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The deformation of plastically non-homogeneous

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M. F. Ashby

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materials, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied
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 [ 399 1

The Deformation of Plastically Non-homogeneous Materials

By M. F. ASHBY
Division of Engineering and Applied Physics, Harvard University,
Cambridge, Massachusetts 02138

[Received 17 May 1969 and in revised form 6 October 19691

ABSTRACT
Many two-phase alloys work-harden much faster than do pure single
crystals. This is because the two phases are not equally easy to deform.
One component (often dispersed as small particles) deforms less than the other,
or not a t all, so that gradients of deformation form with a wavelength equal
to the spacing between the phases or particles. Such alloys are ‘ plastically
non-homogeneous ’, because gradients of plastic deformation are imposed by
the microstructure. Dislocations are stored in them to accommodate the
deformation gradients, and so allow compatible deformation of the two phases.
We call these ‘ geometrically-necessary ’ dislocations to distinguish them from
the ‘statistically-stored’ dislocations which accumulate in pure crystals during
straining and are responsible for the normal 3-stage hardening. Polycrystals
of pure metals are also plastically non-homogeneous.
The density and arrangement of the geometrically-necessary dislocations
can be calculated fairly exactly and checked by electron microscopy and
x-ray techniques. The rate a t which they accumulate with strain is con-
veniently described by the ‘ geometric slip distance ’, a characteristic of the
microstructure. Their arrangement is quite different from that of the
statistically-stored dislocations, which may make them particularly suscep-
tible to recovery effects, even a t low temperatures.
Geometrically-necessary dislocations control the work hardening of the
specimen when their density exceeds that of the statistically-stored ones.
They contribute to hardening in two ways: by acting as individual obstacles
to slip, and (collectively) by creating a long-range back-stress, with wave-
length equal to the particle spacing.
With the exception of single-phase single crystals, almost all metals and
alloys are plastically non-homogeneous to some extent. The model provides
an explanation for the way in which the stress-strain curve is influenced by a
dispersion of particles, and by grain size.

8 1.
INTRODUCTION
MANYtwo-phase alloys work-harden much faster than do those consisting
of a single phase. This is because the two phases are not equally easy t o
deform. One component deforms plastically more than the other, so that
gradients of deformation build up, with a wavelength about equal t o the
spacing between phases. Such alloys are plastically non-homogeneous.
We shall show that the gradients of deformation require that dislocations
be stored (it is convenient t o call them geometrically-necessary dislocations),
that their arrangement and density can be calculated, and that these
geometrically-necessary dislocations contribute to the work-hardening of
the alloy.
400 M. F. Ashby on the
1.1. Geometrically-necessary and Statistically-stored Dislocations
Crystals work-harden when they are strained plastically because
dislocations stop during the straining, and so become stored. They
are stored for two separate reasons : either they are required for the
compatible deformation of various parts of the specimen, or they
accumulate by trapping one another in a random way. Two familiar
examples of the first reason for storage are plastic bending and prismatic
punching. The plastic bending of a crystal to a curvature K can be
accomplished by introducing a density pG = K / b of dislocations of
Burgers vector b (Nye 1953). This array of dislocations has no long-range
stresses associated with it, that is, the internal stress, averaged over a
distance large compared with the separation of dislocations, is zero.

Fig. 1

f BURGERS
GRADIENT OF SLIP
ON THE a SYSTEM

- '\ - XI

Diagram showing that a gradient of slip on the a slip system in the x1 direction
causes a density po = (l/b,)(ay,/as,) of geometrically-necessary dis-
locations to be stored.

Similarly, the displacement u of a punch is accommodated with minimum

long-range stress by building N G = u / b prismatic loops into the crystal
(Seitz 1950). I n both cases a simple geometric relationship exists between
the deformation (described by K or u ) and the number of dislocations
( p or N ) needed to accommodate it with minimum internal stress. These
dislocations we call geometrically-necessary dislocations (Cottrell 1964) ;
the superscript G indicates this.
The density of these dislocations is directly related to the gradients of
plastic deformation with distance. Consider slip on a single slip system,
 Deformation of Plastically Non-homogeneous Materials 40 1

the o( system (fig. l ) , with Burgers vector b, in the x, direction. A gradient

of slip, ayJaxl, on this system requires that a density

of geometrically-necessary dislocations be stored, as a simple Burgers

circuit construction showst. If the gradient is steep, the density p G is
large.
The figure also illustrates that the lattice is rotated by a gradient of
shear on a single slip system. The lattice rotation, E c $ ~ , about a n axis
x3 normal to the diagram, between two points ax, apart is :

Thus the net rotation C$ between two phases, one of which undergoes a
shear on a single slip system while the other does not shear a t all, is
approximately :
C$sy . . . . . . . . . (1.3)
(for 'small y , less than about 0.2). Finally the lattice curvature, K S 1 is
,
given by :

By substituting eqn. ( 1 . 1 ) into this, we recover the result for a bent beam :
K31=pGbl. A similar argument can be made for the case of prismatic
punching, where gradients of normal plastic strain exist. I n this case
the lattice does not rotate.
By contrast, dislocations are not geometrically necessary in the uniform
deformation, such as simple tension, of a pure single crystal. I n spite
of this, dislocations do accumulate ; their presence causes the crystal's
characteristic work-hardening. There is no simple geometric argument
to predict the density of these dislocations ; this is one reason that the
stress-strain relationship for pure metals is so difficult to calculate.
Since their accumulation is probably a result of chance encounters in
the crystal resulting in mutual trapping, we call them statistically-stored
dislocations.

