53

Lecture 4: Glide, Dislocations and Texture

Crystals yield plastically by a process called glide. Twinning can also be a deformation mechanism

(as for example in the ‘cry of tin’), but this is outside the scope of this course. However, twinning is

the subject of Chapter 11 of the book by Kelly and Knowles.

Glide is the translation of one part of a crystal with respect to another without a change in volume.

The translation usually takes place upon a specific crystallographic plane and in a particular

direction in that plane.

The process of glide is illustrated schematically below, e.g., in response to a stress applied normal

to the end faces of the cylinder. When a shearing stress too small to produce glide is applied to the

crystal, the crystal is elastically deformed.

Glide was first discovered in 1867 by Reusch, who recognized steps on the surface of a crystal of

rock salt.

Schematic diagram of glide occurring in the direction in

a rod-shaped crystal of circular cross-section

With the optical microscope the steps are usually seen as lines and were called slip-bands by Ewing

and Rosenhain who reported the first serious investigation of them in many metal crystals in 1899

and 1900. Very careful measurement fails to detect any change in crystal orientation of the parts of

a crystal on either side of a slip line, and there is no evidence for any change in crystal structure as

the two parts of the crystal slide over one another.

It is therefore clear that the translation of one part of the crystal with respect to another must be

equal to an integral number of lattice translation vectors. In view of this, the process is often called

translation glide.

54

Slip lines are easily seen under an optical microscope and sometimes can be observed with the

naked eye. Therefore the translation at a single step can be larger than a micrometre (106 m ),

corresponding to a movement of some thousands of lattice vectors. Glide in a crystal usually occurs

on a well-defined crystallographic plane of low indices, which is called the glide plane or slip

plane, and always in a definite crystallographic direction, the glide direction or slip direction.

Experimental techniques to determine the possible combination of glide planes and glide directions

(together called glide systems) in materials are described in Kelly and Knowles, Chapter 7. A

tabulation of glide systems is given on the next page, taken from this book.

With few exceptions, the glide direction is parallel to the shortest lattice translation vector of the

Bravais lattice. This is so even when the shortest lattice translation vector is considerably larger

than the interatomic distance, as, for example, in bismuth. This rule finds a simple explanation in

terms of the physical mechanism giving rise to glide.

Defects known as dislocations are present in the crystal (to be discussed in the next lecture). Glide

always occurs by the motion of dislocations.

As a rule, the slip planes in simple crystal structures at low temperatures are parallel to the closest

packed atomic planes in the crystal, but this rule has many exceptions. In addition, crystals often

show a number of crystallographically different slip planes, e.g., the h.c.p. metal magnesium glides

on {0001}, { 0101 } and { 1101 }.

In many crystals, at moderate temperatures, while the slip direction remains fixed, the slip plane in

a given slip line varies, so that what is called pencil glide or wavy glide occurs with the slip plane

being any plane with the slip direction as the zone axis. The slip trace then appears to be irrational

on all crystal faces except those parallel to the slip direction.

A schematic of how wavy glide is seen on the surface of a crystal

Crystals can show all variations between very well defined slip planes and wavy glide. For instance,

in NaCl at temperatures less than 200 °C the slip plane is accurately {110} and slip traces on all

crystal faces appear to be very straight. Above a temperature of about 250 °C slip appears to occur

on any plane in the < 011 > type zone. For crystals that show wavy glide, it is the well-defined slip

plane observed at low temperatures that is listed in the table overleaf.

55

56

Independent slip systems

Formal definition:

A slip system is independent of others if the pure strain ij produced by it cannot be produced

by a suitable combination of slip on other systems.

It is evident from the Table on the previous page that different crystal structures have different slip

systems. The number of independent slip systems shown range from two to five, as shown by the

examples in the two tables below. This number has implications for plastic flow.

Independent glide systems for different cubic crystal structures

Slip systems No. of physically

distinct slip systems

No. of independent

slip systems

Crystal structure

< 0 1 1 > {111} 12 5 c.c.p. metals

< 1 1 1 > {110} 12 5 b.c.c. metals

< 0 1 1 > {110} 6 2 NaCl structure

<001> {110} 6 3 CsCl structure

Independent glide systems for hexagonal close packed metals

Slip systems No. of physically

distinct slip systems

No. of independent

slip systems

< 0 2 1 1 > {0001} 3 2

< 0 2 1 1 > { 0 0 1 1 } 3 2

< 0 2 1 1 > { 1 0 1 1 } 6 4

< 3 2 1 1 > { 2 2 1 1 } 6 5

For further examples of slip systems see Kelly and Knowles, pp. 206207.

