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7/25/2019 Independent Slip Systems in Crystals
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This article was downloaded by: [Stanford University Libraries]On: 22 May 2012, At: 20:45Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
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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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http://dx.doi.org/10.1080/14786436308213843http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditionshttp://dx.doi.org/10.1080/14786436308213843http://www.tandfonline.com/loi/tphm197/25/2019 Independent Slip Systems in Crystals
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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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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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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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W. Groves
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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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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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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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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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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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G. W. Groves
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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.
REFERENCES
ASHBEE,K. H. G., 1962, Ph.D. Thesis, Birmingham.
BELL,R . L . , and
CAHN,
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