1.2. Plastically Homogeneous and Non-homogeneous Materials

We define a plastically non-homogeneous material as one in which
gradients of plastic deformation are imposed by the microstructure.
t This result can also be obtained directly from results given by Kroner
(1961, 1962), or Bilby (1960).
402 M. F. Ashby on the
A pure single crystal, correctly oriented and suitably stressed, deforms
in a fairly uniform way. Small deviations from uniform slip do occur,
but these deviations are due to the mutual trapping of dislocations
mentioned above, not the microstructure of the crystal. We call such
crystals ' plastically homogeneous '.
If deformable particles or zones are distributed in the crystal, these
deform when the crystal as a whole is strained ; they do not introduce
gradients of plastic strain. But if strong hard particles or phases are
distributed in the crystal, the deformation immediately becomes non-
uniform. No slip occurs within each particle or within the adjacent layer
of matrix, provided this layer is strongly bonded to the particle ; yet far
from a particle, the slip can be as large as we choose to make it. Such
particles introduce gradients of strain even when the crystal containing
them is deformed in a nominally uniform way, such as simple tension.
Any two-phase alloy in which one phase does not deform plastically,
or deforms less than the other phase does, is plastically non-homogeneous.
Single-phase metals may also be plastically non-homogeneous : each
grain in a polycrystal deforms by a different amount, depending on its
orientation and the constraints imposed on it by its neighbours. Even
a single crystal of a pure metal becomes plastically non-homogeneous
when strained into stage 111, as the rapid increase of Laue asterism
shows. However, it is convenient to restrict most of the following
discussion to crystals containing non-deforming particles, and to
polycrystals, because data are available for such materials.

0 2. THE DENSITYAND ARRANGEMENT OF DISLOCATIONS

IN PLASTICALLY NON-HOMOGENEOUS MATERIALS
It is helpful to consider the pattern of primary and secondary slipt in
plastically non-homogeneous materials in two steps. We imagine the
deformation divided into a general, uniform deformation, during which a
density pa of statistically-stored dislocations accumulate, followed by a
local, non-uniform deformation during which a density pG of geometrically-
necessary dislocations accumulate. This second contribution (pa) is
defined as the dislocation array which provides for compatible deformation
of the parts of plastically non-homogeneous material, and which has stresses
associated with it which nowhere exceed the local yield stress (which may be
much larger than the bulk yield stress). The procedure is to calculate
the displacements which are required for compatible deformation ; to
insert an array of dislocations which produces these displacements ;
and finally to check that the stress field due to these dislocations does
not exceed the local yield stress anywhere. This ' local yield stress ' is,
in fact, the stress required to nucleate some other dislocation array of
t Secondary slip means: slip on any system other than the primary slip
system; cross-slip is therefore secondary slip. A similar definition applies to
secondary dislocations: a dislocation is a primary dislocation only if it has the
primary Burgers vector and glides in the primary slip plane.
 Deformation of Plastically Non-homogeneous bfateriak 403

lower energy ; thus it may be the local stress required to nucleate cross-
slip, or to nucleate slip on some other slip system. This process gives us
the array of geometrically-necessary dislocations introduced by the
non-uniform flow.
Dividing the deformation into these two steps is equivalent t o dividing
the dislocation density stored during a given strain history into two parts.
The first ( p s ) is a characteristic of the material, that is, of the crystal
structure, shear modulus, stacking-fault energy, etc. The second ( p G ) is
a characteristic of the microstructure, that is, the geometric arrangement
and size of grains and phases ; to a first approximation it is independent
of the material.
We shall assume here that the total dislocation density is simply the
sum, ( p s + p G ) . This is obviously an oversimplification valid a t small
dislocation densities ; in general the presence of po will accelerate the
rate of statistical storage. Strictly speaking, the densities of dislocations
calculated in this paper are lower limits.
This procedure is now applied to some particularly simple examples.

2.1. The Dislocation Array required for Compatible Deformation

2.1.1. Equiaxed particles : the shear loop array
Consider the simple shear, y , of a ductile matrix containing strong
cube-shaped particles of side d . Divide the matrix up into imaginary
cells, such that each cell contains one particle, and which fit together
exactly to form the specimen as a whole ; the average cell dimension is
equal to the particle spacing. I n what follows we consider the deformation
of an average cell, which, externally, deforms by the shear y imposed on
the specimen.
Figure 2 ( a ) depicts a cell containing a rigid particle (the cell boundary
is not shown). Imagine the particle to be removed from the matrix cell,
which is then deformed uniformly by a shear y ; during this step
dislocations ( P - ~ are
) statistically stored in the matrix. We now wish t o
replace the particle without deforming it (or only elastically, and thus
by an amount which we neglect in comparison to the plastic strain in the
matrix) in the hole from which it came. Compatibility requires that the
hole be deformed back to its original shape.
Many possible sets of displacements can accomplish this. The simplest
method is to shear the part of the matrix immediately surrounding the
hole back by a n amount of y as shown in fig. 2 ( c ) where the required
displacements are shown as arrows. These displacements are achieved by
inserting n shear loops, surrounding the particles as shown. Neglecting
the elastic displacements (i.e. assuming a rigid particle and a rigid-plastic
matrix) geometry requires that
nb = yd, . . . . . . . . (2.1)
where b is the magnitude of the Burgers vector of the loops. I n practice,
the initial plastic shear will seldom be homogeneous (as we have assumed
404 M. F. Ashby on the
in fig. 2) but will occur on one or a few slip planes intersected by a particle.
The deformation, the restoring displacements and the resulting array of
shear loops then appear as shown in fig. 3 ( a ) and (b). Equation (2.1)
describes the number of loops a t an average particle.

Fig. 2

INITIAL D E F O R M 4
;'

SHEAR MODE

PRISMATIC MODE

I-TYPE LOOPS u* I V-TYPE LOOPS

(d)

Diagram illustrating the uniform deformation of a cube-shaped hole (a,6 ) and

the way in which shear loops (c) or prismatic loops ( d ) can produce
displacements which restore the hole to its original shape.

