ULTRASOUND AS A PROBE OF PLASTICITY?

THE INTERACTION OF ELASTIC WAVES WITH DISLOCATIONS

AGNES MAUREL1, VINCENT PAGNEUX2, FELIPE BARRA3,4 and FERNANDO LUND3,4

1Laboratoire Ondes et Acoustique, UMR CNRS 7587,

Ecole Superieure de Physique et Chimie Industrielles,

10 rue Vauquelin, 75005 Paris, France2Laboratoire d’Acoustique de l’Universite du Maine,

UMR CNRS 6613, Avenue Olivier Messiaen,

72085 Le Mans, France3,4Departamento de Fısica and CIMAT,

Facultad de Ciencias Fısicas y Matematicas,

Universidad de Chile,

Avenida Blanco Encalada 2008, Santiago, Chile

An overview of recent work on the interaction of elastic waves with dislocations is given. Theperspective is provided by the wish to develop non-intrusive tools to probe plastic behavior inmaterials. For simplicity, ideas and methods are first worked out in two dimensions, and results inthree dimensions are then described. These results explain a number of recent, hitherto unexplained,experimental findings. The latter include the frequency dependence of ultrasound attenuation incopper, the visualization of the scattering of surface elastic waves by isolated dislocations in LiNbO3,and the ratio of longitudinal to transverse wave attenuation in a number of materials.

Specific results reviewed include the scattering amplitude for the scattering of an elastic waveby a screw, as well as an edge, dislocation in two dimensions, the scattering amplitudes for anelastic wave by a pinned dislocation segment in an infinite elastic medium, and the wave scatteringby a sub surface dislocation in a semi infinite medium. Also, using a multiple scattering formal-ism, expressions are given for the attenuation coefficient and the effective speed for coherent wavepropagation in the cases of anti-plane waves propagating in a medium filled with many, randomlyplaced screw dislocations; in-plane waves in a medium similarly filled with randomly placed edgedislocations with randomly oriented Burgers vectors; elastic waves in a three dimensional mediumfilled with randomly placed and oriented dislocation line segments, also with randomly orientedBurgers vectors; and elastic waves in a model three dimensional polycrystal, with only low anglegrain boundaries modeled as arrays of dislocation line segments.

Keywords: Dislocations, Elastic waves, Plasticity.

I. INTRODUCTION

The interaction of acoustic waves with dislocations wasthe subject of many detailed studies in the period fromthe early 1950’s to the mid 1980’s, with the theory ofGranato and Lucke [1956 a;b] (hereafter GL) widely heldas the standard theory to this day. To be sure, theGL theory has had many successes that have withstoodthe test of time [Granato & Lucke, 1966]: It describesparticularly well damping, internal friction and modu-lus change in solids. However, it is less successful in theexplanation of, say, thermal conductivity measurements[Anderson, 1983].

From a theoretical point of view, the GL theory buildson the string model of Koehler [1952] in which a pinneddislocation segment is treated as a scalar string that vi-brates around an equilibrium, straight, position. Also,it is a mean-field theory that looks for an effective ve-locity and attenuation coefficient for a coherent scalarwave that propagates in a medium filled with many suchdislocations. Since it is a scalar theory, it cannot treatsimultaneously the three polarizations of an elastic wave,of which the acoustic wave is but one, the longitudinal.

Also, being a mean field theory, it cannot describe theinteraction of an acoustic wave with a single, isolateddislocation. Recently, however, experiments have beenable to distinguish quite clearly the attenuation of longi-tudinal waves from that of transverse waves, using Reso-nant Ultrasound Spectroscopy—RUS [Ledbetter & For-tunko, 1995; Ogi et al., 1999; 2004; Ohtani et al., 2005].Also, stroboscopic X-ray topography experiments havebeen able to visualize quite clearly the scattering of 500MHz surface elastic waves on LiNbO3 off single disloca-tions [Shilo & Zolotoyabbko, 2002; 2003; Zolotoyabko etal., 2001]. How is one to make quantitative sense of theseexperiments? GL theory needs to be extended.

From a different point of view, plasticity in crystals,particularly in metals, is widely understood to depend ondislocation behavior, and there are important problemsin materials science, such as the brittle to ductile tran-sition [Roberts, 2002], and fatigue [Suresh, 1998; Kende-rian et al., 2003a; 2003b], where there is wide agreementthat dislocations play a significant, possibly determinant,role. From an engineering design point of view the issueis well controlled in the sense that structures can be, andare, successfully constructed. From a basic physics point

2

of view, however, those phenomena are far from beingunderstood, in the sense that current theoretical mod-eling has no predictive power: If a new form of steel,say, is fabricated, current theory cannot, to the best ofour knowledge, make quantitative predictions concerningits mechanical properties as a function of temperature orcyclic loading. It is our opinion that one important causefor this lack of basic knowledge is the paucity of experi-mental measurements (as opposed to visualizations) con-cerning dislocations. This is so because dislocations needto be seen through transmission electron microscopy ofsamples that must be specially prepared [Xu et al., 2005;Arakawa et al., 2006; Robertson et al., 2008]. It standsthus to reason that it would be desirable to have quantita-tive measurements carried out with non invasive probes.Is this possible? We believe acoustic, or ultrasonic, mea-surements may provide such quantitative data. In thepursuit of this research program, we found that the com-paratively simple and basic theory problems of the behav-ior of elastic waves traveling through an elastic mediumfull of dislocations, or the interaction of an elastic wavewith a single dislocations, were not well understood.

