# Internal Vibrations of Edge Dislocation Dipoles

Authors: J. J. Kratochvíl; F. Kroupa
Journal: Research Letters in Materials Science
 (2008)

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## Abstract

The resonance frequency of vibrations of dislocation dipoles in fatigued f.c.c. metals is found rather high, in the range of 100 GHz. Because of high attenuation of ultrasound in the GHz range, the contributions of these self-vibrations to degradation of the dipole structures could be expected only in thin layers.


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## Body

## 1. Introduction

A high density of edge
dislocation dipolar loops is formed during plastic deformation of crystalline
materials, especially under cycling loading [1–3]. During
deformation, the loops are clustered becoming main building blocks of the dipolar
deformation microstructure (tangles, veins, dipolar walls). The size of the
prismatic dipolar loops is typically of the order of 10 nm (in Ni crystals
deformed cyclically at room temperatures, the averaged loop height is 4 nm, the
averaged length is 60 nm [2, 3]). The leading factor, which influences the rate
of clustering and hence the degradation process, is the loop mobility expressed
through the drift coefficient. In principle, this coefficient can be decreased
by internal vibration of the loops, in analogy to the effect of temperature
increase. There is experimental evidence that higher temperature accelerates
the clustering process [4].In this note, small internal vibrations in
the slip planes of two long elastically interacting dislocation arms forming a
dipole will be studied as forced vibrations due to an oscillating external
shear stress. The effect of damping will also be briefly discussed. This
approach is similar to the study of vibrations of two partials forming a split
dislocation in f.c.c. metals [5] and of internal vibrations of the core of
screw dislocations in b.c.c. metals modeled by four partials in [6].


## 2. Forced Vibrations

An edge dislocation dipole with the dislocation lines in thez-direction
(Figure 1) is formed by dislocation 1 in the slip plane y
=
h
/
2 with the Burgers vector of length b in the x direction and by dislocation 2 in the slip
plane y
′
=
−
h
/
2 with Burgers vector b
′
=
−
b.  Length l of the dipole in the z direction
will be assumed much larger than the dipole height h. In the equilibrium
configuration of the dipole, x
=
h
/
2 and x
′
=
−
h
/
2,
the force components in the slip planes are zero [6]. The force components in
the direction perpendicular to the slip planes, which may induce the climbing
of dislocations, will not be considered.Figure 1
Dislocation dipole.For changed positionsx and x
′,
the interaction force F
int from dislocation 2 on 1 is equal to F
int
=
b
τ
xy
(
2
,
1
) and that from 1 on 2
F
int
′
=
−
b
τ
x
y
(
2
,
1
)
=
−
F
int. The stress component τ
x
y for an edge dislocation lying in the z-axis is well known [6],
τ
x
y
=
[
G
b
/
2
π
(
1
−
ν
)
]
x
(
x
2
−
y
2
)
/
(
x
2
+
y
2
)
2,  where G is the shear modulus and ν the Poisson’s ratio, so that for the
displaced dislocations (forces will be given on unit lengths of dislocations),
(1)
F
int
⁡
=
−
D
(
x
−
x
′
)
[
(
x
−
x
′
)
2
−
h
2
]
[
(
x
−
x
′
)
2
+
h
2
]
2
,
F
int
′
=
−
D
(
x
′
−
x
)
[
(
x
′
−
x
)
2
−
h
2
]
[
(
x
′
−
x
)
2
+
h
2
]
2
=
−
F
int
⁡
,   where D
=
G
b
2
/
[
2
π
(
1
−
ν
)
]
.Only the vibrations symmetrical to the centraly-z plane will be
considered (which may be called optical mode), so that x
′
=
−
x.  The deviation of dislocation 1 from the equilibrium
position x
=
h
/
2 will be denoted as u
,
x
=
u
+
h
/
2,
so that the width of the dipole L is L
=
h
+
2
u (Figure 1).
For small deviations, u
≪
h
/
2,
the equation for force vibrations is reduced to the equation of a standard
linear undamped oscillator. For oscillating external shear stress
τ
x
y
ext
=
τ
0
⁢
cos
ω
t, the equation of motion reads
as follows:
(2)
m
d
2
u
d
t
2
=
−
D
u
h
2
+
b
τ
0
⁢
⁢
cos
⁢
⁡
ω
t
.  The amplitude A of harmonic solution u
=
A
⁢
cos
ω
t is (3)
A
=
b
τ
0
D
/
h
2
−
m
ω
2
=
b
τ
0
m
(
ω
S
2
−
ω
2
)
. The assumption of small deviations
is violated when the frequency of external oscillations ω approaches the frequency of self-vibrations
ω
S: (4)
ω
S
=
1
h
D
m
=
1
h
G
2
π
(
1
−
ν
)
ρ
. In the last equation the effective mass m of a dislocation per unit length can be estimated [7] as
m
=
ρ
b
2,where ρ is the
mass density. For example, for copper, G
=
45
.
5 GPa, ν
=
0.
35, b
=
0.
256 nm, ρ
=
893
0 kg/m3 [3], it is ω
S
[
s
−
1
]
=
1
.
12
×
1
0
3
(
1
/
h
⁢
[
m
]
)
=
1
.
12
×
1
0
9
(
1
/
h
⁢
[
μ
m
]
). Therefore, for the expected values of the typical dipolar loop
height, h
~
10 nm, the circular self-frequencies are very
high, ω
S
~
1
0
11
⁢
s
−
1,
that is, ω
S is in the range of 100 GHz, that
is, the frequencies f in the range of 10 GHz.