This array is the same as that postulated by Fisher, Hart and Pry (1953)
in their work-hardening model. It is acceptable only if the local yield
stress is not exceeded anywhere. The array of n loops exerts stresses of
order n(Gb/d),that is (from eqn. (2.1)) roughly G y , on the particle and the
material immediately surrounding it. If this local stress is less than the
stress required to nucleate cross-slip, or to generate new dislocations from
the particle-matrix interface, or to shear or fracture the particle itself, then
the shear loop array is stable. But subsequent strain generates more
loops and increases the local stress, which is always of order Gy.
Ultimately, one of these other nucleation stresses is exceeded (this was
called the ' local yield stress ' earlier) and the simple shear mode gives
way to another mode.
 Deformation of Plastically Non-homogeneous Materials 405

Fig. 3
DEFORMATION SHEAR MODE

TTTT

P R I S M A T I C MODE

!-TYPE LOOPS "? 1 V-TYPE LOOPS

u: I
(d)

When a single active slip plane intersects a particle (a),the array of shear
loops ( b ) and prismatic loops (c, d ) which produce the desired displace-
ments are as shown. Note that interstitial (I)or vacancy (V) loops can
occur on either side of a particle according to the scheme shown in the
inset drawing, depending on the sense of the cross-slip which produces
them.

Quantitatively, little is known about the magnitude of the local stress

field required to cause cross-slip a t a particle. However, experiments on
copper or brass containing small ( < 1000 A) particles (Hirsch and
Humphries 1969) show that cross-slip occurs before new dislocations are
generated from the particle-matrix interface. The nucleation stress for
this second process-the generation of new dislocations-has been
measured directly in alloys containing non-coherent particles (Weatherly
1968, Ashby, Gelles and Tanner 1969). It depends on particle size, and
is of order G/lOO for 1000 b particles in copper.
Typically, then, the nucleation stress for either cross-slip or the
generation of new dislocations will be reached before the local stress
4 06 M. F. Ashby on the
(roughly G y ) exceeds a stress of order G/lOO. This occurs at a strain y
of order 1%. Beyond this strain the shear array is unacceptable and
must break down.

2.1.2. Equiaxed particles : the prismatic array

Alternatively, the necessary displacements of fig. 2 can be obtained by
inserting prismatic loops ; these restore the hole to its original shape by
removing material from it and adding to it. One way in which this can
happen is shown in fig. 2 (d). Simple geometry shows that the excess
volume, AV, of matrix material on the left-hand side of the cube-shaped
particle, and which must be removed by prismatic glide, is :
A V = av,y, . . . . . . . (2.2)
where V , is the volume of the particle. This volume has to be injected
into the matrix as ‘ interstitial ’ prismatic loops, whose Burgers vector
is the same as that of the original primary shear. On the other side of
the particle an equal volume has to be supplied from the matrix ; this
is equivalent to generating ‘ vacancy ’ type prismatic loops. Many other
possible patterns of prismatic glide, some not involving the primary
Burgers vector, are able to remove and supply matter to the regions that
require it, and so satisfy compatibility (see, for example, Ashby 1970).
In particular, the microscopic mechanism involved in the formation of
the loops may favour all-interstitial or all-vacancy arrays. The important
point here, however, is that the total volume of material which must be
redistributed, and thus the total number of loops (regardless of their
character), is fixed by geometric considerations. If (as is usually the case)
primary shear occurs on a single plane intersected by the particle, then
the array of loops appears as shown in fig. 3 (c) and (d).
If we again assume that all displacements are plastic, then the
prismatic loops must precisely account for the volume excess and deficit
at the particle. The loops may not all be of the same size, but geometry
requires that their average area be close to +a2, where d is the particle
size. Then the total number of loops per particle nT (interstitial plus
vacancy) is given by :
inTbd2=y V,,
thus
n T = 2Yd
-.
b
. . . . . . . .
For spherical particles the constant 2 is replaced by a slightly smaller
constant.
The loop density, NT (the total number of loops per unit volume),
depends on the number of particles per unit volume Nv ; it is equal to
nTNv. Replacing Nv by flu? we obtain :
 Deformation of Plastically Non-homogeneous Materials 407

where f is the volume fraction of cube-shaped particles of size d. Each

loop has a periphery of about 4d, so that a crude measure of the
dislocation density in the form of loops is :
8fY
-. . . . . . . (2.4)
bd
The numerical factor 8 is obviously approximate ; it is the form of the
expression, which is dimensionally homogeneous, which is important here.
These prismatic arrays form by a microscopic mechanism involving
cross-slip (Hirsch 1957 ; for details see Hirsch and Humphries 1969,
Hirsch and Duesbery, to be published). They first appear when the
local stress a t a particle satisfies the conditions for the nucleation of
cross-slip : one shear loop frequently concentrates stress sufficiently to
do this.
These arrays are acceptable provided that the local yield stress is not
exceeded anywhere. This is true to a much larger strain than for the
array of glide loops : the long-range shear stress at the surface of the
particle due to the column of prismatic loops such as those shown in
fig. 3 ( c ) is maximum on planes which make 45" to the column axis, and
to a sufficient approximation (Appendix I), is given by :
~=Gfy. . . . . . . . . (2.5)
Again taking G/lOO as an illustrative value of the local yield stress required
to nucleate new dislocations, we find that the array is acceptable up to a
strain of order y = 10-2/f, or (for a typical value off = 4%) up t o a strain
of 25 yo. Observations support this conclusion. Humphries and Martin
(1967) and Hirsch and Humphries (1969) show prismatic arrays, stable
up to 15% strain, a t small (500 A) particles in copper.

