Independent Slip Systems in Crystals

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Independent slip systems in crystalsG. W. Groves

a& A. Kelly

a

aDepartment of Metallurgy, University of Cambridge

Available online: 20 Aug 2006

To cite this article:G. W. Groves & A. Kelly (1963): Independent slip systems in crystals,Philosophical Magazine, 8:89, 877-887

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[ 877

1

Independent Slip Systems in Crystals

By

G .

W.

GROVES nd

A .

KELLY

Department of Metallurgy, University of Cambridge

[Received

4

April 19631

ABSTRACT

The slip systems observed in a number of crystal structures common

amongst meta ls and simple ceramic mater ials are examined t o me whether they

allow the crystal to undergo an arbitrary strain without change of volume.

For

most materials, other than f.c.0. and b.c.c. metals, there are insufficient

independent slip systems. The condition under which cross slip can give

rise to extra independent systems is stated. The results explain in a natural

way recent experimental Gndings on the ductility of polycrystals with the

sodium chloride structure.

1.

INTRODUCTION

von Mises 1928) first pointed out tha t for a crystal to be able to undergo

a general homogeneous strain by slip, five independent slip systems are

necessary. This result is often quoted in the literature and it has been

used in studies of face-centred cubic metals in connection with the number

of slip systems found to operate near grain boundaries (Livingston and

Chalmers 1957, Kocks 1958, Hauser and Chalmers 1961) and in theoretical

attempts to deduce the polycrystalline stress-strain curve from that

of

a

single crystal (Taylor

1938, 1956,

Bishop and Hill

1951

a,

b,

Bishop

1953).

von Mises also gave a simple method of determining whether or not slip

systems are independent. Nowhere in the literature

is

there a n account of

the application of this t o crystals with structures other than those of the

f.c.c. metals.

In view of the growing volume of work on plasticity of materials with

crystal structures and slip systems quite different from those of f.c.c.

metals i t seems worth while to describe von Mises method and the results

of testing a number of simple crystal structures to see hoy many indepen-

dent slip systems they usually possess. Further, the number of slip

systems observed in a given crystal structure can alter with temperature.

This can lead to a sudden change in the mechanical properties which

finds a ready explanation in terms of von Mises result. I n the course

of

this work we have also found that the results for f.c.c. metals have a simple

geometrical interpretation.

2. VON

MISES

RESULT

Slip or glide leads on a macroscopic scale (i.e. if one considers

a

volume

containing many slip bands) to the translation of one part of a crystal

relative to another by a motion corresponding to a simple shear. A

single simple shear determines the value

of

one

of

the independent com-

ponents of the strain tensor. Since plastic flow usudly occurs without a

P.M.

3 M

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878

G. W. Groves and A. Kelly on

change in volume, the six independent components of the general strain

tensor are reduced to five because of the condition ;

Ex

+

c y + c 2 =

0.

von Mises

(1928)

noted th at since the operation of one slip system produces

only one independent component of the strain tensor, then five independent

slip systems are needed to produce a general, small, homogeneous strain

without change in volume.

To determine whether a given crystal possesses sufficient slip systems we

proceed as follows (von Mises 1928, Bishop 1953) . Write down the com-

ponents of the strain tensor produced by an arbitrary amount of glide on a

Do

the same thing for four other slip systems, referring the strain tensor to the

same set of axes as before. Finally, since the three tensile strains are not

independent of one another, form the five by five determinant of the

If

this determinant has a value other than zero, then the five chosen slip

systems are independent of one another, since the determinant will equal

zero if any row can be expressed as a linear combination of other rows.

Provided then that the value of the determinant is other th an zero, one has

chosen five slip systems which are independent of one another in t he sense

that the operation of one of them produces components of the strain tensor

which cannot be expressed as linear combinations of the components

produced by the operation of the other slip systems.

Physically, the above amounts to saying that a slip system is independent

of others provided its operation produces a change in shape of a crystal

which cannot be produced by a suitable combination of amounts of slip

on those other systems.