This paper provides a brief overview of our work in thepast few years on the interaction of sound with disloca-tions.

II. THEORETICAL FRAMEWORK

We consider an homogeneous, isotropic, continuouselastic medium of density ρ and elastic constants cijkl =λδijδkl + µ(δikδjl + δilδjk). The experimental results wehave in mind for the theoretical framework to explain areperformed in three dimensions, so indices will take threevalues: i, j, k, · · · = 1, 2, 3. It is instructive however, aswill be discussed below, to develop a theory in the sim-pler case of two dimensional solids, in which case indiceswill take the values a, b, · · · = 1, 2. The state of such asystem is described by a vector field ~u(~x, t), of (small, sothat equations will be linear) displacements ~u, at time t,of points whose equilibrium position is ~x. Dealing witha continuum means that we shall consider length scales,particularly wavelengths, that are long compared withinteratomic spacing. We shall consider both infinite andsemi-infinite media, where wave propagation is governedby the usual wave equation

ρ∂2ui

∂t2− cijkl

∂2uk

∂xj∂xl= 0 (1)

with appropriate boundary conditions.

The elastic medium jus described supports the prop-agation of longitudinal (acoustic) and transverse waves

that propagate with velocities cL =√

(λ + 2µ)/ρ and

cT =√

µ/ρ, respectively. We shall call γ their ratio:γ = cL/cT . It is always greater than one.

Dislocations are modeled as one dimensional objects

(“strings”) ~X(s, t), where s is a Lagrangean parameterthat labels points along the line, and t is time. They

are characterized by a Burgers vector ~b. In the presentwork, in three dimensions, we shall consider only dislo-cations whose motion consists of slight deviations froman equilibrium position that is a straight line, of lengthL, joining two fixed (“pinning”) points, i.e., pinned dislo-

cation segments. When ~b is parallel to the straight line,we have a screw dislocation; when it is perpendicular,an edge dislocation. They are endowed with mass perunit length m, and line tension Γ. Their (unforced) mo-tion will be described by a conventional vibrating stringequation

m∂2Xi

∂t2+ B

∂Xi

∂t− Γ

∂2Xi

∂s2= 0 (2)

with an ad-hoc friction term characterized by a first or-der time derivative with a phenomenological coefficientB. This term takes into account internal losses such asdislocation interaction with phonons and electrons. Thecoefficient B cannot be determined within the frameworkof continuum elasticity, and is a parameter to be adjustedby the data. Expressions for the mass per unit length mand line tension Γ can however be found [Lund, 1988].For an edge dislocation they are

m =ρb2

4π

(

1 + γ−4)

ln

(

δ

δ0

)

Γ =ρb2c2

T

4π

(

1 − γ−2)

ln

(

δ

δ0

)

(3)

where δ and δ0 are external and internal cut-offs. Theinternal cut-off is roughly related to the distance fromthe dislocation core at which linear continuum elasticityceases to be valid and the external cut-off is the distanceto the nearest defect, or the sample size if there are noother defects. There are a number of assumptions neededto get these expressions, they have been spelled out in de-tail in Lund [1988], and will not be violated in the presentwork (see however the discussion in Section IVD) Notethat the speed of propagation of perturbations along thestring

√

Γ/m (when the Lagrangean parameter s has di-mensions of length) does not differ very much from thespeed of propagation of elastic perturbations in the bulk.

Why should a dislocation scatter sound (more gen-erally, elastic waves)? This question was addressed byNabarro [1951] who identified two basic physical mech-anisms: One, the stress associated with the wave wouldload the dislocation, to which the latter would respondby moving, which would in turn generate outgoing radi-ation. This mechanism is sometimes addressed in the lit-erature as “secondary radiation generated by dislocationflutter”. The second mechanism is through the nonlin-ear interaction generated by the probing of the wave ofthe region inside the dislocation core, where Eqns. (1)and (2) no longer hold. We expect this mechanism to benegligible when wavelengths are long compared to coreradius, an assumption we have already made.

3

The motion of the dislocation under external loadingis described by Eqn. (2) with a right hand side [Lund,1988]:

m∂2Xi

∂t2+ B

∂Xi

∂t− Γ

∂2Xi

∂s2= Fk (4)

where

Fk(~x, t) = ǫkjmτmbiσij(~x, t)

is the usual Peach-Koehler [1950] force. All velocities aresupposed to be small by comparison with the speed ofwave propagation, so it is not necessay to worry aboutLorentz-like forces. ǫkjm is the completely antisymmetrictensor of rank three (in two dimensions we shall use ǫab,the antisymmetric tensor of rank two), τm is a unit tan-gent along the (moving!) dislocation line, and σij(~x, t) isthe space- and time-dependent external stress which, inEqn. (4) is supposed to be evaluated at the dislocation

position, ~x = ~X(s, t). As has been noted, in three di-mensions we shall consider pinned dislocation segments,so the boundary conditions needed to solve (4) are fixedendpoints. In two dimensions the dislocation is a point,the equation is an ordinary differential equation and noboundary conditions are needed. The solution is foundusing standard techniques [Maurel et al., 2005a]: go tothe frequency domain and write the solution as a super-position of the normal mode of the string.

