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Research Letters in Materials Science
Volume 2008, Article ID 907895, 3 pages
Research Letter

Internal Vibrations of Edge Dislocation Dipoles

1Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague, Czech Republic
2Institute of Plasma Physics, Academy of Science of Czech Republic, Za Slovankou 3, 182 21 Prague, Czech Republic

Received 12 June 2008; Accepted 8 July 2008

Academic Editor: Korukonda L. Murty

Copyright © 2008 J. J. Kratochvíl and F. Kroupa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

1. Introduction

A high density of edge dislocation dipolar loops is formed during plastic deformation of crystalline materials, especially under cycling loading [13]. 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 the z-direction (Figure 1) is formed by dislocation 1 in the slip plane 𝑦=/2 with the Burgers vector of length 𝑏 in the 𝑥 direction and by dislocation 2 in the slip plane 𝑦=/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=/2 and 𝑥=/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 positions x and 𝑥, the interaction force 𝐹int from dislocation 2 on 1 is equal to 𝐹int=𝑏𝜏(2,1)𝑥𝑦 and that from 1 on 2 𝐹int=𝑏𝜏(2,1)𝑥𝑦=𝐹int. The stress component 𝜏𝑥𝑦 for an edge dislocation lying in the 𝑧-axis is well known [6], 𝜏𝑥𝑦=[𝐺𝑏/2𝜋(1𝜈)]𝑥(𝑥2𝑦2)/(𝑥2+𝑦2)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), 𝐹int=𝐷𝑥𝑥𝑥𝑥22𝑥𝑥2+22,𝐹int𝑥=𝐷𝑥𝑥)𝑥22𝑥𝑥2+22=𝐹int,(1) where 𝐷=𝐺𝑏2/[2𝜋(1𝜈)].

Only the vibrations symmetrical to the central y-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=/2 will be denoted as 𝑢,𝑥=𝑢+/2, so that the width of the dipole 𝐿 is 𝐿=+2𝑢 (Figure 1). For small deviations, 𝑢/2, the equation for force vibrations is reduced to the equation of a standard linear undamped oscillator. For oscillating external shear stress 𝜏ext𝑥𝑦=𝜏0cos𝜔𝑡, the equation of motion reads as follows: 𝑚𝑑2𝑢𝑑𝑡2𝑢=𝐷2+𝑏𝜏0cos𝜔𝑡.(2) The amplitude A of harmonic solution 𝑢=𝐴cos𝜔𝑡 is 𝐴=𝑏𝜏0𝐷/2𝑚𝜔2=𝑏𝜏0𝑚𝜔2𝑆𝜔2.(3) The assumption of small deviations is violated when the frequency of external oscillations ω approaches the frequency of self-vibrations 𝜔𝑆: 𝜔𝑆=1𝐷𝑚=1𝐺2𝜋(1𝜈)𝜌.(4) In the last equation the effective mass 𝑚 of a dislocation per unit length can be estimated [7] as 𝑚=𝜌𝑏2,where ρ is the mass density. For example, for copper, 𝐺=45.5 GPa, 𝜈=0.35, 𝑏=0.256 nm, 𝜌=8930 kg/m3 [3], it is 𝜔𝑆[s1]=1.12×103(1/[𝑚])=1.12×109(1/[𝜇𝑚]). Therefore, for the expected values of the typical dipolar loop height, 10 nm, the circular self-frequencies are very high, 𝜔𝑆1011s1, that is, ωS 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, 𝐹dam=𝑅𝑑𝑢/𝑑𝑡 [5], where R is the damping constant. The equation of motion modifies to𝑚𝑑2𝑢𝑑𝑡2𝑢=𝐷2+𝑏𝜏0𝑐𝑜𝑠𝜔𝑡𝑅𝑑𝑢𝑑𝑡(5) with the solution 𝑢=𝐵cos(𝜔𝑡𝛼), where B is the amplitude of damped vibrations and α is the phase shift, 𝐵=𝑏𝜏0𝑚2𝜔2𝑆𝜔22+𝑅2𝜔2,𝑡𝑔𝛼=𝑅𝜔𝑚𝜔2𝑆𝜔2.(6) The value of the damping constant 𝑅 for metals like Al and Pb was estimated in [5] to be of the order 𝑅106Nm2s (the estimate corresponds to the temperature independent electronic damping, for higher temperature R increases due to the phonon damping). Amplitude B reaches maximum for 𝜔𝑀=(𝜔2𝑆𝑅2/2𝑚2)1/2, however, as 𝑅/𝑚109s1, we have 𝜔𝑀𝜔𝑆 and 𝐵max𝑏𝜏0/𝑅𝜔𝑆. For the validity of the linearized model, the vibrations have to be small, that is, 𝐵max/2, hence 𝜏0𝑅𝜔𝑆/2𝑏2MPa.

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.


This research has been supported by Grant VZ-MMT 840770021.


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