The strain tensor associated with an angle of shear due to glide in a unit direction on a slip plane

whose normal is a unit vector n can be shown to be

3323322

113312

1

32232

12212212

1

31132

121122

111

β)ββ()ββ(

)ββ(β)ββ(

)ββ()ββ(β

γε

nnnnn

nnnnn

nnnnn

ij

where n has components 1n 2n 3n and has components 1β 2β 3β (Kelly and Knowles,

Chapter 6).

57

Independent glide systems in crystals with the NaCl structure: proof of two independent slip

systems

At room temperature the slip systems are {110}< 0 1 1 > in crystals with the NaCl structure.

There are six distinct planes of the type {110}. Within each plane there is only one distinct < 0 1 1 >

direction. Hence, there are six physically distinct slip systems.

It is convenient to label the six slip systems A – F for simplicity:

Label Slip system Label Slip system

A (110)[ 0 1 1 ] D ( 1 1 0 )[011]

B ( 0 1 1 )[110] E (101)[ 1 0 1 ]

C (011)[ 1 1 0 ] F ( 1 0 1 )[101]

Looking at slip system A, the strain tensor produced by shearing an angle is

000

010

001

2

Aε

since

0,

2

1,

2

1n and

0,

2

1,

2

1β (choosing our orthonormal set of axes to be parallel to the

crystal axes).

Evaluating the other five strain tensors in a similar manner, we find:

000

010

001

2

Bε ;

100

010

000

2

Cε ;

100

010

000

2

Dε ;

100

000

001

2

Eε ;

100

000

001

2

Fε

58

It is immediately apparent that

BA εε

DC εε

FE εε

so that there are at most three independent slip systems. Of course the rotation tensors A and B

are different. Likewise, C and D are different from one another, as are E and F.

However, an examination of Eε shows that it can be produced by a linear combination of Aε and

Cε :

CAE εεε

i.e., in words, shear of an amount on the slip system (110)[ 0 1 1 ] combined with shear of the same

amount on (011)[ 1 1 0 ] together produce a shape strain equivalent to shear of an amount on the

slip system (101)[ 1 0 1 ].

Hence there are only two independent slip systems in crystals with the NaCl structure, e.g., A and

C.

Similarly, it can be shown that in c.c.p. and b.c.c. crystals there are 5 independent slip systems – a

proof is given in Appendix 5 of Kelly and Knowles.

This is important in the context of polycrystalline metals being able to accommodate general

changes of shape von Mises in 1928 showed that a general pure strain can be produced in a

crystal by glide, provided the crystal can slip on five independent systems.

59

Large strains of single crystals: the choice of glide system

In many experiments on glide a single crystal of the substance under investigation is pulled in

tension or compressed along a given direction, as in the diagram below. It is easy to find the shear

stress on the slip plane and resolved in the slip direction (called the resolved shear stress) by

altering the axes of reference of the tensor representing the applied stress.

However, this can be very simply derived directly from the diagram below. If a force F is applied to

the crystal of cross-sectional area 0A , the tensile stress parallel to F is ./ 0AF The force F has a

component 0cos F in the slip direction, where 0 is the angle between F and the slip direction;

this force acts over an area 00 cos/ A , so that the resolved shear stress is

0000

0 coscoscos/

cos

A

F

where 0 is the angle between F and the normal to the glide plane. The factor 00 coscos is

often known as the Schmid factor.

To a very good approximation, it is always found that when a crystal is subjected to an increasing

uniaxial tensile or compressive load, as in the diagram above, slip always occurs first on that glide

system on which the resolved shear stress is greatest – the Schmid criterion.

To a less good approximation, it is found that in a given pure crystal at a given temperature, glide

starts when the resolved shear stress reaches a certain critical value. This last approximation, called

the law of critical resolved shear stress, is moderately well obeyed in crystals of metals, where

there is usually a high density of mobile dislocations present (see the next lecture), but in non-

metals the resolved shear stress at which glide occurs is so dependent on the previous history of the

crystal that such a rule is seldom of use.

60

The particular glide system with the highest resolved shear stress for a particular orientation of F

with respect to the crystal axes may be found by applying the Schmid criterion knowing the

direction of applied uniaxial stress and the possible slip systems.