Once the dislocation moves, it will generate waves.How will it do it? By way of Eqn. (1) suitably modi-fied by a right hand side source term. It was found byMura [1963] that this behavior was best described not interms of particle displacement ~u but in terms of particlevelocity ~v = ∂~u/∂t:

ρ∂2vi

∂t2− cijkl

∂2vk

∂xj∂xl= si (5)

where the source term si is, when a single dislocation ispresent,

si(~x, t) = cijklǫmnk

∫

L

ds Xm(s, t)τnbl∂

∂xjδ(~x − ~X(s, t)).

When many dislocations are present, it is enough to takea sum over all of them.

In the deterministic case, i.e. when the right handside of (5) is precisely known, its solution is given bya convolution of the Green’s function G0

im (also calledimpulse response) of the medium with the source si:

vi(~x, t) =

∫

d3x′

∫

dt′G0im(~x − ~x′, t − t′)sm(~x′, t′), (6)

the Green function G0im being the solution to the equa-

tion

ρ∂2

∂t2G0

im(~x, t) − cijkl∂2

∂xj∂xlG0

km = δimδ(~x)δ(t) (7)

with appropriate boundary conditions. We shall considerinfinite two dimensional space, infinite three dimensionalspace, and semi-infinite three dimensional space with astress-free boundary.

When very many dislocations are present, it is notpractical to consider a deterministic approach, a statisti-cal one been more profitable and it will be discussed inSection IV.

III. SCATTERING BY A SINGLE

DISLOCATION

As it was mentioned above, the ideas and methodsare best explained in a step-by-step way, starting in twodimensions. There is one caveat in this case, namelythat wave propagation is qualitatively different in twodimensions from wave propagation in three dimensions:the impulse response in two dimensions has an infinitelylong tail in time.

A. Two dimensions, anti-plane case

The anti-plane case is a two dimensional elastic solidwhose displacements have only one nonvanishing compo-nent, u(xa, t), a = 1, 2, perpendicular to the (flat) equi-librium position. There is a screw dislocation that is alsoperpendicular to the plane, as is its Burgers vector (seeFigure 1), with only one nonvanishing component b. Thecut of the dislocation line by the plane is a point Xa(xb, t)[Maurel et al., 2004a].

u

b

k

FIG. 1: Scattering of an anti-plane wave by a single screwdislocation: The motion occurs on a plane, along which thewave propagates with wave vector k. The motion u howeveris perpendicular to the plane, as is the Burgers vector of thescrew dislocation

This is a scalar wave problem, with the dynamics of thedislocation being given by the appropriate specializationof Eqn. (2):

md2Xa

dt2= −µǫab

∂u

∂xb( ~X(t), t) (8)

To keep things as simple as possible, we have omitted theinternal friction term.

4

Concerning the anti-plane wave, it is described by theappropriate specialization of (5), which is

ρ∂2v

∂t2− µ

∂2v

∂xa∂xa= µbǫabXb

∂

∂xaδ(~x − ~X(t)) (9)

Its solution is found by the appropriate special case ofEqn. (6):

v(~x, ω) = µbǫabXb(ω)∂

∂xaG0(~x, ω) (10)

where an overdot means time derivative and the Green’sfunction G0(~x, ω) for the wave equation in two dimen-sions in infinite space with outgoing wave boundary con-ditions is well known to be a Hankel function of the firstkind:

G0(~x, ω) =i

4µH

(1)0

(

ω|~x|cT

)

We are assuming small displacements, so the disloca-tion will move little, and the right hand side of Eqn. (8)will be evaluated at the equilibrium position of the dis-location, which we shall take to be the origin. Also, theright hand side of that equation involves particle displace-ment u. As mentioned above, the generation of waves bya moving dislocation is best analyzed in terms of particlevelocity v = ∂u/∂t. So, taking the time derivative of (8)and going to the frequency (ω) domain we obtain

Xa(ω) = −ǫabµb

mω2

∂v

∂xb(0, ω) (11)

Now comes another approximation: it is assumed thatthe sound-dislocation interaction is sufficiently weak thatthe scattered wave will be small compared to the incidentwave. In this case a Born approximation is appropriateand the wave in the right hand side of (11) can be takento be the incident wave:

Xa(ω) = −ǫabµb

mω2

∂vinc

∂xb(0, ω) (12)

Substitution of (12) into (10) gives the solution forthe wave scattered by the screw dislocation. Taking theasymptotic behavior of the Hankel function for distancesfrom the dislocation large compared to wavelength, wefinally get the scattered wave vs:

vs(~x, t) = f(θ)eiωx/cT

√x

e−iωt (13)

where we have taken an incident plane wave of frequencyω propagating along the x1 direction:

vinc(~x, t) = eiω(x1/cT−t)

and the scattering amplitude comes out to be

f(θ) = −µb2

2m

eiπ/4

√

2πωc3T

cos θ (14)

with θ the angle between the direction of incidence x1

and the direction of observation ~x. Expression (14) wasfound by Eshelby [1949] and Nabarro [1951] on the basisof thinking by analogy with electromagnetism. This anal-ogy is no longer available in the more complicated casesto be examined below. Note the, at first sight, unintu-itive behavior of scattering: it grows with wavelength.This behavior does not carry over to three dimensions,where the finite length of a dislocation segment providesan additional length scale to compare wavelength with.

