Introduction to and fundamentals of discrete dislocations ...Introduction to and fundamentals of...
Transcript of Introduction to and fundamentals of discrete dislocations ...Introduction to and fundamentals of...
Summer school
Generalized Continua
and Dislocation Theory
Theoretical Concepts,
Computational Methods
And Experimental Verification
July 9-13, 2007
International Centre for Mechanical Science
Udine, Italy
Lectures on:
Introduction to and fundamentals of
discrete dislocations and dislocation
dynamics. Theoretical concepts and
computational methods
Hussein M. ZbibSchool of Mechanical and Materials Engineering
Washington State University
Pullman, WA
Contents
Lecture 1: The Theory of Straight Dislocations – Zbib
Lecture 2: The Theory of Curved Dislocations –Zbib
Lecture 3: Dislocation-Dislocation & Dislocation-Defect Interactions -Zbib
Lecture 4: Dislocations in Crystal Structures - Zbib
Lecture 5: Dislocation Dynamics - I: Equation of Motion, effective mass - Zbib
Lecture 6: Dislocation Dynamics - II: Computational Methods - Zbib
Lecture 7 : Dislocation Dynamics - Classes of Problems – Zbib
Lecture 2: The Theory of Curved Dislocations
The Burgers Displacement Equation
Stress Field of a curved dislocation
Volterra Dislocations
• Circular Dislocation Loop
• Rectangular Dislocation Loop
• The stress filed about a straight segment dislocation
The Somigliana Ring Dislocation
Referecences
• I. Demir, J.P. Hirth and H.H. Zbib, The Somigliana Ring Dislocation, J. Elasticity, 28, 223-246, 1992.
• Khraishi, T.A., Zbib, H.M., Hirth, J.P. and de La Rubia, T.D., “The stress Field of a General Volterra Dislocation
Loop: Analytical and Numerical Approaches”, Philosophical Magazine, 80, 95-105, 2000.
• Khraishi, T. and Zbib, H.M., The Displacement Field of a Rectangular Volterra Dfislocation Loop, Phil Mag,82,
265-277, 2002.
Mixed Dislocation
The Peach-Koehler equation for the self-stresses of any curved, closed dislocation loop, is given by the following line integral (see Hirth and Lothe 1982):
k
C ii
imkm
iC
imm
iC
imm
xdRxxxx
Rb
G
xdRx
bG
xdRx
bG
23
22
14
88
)(
The self-stress
“1” P
Rjp
dl’
C1
Field point
b
k
C jm
ijkik
C
mikij
A j
mm xdxx
RbxRdbRdA
xbu
222
)1(8
1
8
1
8
1)(
r
The Burgers equation for displacements,
for any closed curved dislocation loop,
is given in terms of line and area integrals as (see Hirth and Lothe [21]):
Alternatively, Burgers equation has the following vector form:
CCR
dgrad
R
d lRblbbru
)(
)()(
18
1
4
1
4
A
R
d3
AR
is the solid angle through which the positive side of A is seen from is seen from r , and is defined as
The displacement field is important in interaction problems between dislocations and
embedded particles in a softer metallic matrix, e.g. in metal-matrix composites (MMCs), the
boundary condition in the problem is that of zero displacement at a set of collocation points on the
interface. The condition is enforced by annulling the undesired displacements caused by the crystal
or matrix dislocations with the displacements coming from the distribution of rectangular
dislocation loops. In order for this to happen, a quantification of the displacement field of each of
the rectangular dislocation loops is necessary and is therefore developed in this paper.
The integration of the Burgers equation to obtain the displacements turns out to be rather
involved. It is carried out with respect to the global xyz system. To perform the integration, the
indices in (1) are first expanded. This gives three independent equations, one for each displacement
component ),and,,( wvu corresponding to subscript or index m values of 1, 2 and 3, respectively.
Note that u is the displacement in the x-direction, v in the y-directions and w in the z-direction.
Before continuing along with the integration, few steps are in order. First, note that the elevation of
the dislocation loop is fixed with respect to the global coordinate system xyz. This means that the
dislocation loop lies in a plane parallel to the xy planes such that z is a constant. Second, note that
the following holds: RR /22 . Third, along segment 1, ax and 0xd , along segment 3,
ax and 0xd , along segment 2, by and 0yd , and along segment 4, by and
0yd .