## 3. Forced Vibrations with Damping

The damping decreases
the amplitude accompanied by a phase shift. The main part of the damping force
can be taken proportional to the dislocation velocity,F
dam
=
−
R
d
u
/
d
t [5], where R is the damping constant. The equation of motion modifies
to(5)
m
d
2
u
d
t
2
=
−
D
u
h
2
+
b
τ
0
⁢
c
o
s
⁢
ω
t
−
R
d
u
d
t     with the solution u
=
B
cos
(
ω
t
−
α
)
, where B is the amplitude of damped vibrations and α is the phase shift,
(6)
B
=
b
τ
0
m
2
(
ω
S
2
−
ω
2
)
2
+
R
2
ω
2
,
t
g
α
=
R
ω
m
(
ω
S
2
−
ω
2
)
. The value of the damping constant R
for metals like Al and Pb was estimated in [5] to be of the order R
≅
1
0
−
6
⁢
Nm
−
2
⁢
s (the estimate corresponds to the temperature independent
electronic damping, for higher temperature R increases due to the phonon
damping).  Amplitude B reaches maximum for
ω
M
=
(
ω
S
2
−
R
2
/
2
m
2
)
1
/
2, however, as
R
/
m
≈
1
0
9
⁢
s
−
1,
we have	ω
M
≈
ω
S  and
B
max
≈
b
τ
0
/
R
ω
S. For the validity of the linearized
model, the vibrations have to be small, that is, B
max
≪
h
/
2,
hence τ
0
≪
h
R
ω
S
/
2
b
≈
2
⁢
MPa.


## 4. Discussion

According to the presented estimate,
the mobility of a typical dipolar loop approximated as the edge dipole of the
height ~10 nm can be facilitated by ultrasound waves in the 100 GHz range of
frequencies. The commercial apparatus works with the frequencies up to 10 GHz,
using special equipment, the ultrasound frequency 600 GHz was reached in thin
foils [7]. The main problem is a strong attenuation increase at higher
frequencies, especially in metals. Therefore, the effect of ultrasound in the
GHz range on dislocation dipole vibrations can be reached in principle,
however, only in thin specimens; for example, in fatigued specimens exposed to
GHz ultrasound waves, the degradation process could be accelerated. To our
knowledge, no experiments of that kind have been done, as a verification of
this effect seems to be rather difficult.
The contribution
of dislocations dipoles to distortion of ultrasound waves was analyzed and
measured in [8] at frequency 5 MHz, that is, far below the resonance frequency
estimated in the present note.



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*Source: 907895-2008-08-24.xml*