B. Two dimensions, in-plane case

The two dimensional in-plane case corresponds to anelastic wave propagating in two dimensions and inter-acting with an edge dislocation whose Burgers vectorba lies along the plane. Particle displacements ua(xb, t)now occur along the plane as well (see Figure 2). Thereare two possible polarizations for the wave: longitudinal,with particle displacements along the direction of mo-tion, with speed of propagation cL, and transverse, withparticle displacement perpendicular to the direction ofpropagation, with speed of propagation cT .

b

kL

kT

FIG. 2: Scattering of an in-plane wave by a single edge dis-location. The wave propagates along a plane, with two wavevectors: kL for longitudinal polarization, and kT for trans-verse polarization. The Burgers vector of the edge dislocationlies within the plane

We only consider dislocation glide: that is, dislocationmotion along the direction of the Burgers vector. Dislo-cation climb would involve mass transfer and falls outsidethe scope of the problems for which the response of a dis-location to an external load can be discussed within theframework of continuum elasticity.

The main difference between the physics of anti-planeand in-plane wave scattering from a dislocation is modeconversion: a longitudinal incident wave can be partiallyscattered as a longitudinal wave, and partially as a trans-verse wave. Similarly, a transverse incident wave can alsobe partially scattered as a longitudinal wave, and par-tially as a transverse wave. The vector nature of theproblem introduces a need for a more cumbersome nota-tion but does not add other physical difficulties. Thereare thus four scattering amplitudes. The algebra was

5

worked out in detail by Maurel et al., [2004a] and weshall give here their expressions:

fLL(θ) =µb2

2m

eiπ/4

√

2πωc3L

(

cT

cL

)2

sin 2θ0 sin(2θ − 2θ0),

fLT (θ) =µb2

2m

eiπ/4

√

2πωc3L

cos 2θ0 sin(2θ − 2θ0),

fTL(θ) = −µb2

2m

eiπ/4

√

2πωc3T

(

cT

cL

)2

sin 2θ0 cos(2θ − 2θ0),

fTT (θ) = −µb2

2m

eiπ/4

√

2πωc3T

cos 2θ0 cos(2θ − 2θ0), (15)

where the first sub index indicates the polarization of thescattered wave, and the second indicates the polarizationof the incident wave. For example, fTL is the amplitudefor the scattering of an incident longitudinal wave into atransverse wave. The angle θ0 is the angle between thedirection of propagation of the incident (plane) wave andthe Burgers vector. Note that Eqns. (15) indicate a scal-ing, particularly with wavelength, much the same as inthe anti-plane case. Since cT < cL, both longitudinal andtransverse waves have a larger amplitude for scatteringinto transverse waves.

C. Three dimensions, pinned dislocation segment

in infinite space

We now go back to Eqn. (4) for a line segment ~X(s, t)of length L (|s| ≤ L/2) that represents small deviations ofthe position of an edge dislocation from an equilibrium

position ~X0(s, t) that is a straight line, with fixed end

points: ~X(±L/2, t) = 0 [Maurel et al., 2005a].

X(s,t)

kT

kL

kL kT

FIG. 3: An elastic plane wave is incident upon a pinned dis-location segment. The latter responds by oscillating like astring and re-radiating scattered elastic waves. Both incidentas well as scattered waves have longitudinal (kL) and trans-verse (kT ) polarizations

As in Section III B only glide motion is allowed so Eqn.(4) is projected along the glide direction t, which is a unit

vector along the direction of the Burgers vector ~b = bt.We shall also use a unit (binormal) vector n ≡ τ × t,overdots for time derivatives and primes for derivativeswith respect to s so that Eqn. (4) for X ≡ Xktk becomes

mX(s, t) + BX(s, t) − ΓX ′′(s, t) = µb Mlk∂luk( ~X, t),(16)

with

Mlk ≡ tlnk + tknl. (17)

In this subsection we shall assume that the wavelengthis large compared to the length of the dislocation seg-ment: kL ≪ 1. In this case it is possible to solveEqn. (16) evaluating the Peach-Koehler force at a sin-

gle point of the dislocation, say its middle point ~X0:

~u( ~X, t) ≃ ~u( ~X0, t). In the frequency domain, this equa-tion becomes

−(mω2 + iωB)X(s, ω)−ΓX ′′(s, ω) = µbMlk∂luk( ~X0, ω),(18)

whose solution, using the particle velocity ~v(~x, ω) =−iω~u(~x, t), is

X(s, ω) = − 4

π

µb

mS(s, ω)Mlk∂lvk( ~X0, ω), (19)

with

S(s, ω) =∑

oddN

1

N(ω2 − ω2N + iωB/m)

sin

[

Nπ

L(s + L/2)

]

,

(20)and

ωN ≡ Nπ

L

√

Γ

m(21)

Taking Eqn. (6) to the frequency domain and substi-tuting (19) into it gives

vsm(~x, ω) =

8L

π2

µb

m

S(ω)

ω2cijkl ti nj

× ∂

∂xlG0

km (~x, ω)Mnp∂nvp( ~X0, ω), (22)

where we have used ǫjnhXnτh = −Xnj and with

S(ω) ≡ πω2

2L

∫ L/2

−L/2

ds S(s, ω) ≃ ω2

ω2 − ω21 + iωB/m

.

