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Linear dislocation arrays in heterogeneous materials
LINEAR
DISLOCATION
ARRAYS
IN
HETEROGENEOUS
MATERIALS*
Y. T. CHOUt A simple class of dislocation pile-ups in materials composed of soft and hard phases is examined. The analysis is based on the assumption that an array of discrete dislocations may be replaced by a oontinuous distribution of elementary dislocations. Two special oases considered are (i) a pile-up of screw dislocations in the soft matrix against a welded boundary, (ii) a pile-up of edge dislocations against a slipping boundary, i.e. a boundary which can transmit normal but not shear stresses. It is shown that, in a heterogeneous material, the length of the piled-up array depends linearly on the number of dislocations in the pile-up, as is the case in a homogeneous material. The number of dislocations in a given array, however, is generally less in the heterogeneous then in the pure matrix material. The results indicate that the two-phase material would work harden more than the pure matrix, provided the mean free path is not small. ARRANGEMENT
LINEAIRE MATERIAUX
DE DISLOCATIONS HETEROGENES
DANS
LES
L’auteur examine un cas simple d’empilements de dislocations dans les mat&iaux composes de phases dures et mains dures. L’analyse est basee sur l’hypothese qu’un arrangement de dislocations disc&es puet Btre assimile 8. une distribution continue de dislocations elementaims. Deux cas speciaux sont consider&, a savoir (i) un empilement de dislocations-vi8 dans la matrice non dure contre une frontiere soudee, (ii) un empilement de dislocations-coins contre une frontier-e de glissement, c’est-Q-dire une frontiere qui peut transmettre des tensions normales mais non de cisaillement. L’auteur montre que, dans un materiau heterogene, la longueur de l’arrangement empile depend lineairement du nombre de dislocations dans l’empilement, comme c’est le cas dans un materiau homogene. Le nombre de dislocations dans un arrangement don&, cependant, est gen&alement moindre dans le materiau heterogene que dans la matrice pure. Les resultats indiquent que le materiau 8. deux phases serait plus sensible au durcissement d’ecrouissage que la matrice pure, pour autant que le libre parcours moyen ne soit pas petit. LINEARE
VERSETZUNGSANORDNUNGEN
IN HETEROGENEN
MATERIALIEN
Eine einfache Klasse van Versetzungsaufstauungen in Materialien, die aus weichen und harten Phasen zusammengesetzt sind, wird untersucht. Die Untersuchung geht van der Annahme aus, da13eine Anordnung diskreter Versetzungen durch eine kontinuierliche Verteilung elementarer Versetzungen ersetzt werden kann. Zwei Spezialfalle werden betrachtet: (i) eine Aufstauung van Sohraubenversetzungen in einer weiohen Matrix an einer SchweiOgrenze, (ii) eine Aufstauung van Stufenversetzungen an einer Grenzflache, die Normalspannungen, aber keine Schubspannungen iibertragen kann. Es wird gezeigt, dal3 in einem heterogenen Material die Lange der Aufstauung linear van der Zahl der Versetzungen in der Aufstauung abhangt, wie es in einem homogenen Material der Fall ist. Jedoch ist die Zahl der Versetzungen in einer Aufstauung im heterogenen Material allgemein kleiner als in der reinen Matrix unter sonst gleichen Bedingungen. Die Ergebnisse besagen, da13das zweiphasige Material sich starker verfestigt als die reine Matrix, vorausgesetzt, da9 der mittlere Laufweg nicht klein ist.
INTRODUCTION
With
the increasing
dislocations
need for ultra high-strength
materials in modern technology,
it becomes necessary
to obtain a better understanding
of the properties
of
composite materials and of the theoretical background underlying these properties. For crystalline materials in particular, and
their
the study of the behavior
interactions
elements in these importance.
with
materials
various
basic
treatment
In the present paper
we treat a simple case of the problem
is useful in understanding
of
havior of composite
analytically
by
to arrays conThe information
the flow and fracture
be-
materials. ANALYSIS
The Hookean
elastic field of
Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corporation Research Center, Monroeville, Pennsylvania.
6
is concerned with small arrayst4).
heterogeneous
is evidently
Head’s
of linear arrays, however, is essentially numerical and
of dislocations
* Received December 14, 1964. t Divisionof Engineering, BrownUniversity. On leave from
ACTA METALLURGICA,
arrays in inhomogeneous
by Headtzp4).
an approximate method applicable taining large number of dislocations.