Since five independent slip systems suffice to produce an arbitrary strain

without change in volume, it is apparent that a crystal cannot possess more

than five independent systems.

given slip system. Call these components c Z , ey e Z ,

eXy

,

eX2

,

quantities ex' ey' Z1 , cX y1 , eZBI c y z l ; e x a

-

yU

E ~

yl ' *

3.

EXPRESSIONOR THE

STRAIN

OMPONENTS

We need a simple method for writing down the components of the strain

tensor produced by glide on

a

given slip system. The following method is

convenient (Bishop

1953,

Livingston and Chalmers

1957).

We define a

glide system by a unit vector n normal to th e glide plane, with components

nx,ny,n, parallel to an orthogonal set of axes x, , z , and by a unit vector

p in the glide direction with corresponding components p x , ,, pa The

components of the strain tensor are then:

ey=

an,

;

x

=m X p x

c, = olnzpz;

a a

a

2

Exy =

2By

+ n,Bx) ;

E x z

=

(nxB2

+ Px)

and

E V E =

- (n,Bz

+

nsB, *

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Independent Sl ip Systems in Crystals

879

Using these relations we can write down immediately the components of

the strain tensor produced by a simple shear of magnitude t a n a by the

operation of the glide system defined by n and p. Since the relations are

symmetrical in

n

and

p

it is immediately clear that the strain produced by

slip on the plane n in the direction

p

is the same as tha t produced by slip

on the plane p in the direction n.

The directions

of

n and

p

are usually given in terms of Miller indices.

These must be referred to orthogonal axes of equal measure when the above

relations are used in non-cubic crystals. When the directions of n and p

are given in terms of Miller indices

(or

Miller-Bravais indices) we call a

family of slip systems all those combinations of slip plane and slip direction

which must arise from the point group symmetry of the crystal if one slip

plane and one slip direction are given.

4.

RESULTS

4.1. F.C.C. Metals

The usual slip systems are the family

{

1 1 1}( 110 .

There are

12

physically different slip systems. We evaluate

the components of the strain tensor referred to axes parallel to the conven-

tional cell edges. For the slip system

( l l l ) [ l IO]

we have for the com-

ponents of

n

and

p :

Consider a f.c.c. metal crystal.

Using the relations of $ 3

61

E y Z =

2/6

s y = o E x , = --

22/6

Proceeding similarly one can write down the components of the strain

tensor produced by operation

of

any

of

the physically different slip systems

in the { 11 1}( 110) family.

Only five of these are independent. The

selection of five independent ones can be made in a number of ways. This

selection can be illustrated in terms of Thompson s tetrahedron, which is a

regular tetrahedron with vertices A, B, C, D and

a, 8

y 6 the midpoints

of the opposite faces respectively. A slip direction corresponds to an edge

of the tetrahedron and a slip plane normal can be designatsd by one of the

letters 01, /3,

y

6. Thus (AB), corresponds to slip in the

[Oll]

direction on

the ( i i l )plane. A possible set of five independent slip systems is (AC),,

(AB),, (AD),, (AC),, (AB ),. I n this crystal structure each slip direction lies

in two slip planes

;

we might designate these the primary and the cross-

slip plane.

P . M . 3

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G.

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Any slip system can be expressed in terms of two others with either the

same primary slip plane or with the same cross-slip plane. Thus, for

example, we can write :

(DB),+(BC),+(CD),=O

.

(4.1a)

and

(DB),+(BC),+(CD),=O. . . . (4.1b)

These equations mean that equal shears on the three slip systems produce

zero net distortion of the crystal. It can be visualized that the operation

of the first set of shears (4-1

a)

produces no net displacement at any point,

while the operation of the second set

(4-1 )

merely rotates the crystal about

the direction ALY,he normal t o the plane containing the slip vectors.