Rectangular Dislocation Loop
A rectangular dislocation loop is composed of four linear dislocation segments (see figure 1). For this work, the Burgers vector b is assumed to be constant, in magnitude and direction, from one point to another along the loop.
x
z
yO
2a
2b
bz
bx
by
r´
rR
1
2
pt P = (x,y,z)
3
4
z´
dl´
Khraishi, T. and Zbib, H.M., The Displacement Field of a Rectangular Volterra Dfislocation Loop, Phil
Mag,82, 265-277, 2002.
The displacement field of a rectangular dislocation loop
bu
z
y
x
b
b
b
w
v
u
][
333231
232221
131211
K
KKK
KKK
KKK
22
222222
22
222222
11
)()(
)()()()()()(
)(
)()(
)()()()()()(
)(
)1(8
)(
4
),,(
zzxa
zzybxa
yb
zzybxa
ybxa
zzxa
zzybxa
yb
zzybxa
ybxa
zz
zyx
K
222222
222222
12
)()()(
1
)()()(
1
)()()(
1
)()()(
1
)1(8
)(
zzybxazzybxa
zzybxazzybxazz
K
32
10
12
xa 0
0.5
1
1.5
2
ya
0.060.040.02
00.020.040.06
ubz
32
10
12
xa
Figure 3. Three-dimensional carpet plot of u(x,y,z) normalized by the Burgers vector magnitude |b|
, where b=(bx=0,by=0,bz0). Here, =1/3, a=b=100bz, z=0, and z=20bz.
Khraishi, T. and Zbib, H.M., Dislocation Dynamics Simulations of the Interaction Between a Short Rigid Fiber and a Glide Dislocation Pile-up. Comp. Mater. Sci. 24, 310-322 2002.
Surface pt, P
B
C
B. C.:
u=0 A
Surface pt, P
B
C
B. C.:
u=0 A x'
z'
y'
Loop i
Oz
Ox
z
y
i
R
Loop A
pt P (x,y,z)
by'
bx'
bz'
x'
z'
y'
Loop i
O
x'
z'
y'
Loop i
Oz
Ox
z
y
i
R
Loop A
pt P (x,y,z)
z
Ox
z
y
Ox
z
y
ii
R
Loop A
R
Loop A
pt P (x,y,z)
pt P (x,y,z)
by'
bx'
bz'
by'
bx'
bz'
A rigid fiber surrounded by a glide
dislocation loop. The fiber surface is
meshed with
rectangular elements representing
rectangular dislocation loops (see
inset). Point P is just a field point.
A schematic of a rigid
fiber close to elastic
displacement sources
(e.g. a dislocation loop
and/or segment). The
correct boundary
condition to satisfy is
zero elastic
displacement on the
fiber's surface.
R
r
r
pt. P
x
loop A
zy
x
y
zz ,
OrA
C
xb
yb
zz bb ,Rr
r
pt. P
x
loop A
z
y
x
y
zz ,
OrA
C
dA
Circular Dislocation loop
Prismatic Dislocation Loops
• Prismatic dislocation loops with a Burgers vector normal to the plane of the loop can form in a material subject to irradiation or quenching by the precipitation of vacancies or interstitial atoms.
• Glide or shear loops can form readily, e.g., in the glide plane of a Frank-Read source as it continues to emit propagating dislocations. Hence, it is important to quantify the stress field of these loops in order to account for their interaction with other dislocation curves or defects in the crystal.
b
b
The stress field of a circular Volterra dislocation loop
zyzxyz
zyzxxz
yxyyxxxy
zz
yyyxxxyy
yyyxxxxx
xy
yx
yxxy
xxyy
yxyx
zz
2
22
2
2
2
22
2
2
2
22
2
2
2
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
kCkCGb
yy
yx
zxxzx
yy
zy
zxzz
zx
zxyy
zxzxx
KEKE
KEKEKE
KEKE
KEKE
KEKE
KEKE
KEKEKE
32313029
282726252423
22212019
18171615
14131211
10987
654321
122
12142
122
1212
1212
)4(1212
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
x/R
zz/(
Gb
z/2
(1
- ))
N=6
N=12
N=18
N=24
szz, analy.