A detailed study of the non-Hookean interactions of dislocations with lattice inhomogeneities was made by Jaswon and Foreman(r).
and dislocation
media was analyzed
VOL. 13, JULY
1965
Consider an infinite isotropic medium of shear modulus G, and Poisson’s ratio y1 for x > 0, and G,, y2 for x < 0. The boundary at x = 0 can be either a welded interface, or a slipping interface which can transmit normal but not shear stresses. For a positive screw dislocation with Burgers vector b, sitting on the x-axis at a distance a from a welded boundary, 779
Headc2) gives
ACTA
780
METALLURGICA,
VOL.
(i) For x > 0
13,
1965
Under the assumption
K = 1, equations
(5) reduce
to
x+a
SK
(x + aI2 + Y2
)
(la)
+K ’ (x - d2 + Y2 (x + aY + y2 ’ (lb)
where
A, = G,b/27r, K = (G, -
GJCG,
(6) Equations
(6) also hold for edge arrays piled up against
a slipping
boundary(3)*,
(3)
+ G,),
with A, being replaced
A, = G,b/277(1 Note that o refers to the oI, Mathematically,
x-a (x
a)2
_
equations
given by equations +
y2
(4a)
J
rium equations
-41
=
For a positive
+ K)
’
edge dislocation
W)
a)2 + y2 .
(x -
at (a, 0) the expres-
However,
in the limiting case, K = 1, as shown below
the magnitudes of the shear stresses on the slip plane, o,, of a screw dislocation in front of a welded interface and cr,, of an edge in front of a slipping interface, identical
except for a constant
Now suppose that n parallel positive
axis.
for a homogeneous
array which contains under
0,
a
stress,
of the dislocations
By the approximation
At present, numerical
3
,...,
n
distribution,
to the following
A
L f(x) s -Lx-t
singular
dx = o(t),
(8) function
of the
L), a(t) is the stress function
with (5)
a(t) = o
for t > 0,
equations
(5) can be solved
as has been
only
by
a(t) = ---CT
for t < 0,
demonstrated
by
Headc4). If, however, our concern is with the nature of these arrays one can solve a limiting case analytically by an approximate
The approximation
is
made by replacing an array of discrete dislocations
by
method.
distribution
of elementary
with the same total Burgers vector(5-g). an analytic
be
integral equation
simulated array in (-L,
z
if K <
must
(0, L).
methods,
a continuous
which
of a continuous
(6) are equivalent
are given by j=l,2
i#j
in the domain
oim(x, o, K),
where f (x) is the unknown distribution
5 L+K{$l*,=o, +lxj - xi
of screws of
if K > 0 or negative
determined.
under an applied stress o, then the equilib-
rium positions
sets of screw dislocations
the same number
strength lK\ A,, positive
points 5r, x2, . . . , x, are piled up against a welded boundary
to the equilib-
medium of G, and
n positive screws of strength A, under the The other of span (-L, 0) is an imaginary
equations screws at the
(5) with the special case
The one with a span (0, L) is the real array
stress -0.
are
factor.
for screw
for edge arrays.
are piled up against each other at the origin on the xcontaining
sions for the stress field are much more complex(3).
by (7)
(6) are equivalent
v1 in which two symmetrical u zz
YJ.
component
arrays and to the o,, component
41 + K)
=
izi
(2)
(ii) For z < 0
u ?Jz
j = 1, 2, . . . , n.
expression
can be obtained
dislocations On this basis, by applying
the inversion theorem of singular integral equationsoO). Let us consider the case in which G, > G,, such that the parameter K approaches unity. This is an extreme case which applies to a heterogeneous material consisting of a rigid phase embedded in a soft matrix. A practical example close to the limiting case is the WCCo system studied by Gurland and co-workers(11*12).
(9)
A = A, for screw arrays and A = A, for edge arrays. The solution of equation (8) can readily be obtained by
using
the
Muskhelishvili
inversion
theorem
and co-workers
formulated
(see Appendix).
by For
* In the case K = 1, the shear stress uw of a positive edge dislocation lying on the z-axis at a distance a from a slipping boundary is given by@’ CTa-
_
(z - a)[(x - a)2 - Y21
A
e i
[(5 - a)” + YT
+ (z + a)[@ + a)” - Y21 , [(z + aj2 + Y212 1 Here, K is defined by
2
> 0.