For common families of slip systems the result of examining all members to find which one is

subject to the largest resolved shear stress is easily shown on a stereogram. A stereogram of a cubic

crystal centred on 001 is shown in below. If a cubic close-packed metal crystal is considered, the

point group is m3 m and slip occurs on the {111}< 011 > system. There are twelve physically

distinct glide systems.

If F is the direction of a uniaxial force applied to the crystal, the direction of F can be plotted on the

stereogram, say as 0F in the figure below. The particular glide system having the largest resolved

shear stress is indicated by the lettering of the unit triangle on the stereogram, within which F falls.

For instance, for the case shown below the triangle is lettered B IV, meaning that slip will occur

first on the octahedral plane B, i.e. (111), in the direction IV, i.e. ].101[ If F lies on a boundary

between two unit triangles, then clearly two slip systems are equally stressed.

If F is exactly along <110>, <111> or <001>, then there are four, six and eight slip systems

respectively with the same largest resolved shear stress.

Standard stereogram of a cubic close packed metal crystal where slip occurs on the {111}< 011 >

to show the particular slip system with the maximum resolved shear stress for any orientation of the

tensile axis (after Schmid and Boas, who redrew a stereographic diagram of Taylor and Elam).

In cubic crystals slipping on {111}< 011 > a simple rule has been given by Diehl to identify the

primary glide system when the direction of F lies in any of the unit triangles shown in the above

figure. This states: ‘Reflect the <110> pole of the triangle in question in the opposite side of the

triangle to find the glide direction, and reflect the {111} pole of the triangle in the opposite side to

find the glide plane normal.’

An equivalent rule to this rule is OILS rule devised by Hutchings as a non-graphical method for

deducing the slip system with the highest Schmid factor. A mathematical proof of the validity of

these two rules is given in Appendix 5 of Kelly and Knowles.

61

Dislocations and Dislocations in Crystals

Fundamentals:

(a) (b)

(c) (d)

(a) screw dislocation, (b) crystal after dislocation has produced slip in the entire crystal, (c) an edge

dislocation and (d) a mixed dislocation. These are all in a simple cubic lattice.

The extra half plane associated with an edge dislocation in a simple cubic lattice.

62

A Burgers circuit in the FS/RH convention. Note the need to define the

Burgers vector with respect to the lattice of the perfect crystal.

A closed loop of dislocation lying in a slip plane: b is maintained along the

line; the line directions at E1 and E2 are antiparallel.

Force per unit length acting on a dislocation: lbF )( ; it is evident from this equation that the

force acts normal to the dislocation line l.

63

Methodology for determining the strain energy of a screw dislocation

For a screw dislocation the appropriate element is a thin cylindrical shell of radius r, thickness dr,

because the shear strain γ within such a thin shell is constant:

r

b

π2γ

An elastically isotropic medium is assumed. The strain energy per unit length of shell is then

rr

brrE d

π4

μdπ2 )μγ(d

22

21

where is the shear modulus. Therefore, the energy per unit length of dislocation in the region

from the core, radius 0r , out to a radius R is

R

rr

r

bE

2

0

d π4

μ

or

0

2

lnπ4

μ

r

RbE

To obtain the total energy of a dislocation, the energy stored within the dislocation core must be

added to this energy. The core energy can be roughly estimated in various ways, which agree in

giving its order of magnitude as π4/μ 2b J m1, much less than the above.

A convenient estimate of the total energy per unit length of a dislocation which is often used is that

it is 2/μ 2b per unit length; this is taken to be E.

64

It is always geometrically possible for two dislocations to combine to form a single dislocation, e.g.,

here:

213 bbb

A simple rule (Frank’s rule) is that reduction will occur if

22

21

23 bbb

.. but there are a few caveats to this – see Kelly and Knowles for a discussion.

65

For a piece of dislocation line curved as shown to a radius r, the line tension T (in N) is, to a first

approximation, E (in J m1), the energy per unit length of the dislocation (ignoring the core

contribution).

This diagram which helps in the calculation of the stress needed to extrude a dislocation between

obstacles a distance l apart. The shear stress required is

l

bμτ

and this is also the shear stress needed to operate a FrankRead source whose pinning points are a

distance l apart (see Kelly and Knowles for further details).

66

Partial dislocations

It is now well established that <110> dislocations in c.c.p. metals dissociate into partials on the

{111} slip plane. The fault that lies between the partials is a fault in the stacking sequence of the

{111} planes, which does not change the relative positions of atoms that are nearest neighbours of

one another.