(23)The Green function G0

km for an infinite, homogeneousand isotropic elastic medium in three dimensions is wellknown [Love, 1944]. In a Born approximation, the in-cident wave is replaced on the right hand side of Eqn.(22) and to obtain the scattered wave the limit of G0

kmfor distances large (compared to both wavelength anddislocation size L) from the dislocation is used.

6

We shall consider an incident plane wave of frequencyω that is the superposition of a wave with longitudinal

polarization, along the propagation direction kL, and awave with linear transverse polarization along a direction

kT , kL · kT = 0, with amplitudes AL and AT respectively:

~vinc(~x, ω) = ALei~kL·~xkL + AT ei~kT ·~xkT (24)

where kL = ω/cL and kT = ω/cT . Substitution of thisexpression into (22) leads to longitudinal, ~vs

L, and trans-verse, ~vs

T , scattered waves, far away from the dislocation:

~vsL(~x, ω) = [fLLAL + fLT AT ]

eikLx

xx,

~vsT (~x, ω) = [fTLAL + fTT AT ]

eikT x

xy, (25)

with the scattering amplitudes

fLL = − 2

π3

(

ρb2

m

)

γ−4 L S(ω) fL(k0)gL(x)

fLT = − 2

π3

(

ρb2

m

)

γ−3 L S(ω) fT (k0)gL(x)

fTL = − 2

π3

(

ρb2

m

)

γ−1 L S(ω) fL(k0)gT (x)

fTT = − 2

π3

(

ρb2

m

)

L S(ω) fT (k0)gT (x)

(26)

where fL, fT , gL and gT are dimensionless functions thatdepend on the geometry. Note that for long wavelengthsλ, all four scattering amplitudes scale like

f ∼ L

(

L

λ

)2

so that they vanish as the ratio of dislocation length towavelength, squared, leading to scattering cross sectionsinversely proportional to wavelength to the fourth poweras in Rayleigh scattering. The direction of the transversepolarization is found from the geometry of the problem.Since γ > 1, both longitudinal and transverse incidentwaves prefer to be scattered as transverse waves, just asin two dimensions.

D. Three dimensions, pinned dislocation segment

in a half space

The experiments of Shilo and Zolotoyabko, [2002; 2003]and Zolotoyabko et al., [2001] provide strong motivationto tackle this problem. They have visualized the sur-face waves scattered by a subsurface dislocation segmentin LiNbO3. From a theoretical point of view, Eqn. (6)holds just as well for a half space as for a whole space,the difference being that G0

im is now the impulse responsefunction of an elastic half-space with a stress-free bound-ary. Finding this function is a well posed problem and hasbeen widely studied (see Maurel et al. [2007a] and refer-ences therein). However, there is no expression available

that compares in simplicity or ease of analytical manip-ulation with the expression for the impulse response fora whole space. The differences with the whole-space re-sponse are obviously especially important near the freesurface. Notably, in that region, in addition to polescorresponding to the usual transverse and longitudinalwaves, there is an additional singularity that correspondsto surface (Rayleigh) waves, whose speed of propagationis cR = ζcT , where ζ ∼ 0.9 is the zero of the Rayleighpolynomial P (ζ) = ζ6−8ζ4+8ζ2(3−2/γ2)−16(1−1/γ2).These waves are not dispersive, and they have an ellip-tical polarization, a combination of the longitudinal andtransverse perpendicular to the free surface [Landau &Lifshitz, 1981]. They penetrate the medium for a dis-tance comparable to their wavelength. Consequently, fora dislocation to feel their presence it must be located ata distance from the free surface that is less than a fewwavelengths.

In this subsection we are interested in the behavior ofthe secondary radiation emitted by a dislocation excitedby the incident surface wave near the dislocation itself.So the concepts of scattering amplitude and cross sec-tion will not be used, they being tailored to the far-fieldbehavior of the secondary radiation.

u inc

z0

x1

x2

z

S

b α

α

ξ

FIG. 4: A subsurface dislocation segment lying in a semi-infinite half space with a stress-free boundary is excited by asurface Rayleigh wave. It responds by oscillating, and reradi-ating. At the free surface, these secondary waves will interferewith the incident wave [Maurel et al. 2007a].

We consider the configuration of Figure 4, and the firstorder of business is to find the response of the disloca-tion to the surface wave. This is given by the solutionof Eqn. (4) with the Peach-Koehler right-hand side com-puted from the resolved shear stress generated by theincoming surface wave at the position of the dislocation.Importantly, it is possible to find such a solution withoutassuming that wavelength is large compared to disloca-tion length. Indeed, in the case at hand the wavelength issmall compared to dislocation length. The said responseis a straightforward but cumbersome expression that canbe found in Eqn. (3.7) of Maurel et al. [2007a]. In partic-ular, we find that, for the conditions of Shilo & Zolotoy-abko [2003] (wavelength λ ∼ 6 µm, dislocation depth of2/kR) the amplitude of dislocation motion is of order 500times the amplitude of the incident wave: X ∼ 500uinc

z

which gives about 25 nm for an incident wave of ampli-tude 0.05 nm. The dislocation velocity X ∼ ωX thuscomes out around 90 m s−1, and we infer a, not unusual,

7

drag coefficient B ≃ 10−5 Pa s.