2.1.3. Equiaxed particles : multiple slip, decohesion and fracture

The simple displacements associated with cross-slip do not reduce the
long-range stresses a t the particle to zero, they merely give lower stresses
than do the displacements associated with primary shear alone (roughly
Gfy instead of G y ) . The larger the strain, the larger these accumulating
stresses become. Ultimately the nucleation stress for slip on new slip
systems, or to cause decohesion of fracture, will be reached.
Except in the region close to a dislocation core, the largest stresses in
a plastic crystal containing strong particles generally occur a t the
interface between particle and matrix, because of the discontinuity of
elastic properties there. Nucleation of dislocations with new Burgers
vectors will occur here (assuming that dislocation sources with the
appropriate Burgers vector are not available locally in the matrix).
Prismatic generation, or local shear, which must now involve non-primary
Burgers vectors, will take place. These arrays, though complicated, still
satisfy compatibility and must therefore satisfy eqns. (2.3) and (2.4), a
408 M. F. Ashby on the
fact which is important in calculating the effect of particles on t h e
work-hardening rate. This, and its ability to cope with deformations
more general than simple shear (Appendix 11),illustrates the power of
this combined continuum and dislocation-mechanics approach.
Studies of this sort of generation of new dislocations from a n interface
(Gleiter 1967, Weatherly 1968, Brown, Woolhouse and ValdrB 1968,
Ashby et al. 1969) have shown the following:
( a ) Nucleation of new dislocations from a coherent interface is extremely
dificult and requires local stresses of order Q/6 ; we therefore anticipate
the cross-slip mode at such particles.
( b ) Nucleation of new dislocations a t an incoherent or weak interface
is relatively easy, requiring a stress of order a l l 0 0 decreasing as the
particle becomes larger. This means that a t large particles, multiple
slip should be easier than cross-slip, in agreement with the observation
that prismatic arrays are found a t small particles ( < 1000 A ; Humphries
and Martin 1967, Hirsch and Humphries 1969), but that multiple slip
arrays are found a t large ones ( > 1000 d ; Stobbs and Brown 1968).

Multiple
Shear mode Cross-slip mode slip mode

Strength of the
particlematrix
interface
Coherent or
strong
Coherent or
strong 1 Incoherent or
weak

Stacking-fault
energy
I Low 1
I
High I Low

Particle size Small Small Large

-
Volume fraction - Small Large
Small, usually Intermediate Large
Plastic strain 0-30% 5-1 00%
< 1%

Particle shape plates andat

needles
Favoured at
equiaxed I -
Of course, shear within the particle, decohesion at the particle-matrix
interface or fracture of the particle or matrix may intervene at any stage
of the deformation process if the local stress exceeds that required to
nucleate one of these failure modes.
The table summarizes how structural features influence the nucleation
stress for cross-slip and multiple slip, and thus determine the array of
geometrically-necessary dislocations which form. The strength of the
interface, stacking-fault energy and particle size influence the relative
ease of cross-slip and multiple slip. Volume fraction, f , and strain, y ,
 Deformation of Plastically Non-homogeneous Materials 409

influence the breakdown of the prismatic glide and start of multiple slip,
since the local stress a t a particle during the prismatic glide depends on
these two quantities (it is roughly Gfy). Particle shape is considered in
the next section.
Prismatic arrays involve no lattice rotation. Secondary shear arrays
(as opposed to prismatic arrays) do, however, produce local lattice rotation.

2.1.4. Plate-like and needle-like particles

When non-deforming plates or needles intersect the primary slip plane,
the array of geometrically-necessary dislocations changes. As the aspect
ratio of the particles (the ratio of length to thickness) increases, the
prismatic mode becomes less and less useful as a general way of achieving
compatible deformation.

Fig. 4

--X

i
In*
,PN

--XI

(bl

(a)A model composite consisting of non-deforming plates strongly bonded to

a single-crystal matrix. A t ( b ) the matrix is sheared on a single slip
system; close to the plates it cannot shear and must therefore rotate.

Consider first the idealized, composite structure shown in fig. 4 (a).

Plates are spaced 1 apart in a ductile single-crystal matrix. Suppose
that the plates are rigid (i.e. non-deforming), and that the matrix is
P.M. 2 D
410 M. F. Ashby on the
strongly bonded to them ; this means that no dislocations can enter the
particle-matrix interface, and that no sliding occurs in the plane of the
interface. Suppose the composite shears by slip in the matrix on a single
slip system, in a direction normal to the plate surface as fig. 4 shows.
The central part of each slip plane shears by an amount y. But the ends
of each slip plane are bonded to the plates which prevent their shearing :
the shear in an element of matrix immediately adjacent to a plate is
therefore zero, and the average gradient of shear on the single slip system
is y / ( l / 2 ) . Application of eqn. ( 1 . 1 ) immediately yields the average
density of geometrically-necessary edge dislocations :

As pointed out in 9 1.1, the gradient of shear on a single slip system

has a second consequence : it requires that the plates rotate through an
angle relative to the original slip plane normal N N ' , and the slip planes
themselves are bent from y through zero t o y again, as shown in fig. 4 ( b ) .
The matrix lattice acquires a curvature whose mean magnitude is 2yll.
X-ray techniques can measure this lattice rotation : for example, a
Laue reflection from the slip planes of the composite, which was sharp
before the deformation, should develop asterism of angular width of 2y
(for the same reason that a light beam, reflected from a mirror which is
bent through an angle 4, has an angular width of 24) about an axis in
the slip plane and normal to the slip direction. The rotation 4 must
have a wavelength equal to the particle spacing, I , so that the mean
lattice curvature is given by :

Since the geometrically-necessary dislocation density is Klb, we can

measure the geometrically-necessary component of the dislocation
density directly by measuring asterism and particle spacing.
Anomalous asterism, of the sort described above, is typical of alloys
containing plates and needles (Price and Kelly 1963, Carlsen and
Honeycombe 1954, Matsuura and Koda 1960, DeLuca and Byrne 1968,
Byme, Fine and Kelly 1961). One detailed study (Russell and Ashby
1968) has confirmed that the angular width of the asterism is indeed equal
to twice the shear strain, and that the density of geometrically-necessary
dislocations is given by 2ylb.l.
In the simple, two-dimensional case depicted in fig. 4, the dislocation
array has no long-range stress associated with it, and is acceptable up
to strains of order unity. Of course, this simple case is not very realistic.
But the two most important features of its deformation (namely that pG
is proportional to y / b l , and that relative rotations of parts of the lattice
by an amount y occur and can be measured) appear to be true of more
realistic materials containing plates and needles whose spacing is com-
parable with their length or width. Plates in a real dispersion-hardened
 Deformation of Plastically Non-homogeneous Materials 41 1

alloy differ from the idealized model in two important ways. First, they
form a three-dimensional pattern, each plate totally surrounded by
matrix materials as shown in fig. 5, and, second, they do not lie normal
to the primary slip direction. Both differences require modifications t o
the dislocation array t o obtain a stress field which is everywhere below
the local yield stress.