CHOU
: DISLOCATION
ARRAYS
IN
the simulated arrayf(x) is bounded at both ends of the segment (-I;, L). Thus, the inversion formula gives
f(x) =
$
Gosh-f /4/ .
W)
Equation (lo), with the range of x specified in (0, L), gives the distribution of dislocations in a linear array piled up at the interface in a heterogeneous medium. The number of d~locatio~, n, in the range (0, L) is therefore
n, =
s)(x)
ax=
2
)
a
(12) \ I
The corresponding expressions for the array in a homogeneous material* are(13*@
[gyZ,
f(x) = f
no=-,
and
MATERIALS
781
DISCUSSION
The above analysis of the limiting case, when coupled with the results of Head, may lead to a more general relation. Head’s data for small screw arrays (see Fig. 2) clearly indicate a linear relationship between L/nk and K, i.e. a i A, where
L - = tt + p, nk
1
(17)
(11)
where the subscript attached to n refers to the K values. The range of the army, L, may be expressed in terms of n, and TVby L=&nlA.
HETEROGENEOUS
(13)
La 2A
L1-2n,B* a
and
in,
A,
n,l
’
It is not difficult to see that equation (16) holds for large n’s. As the number of dislocations increases, the direct effect of the image array becomes more remote on the dislocations near the tail of the array. Thus, as n increases, the nth ~location senses much less the change in K. This indicates that, if the linear relation is valid for small arrays, it should remain valid for large arrays. Furthermore, the slope of the lines in Fig. 2 for the nth dislocations should approach the limiting slope of (rr - 2). This trend is observed in Fig. 2. It is therefore quite plausible that equation (16) is valid in general.
Figure 1 shows the curves of the distribution functions given by equations (10) and (13). UNIT
3
OF
LENGTH: 1
/
2
As/o-
0
/
/MT /
L
c” 2
I
0.5
[=
I .o
X/L
Fro. I. Distribution of s lmear dislocation array under a constant applied stress, o. (a) Homogeneous medium, equation (13). (b) Heterogeneous medium, K = 1, equation (10). * In this case the head dislocation is assumed to be locked in a stress field proportional to 6’(z), where 6’ is the derivative of Dime’s 6 function.
’
L 0
V.5
_ I.U
K FIG. 2. Relation between L/m%and K for screw arrays.
The solid lines are reproduced from the work of Head.“) The dashed line is obtained from the present analysis valid for all sufficientiy large 12’s.
ACTA
782
METALLURGICA,
Now, for large n where the approximation of the continuous distribution is warranted, equation (11) and (14) give u = 2, /3 = (77- 2)
= 2 + (7r -
2)K,
0
SK
5
1.
1965
Thus, we have (i) Near the tip of the array the stress distribution is approximately
0>2>-L.
(19)
It seems likely that a similar general relation may hold for edge arrays, since the extra terms in the equilibrium equations have much less effect on x,. More numerical computations are worth doing to support these points. In comparing equations (19) and (14), it is seen for the same applied stress and span of distribution that the number of dislocations in a piled-up array is less for a material containing hard particles than for the pure matrix material. * In the limiting case the ratio is 2177. Thus, the heterogeneous material would accomodate less plastic flow under loading, provided that the mean free path through the matrix is not too smal1.t This may probably constitute the primary hardening mechanism in the early stage of plastic deformation(14). The tip stress of an array of n dislocations piled up at an interface in a material with hard phases is simply n(~ as given also for a homogeneous material by the virtual work principle(15). The stress concentration factor due to an array is therefore smaller in a heterogeneous material than in a homogeneous material. The ratio of the factors is again 2177in the limiting case. If crack nuclei once initiated from the pile-ups grow as Griffith cracks, a heterogeneous material would exhibit a higher fracture strength. Strictly speaking, in the limiting case (K = 1) the stress is not uniquely determined in the half plane x < 0. It is nevertheless interesting and informative to see the characteristics of the stress distribution ahead of the array as the limiting condition is approached. As K -+ 1, the shear stress o,, at any point on the negative x-axis due to a screw pile-up approaches ~~~(x,O)=-~[(eosh-l/~~]z+%),
13,
agz(x,O)f --~(~n~l]“+%~,
and we have the general relation for screw arrays -
VOL.