The production of this stacking fault can best be visualized with the aid of sheets of closely packed

balls, stacked one on top of another. By sliding an upper block of sheets through a/6<211> over

those underneath, the so-called intrinsic stacking fault is produced, as shown below.

For example, if the second ABC block in the sequence ABC ABC is slid over the first, the sequence

becomes ABCBCA. The sheets at the fault are now in the sequence BCBC which, if continued,

produces a close packed hexagonal structure. Such a fault occurs when the 21 <110> dislocation

dissociates into 6

1 <211> dislocations, which are called Shockley partials or HeidenreichShockley

partials:

]121[]211[]110[61

61

21

for dissociation of the 21 <110> dislocation on the (111) plane. The pattern of atoms at the (111)

slip plane around a 21 <110> 60º dislocation that has dissociated into Shockley partials is shown in

this figure. (For simplicity in drawing, the partials are shown to be very narrow; in a real crystal

they would be wider.)

Dislocation on a (111) plane of a c.c.p. metal of a perfect dislocation

which has split into two Shockley partials:

]121[]211[]110[61

61

21

67

The separation r of the Shockley partials is determined by the energy per unit area, γ , of the

stacking fault between them. The result is

ν1

θsinθsinθcosθcos

πγ2

μ 2121

21bbr

where the two Shockley partials have magnitudes of Burgers vectors 1b and 2b , is the shear

modulus of the metal and Poisson’s ratio. 1θ and 2θ are the angles between the relevant partial

dislocation line and its Burgers vector. In applying this formula, care must be taken to measure 1θ

and ,θ2 the angles between the partial dislocation lines and their Burgers vectors, in the same

sense.

The substitution of typical values for the constants in the above equation shows that a relatively

wide separation of the partials of the order of 100 Å occurs only when the stacking fault energy is

as low as 10 mJ m2. Most c.c.p. metals and alloys have a higher stacking fault energy than this, so

the separation of partials will only be seen in the transmission electron microscope using weak

beam imaging conditions.

Vacancy loops and interstitial loops

(a) Faulted vacancy loop in a c.c.p. metal, showing traces of the

(111) planes in a plane normal to the loop. (b) Extra atom loop.

68

Dislocation reactions

Thompson’s tetrahedron for specifying dislocation reactions in c.c.p. crystals

Reaction of two dislocations on a common slip plane. The

dislocations are assumed to be pinned at the points p, q, r and s

Same reaction as above but showing splitting into partials

69

Dislocation ‘locks’

Lomer–Cottrell lock: αγγBBα

The edge dislocation is called a stair-rod, a name suggested to Thompson by Nabarro (quoted

by Thompson in a footnote to his 1953 paper), from the way in which it joins two stacking faults at

a bend rather like a stair-rod holding down a stair-carpet.

Hirth lock: the reactions are as follows:

CAAC

DBBD

(Dissociation of perfect dislocations)

]010[]112[]112[31

61

61 (reaction of the leading partials)

This is called the Hirth lock even though, as Nabarro notes in his book Theory of Crystal

Dislocations, it appears in a tabulation given by Jacques Friedel in an article in Philosophical

Magazine in 1955.

70

Dislocations in rock salt

Edge dislocation in NaCl. The slip plane and the two sheets

of ions constituting the extra plane are shown dotted.

71

Dislocations in h.c.p. metals

Further details of dislocations in h.c.p. metals are discussed in Kelly and Knowles. Here we choose

three aspects of dislocations in h.c.p. metals to consider pictorially:

Partial dislocations

As with c.c.p. metals, stacking faults can arise as a result of point defect condensation. Such a fault

is different from the fault produced by the splitting of a perfect 021131 dislocation, e.g.:

]0110[]0011[]0121[

BAAB

31

31

31

Vacancy loops

When a disc of vacancies collects on a basal plane, a prismatic dislocation loop forms at its

perimeter. If the disc is one vacancy thick, and if A–A stacking is to be avoided, the loop may be

faulted in one of two ways:

Two possible structures for a vacancy loop in a hexagonal metal.

72

Either a single basal layer or the whole crystal on one side of the loop must be sheared over by a

vector of the A type. These alternatives are shown on the previous page.

In case (a), the single layer can be shifted from an A into a C position by passing A partials of

opposite sign on either side of it. The Burgers vector of the prismatic dislocation remains

]1000[21 , S in the top figure on the previous page, at the cost of a stacking fault which is triple,

in terms of the number of next-nearest neighbour stacking violations: ‘ACB’.