Fig. 5

A three-dimensionalpattern of plates whose spacing 1 is comparable with their

width L.

Figure 5 shows a three-dimensional arrangement of plates which

effectively divides the matrix up into boxes of side 1. Not only are edge
dislocations prevented from leaving the box (as in the two-dimensional
model sketch in fig. 4 ( b ) ) ,but screw dislocations, which were free t o run
out of the front and back surfaces in fig. 4 ( b ) , are trapped also. This
precisely doubles the length of geometrically-necessary dislocations
stored during a shear y on the primary slip plane, giving, in place of
eqn. (2.6) :

To proceed further, we apply the general method outlined in tj 2.

Remove a particle from its hole and apply a uniform primary shear y
to the matrix as shown in fig. 6 ( a )and ( 6 ) . Because it did not lie normal
t o the slip direction, the hole becomes longer, by an amount L y cos dsin 8,
and thinner by an amount - t y cos 0 sin 8. Since the particle is non-
deforming, we must seek displacements in the matrix which will precisely
restore the hole to its former shape. One of the many possible sets of
displacements, and a section through the dislocation array which produces
it, is shown in fig. 6.
This array is acceptable only if its stress field is less than the local
yield stress. The array shown a t fig. 6 ( c ) has much in common with
a pair of finite, low-angle dislocation boundaries of separation w , whose
stress field can be obtained by superposition from the result for a single,
finite boundary given by Li (1963). Except near the edges of the plate,
the stresses are low. Within a distance w / 2 of the edges, large stresses
2D2
412 M. F. Ashby on the
Fig. 6
t - t y cos 8 gin8

(C)

When the plate shown at (a)is removed from its hole, and the matrix sheared
uniformly ( b ) , the hole becomes longer and thinner. The displacements
and array of dislocations shown at (c) restore the hole to its former shape.

exist, and further slip, modifying the array, must occur here. But the
affected region decreases in size as the aspect ratio l / t of the plate
increases, and so edge-effects at thin plates will not contribute in an
important way to pO. Indeed, as the plate is made thinner, so that w
decreases, the array becomes a stack of edge dipoles. A t particles whose
broad faces lie more or less parallel to the slip direction, a corresponding
stack of screw dipoles forms. It thus appears that the energy of shear
arrays decreases as the aspect ratio E/t of the plates increases, while it is
trivial to show that the size of the equivalent prismatic loops, and thus
their energy, increases. In summasy, shear arrays are favoured at thin,
broad plates.

2.1.5. Other plastically non-homogeneous materials : polycrystals, eutectics,

etc .
A polycrystalline specimen, even of a pure metal, deforms plastically
in a non-uniform way. Laue reflectiohs from a, single grain rapidly
acquire asterism. The interior of each grain deforms predominantly by
single slip, and in a region of either side of the grain boundary the lattice
is rotated (indicating gradients of simple shear) and secondary slip occurs
(Essmann, Rapp and Wilkins 1968).
Consider the tensile deformation of a polycrystal (fig. 7 ( a ) ) or of a
two-phase alloy in which both phases are able to deform. As before,
we will separate the deformation of each grain into a uniform deformation,
 Deformation of Plastically Non-homogeneous Materials 413

achieved by shear on one or more slip systems, and a set of local, non-
uniform deformations.
Consider the strain in the nth grain, which shears by yen) on a slip
system of unit normal hi in the direction b j . If coordinates are chosen
parallel to, and a t right angles to the tensile axis, then

Suppose the tensile strain is E, then on average each grain elongates by

this amount and eij(n)becomes eij(n)= E Aij(n),where Aij(n) is a function
of direction cosines hi and bj only, and relates the orientation of the grain
to the orientation of the specimen.
Fig. 7

If each grain of a polycrystal, shown at (a),deforms in a uniform manner,

overlap end voids appear ( 6 ) . These can be corrected by introducing
geometrically-necessary dislocations, as shown at ( c ) and (d).

Each grain is characterized by a different Aij. If each grain is caused

to undergo its uniform strain, the result, as shown in fig. 7 ( b ) , is that
overlap of material occurs in some places, and voids appear in others.
Note that, since each grain was deformed uniformly, the dislocation
density pS and the state of work-hardening within i t must be the same
414 M. I?. Ashby on the
as that of an ' equivalent single crystal ', i.e. a single crystal so oriented
that it deforms on the same slip system or systems as the grain under
consideration. (The grain-size-independent features of the polycrystal
stress-strain curve can be explained reasonably as a suitable average over
many ' equivalent single crystals ' (Kocks) 1970.)
The difference in strain between neighbouring grains, the nth and the
(n+ l)th, is then Aeij = < (Aii(n)-Aii"+'). The origin is taken at the centre
of the nth grain, of diameter D. Then the average relative displacement
of a point xi = (D/2)5$on the surface of this grain, relative to ,the point
on the surface of the neighbouring (n+ 1)th grain which, before straining,
coincided with xi, is a suitable average of the modulus of Aui=Aegixi
(since the strains were uniform). Substituting gives for the average
overlap or mismatch between the grains
- ZD-
lAu*l= 2 {A4+f,h
where AAii = Aij(n) - Aii("+1). The bar indicates an average value, the
average being taken over the surface of one grain (i.e. over all Z,) and
over all grains (i.e. over all A$i). The term in curly brackets is a purely
geometrical constant. It is the mean value of a function of direction
cosines only, and thus is independent of strain and grain size. For the
purpose of further discussion we shall assume i t to be equal to unity.
As in earlier sections of this paper, we now treat the grains as rigid-
plastic and restore compatibility according to the scheme of non-uniform
deformation illustrated in fig. 7 (c). When overlap can be corrected by
shear displacements, introduce local shear by running dislocations in
from the boundary. When normal displacements are required, introduce
dislocation pairs (prismatic loops). Voids are corrected by the same
process, using dislocations of opposite sign. A combination of the two
processes can correct overlap or voids of any shape ; the dislocations
involved are geometrically necessary. This local, non-uniform deformation
occurs on each grain face, and is different on each, leading to the
complicated pattern shown in fig. 7 (d).
Since the amount of overlap or void is always proportional to DZ/2,
the number of geometrically-necessary dislocations pumped into each
grain during this part of the deformation is roughly D ~ / 4 bwhere
, b is
the magnitude of the Burgers vector. The area of a grain in this two-
dimensional analysis is roughly D2, so that the density of geometrically-
necessary dislocations is :