(21) (ii) At a large distance from the array the stress is proportional to x-l, aJx,
0)
5
2ak )
(22)
xg-L,
77X
as is expected. In a homogeneous material the stress induced in front of the array is given byt6) ..,(X,O)=-~(~~]“‘-l),
x
(23)
where the applied stress -a is being excluded. Corresponding to equations (21) and (22) we have near the tip
(24) and at a large distance a2/&x, 0) i
%4 ,
These results indicate that hard particles the stress very is generally smaller than in However, at a large distance slightly greater.
x<--L.
in materials containing near the tip of the array homogeneous materials. from the tip the stress is
ACKNOWLEDGMENTS
The author wishes to thank Professors D. C. Drucker, C. Elbaum and J. Gurland for helpful discussions. This work was supported by the United States Atomic Energy Commission.
O>x>--L REFERENCES
=-;(%-
[cos-l@z~,
x 5 -L,O
s cos-14 22;. (20) I I
* It is understood here and throughout the discussion that in the heterogeneous material the array lies in the soft matrix with its tip at the interface. t The statement may not. be true if the mean free path is very small and the dislocations are generated at the interfaces of the neighboring particles (see Ref. 7).
1. M. A. JASWON and A. J. E. FOREMAN, Phil. Mag. 43,201 (1952). 2. A. K. HEAD, Phil. Mug. 44, 92 (1953). 3. A. K. HEAD, Proc. Phys. Sot. Lord 66B, 793 (1953). 4. A. K. HEAD, Au&. J. Phys. 13,278 (1960). 5. G. LEIBFRIED, 2. Physik 130, 214 (1961). 6. A. K. HEAD and N. LOUAT, Amt. J. Phys. 8, 1 (1955). 7. Y. T. CHOU and N. LOUAT, J. A&. Phys. 38,3312 (1962). 8. B. A. BILBY, A. H. COTTRELL and K. H. SWINDEN, Proc. Roy. Sot. A272, 304 (1963). 9. N. LOUAT, Phil. Mag. 8, 1219 (1963). 10. N. I. MUSKHELISHVILI, Singular Integral Equations. Noordhoff Ltd., Groningon, The Netherlands (1953).
I?.
CHOU:
DISLOCATION
ARRAYS
IN
NISHIMATSU and J. GURLAND, Trans. Amer. Sot. Metals 52, 469 (1960). 12. J. GURLAND, Trans. A.I.M.E. 227, 1146 (1963). 13. J. D.EsHELBY,F.C.FRANK~~~F.R.N.NABARRO,PM. Mag. 42, 351 (1951). 14. C. W. SHAW, L. A. SHEPARD andJ. WULFF, Trans. Am. Sm. Metals 57, 94 (1964). 15. A. H. COTTRELL, Progress in Metal Physics 1, 77 (1949).
11. C.
APPENDIX For convenience, the inversion theorem of singular integral equations is given in the following’lo):
HETEROGENEOUS
provided p -
R,(x) =
sSt
ax=
D
o(t)
D
(Al)
the unknown functionf(x) and the given function u(t) belong to the Holder classes H* and H in the region D, respectively, and that D consists of p finite segments of which at q of the 2p end pointsf(x) is bounded and at the remaining 2p - q end points f(x) it unbounded. Then
+[
R,(x) 1
R,(z)
2 + [
z,-a-~(x), (AZ)
fi
(x -
e,),
(A3)
i=q+1
u2
xW(x)
ax = 0,
m = 0, 1, .
. . . (n - P -
1).
(A4)
It is clear that the form of this inversion depends on the proper interpretation of the end points in accordance with the physical conditions. For the case given in the text D contains a single segment with bounded end points at -L and L. Evidently, p = 1, q = 2 and p - q -c 0. Equation (A4) is also satisfied. Hence, f(x) =
R,(x) ‘I2 p
R,(x) =
P V_-Q_-l(x)is an arbitrary polynomial of degree not greater than p - q - 1 with P_l = 0, and es’s are the end points. If p - q < 0, the same solution holds with P a_o_l = 0 and with the necessary and sufficient condition
sL-1
Suppose that in the singular integral equation
783
q 2 0, where
el(x -e,),
R,(x)
The inversion theorem
MATERIALS
(L2 -
x2)1’2 ( JOh (A2)
&)1’254
1’2 sx
.