In case (b), a single A partial is passed, so that the loop encloses a single stacking fault, but its

Burgers vector increases to AS, as follows:

]3022[]1000[]0011[

ASSA

61

21

31

}

Similar reasoning applies to faulted loops produced by discs of extra atoms.

Pyramidal glide

Pyramidal glide in h.c.p. metals.

Pyramidal glide occurs in Zn, Cd, Mg and Be, for example. Each slip plane contains only one slip

vector and each slip vector lies in only one slip plane. This system operates second only to basal

slip in Zn and Cd, which have the largest c/a ratios.

Further aspects of dislocations in other structures are considered in the book by Kelly and Knowles.

73

Texture

Because large strains can cause a reorientation of the lattice of a crystal it follows that, in a

polycrystalline material, continued plastic flow, such as that which occurs during the working of

metals, tends to develop a texture or a preferred orientation of the lattice within the grains, as well

as a preferred change of shape of the grains. Working involves very large plastic strains, often

hundreds of per cent. The development of a texture significantly changes the crystallographic

properties of the material.

A fine-grained material in which the grains have random lattice orientations will have isotropic

properties (provided that other crystal defects such as inclusions and boundaries are uniformly

distributed). However, a specimen with a preferred orientation will have anisotropic properties,

which may be desirable or undesirable, depending upon the intended use of the material.

The final texture that develops may resemble a single orientation or may comprise crystals

distributed between two or more preferred orientations. During plastic flow the process of

reorientation is gradual. In theory the orientation change proceeds until a texture is reached that is

stable against indefinitely continued flow of a given type.

The theoretical end distribution of orientations and the manner in which it develops are a

characteristic of the material, its crystal structure, its microstructure and the nature of the forces

arising from the deformation process. Whether a material actually reaches its stable orientation

depends upon many variables, e.g. the rate of working and the temperature at which it is carried out.

Textural changes may correspond with marked changes in the ductility of a material.

Thus there is an interplay between the role of deformation in creating texture and the role of texture

in facilitating deformation. The prediction of actual textures during forming processes is necessarily

very complicated.

The various processes of texture development have been categorized according to their final state

and to the kind of working involved, e.g. wire and fibre textures, compression textures, sheet and

rolling textures, torsion textures and deep drawing textures.

Some examples of ideal end textures are presented in the table on the next page.

In addition to deformation textures, anisotropic properties of a polycrystalline material can also

arise as a result of recrystallisation following working or as a result of material fabrication

processes that do not involve plastic deformation.

For example, the vapour deposition of layers on substrates often results in the layer having a

polycrystalline texture with a preferred orientation in the growth direction and random orientations

perpendicular to the growth direction.

74

Examples of some common textures are shown in the table below, taken from Kelly and Knowles:

75

There is usually a statistical distribution of orientations of the various crystals in a sample. This

distribution can be measured and represented on a pole figure, such as below. A pole figure is

similar to a stereographic projection. On it is presented the statistical distribution of the orientations

of the plane normals from a particular set of planes. A pole figure is usually obtained using

diffraction from X-rays, neutrons or electrons.

In the typical case of X-ray diffraction in the laboratory the detector position is fixed to collect the

diffraction from one particular set of planes. The sample is rotated about two axes so as to scan

over all possible orientations.

The corresponding pole figure is usually presented as a contour map, where the contours are

labelled according to the relative strength of the diffraction signal. For example, the figure below is

a pole figure showing the relative intensities of the reflections from 111 planes in a cold-rolled

copper sample. The relative populations of planes in the various orientations are proportional to the

numbers against the contour lines. The pole figure is interpreted in terms of an ideal end

orientation.

In uniaxial textures, such as those obtained in tension, compression and wire- and rod-forming

processes, it is often sufficient to specify which crystallographic direction or directions lie parallel

to the unique axis.

For sheet textures, ideal orientations are given by a plane or planes lying parallel to the plane of the

sheet and a direction or directions in the plane lying parallel to a significant direction in the sheet,

e.g. the rolling direction (RD).

Hence, the nomenclature (110)[ 121 ] in the {111} pole figure below represents a texture where RD

is [ 121 ] and the preferred plane whose normal (110) should be at the centre of the stereogram.

In this diagram CD is the cross direction (or transverse direction, TD) to RD lying in the plane of

the sheet.

Further details of texture and its measurement are given in Chapter 7 of Kelly and Knowles and in

the references in this Chapter.