There is a third group of dependent sets, similar to those of (4.1

b ) ,

for

example

Again the slip vectors are co-planar and sum to zero, and their slip planes

are equally inclined to the plane containing the slip vectors. The appli-

cation of this set merely rotates the crystal about the normal t o this plane,

i.e. about a (100).

In choosing five independent slip systems the above sets, and com-

binations of them, must be avoided. There are 384 different ways of doing

this (BishopandHill 1951b).

(CD),

+

(DC),

+

(AB),

+

(BA),

=

0.

. . .

(4 . lc )

4.2. B.C.C. Metals

Since the components of the strain tensor produced by slip on a plane

of normal

n

n a directionp are the same as those produced by slip on a plane

of normal p in a direction n, von Mises analysis for a f.c.c. crystal applies

equally well to a b.c.c. metal crystal which possesses the family of slip

systems {110) ( 1 11 ) and a general strain can be produced by slip. These

slip systems can also be represented in terms of a modified tetrahedron

shown in the figure. The letters

LY3

y , 6 now represent slip directions and

the pairs of letters, e.g. AB, now represent slip planes, e.g. the plane AOB

in the figure.

I n this system

then

a

corresponds

to

slip in the

[ T T l ]

direction on the

(011)

plane.

A possible set of five independent slip systems is then a BAD PAC,

yAB,

nd again there are 384 different ways of choosing five of these from

the

12

physically distinct slip systems. Again combinations involving any

of the relations analogous to 4.1) must be avoided.

Each slip direction lies in three slip planes.

4.3. NaCl Structure

Crystals with the sodium chloride structure possess the family of slip

There are six physically distinct slip systems.

Consider first ( loi)[ ioi] .

systems {110}(

1 T O ) .

However, there are only two independent ones.

Following the procedure in

3,

and taking axes parallel to (001 ),

nz=

1 / 4 2 ,

n y =O

ns= 1 / 4 2 , Pz= 1 / 4 2 , = O , jgB= 1 / 4 2 .

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Independent Slip Systems i n Crystals

881

All other components of the strain tensor

Since

01

is an arbitrary constant defining th e amount of slip we

Whence

e z=

-

12 e z = 0 ~ 1 2 .

are zero.

C

Figure illustrating slip planes and directions

for

{ l l O ) t l T l >

slip in a cubic

crystal.

Slip

planes are planes such as AOC, COD, etc. Slip directions are directions

Aa, Bp,

etc., the normals to the faces

of

the tetrahedron. If

(110)

n o )

slip is considered, the slip planes are the same but the slip directions are

the edges of the tetrahedron,

AB,

BC, etc.

can put i t equal to 2. Proceeding similarly for the systems

( o l l ) [ o i i ]

and

( ~ l o ) [ i l o ] e find for the respective components of the strain tensor :

0 is the centroid of the reguIar tetrahedron ABCD.

It

is clear th at only two of these are independent and thus the change of

shape produced by slip on any one of these systems can be produced by an

appropriate amount of slip on the other two. Since the three remaining

physically distinct slip systems merely correepond to the interchange of

slip plane and direction with those we have just considered, there are in all

only two independent systems. A general deformation is thus not possible

by slip. I n particular the off-diagonal terms of the strain tensor are always

zero,

so

a deformation tending to change the angle between the crystal axes

cannot occur, e.g. a crystal slipping only on these systems cannot be twisted

about a (001 ) axis, nor extended along (1 11 ).

3 N Z

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G . W . Groves

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on

The equivalence of slip on the various systems which reduces the number

of independent slip systems can again be seen from the figure, where now

the slip directions are the edges of the tetrahedron and the slip planes are

planes such as AOB, COD. There is only one slip direction in each slip

plane. Thus a symbol such as (AB),, represents slip in the direction AB

on the plane AOB, of which the normal is CD.

It

is immediately verified by

substituting in the relations of 3 that

and also that

(4.3)

is the analogue in this structure of

(4 . lb )

for f.c.c. metals. The

remaining relation, which with (4.2)and (4.3) ensures that no more than

two independent systems can be chosen, is :

(4.4) is obtained by interchanging slip plane normal and slip direction in

(4.3).