czt
(A5)
The final result is given by equation (10) in the text
DISLOCATION
ARRAYS
IN
HETEROGENEOUS
MATERIALS*
Y. T. CHOUt A simple class of dislocation pile-ups in materials composed of soft and hard phases is examined. The analysis is based on the assumption that an array of discrete dislocations may be replaced by a oontinuous distribution of elementary dislocations. Two special oases considered are (i) a pile-up of screw dislocations in the soft matrix against a welded boundary, (ii) a pile-up of edge dislocations against a slipping boundary, i.e. a boundary which can transmit normal but not shear stresses. It is shown that, in a heterogeneous material, the length of the piled-up array depends linearly on the number of dislocations in the pile-up, as is the case in a homogeneous material. The number of dislocations in a given array, however, is generally less in the heterogeneous then in the pure matrix material. The results indicate that the two-phase material would work harden more than the pure matrix, provided the mean free path is not small. ARRANGEMENT
LINEAIRE MATERIAUX
DE DISLOCATIONS HETEROGENES
DANS
LES
L’auteur examine un cas simple d’empilements de dislocations dans les mat&iaux composes de phases dures et mains dures. L’analyse est basee sur l’hypothese qu’un arrangement de dislocations disc&es puet Btre assimile 8. une distribution continue de dislocations elementaims. Deux cas speciaux sont consider&, a savoir (i) un empilement de dislocations-vi8 dans la matrice non dure contre une frontiere soudee, (ii) un empilement de dislocations-coins contre une frontier-e de glissement, c’est-Q-dire une frontiere qui peut transmettre des tensions normales mais non de cisaillement. L’auteur montre que, dans un materiau heterogene, la longueur de l’arrangement empile depend lineairement du nombre de dislocations dans l’empilement, comme c’est le cas dans un materiau homogene. Le nombre de dislocations dans un arrangement don&, cependant, est gen&alement moindre dans le materiau heterogene que dans la matrice pure. Les resultats indiquent que le materiau 8. deux phases serait plus sensible au durcissement d’ecrouissage que la matrice pure, pour autant que le libre parcours moyen ne soit pas petit. LINEARE
VERSETZUNGSANORDNUNGEN
IN HETEROGENEN
MATERIALIEN
Eine einfache Klasse van Versetzungsaufstauungen in Materialien, die aus weichen und harten Phasen zusammengesetzt sind, wird untersucht. Die Untersuchung geht van der Annahme aus, da13eine Anordnung diskreter Versetzungen durch eine kontinuierliche Verteilung elementarer Versetzungen ersetzt werden kann. Zwei Spezialfalle werden betrachtet: (i) eine Aufstauung van Sohraubenversetzungen in einer weiohen Matrix an einer SchweiOgrenze, (ii) eine Aufstauung van Stufenversetzungen an einer Grenzflache, die Normalspannungen, aber keine Schubspannungen iibertragen kann. Es wird gezeigt, dal3 in einem heterogenen Material die Lange der Aufstauung linear van der Zahl der Versetzungen in der Aufstauung abhangt, wie es in einem homogenen Material der Fall ist. Jedoch ist die Zahl der Versetzungen in einer Aufstauung im heterogenen Material allgemein kleiner als in der reinen Matrix unter sonst gleichen Bedingungen. Die Ergebnisse besagen, da13das zweiphasige Material sich starker verfestigt als die reine Matrix, vorausgesetzt, da9 der mittlere Laufweg nicht klein ist.
INTRODUCTION
With
the increasing
dislocations
need for ultra high-strength
materials in modern technology,
it becomes necessary
to obtain a better understanding
of the properties
of
composite materials and of the theoretical background underlying these properties. For crystalline materials in particular, and
their
the study of the behavior
interactions
elements in these importance.
with
materials
various
basic
treatment
In the present paper
we treat a simple case of the problem
is useful in understanding
of
havior of composite
analytically
by
to arrays conThe information
the flow and fracture
be-
materials. ANALYSIS
The Hookean
elastic field of
Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corporation Research Center, Monroeville, Pennsylvania.
6
is concerned with small arrayst4).
heterogeneous
is evidently
Head’s
of linear arrays, however, is essentially numerical and
of dislocations
* Received December 14, 1964. t Divisionof Engineering, BrownUniversity. On leave from
ACTA METALLURGICA,
arrays in inhomogeneous
by Headtzp4).
an approximate method applicable taining large number of dislocations.