Once the motion of the dislocation has been found, thefree surface vertical displacement us

z(~x)can be computedfrom Eqn. (6). As already mentioned this involves theGreen function for a half space and the task is best donenumerically. This result can then be superposed withthe incident wave uinc

z (~x) to obtain the total displacementuinc

z (~x)+uincz (~x), as is shown on Figure 5, and compare it

with the X-ray images of Shilo & Zolotoyabko [2003]. Thecomparison appears satisfactory [Maurel et al. 2007a].

FIG. 5: The upper panel shows the interference pattern be-tween incident and scattered waves generated at the free sur-face by subsurface dislocation segments whose surface traceis given by the dashed lines. The lower panel is a close up atthree times the magnification of the upper panel. The dashedline is the direction of the main dislocation segment and thedotted line indicates the direction of propagation of the inci-dent wave [Maurel et al. 2007a].

IV. MULTIPLE SCATTERING

Acoustic attenuation experiments are typically per-formed with samples that have a high dislocation density.A wave that is incident upon a dislocation-filled mediumwill interact with many dislocations and, in general, theresulting wave behavior will be quite complex. Under cer-tain circumstances however, special cases may arise: the(multiply) scattered waves may interfere so as to gen-erate a coherent, forward propagating, wave describedby a complex index of refraction whose real part gives arenormalized speed of propagation, and whose imaginarypart gives an attenuation (see Figure 6). The origin ofthis attenuation is that energy is taken away from theincident wave by the radiation that is redirected in alldirections other than the incident one by the scattering:this incoherent radiation may obey, in turn, a diffusion

equation characterized by a diffusion coefficient D, in-troducing thus a different character for energy transport.And further, it may happen that the diffusion coefficientvanish, D = 0, in which case, we are in the presence ofAnderson localization [Sheng, 1990; 1995]. In the remain-ing of this paper we shall deal with coherent elastic wavepropagation in an elastic continuum with many pinneddislocation segments.

k

K

Coherent radiation("Propagation")

n = nr + i ni

Incoherent radiation("Diffusion")

D(D = 0, "Localization")

MULTIPLE SCATTERING

FIG. 6: An elastic plane wave is incident on a medium filledwith randomly located scatterers. Part of it will propagatecoherently with a complex index of refraction n = nr + ini.This implies a renormalized velocity, and an amplitude thatexponentially decreases with distance. Another part of it willpropagate incoherently, and may obey a diffusion equationwith a diffusion constant D, whose value may, under certaincircumstances, vanish, signaling Anderson localization.

When many dislocations are present, the behavior ofelastic disturbances is governed by the wave equation (5)with a source term si that now reflects the presence ofmany dislocations:

si(~x, t) = cijklǫmnk

N∑

n=1

∫

L

ds Xnm(s, t)τnbl

× ∂

∂xjδ(~x − ~Xn(s, t)). (27)

Replacing the solution of the string equation (4) in thissource term leads to an equation of the form

ρ∂2vi

∂t2− cijkl

∂2vk

∂xj∂xl= Vikvk (28)

where

Vik =8

π2

(µb)2

m

S(ω)

ω2L

N∑

n=1

Mnij

∂

∂xjδ(~x − ~Xn

0 ) Mnlk

∂

∂xl

∣

∣

∣

∣

∣

~x= ~Xn

0

.

(29)Equations of this type, when V describes a random indexof refraction have been much studied [Ishimaru, 1997].Our case, however, is considerably different.

8

A. Two dimensions, anti-plane case

In the two-dimensional, anti-plane case, the specialcase of Eqn. (28) is

ρv(~x, t) − µ∇2v(~x, t) = µ

N∑

n=1

bnǫabXnb

∂

∂xaδ(~x − ~Xn).

(30)where Xn is the solution of Eqn. (11) for the n-th dislo-cation. Going to the frequency domain this gives

(

∇2 + k2T

)

v(~x, ω) = −V (~x, ω)v(~x, ω), (31)

where the “potential” V is the appropriate special caseof (29):

V (~x, ω) =

N∑

n=1

µ

M

(

bn

ω

)2∂

∂xaδ(~x − ~Xn

0 )∂

∂xa

∣

∣

∣

∣

~Xn

0

. (32)

where kT ≡ ω/cT . In this expression, the positions ~Xn(t)of the screw dislocations have been replaced by their

mean values (positions at rest) ~Xn0 . The screw dislo-

cations in the right hand side of (31) are randomly, anduniformly, placed. This is all the randomness we shallconsider in this subsection.

x

x

k0

1

2

v

σ

K

X0

bn

n

FIG. 7: Anti-plane wave propagating through a medium filledwith many screw dislocations [Maurel et al., 2004b]. k isthe wave vector in the abscence of dislocations, and K theeffective wave vector for coherent propagation.

In the absence of scatterers, plane waves propagatewith a wave vector kT . The exercise is to find an effec-tive wave vector KT , if it exists, for the propagation ofcoherent plane waves.