There are

12

different pairs of independent slip systems in the NaCl

(AB),,=(CD),,, . . (4.2)

(DB),,+(BC),D+(CD),,=O. , . . . . (4.3)

(AC),,+(AD),,+(,4B),,=O.

. .

(4.4)

structure.

4.4. CsCl Structure

The family of slip systems is

{100}(010)

(Rachinger and Cottrell

1956).

A general deformation is impossi-

I n particular extensions parallel to the crystal axes cannot be pro-

There are eight different ways of choosing three independent slip

These yield three independent systems.

ble.

duced.

systems.

4.5.

C a p Structure

Two families of slip systems have been observed, viz. (001}( 110) and at

higher temperatures

{ 1

lo}( 110) (Roy 1962). The first family yields three

independent slip systems and cannot produce a general deformation. I n

particular extensions parallel to the crystal axis are not possible. There

are

16

different ways of choosing three independent systems. The second

family

is

that found in NaCl and permits only extensions parallel to the

crystal axes. It yields two independent systems which are independent of

those in the first family. A general strain can be produced if, and only

if, both families of slip systems operate simultaneously and independently.

Five independent systems can be chosen in 192 different ways.

4.6. Rutile T iO ,) Structure

Two families of slip systems are reported

by Ashbee (1962). These are

{ l O l } l O T )

and (110}(001). The first

yields four independent slip systems. This is different from the sodium

chloride structure in which the family of slip systems has the same indices.

The difference arises because rutile is tetragonal and hence slip on (101)

in the direction

[

1011 does not produce the same shear as slip on (101)in the

The structure is tetragonal.

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Independent Sli p Systems i n Crystals 883

[ l o l l

direction. The slip systems in the family

{110}(001)

produce

nothing new. All the components of the strain tensor produced by shear

on the two planes of this family can be produced by a linear combination

of members of the

{ l O l } l O T )

family. There are in all four independent

slip systems. The component of the strain tensor eZy (with the z axis

parallel to the tetrad axis) is always zero. A general deformation is not

possible.

4.7.

Hexagonal Materials

A common family of slip systems is

{0001}( 1120).

There are only two

independent slip systems and three different ways of choosing these.

A

crystal cannot be extended parallel to the conventional crystallographic

axes, nor can the angle between the axes lying normal to the hexagonal axis

be altered. This family

of

slip systems is the only one possessed by graphite

(Freise 1962). Other hexagonal materials often show slip on other families

of slip systems.

Zirconium (Rapperport and Hartley 1960), and tellurium (Stokes et

al.

1961)exhibit slip on the {lOTO}( 1120) family. There are only two indepen-

dent slip systems and again these can be chosen in three different ways.

These slip systems allow extensions parallel to the two crystallographic

axes lying in the basal plane and alteration of the angle between these. A

crystal which possesses the two families of slip systems, {0001}(

1120)

and

{ l O Y O }

1120), possesses therefore four independent slip systems, subject

to the proviso in 5 . An example is magnesium a t

a

temperature greater

than 18O c (Flynn

et al.

1961) and aluminium oxide a t high temperature

(Scheuplein and Gibbs 1962).

In addition many hexagonal metals show pyramidal slip on the family

{lOil}( l l20), e.g. zinc, cadmium (Schmid and Boas 1935) and titanium

(Churchman 1954). There are six physically different slip systems of this

family. They produce four independent systems, which can be chosen in

nine different ways. The changes of shape which can be produced by

{lOil} 1120)

slip are precisely the same as those which can be produced

by the simultaneous and independent operation of both the

{0001}( 1120)

and (lOiO}(ll20) families. In no case is an extension parallel to the

hexagonal axis possible.