A detailed study of the non-Hookean interactions of dislocations with lattice inhomogeneities was made by Jaswon and Foreman(r).
and dislocation
media was analyzed
VOL. 13, JULY
1965
Consider an infinite isotropic medium of shear modulus G, and Poisson’s ratio y1 for x > 0, and G,, y2 for x < 0. The boundary at x = 0 can be either a welded interface, or a slipping interface which can transmit normal but not shear stresses. For a positive screw dislocation with Burgers vector b, sitting on the x-axis at a distance a from a welded boundary, 779
Headc2) gives
ACTA
780
METALLURGICA,
VOL.
(i) For x > 0
13,
1965
Under the assumption
K = 1, equations
(5) reduce
to
x+a
SK
(x + aI2 + Y2
)
(la)
+K ’ (x - d2 + Y2 (x + aY + y2 ’ (lb)
where
A, = G,b/27r, K = (G, -
GJCG,
(6) Equations
(6) also hold for edge arrays piled up against
a slipping
boundary(3)*,
(3)
+ G,),
with A, being replaced
A, = G,b/277(1 Note that o refers to the oI, Mathematically,
x-a (x
a)2
_
equations
given by equations +
y2
(4a)
J
rium equations
-41
=
For a positive
+ K)
’
edge dislocation
W)
a)2 + y2 .
(x -
at (a, 0) the expres-
However,
in the limiting case, K = 1, as shown below
the magnitudes of the shear stresses on the slip plane, o,, of a screw dislocation in front of a welded interface and cr,, of an edge in front of a slipping interface, identical
except for a constant
Now suppose that n parallel positive
axis.
for a homogeneous
array which contains under
0,
a
stress,
of the dislocations
By the approximation
At present, numerical
3
,...,
n
distribution,
to the following
A
L f(x) s -Lx-t
singular
dx = o(t),
(8) function
of the
L), a(t) is the stress function
with (5)
a(t) = o
for t > 0,
equations
(5) can be solved
as has been
only
by
a(t) = ---CT
for t < 0,
demonstrated
by
Headc4). If, however, our concern is with the nature of these arrays one can solve a limiting case analytically by an approximate
The approximation
is
made by replacing an array of discrete dislocations
by
method.
distribution
of elementary
with the same total Burgers vector(5-g). an analytic
be
integral equation
simulated array in (-L,
z
if K <
must
(0, L).
methods,
a continuous
which
of a continuous
(6) are equivalent
are given by j=l,2
i#j
in the domain
oim(x, o, K),
where f (x) is the unknown distribution
5 L+K{$l*,=o, +lxj - xi
of screws of
if K > 0 or negative
determined.
under an applied stress o, then the equilib-
rium positions
sets of screw dislocations
the same number
strength lK\ A,, positive
points 5r, x2, . . . , x, are piled up against a welded boundary
to the equilib-
medium of G, and
n positive screws of strength A, under the The other of span (-L, 0) is an imaginary
equations screws at the
(5) with the special case
The one with a span (0, L) is the real array
stress -0.
are
factor.
for screw
for edge arrays.
are piled up against each other at the origin on the xcontaining
sions for the stress field are much more complex(3).
by (7)
(6) are equivalent
v1 in which two symmetrical u zz
YJ.
component
arrays and to the o,, component
41 + K)
=
izi
(2)
(ii) For z < 0
u ?Jz
j = 1, 2, . . . , n.
expression
can be obtained
dislocations On this basis, by applying
the inversion theorem of singular integral equationsoO). Let us consider the case in which G, > G,, such that the parameter K approaches unity. This is an extreme case which applies to a heterogeneous material consisting of a rigid phase embedded in a soft matrix. A practical example close to the limiting case is the WCCo system studied by Gurland and co-workers(11*12).
(9)
A = A, for screw arrays and A = A, for edge arrays. The solution of equation (8) can readily be obtained by
using
the
Muskhelishvili
inversion
theorem
and co-workers
formulated
(see Appendix).
by For
* In the case K = 1, the shear stress uw of a positive edge dislocation lying on the z-axis at a distance a from a slipping boundary is given by@’ CTa-
_
(z - a)[(x - a)2 - Y21
A
e i
[(5 - a)” + YT
+ (z + a)[@ + a)” - Y21 , [(z + aj2 + Y212 1 Here, K is defined by
2
> 0.
