Physics, 2005
Crystallographic Slip
Metals are crystalline except for very special conditions. Metals and alloys generally
crystallize in a face-centered cubic (f.c.c.), body-centered cubic (b.c.c.), and hexagonal
close-packed (h.c.p.) crystal structure. There is unambiguous evidence that metals
retain their crystal structure during plastic deformation. This conservation principle
of crystal plasticity has serious consequences for the deformation process and
material properties. In a noncrystalline material, a macroscopic shape change can
be accommodated on an atomistic level by a respective atomic rearrangement. If
a macroscopic deformation of a crystal is also copied on an atomistic level, the
crystal structure would change (Figure 1a); this is however contrary to observation.
The crystal structure can be conserved, if the shape change on an atomistic level
is accommodated by a displacement along crystallographic planes in multiples of
translation vectors of the crystal lattice (Figure 1b). This phenomenon is referred
to as crystallographic glide (or crystallographic slip). As a result, crystal plasticity
proceeds by a structure conserving simple shear deformation. In principle, there
are an infinite number of planes and translation vectors to accomplish an imposed
shape change. Due to energetic reasons, crystallographic glide is confined only to
a few crystallographic planes (slip planes) and directions (slip directions), normally
the most densely packed planes and directions in a crystal. A slip plane and a
slip direction constitute a slip system and the most prominent slip systems of the
three major crystal structures of metals are given in Table 1. The mechanism of
crystallographic glide is related to the motion of crystal dislocations. The glide plane
of a dislocation and the direction of its Burgers vector correspond to the slip plane
and slip direction, respectively. On an atomistic level, therefore, plastic deformation
of a metal requires the generation and motion of crystal dislocations.
Figure 1. Plastic deformation of crystals (dotted lines=unit cell): (a) with change of
crystal structure; (b) by crystallographic glide; and (c) by deformation twinning.

Table 1. a Slip systems of the major metallic crystal structures

Crystal struc- Slip plane Slip direction Number of Slip direc- Number of slip
ture nonparallel tions per systems
planes plane
f.c.c. {1 1 1} 4 3 12=(4×3)
{1 1 0} 6 2 12=(6×2)
b.c.c. {1 1 2} 12 1 12=(12×1)
{1 2 3} 24 1 24=(24×1)
{0 0 0 1} 1 3 3=(1×3)
h.c.p. 3 1 3=(3×1)
6 1 6=(6×1)

Table 1b Twinning systems of the major metallic crystal structures

Crystal Twinning Shear direc- Displace- Prototype

structure plane tion ment plane
f.c.c. {1 1 1} Ag, Cu
b.c.c. {1 1 2} -Fe

h.c.p. Cd, Zn

Dislocations are crystal defects and, therefore, cause a distortion of the atomic
arrangement, which manifests itself into a long-range stress field. Via their stress
field, dislocations interact with each other. This constitutes a glide resistance, which
is macroscopically felt as a yield stress and eventually causes immobilization of
moving dislocations. The traveling distance (slip length) of a moving dislocation
is normally much smaller than the macroscopic dimension of the deformed body;
therefore, dislocations are stored in the crystal and the dislocation density grows
with progressing deformation which in turn increases dislocation interaction and
thus the flow stress. The relation between flow stress and dislocation density can
be represented by the Taylor equation

[1]

where the dislocation density (m−2) is defined as the total dislocation line length
per unit volume – typically 1010m−2 for annealed and 1016m−2 for heavily deformed
metals, b is the Burgers vector (slip vector), G is the shear modulus, and M is
the Taylor factor which relates the applied stress to the shear stress in the glide
system and depends on crystallographic texture (see below). Typically, M 3, and the
geometrical constant 0.5.

Any change in the glide resistance of the moving dislocations will affect the flow
stress. This is the reason for work hardening (increasing dislocation density), other
strengthening mechanisms (grain size, solute atoms, particles, precipitates, etc.),
and softening by recovery and recrystallization (decreasing dislocation density). It is
important to note that the macroscopic deformation behavior can be related to the
microscopic properties of the crystal dislocations. The force K on a dislocation due
to a stress applied to a crystal is given by the Peach–Koehler equation

[2a]

or the magnitude of the force K in the slip system due to the resolved shear stress

[2b]

The imposed strain rate is reflected by the dislocation velocity through the Orowan
equation

[3]

where m is the density of the mobile dislocations moving with an average velocity
. The velocity, of course, depends on the applied stress . One can imagine the
 dislocations as rigid rods, moving with velocity due to a force K, and this process
is experienced macroscopically as a plastic deformation with strain rate at a flow
stress .

> Read full chapter

Internal Stresses in Coherent Multilay-

ers
G. Saada, in Encyclopedia of Materials: Science and Technology, 2008

2.2 Surface Dislocation Density Tensor

Following Kröner (1958, 1964, 1981) we consider that the multilayer is built starting
from a stack made of n lamellae, all with the lattice vectors in P identical to those of
lamella n. The matching is then recovered by giving each lamella a homogeneous
plastic (stress-free) distortion Bq. This results in the creation of a homogeneous dis-
location distribution at the interfaces. These dislocations, named quasi-dislocations,
are a mathematical device, not crystal dislocations. Their distribution is defined by a
homogeneous surface dislocation density tensor (SDDT) a (Kröner 1958, 1964, 1981,
Nye 1953, Bilby 1955), whose general expression on the plane Pq is (Saada 1998, see
Planar Dislocation Arrays and Crystal Plasticity)

(3a)

n is the unit vector normal to P; ikl is the Levi-Civita symbol whose nonzero
components are

(3b)

In the case of a bilayer, Eqn. (3a) become

(4)

Using the coordinate axes defined in Fig. 1, the components of a are written as

(5)

A detailed account of the calculation and use of SDDTs adapted to this specific
problem is found else where (see Planar Dislocation Arrays and Crystal Plasticity).

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