I n zinc (Bell and Cahn

1957)

and in cadmium (Price

1961)

the operation

of the family of slip systems {1122}( 1121) has been reported. There are

six physically different members of this family and they provide five

independent slip systems, so that a general deformation is possible. The

operation of one member of the

{1122}(1121)

family allows a crystal to

extend parallel to the hexagonal axis.

5.

CROSS

SLIPOR PENCIL

LIDE

At low temperatures slip usually occurs in crystals upon well-defined

A s

the temperature is raised slip planes often become

lip planes.

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884

G. W. Groves and A. Kelly on

ill-defined and slip appears to be taking place upon any plane of which

the slip direction is zone axis. We shall call such a situation pencil glide.

Then it is easily seen by substitution in the relations given in

3

that glide

in any given slip direction will produce two independent components of the

strain tensor. There is, however, an important physical stipulation which

must be obeyed if two truly independent components of the strain tensor

are to be produced.

Suppose

pencil glide occurs in a crystal with the NaCl structure. Consider the slip

direction

[ O l i ] .

At low temperatures this possesses the slip plane (011)

and the components of the strain tensor produced are, for

a

strength of

0 0 0

eformation a

This is best illustrated by an example.

[:

;

:I

hen pencil glide occurs we can resolve the shears on t o

(01

1 ) and any other

plane in the zone with axis [OlT]. Suppose we take the second plane to

be

(100). Then the components of the strain tensor due to the second shear

will be :

0

a1 /2 -a /2

all

0

I a l p

0 0 1

These two slip systems produce then two independent components of the

strain tensor, if and only if there is no necessary connection between a and

a .

Whether

or

not there is a connection between

a

and

a

depends upon

the physical process by which pencil glide is produced. This may be

important for instance in deciding whether prismatic glide and basal glide

are independent in, for instance, magnesium.

There is a good deal of evidence from electron microscope and etch-pit

observations of the occurrence of a microscopic form of cross slip involving

essentially the motion of a screw dislocation out of its primary slip plane

for a short distance upon another slip plane, of indeterminate indices but

with zone axis along the screw, and subsequent cross slip of the screw

back onto the primary slip plane. Gilman and Johnston

(1960)

first

discovered this and called it multiple cross glide.

As

far as one can tell

at

present the amounts of slip on primary and cross-slip plane are not indepen-

dent of one another and hence there is a connection between a and 01 .

Such a process does not appear to provide two slip systems corresponding

to each slip direction.

When pencil glide occurs then a crystal need only possess, in general,

three non-coplanar slip directions in order for a general strain to be

possible.

6.

DISCUSSION

It

may still

be applied to the large strains produced in plastic deformation by viewing

The analysis in

2

is

valid for infinitesimally small strains.

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Independent Slip Systems in Crystals

885

the components of the strain tensor as strain rates multiplied by an infini-

tesimal length of time (von Mises

1928).

This device avoids comparing a n

instantaneous configuration with an initial one and merely compares the

instantaneous configuration with one neighbouring in time. This is

usually done in theories of plasticity to avoid the mathematical difficulties

associated with

a

treatment of large strains.

The most obvious application of these results is to the deformation of

polycrystalline specimens. It is usually assumed, following Taylor, that

the deformation of a polycrystalline specimen can only proceed in general

without the production of voids provided the grains can undergo a general

strain. If this change in shape is to be produced by slip then five indepen-

dent slip systems are necessary.

A s

a very general rule it appears to be

found that when five slip systems are not available voids are formed during

plastic deformation of a polycrystalline aggregate.

For

example, a t low

temperature magnesium slips mainly on OOOl} in a 1120) direction with

some lOiO}

1120)

slip. Voids are formed a t very small strains (Hauser

et al 1955).

The most convincing experimental evidence that five independent slip

systems are necessary for polycrystalline ductility comes from experiments

on crystals with the NaCl structure. At low temperatures where only

1 lo}

110)

slip operates, polycrystals and even bicrystals are found to be

very brittle (Stokes and Li

1963).