CHOU
: DISLOCATION
ARRAYS
IN
the simulated arrayf(x) is bounded at both ends of the segment (-I;, L). Thus, the inversion formula gives
f(x) =
$
Gosh-f /4/ .
W)
Equation (lo), with the range of x specified in (0, L), gives the distribution of dislocations in a linear array piled up at the interface in a heterogeneous medium. The number of d~locatio~, n, in the range (0, L) is therefore
n, =
s)(x)
ax=
2
)
a
(12) \ I
The corresponding expressions for the array in a homogeneous material* are(13*@
[gyZ,
f(x) = f
no=-,
and
MATERIALS
781
DISCUSSION
The above analysis of the limiting case, when coupled with the results of Head, may lead to a more general relation. Head’s data for small screw arrays (see Fig. 2) clearly indicate a linear relationship between L/nk and K, i.e. a i A, where
L - = tt + p, nk
1
(17)
(11)
where the subscript attached to n refers to the K values. The range of the army, L, may be expressed in terms of n, and TVby L=&nlA.
HETEROGENEOUS
(13)
La 2A
L1-2n,B* a
and
in,
A,
n,l
’
It is not difficult to see that equation (16) holds for large n’s. As the number of dislocations increases, the direct effect of the image array becomes more remote on the dislocations near the tail of the array. Thus, as n increases, the nth ~location senses much less the change in K. This indicates that, if the linear relation is valid for small arrays, it should remain valid for large arrays. Furthermore, the slope of the lines in Fig. 2 for the nth dislocations should approach the limiting slope of (rr - 2). This trend is observed in Fig. 2. It is therefore quite plausible that equation (16) is valid in general.
Figure 1 shows the curves of the distribution functions given by equations (10) and (13). UNIT
3
OF
LENGTH: 1
/
2
As/o-
0
/
/MT /
L
c” 2
I
0.5
[=
I .o
X/L
Fro. I. Distribution of s lmear dislocation array under a constant applied stress, o. (a) Homogeneous medium, equation (13). (b) Heterogeneous medium, K = 1, equation (10). * In this case the head dislocation is assumed to be locked in a stress field proportional to 6’(z), where 6’ is the derivative of Dime’s 6 function.
’
L 0
V.5
_ I.U
K FIG. 2. Relation between L/m%and K for screw arrays.
The solid lines are reproduced from the work of Head.“) The dashed line is obtained from the present analysis valid for all sufficientiy large 12’s.
ACTA
782
METALLURGICA,
Now, for large n where the approximation of the continuous distribution is warranted, equation (11) and (14) give u = 2, /3 = (77- 2)
= 2 + (7r -
2)K,
0
SK
5
1.
1965
Thus, we have (i) Near the tip of the array the stress distribution is approximately
0>2>-L.
(19)
It seems likely that a similar general relation may hold for edge arrays, since the extra terms in the equilibrium equations have much less effect on x,. More numerical computations are worth doing to support these points. In comparing equations (19) and (14), it is seen for the same applied stress and span of distribution that the number of dislocations in a piled-up array is less for a material containing hard particles than for the pure matrix material. * In the limiting case the ratio is 2177. Thus, the heterogeneous material would accomodate less plastic flow under loading, provided that the mean free path through the matrix is not too smal1.t This may probably constitute the primary hardening mechanism in the early stage of plastic deformation(14). The tip stress of an array of n dislocations piled up at an interface in a material with hard phases is simply n(~ as given also for a homogeneous material by the virtual work principle(15). The stress concentration factor due to an array is therefore smaller in a heterogeneous material than in a homogeneous material. The ratio of the factors is again 2177in the limiting case. If crack nuclei once initiated from the pile-ups grow as Griffith cracks, a heterogeneous material would exhibit a higher fracture strength. Strictly speaking, in the limiting case (K = 1) the stress is not uniquely determined in the half plane x < 0. It is nevertheless interesting and informative to see the characteristics of the stress distribution ahead of the array as the limiting condition is approached. As K -+ 1, the shear stress o,, at any point on the negative x-axis due to a screw pile-up approaches ~~~(x,O)=-~[(eosh-l/~~]z+%),
13,
agz(x,O)f --~(~n~l]“+%~,
and we have the general relation for screw arrays -
VOL.