A result that goes back at least to Foldy [1945] statesthat, for low densities n of scatterers, this many-bodyproblem can be reduced to a single body one:

KT = kT + n

√

2π

k〈f(0)〉e−iπ/4 (33)

where the effective wave vector is given in terms of theforward scattering amplitude for single-body scattering,and the brackets denote an average over all internal de-grees of freedom (absent in the present case). We already

know the scattering amplitude, Eqn. (14), so it is easyto replace in (33) to get

KT = kT

(

1 − µnb2

2mω2

)

. (34)

A simple minded expression for the attenuation lengthΛ is given by

Λ−1 = nσ

where n is the dislocation density and σ is the total scat-tering cross section for single dislocation scattering de-fined as:

σ =

∫ 2π

0

dθ|f(θ)|2

Since we know the scattering amplitude for all angles,f(θ) Eqn. (14), not only in the forward direction, we cancompute the total cross section to get an expression forthe attenuation length:

Λ =8m2c4

T k

nµ2b4. (35)

This simple minded approach is not available in morecomplicated situations so we describe a more system-atic approach, based on Green functions [Maurel et al.,

2004b].

The Green function approach is based on the fact that,in the case when no scatterers are present, the propaga-tion character of the wave is determined by the poles of

the corresponding Green function G0(~k, ω) in wavenum-ber space:

G0(~k, ω) =1

k2T − k2

The strategy provided by Multiple Scattering theoryis to focus on the impulse response function G of themedium filled with scatterers, in our case screw disloca-tions:

(

∇2 + k2T + V (~x)

)

G(~x, ~x′) = −δ(~x − ~x′). (36)

Then, since this equation involves a random operator V ,

consider the average < G(~k, ω) >, and look for its polesin the form

⟨

G(~k, ω)⟩

=1

k2T − Σ(k) − k2

(37)

where Σ(k), the mass operator, is a functional of the “po-tential” V . Assuming (a big assumption) that there willbe a coherent wave whose (effective) wave vector does notdiffer very much from the wave vector in the absence ofscatterers, the problem becomes finding the mass oper-ator Σ perturbatively, treating the “potential” V as theperturbation. To second order in V , the result is

Σ = 〈V 〉 + 〈V G0V 〉 − 〈V 〉G0〈V 〉. (38)

9

The result of this calculation in the current, antiplanecase, is

Σ(k) =µnb2

mω2k2

[

−1 +iρb2

8m

]

. (39)

to leading order in the real part, as well as the imaginarypart, of the mass operator. This expression can be read-ily generalized [Maurel et al., 2004b] to the case whenthe dislocations have random Burgers vectors, as well asrandom masses.

We are now in a position to obtain the effective wavenumber KT , the pole of 〈G〉 in Eqn. (37). Since KT issupposed to be close to kT we get

KT ≃ kT

(

1 +Σ(kT )

2k2T

)

, (40)

≃ kT

(

1 +µnb2

2mω2

[

−1 +iρb2

8m

])

.

The real part of this expression coincides with (34), andthe imaginary part gives the attenuation length (35)through:

Λ−1 = 2ℑ(KT )

The effective velocity veff ≡ ω/ℜ(KT ) for coherent wavepropagation is thus

veff = cT

(

1 +µnb2

2mω2

)

Note that the effective phase velocity is greater than cT ,although the group velocity is smaller. This is probablyan effect of the bidimensionality, and just like the de-pendence on wavelength, it does not carry over to threedimensions.

B. Two dimensions, in-plane case

As already mentioned, the main difference between theanti-plane and in-plane cases is that the latter is of a vec-tor nature: a wave with both longitudinal and transversepolarizations travels in a medium filled with randomlyplaced edge dislocations, whose Burgers vectors are ran-domly oriented (see Figure 8. There will be two effectivevelocities and two attenuation lengths, one for each po-larization. On the technical side, the results of Foldy,and of the Green function approach, mentioned in theprevious subsection, have to be generalized to this vectorcase. The “potential” V , the Green function G and massoperator Σ become rank-two matrices, and the effectivewave vectors are given by the zeros of the determinant of< G >−1.

The generalization of the simple-minded approach of

x2

x1

vT

vL

X0

n

bn

θnkL,kT

KL ,KT

FIG. 8: An inplane wave is incident on a medium with manyedge dislocations whose Burgers vectors are randomly ori-ented. kL and kT are the longitudinal and transverse wavevectors, respectively, in the absence of dislocations. KL andKT are the effective longitudinal and transverse wave vectorsof the coherently propagating wave [Maurel et al., 2004b].

Foldy to the anti-plane case gives [Maurel et al., 2004b]

Kc = kc + n

√

2π

kc〈fcc〉b,θ(0)e−iπ/4,

= kc

(

1 − nµb2Ac

4mω2

)

, (41)

with

AL = γ−2,

AT = 1. (42)

There is no mode conversion into the forward directionthrough scattering. The brackets 〈〉b,θ mean taking theaverage for the magnitudes and orientations of the Burg-ers vectors. The second line in Eqn. (41) is obtainedwhen all Burgers vectors have the same magnitude andall orientations are equally likely.

The generalization of the Green function method leadsto

Kc = kc

{

1 − µnb2Ac

4mω2

[

1 −(

1 +1

γ4

)

iρb2

8m

]}

, (43)

The real part of this expression coincides with (41) andthe imaginary part gives two attenuation lengths, one forlongitudinal and one for transverse wave propagation

ΛL = 16m2

nρ2b4

γ8

γ4 + 1kL.