In

a

compressed bicrystal there are in

general five conditions to be satisfied by the strains within the grains if they

are to remain in contact whilst deforming homogeneously : the two tensile

strains and the shear strain in the plane of the boundary must match in each

grain, and the strains along the compression axis must equal the imposed

strain (Livingston and Chalmers

1957).

Since only four independent slip

systems are available, two from each grain, a general bicrystal cannot

deform so as to maintain contact at the grain boundary. In practice,

both bicrystals of MgO (Johnston

e t al. 1962,

Westwood

1961)

and poly-

crystals (Stokes and Li

1963,

private communication) are observed to

fracture from a crack which forms at a grain boundary where a slip band

runs into it . The deformation of a general boundary produced by a slip

band in one grain cannot be accommodated by {llO} liO) slip in the

neighbouring grain, since the three strain components in the boundary

plane, which must be matched, cannot, in general, be produced by the two

independent slip systems which are available. Stokes and Li

1963)

lso

show clearly that in silver chloride, sodium chloride and magnesium oxide,

a t low temperatures, the rate of strain hardening after the yieldpoint (which

corresponds to plastic flow occurring in some of the grains) is extremely

rapid. This can be understood since the total plastic strain obtainable

before fracture will only be of the same order as the elastic strain. Similar

behaviour is found for LiF (Budworth and Pask

1963).

Stokes and Li

1962)

ave also shown that when large amounts of pencil glide are observed

in silver chloride and sodium chloride then appreciable plastic extensions

of a polycrystal are obtained.

Allhe same is true of LiF (loc. cit.).

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886

G. W. Groves

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A. Kelly

on

these results receive

a

completely general explanation from the above

analysis. A t temperatures below those a t which pencil glide can occur the

operation of the six physically distinct slip systems of the

{ l l O } l T O )

family will lead to no change in shape which cannot be produced by the

operation of only two of these, e.g.

l l O ) [ l T O ]

and l O l ) [ l O i ] . The onset

of pencil glide, having three non-coplanar slip vectors, produces five

independent slip systems and a general strain is then possible. Poly-

crystals then become ductile.

It must be noted that the requirement of five independent slip systems

to allow an arbitrary strain is

so

general that by itself it does not allow

particular predictions to be made as to what will happen if sufficient systems

are not available. Fracture may occur in a polycrystal as soon as slip occurs

within a few grains and the slip bands intersect the boundary, as in

MgO.

Alternatively, pencil glide might be forced to occur in regions of very high

stress, or slip could occur on planes not normally functioning as slip planes ;

this last seems to occur in LiF at room temperature (Budworth and Pask

1963 . Fracture does not

necessarily

occur if five independent slip systems

are not available. Whether

or

not fracture occurs will depend on whether

in a particular stress state cracks are opened at lower stresses than are

required to produce slip on new slip systems.

4 show that polycrystals of materials of

CsCl type, graphite, and TiO, type will not deform without the opening of

voids unless other slip systems operate.

Polycrystals of CaF, (orUO, will

not do

so

unless both the observed families of slip systems operate.

The other results listed in

ACKNOWLEDGMENTS

We would like to thank Professor A. H. Cottrell, F.R.S., and

Dr. F.

J. P.

Clarke, for stimulating this work, and for useful discussions. A special

debt of gratitude is due to

Dr.

J. D. Eshelby for pointing out to us the

relations used in 3.

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CAHN,

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BISHOP,

.

F.

W . ,

1953,

Phil . Mag. ,

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W.,

and PASIC,

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CHURCHMAN, . T., 1954,

Proc. roy. SOC . ,

226, 216.

FREISE,.

J .

1962, Ph.D. Thesis, Cambridge.

FLYNN,. W., MOTE, J. and DORN, . E., 1961, Trans. Amer. Inst . min.

GILMAN, . J. , and JOHNSTON

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G . , 1960, J . a p p l . Phys. , 31,

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HAUSER, .

E.,

STARR,. D., TIETZ,L. , and DORN, . E., 1955, Trans. Amer.

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