x
y
(a) (b) (c)
bz
RR
31 r/
The stress field of a circular loop, and straight segments approximations
Interaction between a defect (Frank sessile loop; defect
clusters in irradiated materials) and a glide edge
dislocation
d=2R
ly
b z
Fig. 5. (a) A TEM picture showing dislocation pinning by the dispersed hardening
mechanism. (b) A DD picture showing the same mechanism at work as captured in the
simulations. The dots in the figure represent defect clusters.
Interaction is
weak31 r/
Somigliana Dislocation
Many internal defect, such as broken fiber, debonding etc., can be
modeled as a set of discontinuities In the continuum resulting in
perturbations (of small or large magnitudes) in an otherwise simple
elastic field. The Somigliana dislocation can be considers as one
of the most general kins of such discontinuities an, thus, can be
used to model many imperfections.
Burgers vector is NOT conserved along the dislocation line
The Somigliana Ring Dislocation
used to model cylindrical cracks
432422412
332322312
232222212
132122112
2
431421411
331321311
231221211
131121111
1
1212
,,,
,,,
,,,
,,,
,,,
,,,
,,,
,,,
)(
)(
)(
AAA
AAA
AAA
AAA
Gb
k
kE
kK
AAA
AAA
AAA
AAA
Gb
drz
z
r
Rr
R
Gb
for
0
2
1
1
12
1
Rr
r
RGb
for
0
0
1
1
12 2
1
,,
22222222
32111
2bRrRrbbpg
pRgr
bA
,,
2222222
32
24222222
212144
2223 pgRrRrb
gp
bRpgbRrRrb
Rpgr
bA
,,
22222
2211232 rRbrb
Rgr
bA
Etc…
1. Demir,I., Hirth,J.P. and Zbib,H.M., "The Extended Stress Field Around the
Cylindrical Crack Using the Theory of Dislocation Pile-ups", Int. J. Engrg
Sci., 30, 829-845, 1992.
2. Zbib, H.M., Hirth, J.P. and Demir, I., ``The Stress Intensity Factor of
Cylindrical Cracks", Int. J. Engrg. Sci., 33,247-253, 1995.
3. Zbib, H.M. and Demir, I., "On the Micromechanics of Defects in
Fiber-Matrix Composites", review article, in: Damage in Composite
Materials, Voyiadjis, G.Z., ed., pp. 103-133, Elsevier Science Publisher,
N.Y., 1993.
4. Zbib, H.M. and Demir. I.,``The Interface Ring Dislocation in Fiber-Matrix
Composites: Approximate Analytical Solution", ASME Journal of
Engineering Materials and Technology, 116, 279-285, 1994.
5. Close, S. and Zbib, H.M., “The Stress Intensity Factors and Interactions
Between Cylindrical Cracks in Fiber Matrix Composites”, an invited
article, in: Damage and Interfacial Debonding in Composites, ed.
Voyiadjis,G., Z., and Allen, D.H., Elsevier Science, The Netherlands,
1996, pp. 3-27.
1. Demir,I., Hirth,J.P. and Zbib,H.M., "The Extended Stress Field Around the Cylindrical Crack Using the Theory of
Dislocation Pile-ups", Int. J. Engrg Sci., 30, 829-845, 1992.
2. Close, S. and Zbib, H.M., “The Stress Intensity Factors and Interactions Between Cylindrical Cracks in Fiber Matrix
Composites”, an invited article, in: Damage and Interfacial Debonding in Composites, ed. Voyiadjis,G., Z., and Allen,
D.H., Elsevier Science, The Netherlands, 1996, pp. 3-27.