Generalization of this calculation to a three-dimensional array of

grains or phases leads to the same result. p G measures the dislocation
density over and above that found in an equivalent single crystal, and
thus (on this model) is the origin of the grain-size-dependent part of the
work-hardening in polycrystals. It can lead to work-hardening which
 Deformation of Plastically Non-homogeneous Materials 4 15

varies as D-ll2, as shown in 3. Conrad, Feuerstein and Rice (1968)

have obtained an expression of this form by a different argument,
involving the assumption that the slip distance for dislocations is
proportional t o the grain size.
Measurements of dislocation densities in deformed polycrystals do
seem to follow this law. Unlike that in single crystals, the density is
found to increase linearly with tensile strain, and a t a given strain, t o
increase as the reciprocal of the grain size (Essmann et al. 1968, Keh and
Weissman 1963, McLean 1967, Conrad et al. 1967, 1968, Jones and
Conrad 1969).

2.2. The Geometric Slip Distance and the Relative Magnitude of pG and ps
The total dislocation density in a crystal is the sum of the geometrically-
necessary dislocations ( p G ) and the statistically-stored dislocations ( p s ) .
The two dislocation storing mechanisms may interact, so that the
presence of pG influences ps, but for simplicity we assume that this does
not happen. Then qualitatively, we expect properties which depend
on p , such as the flow stress, to reflect the presence of the geometrically-
necessary dislocations when their density pG dominates the total density.

2.2.1. The geometric slip distance

The effectiveness of particles, or plates, or grain or phase boundaries
in causing dislocations to be stored is conveniently described by the
geometric slip distance. Non-deforming plates, for example, cause a
density

of geometrically-necessary dislocations to be stored (eqn. (2.8)). Equiaxed

particles require instead a density of approximately :

(f)
,;
where r = d / 2 is the radius of a spherical particle (eqn. (2.4)).
The quantity 1 for plates, and rIf for equiaxed particles, we call the
geometric slip distance, hG. It is analogous to the quantity As, the slip
distance for statistical storage defined by the analogous (differential)
equation :

(As is often regarded as the diameter, in a pure crystal, to which a dislocation

ring, generated by a source, expands before it stops for good.) The
accumulation of geometrically-necessary dislocations is then described
in a general way by :

(2.9)
416 M.F.Ashby on the
In polycrystals, or structures consisting of two primary phases, AG is
proportional to the grain size, or phase separation.
The geometric slip distance, AQ, is a characteristic of the microstructure,
and is independent of strain. As, on the other hand, does vary with
strain, and can be measured from the length of slip lines (AG cannot).
Published measurements of As in pure copper single crystals show it to
vary from 1OOp or more in stage I to less than l o p in stage 111 (e.g.
Mader 1963). In alloys in which the microstructure imposes a geometric
slip distance which is larger than As, we expect little effect of the

Fig. 8

The statistically-stored (shaded band) and geometrically-necessary dislocation

density, plotted against strain. Note that p' can dominate the total
density at small strains but can be swamped by the statistically-stored
dislocations at larger strains.
 Deformation of Plastically Non-homogeneous Materials 417

microstructure on the work-hardening. But when hG is much less than As

(in practice this means h G is less than a few microns), we expect the
microstructure to have a strong effect on dislocation storage, and thus
on work-hardening.
2.2.2. The relative magnitude of pQ and p s
Figure 8 shows schematically, as a shaded band, the increase of
statistically-stored dislocations with tensile strain in pure copper single
crystals. The band was constructed from published measurements
(Basinski and Basinski 1966). In such crystals no geometrically-necessary
dislocations are stored.
The variation of pG with strain for a number of values of hG is also
shown. Because pG increases linearly with strain, it may dominate the
total density at small strains, only to be swamped later by the
statistically-stored dislocations whose density in stage I1 increases roughly
as y 2 . This reversion to pure single-crystal behaviour a t larger strains is
apparent in the stress-strain of certain Cu-SiO, alloys (AG NN 20 p )
discussed in 5 3. When XG is small, less than 2 p , then pG dominates
the total dislocation density a t all strains. At the other extreme, values
of hG greater than l o o p will contribute almost nothing to the dislocation
density. Recovery reduces the value of pG.

5 3. WORK-HARDENINQ
Work-hardening reflects the way in which the arrays of stored
dislocations, both the geometrically-necessary and the statistically-stored
ones, obstruct the motion of other moving dislocations. Dislocations of
the array act as individual obstacles (a ' short-range ' interaction) and,
collectively, as sources of long-range internal stress.