(21) (ii) At a large distance from the array the stress is proportional to x-l, aJx,
0)
5
2ak )
(22)
xg-L,
77X
as is expected. In a homogeneous material the stress induced in front of the array is given byt6) ..,(X,O)=-~(~~]“‘-l),
x
(23)
where the applied stress -a is being excluded. Corresponding to equations (21) and (22) we have near the tip
(24) and at a large distance a2/&x, 0) i
%4 ,
These results indicate that hard particles the stress very is generally smaller than in However, at a large distance slightly greater.
x<--L.
in materials containing near the tip of the array homogeneous materials. from the tip the stress is
ACKNOWLEDGMENTS
The author wishes to thank Professors D. C. Drucker, C. Elbaum and J. Gurland for helpful discussions. This work was supported by the United States Atomic Energy Commission.
O>x>--L REFERENCES
=-;(%-
[cos-l@z~,
x 5 -L,O
s cos-14 22;. (20) I I
* It is understood here and throughout the discussion that in the heterogeneous material the array lies in the soft matrix with its tip at the interface. t The statement may not. be true if the mean free path is very small and the dislocations are generated at the interfaces of the neighboring particles (see Ref. 7).
1. M. A. JASWON and A. J. E. FOREMAN, Phil. Mag. 43,201 (1952). 2. A. K. HEAD, Phil. Mug. 44, 92 (1953). 3. A. K. HEAD, Proc. Phys. Sot. Lord 66B, 793 (1953). 4. A. K. HEAD, Au&. J. Phys. 13,278 (1960). 5. G. LEIBFRIED, 2. Physik 130, 214 (1961). 6. A. K. HEAD and N. LOUAT, Amt. J. Phys. 8, 1 (1955). 7. Y. T. CHOU and N. LOUAT, J. A&. Phys. 38,3312 (1962). 8. B. A. BILBY, A. H. COTTRELL and K. H. SWINDEN, Proc. Roy. Sot. A272, 304 (1963). 9. N. LOUAT, Phil. Mag. 8, 1219 (1963). 10. N. I. MUSKHELISHVILI, Singular Integral Equations. Noordhoff Ltd., Groningon, The Netherlands (1953).
I?.
CHOU:
DISLOCATION
ARRAYS
IN
NISHIMATSU and J. GURLAND, Trans. Amer. Sot. Metals 52, 469 (1960). 12. J. GURLAND, Trans. A.I.M.E. 227, 1146 (1963). 13. J. D.EsHELBY,F.C.FRANK~~~F.R.N.NABARRO,PM. Mag. 42, 351 (1951). 14. C. W. SHAW, L. A. SHEPARD andJ. WULFF, Trans. Am. Sm. Metals 57, 94 (1964). 15. A. H. COTTRELL, Progress in Metal Physics 1, 77 (1949).
11. C.
APPENDIX For convenience, the inversion theorem of singular integral equations is given in the following’lo):
HETEROGENEOUS
provided p -
R,(x) =
sSt
ax=
D
o(t)
D
(Al)
the unknown functionf(x) and the given function u(t) belong to the Holder classes H* and H in the region D, respectively, and that D consists of p finite segments of which at q of the 2p end pointsf(x) is bounded and at the remaining 2p - q end points f(x) it unbounded. Then
+[
R,(x) 1
R,(z)
2 + [
z,-a-~(x), (AZ)
fi
(x -
e,),
(A3)
i=q+1
u2
xW(x)
ax = 0,
m = 0, 1, .
. . . (n - P -
1).
(A4)
It is clear that the form of this inversion depends on the proper interpretation of the end points in accordance with the physical conditions. For the case given in the text D contains a single segment with bounded end points at -L and L. Evidently, p = 1, q = 2 and p - q -c 0. Equation (A4) is also satisfied. Hence, f(x) =
R,(x) ‘I2 p
R,(x) =
P V_-Q_-l(x)is an arbitrary polynomial of degree not greater than p - q - 1 with P_l = 0, and es’s are the end points. If p - q < 0, the same solution holds with P a_o_l = 0 and with the necessary and sufficient condition
sL-1
Suppose that in the singular integral equation
783
q 2 0, where
el(x -e,),
R,(x)
The inversion theorem
MATERIALS
(L2 -
x2)1’2 ( JOh (A2)
&)1’254
1’2 sx
.
czt
(A5)
The final result is given by equation (10) in the text