Modeling the cylindrical crack around a fiber as a pileup of Somigliana ring dislocations
A curved dislocation can be approximated as a set of straight dislocation segments
x
y
(a) (b) (c)
bz
RR
Arbitrary shape
circular shape
Discrete segments
Continuous curve
N
j
D
jjp1
1,)(
J+1J
Nodes and collocation points on dislocation
loops and curves“1”
“3”
P
Rjpj
j+1
j-1
dl’
C1
C2
C3
i
i+1
i-1i-2
jj+1
“2”
vj+1
vj
v
Field point
Velocity
vector
kC
2
i
αβ
βαi
3
imkm
α
2
iC
imβmβ
2
iC
imαmαβ
xdRx
δxxx
Rb
ν)π(14
G
xdRx
bπ8
GxdR
xb
π8
G)(σ
p
Arbitrary shape
z
x
y A
B
b
x,y,zp
)()()( ABP ijijij
zzzRyx
RR
xb
R
xb
R
y
Rb
RR
yb
R
x
Rb
R
xb
x
R
x
R
yb
x
R
x
R
xb
R
x
R
xb
R
y
R
yb
y
R
y
R
xb
y
R
y
R
yb
x
R
x
R
xb
x
R
x
R
yb
zyx
yz
zyxxz
yx
xy
yxzz
yx
yy
yxxx
,,222222
33
2
0
3
2
30
2
2
2
2
22
2
2
2
20
32320
2
2
2
2
22
2
2
2
20
2
2
2
2
22
2
2
2
20
1
1
21
21
22
21
21
21
21
Other forms are given in
Hirth and Lothe (1982, p. 134)
This form is most convenient to use
RA
RB
The stress field of a dislocation segment
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
x/R
zz/(
Gb
z/2
(1
- ))
N=6
N=12
N=18
N=24
szz, analy.
x
y
(a) (b) (c)
bz
RR
Discrete Circular Volterra Dislocation Loop
Stacking-Fault Tetrahedra
A
B
C
D
SFT
Observed in quenched metals and metals and alloys of low stacking-fault energy. It consists of a tetahedron of intrinsic staking faults on {111} planes with 1/6<110> type stair-rod dislocations along the edges of the tetrahedron. A dislocation loop (of Frank partial dislocation formed by the collapse of vacancies), may dissociate into a low-energy stair-rod dislocation and a Shockly partial on an intersecting slip plane, leading to the formation of a SFT.
R=10b, y=5
-8.00E+08
-6.00E+08
-4.00E+08
-2.00E+08
0.00E+00
2.00E+08
4.00E+08
6.00E+08
75 80 85 90 95 100 105 110 115 120 125
x (b)
glid
e P
F-fo
rce (
N/m
)
SFT
FS loop
Defect:
a)Loop,
b) dissociate
into SFT
Edge
disloc
ation
Xy
.Comparison between the Stress Fields of Unit Defects
To contrast the differing stress fields of a FS loop and a SFT, we calculate the Peach-Koehler force that both exert at
a point on a straight edge dislocation situated an x-distance from the loop plane (or correspondingly, from the
tetrahedron's base plane). An edge view of the loop and its possible SFT offspring (shown in dashed line) is exhibited
in this figure along with a nearby edge dislocation. In this figure, the glide force (in the direction of the dislocation's
Burgers vector) is plotted versus the y-coordinate of the dislocation at a given x. Note that in the force exerted by the
SFT is smaller than that caused by the FS loop. An exception to this is the case when x=9b where the edge
dislocation almost cuts the tip of the tetrahedron in question. In such a case, the SFT force is stronger than the
loop's. Moreover, if the dislocation line collides with a SFT (a more probable event than a loop collision, considering
volume differences), then their inelastic interaction is also expected to be strong. In this figure, the edge length of the
triangular FS loop, or correspondingly the SFT, is equal to 10b, which is a typical defect dimension as measured
experimentally. Now, if one fixes the y value and varies the x-coordinate instead, it would still be observed that
overall, and as long as elastic interactions are concerned, the FS loop causes a stronger glide force on the
dislocation. The results demonstrate a higher stress field associated with the FS loop, which should be in-line with
physical reasoning. Since the SFT forms from a FS loop (i.e. it’s a preferred lower energy state), it is then expected
that a more relaxed strain-field accompany such geometry over that of a loop.
Dislocation multiplication
Frank-Read sources