3.1. The Relative Contributions of ps and pG

The stress-strain curve of a pure copper single crystal is shown in fig. 9
a t 1. This work-hardening is due entirely to statistically-stored
dislocations. The curves labelled 2, 3 and 4 in this figure show the
effect of increasing amounts of hard spherical SiO, particles. The
low-strain part of the curve is affected first, as curve 2 shows. I n this
crystal the geometric slip distance XG was about 2 0 p and, as expected
from fig. 8, the crystal reverts to ' normal ' single-crystal behaviour after
a strain of about 15%. Increasing the volume fraction of particles
reduces ha to about 2 p (curve 4). Both stage I and stage I1 have now
disappeared, and the entire stress-strain curve is controlled by the
presence of geometrically-necessary dislocations.
This is shown in a more general way in fig. 10. Here the rates of
work-hardening of a number of dispersion-hardened copper single crystals
are plotted against the geometric slip distance, Xu. Note how the work-
hardening rate climbs away from that characteristic of pure f.c.c. crystals
as the geometric slip distance Xu is reduced below the value of As.
418 M. F. Ashby on the
Fig. 9

SHEAR STRAIN,

Stress-strain curves of a pure copper single crystal (1) and of identically

oriented crystals containing, respectively, +yo,4% and 1% of spherical
SiOa particles, of diameter about 900 A. From Ebeling and Ashby
(1966).

3.2. The One-parameter Work-hardening Theory

The theory of 5 2 predicts that long-range internal stresses, with a
wavelength equal to the particle spacing, form during the first few per
cent of straining in plastically non-homogeneous alloys. Their presence
is confirmed by the x-ray measurements of internal stress in dispersion
strengthened alloys (Wilson and Konnan 1964, Wilson 1965) and by the
large Bauschinger effect found in such alloys (Wilson 1965, Abel and Ham
1966). However, equilibrium requires that the average internal stress
be zero, and it is not, at present, clear which moment of the long-range
internal stress best describes its contribution to the flow stresst. As
pointed out in 5 2, the amplitude of the internal stress, which may
increase rapidly during the first 1 % of plastic strain, is limited by the
onset of cross-slip or multiple slip. Thus the contribution of long-range
internal stress of the flow stress should be greatest at small strains.
The short-range interaction between an array of stored dislocations and
a gliding dislocation depends on the density, type and arrangement of
dislocations in the array. But to get at least a rough idea of the short-
range contribution to work-hardening, and retain simplicity and
generality, it is convenient to describe the array by one parameter only :
its average density. To some extent the arrangement can then be

t I am indebted to Dr. L. M. Brown for a helpful correspondence on this point.

 Deformation of Plastically Non-homogeneous Materials 419

allowed for later by adding an appropriate contribution describing the

long-range back stress, if it can be estimated.

Fig. 10

o JONES AND KELLY,JONES

- 0 HUMPHRIES AND MARTIN
A EEELINC AND ASHEY

-
+SLIP DISTANCE -A'
IN PURE FCC METALS
- - ---- - -__ _ _ __ - -
Q

T- *o O
- I N PURE FCC METALS
0 O
B A A
O

1 000

GEOMETRIC SLIP DISTANCE A' (cms)

The rate of work hardening, 0, at 5% shear strain, in units of the shear modulus,
plotted against the geometric slip distance for a number of dispersion-
hardened alloys. The results are plotted from data of Jones and Kelly
(1968), Jones (1969), Humphries and Martin (1967) and Ebeling and
Ashby (1966).

Within the framework of this one-parameter approximation, only

one dimensionally homogeneous relationship between flow stress, T , and
the total dislocation density, pT, is possible (Nabarro, Basinski and Holt
1964), namely :
T =To +C G b d p T . . . . . . . (3.1)
r0 includes other contributions to the flow stress (e.g. solution hardening),
G is the shear modulus, b the magnitude of the Burgers vector, and C is
a constant which more detailed calculations show t o be about 0.3.
The total dislocation density, pT, is simply ( p G + p s ) . When p'
dominates, as it does when the geometric slip distance, XG, is sufficiently
small, the total dislocation density is given by eqn. (2.9). Inserting this
into the one-parameter work-hardening expression yields :

T = T o + 2 ~ ~ J(g). , . . . . (3.2)
420 M. F. Ashby on the
Since hG is independent of strain (it is a characteristic of the micro-
structure), this equation predicts that plastically non-homogeneous
materials (in which A" is less than As) should show a parabolic stress-strain
curve. Experimentally, both dispersion-hardened alloys and polycrystals
do show roughly parabolic hardening.

3.3. Stress-strain Equations

By substituting the appropriate geometric slip distance into eqn. (3.2),
we obtain stress-strain expressions from the one-parameter theory.
Assuming no recovery effects, XG for well-separated, equiaxed inclusions
is given by r / f , thus ;

where f is the volume fraction, r the particle radius, b the Burgers vector,
y the shear strain and G the matrix shear modulus. Note that both size
and volume fraction of particles influences the flow stress.
For plate-like particles, the geometric slip distance is equal to the plate
spacing, 1 ; thus :

The work-hardening increases as 1, the plate spacing, decreases.

In polycrystals, or in alloys containing two primary phases, the appro-
priate geometric slip distance is proportional to the grain size, D , giving :

Note that this yields a ' ' law for work-hardening in polycrystals.
Experimental support for these equations is reviewed elsewhere (Ashby
1970). Equations (3.3) and (3.4) do seem to describe the stress-strain
behaviour of dispersion-hardened alloys a t low temperatures. A t room
temperature and above, recovery can lead to time and temperature
dependent stress-strain behaviour.

5 4 . SUMMARY
1. Many two-phase alloys deform plastically in a non-uniform way,
because slip in the two phases is not equally easy. One phase, often
dispersed as discrete particles, deforms less than the other, or not at all.
This means that gradients of deformation are set up, with a wavelength
equal to the particle or phase spacing.
2. We define a ' plastically non-homogeneous alloy ' as one in which
gradients of deformation are imposed by the microstructure.
3. As such an alloy is deformed, an increasingly dense array of dislo-
cations has to be stored in it to accommodate the gradients of deformation
and thus allow the two phases to deform in a compatible way. We call
 Deformation of Plastically Non-homogeneous Materials 421

these dislocations ' geometrically-necessary ' dislocations t o distinguish

them from the ' statistically-stored ' dislocations which accumulated by
randon mutual trapping in a pure crystal. The former are characteristic
of the microstructure, the latter are characteristic of the material.
4, The arrangement and density of the array of geometrically-necessary
dislocations can be derived. The procedure is to calculate the displace-
ments which are required for compatible deformation ; t o insert an array
of dislocations which produces this displacements ; to calculate the stress
field of this array ; and finally to check that this stress field does not
exceed the ' local yield stress ' anywhere. This ' local yield stress ' is
usually the local stress required to nucleate cross-slip, or t o generate
dislocations with new Burgers vectors a t the particle-matrix interface,
whichever is lower. When this local yield stress is exceeded, a new array
must be sought which gives the same displacements but has lower stresses
associated with it.
5. This procedure is applied, in the text, to materials containing non-
deforming particles and plates, and t o polycrystals. It leads to an
understanding of the dislocation arrays observed by transmission electron
microscopy, and of the lattice rotations observed by x-ray techniques.
6. Plastically non-homogeneous materials are characterized by a
'geometric slip distance ' hG, analogous to the slip distance in pure
crystals, but determined only by the microstructure. The density p G of
the geometrically-necessary dislocations is then given by :

where b is the Burgers vector and y the shear strain. For alloys
containing more or less spherical particles, XG=rlf, where r is the
particle radius and f the volume fraction. For alloys containing broad,
plate-like particles, XG is equal to the plate spacing. I n pure polycrystals,
it is proportional t o the grain size.
7 . When the density of geometrically-necessary dislocations exceeds
that of statistically-stored ones, the former control the stress-strain
curve. They contribute to hardening in two ways. First, geometrically-
necessary arrays in general have a long-range stress field, with wavelength
equal t o the particle spacing associated with them. There is evidence
that this long-range stress field controls the hardening during the first
1 or 2% strain, but thereafter remains more or less constant. The rest
of the stress-strain curve is then controlled by short-range interaction
between moving dislocations, and the steadily increasing density of
geometrically-necessary dislocations. This leads t o a characteristic
parabolic stress-strain curve, whose slope (at a given strain) depends on
the size and spacing of the particles in a predictable way.
8. This sequence can be interrupted at any point by shear or fracture
of the particle, or decohesion a t the particle-matrix interface.
422 M. F. Ashby on the

ACKNOWLEDGMENTS
This work was supported in part by the Advanced Research Projects
Agency under Contract SD-88, in part by the Office of Naval Research
under Contract N00014-67-A-0298-0010
' and by the Division of
Engineering and Applied Physics, Harvard University.
I particularly wish to thank Professor P. B. Hirsch, F.R.S., Dr. F. J.
Humphries, Dr. L. M. Brown, Dr. W. M. Stobbs for permitting me
to quote their unpublished results, and to Professor P. B. Hirsch,
Professor F. A. McClintock, Dr. L. M. Brown, Mr. H. Frost and
Miss K. Russell for valuable comments and criticism.

APPENDIX I
We calculate the average principal stresses a t the particle surface
caused by a column of length I' of prismatic loops, each of radius d d 2 .
(Here I' is, strictly speaking, the linear separation of particles since the
loops lie along crystallographic directions in the matrix ; it is related to
volume fraction f by the expression dll'=f.) To obtain a lower bound
for the principal stresses, imagine that the tube is compressed longitu-
dinally by a displacement u=nb, where n is the number of prismatic
loops encircling the tube, that the cylindrical surface of the tube is free
to slide, but is constrained by the matrix so that it cannot expand
radially. Then Hooke's law gives the principal stresses :
an axial stress :

ug= * 2G( 1 - V ) nb
(1-2v) 7'
and two equal radial stresses :
2Gv nb,
or= 5 --
1' '
(1-2v)
where G is the shear modulus and v Poisson's ratio ; the positive sign
refers to vacancy loops, and the negative sign to interstitial ones. The
maximum shear stress at the particle surface lies at 45' to the cylinder
axis and, substituting n =inT and dll'=f, we obtain
nb
T~~~ =G -
I' '
= Gfy.

An upper bound is obtained by assuming the tube is free to expand

laterally when it is compressed longitudinally ; the resulting shear stress
+
is larger by a factor of only (1 v ) .
 Deformation of Plastically Non-homogeneous Materials 423

A P P E N D I X I1

Suppose a cell of matrix, containing one particle, is subjccted t o a

uniform general plastic deformation, eii, large compared to the elastic
deformation of the particle. Choose a Cartesian coordinate system given
by the principal axes of the strain. Assume that the particle is cube-
shaped of side d and happens to lie with its flat faces normal to these
axes (a mathematical convenience only). Suppose, of the principal
strains, e l l * , e2,*, e3,*, the first is negative (a compression) and the
other two are positive. To restore the hole to its original shape, a volume
e,,*d3 must be injected into the matrix, and a volume (e,,*+e,,*)d3
must be withdrawn from it, in the form of interstitial or vacancy loops
of area d2. The total number of loops is therefore :

Suppose, in addition, that the particle can also deform either elastically
or plastically b u t by an amount which differs from the deformation of
the matrix. Suppose its principal strains are E l l * , E,,*, E3,*, then
d
n T = 6( l e l l * - E l l * l + Je,,*-E,,*(+ le,,*-E,,*I).

For simple shear (ell* = -e2,* = y , e3,* = 0 ) and a non-deforming particle

( E l l * = E z 2 *= E3,*= 0) the expression reduces to the one we had before.
But for equal biaxial plastic compression (ell* =e22*, e3,*= -2el,*) the
result is four times greater.

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