ΛT = 16m2

nρ2b4

γ4

γ4 + 1kT .

(44)

The above expression for attenuation length also sat-isfy

Λ−1L = n(σLL + σLT )

Λ−1T = n(σTL + σTT ) (45)

where the scattering cross sections σab are given in termsof the scattering amplitudes (15) by

σab =

∫ 2π

0

dθ < |fab|2 > (46)

10

The effective velocities vceff ≡ ω/ℜ(Kc) for coherent

wave propagation are easily read off from (43):

vceff = cc

(

1 +µnb2

4mω2Ac

)

.

Just as in the antiplane case, the effective phase veloc-ity, but not the effective group velocity, is larger in thepresence than in the absence of dislocations, an effectprobably due to the bidimensionality of the situation.

C. Three dimensions, many dislocation segments

We now study the configuration illustrated on Fig. 9[Maurel et al., 2005b]. A plane elastic wave propagatesthrough an elastic medium that is filled with edge dis-location segments whose end points are fixed, and theyare straight lines in equilibrium. The position of thesesegments is random, with a uniform distribution throughspace. The orientation of the lines is also random, alsouniformly distributed throughout the solid angle. All seg-ments have the same length L and magnitude of Burgersvector b. The latter is perpendicular to the line, but,within that plane, the orientation is random, uniformlydistributed in angle. Dropping the assumptions of uni-form distribution for the orientation of the lines and ofthe Burgers vectors does not lead to significant complica-tions. Dropping the assumption of constant L, however,would lead to significant changes, as would dropping theassumption of uniform distribution in space. Through-out this subsection, wavelengths will be assumed to belarge compared to dislocation line segment length L.

k , kL TvLinc

vTinc

x3

x2

x1

FIG. 9: An elastic wave with both longitudinal (L) and trans-verse (T ) polarizations is incident upon a medium filled withmany, randomly placed and oriented, edge dislocations seg-ments whose Burgers vector are randomly oriented as well.[Maurel et al., 2005b].

In the Green function approach to multiple scatteringfor this system, the starting point for the algebra is the

set of equations

ρω2G0im(~x, ω)+cijkl

∂2

∂xj∂xlG0

km(~x, ω) = −δimδ(~x) (47)

for the rank-three Green tensor G0im that describes the

impulse response function of the elastic medium in theabsence of dislocations, and

ρω2Gim(~x, ω) + cijkl∂2

∂xj∂xlGkm(~x, ω) =

−VikGkm(~x, ω) − δimδ(~x), (48)

where V is the “potential” of Eqn. (28), for the rank-three Green tensor Gim that describes the impulse re-sponse function of the elastic medium with a randomdistribution of dislocations. Both equations have beenwritten in the frequency domain.

The exercise now is, as in two dimensions, to find ef-fective wave numbers Ka (a = L, T ) for longitudinal aswell as transverse coherent wave propagation. In gen-eral these effective wave numbers will be complex, withthe real part leading to effective speeds of wave prop-agation, and the imaginary part leading to attenuationlengths. They are found as the zeroes of the determi-nant of the three by three matrix for the inverse of theaveraged Green tensor G in wave number space:

< G(~k, ω) >−1= G0(~k, ω)−1 − Σ(~k, ω) (49)

where the mass operator Σ is a functional of the “poten-tial” V that can be found perturbatively. The result is,to leading order for both the real and imaginary parts

KL ≃ kL

[

1 − 16

15π2

1

γ4

ρb2

m

S(ω)

ω2nLc2

L

+i64

225π5

3γ5 + 2

γ8

(

ρb2

m

)2 ℜ[

S2(ω)]

ωnL2cL

]

,

KT ≃ kT

[

1 − 4

5π2

ρb2

m

S(ω)

ω2nLc2

T

+i16

75π5

3γ5 + 2

γ5

(

ρb2

m

)2 ℜ[

S2(ω)]

ωnL2cT

]

.(50)

where S(ω) is given by (23) and n is the number of dislo-cation segments per unit volume. The magnitude of theseeffective wavenumbers is independent of orientation sincethe effective medium is isotropic. This is a consequencethat, whenever random vector quantities appear, all di-rections are equally likely.

The real part of these expressions leads to effectivewave velocities va = ω/ℜ(Ka) that are straightforwardto find. In the long wavelength limit they are

vL ≃ cL

(

1 − 16

15π4

1

γ2

µb2

Γ

nL3

1 + (ω/ωB)2

)

vT ≃ cT

(

1 − 4

5π4

µb2

Γ

nL3

1 + (ω/ωB)2

)

, (51)

11

where ωB = mω21/B, and ω1 is the fundamental mode for

oscillations of the string-like dislocation segment, givenby (21), the only mode of importance for long wavelengths.

The imaginary part leads similarly to attenuation co-efficients αa = ℑ(Ka):

αL =16

15π6

1

γ2

µb2nL5

cLΓ2

(

Bω2

1 + (ω/ωB)2

+4(3γ5 + 2)

15π3γ4

ρb2L

cL

ω4[1 − (ω/ωB)2]

[1 + (ω/ωB)2]2

)

,

αT =4

5π6

µb2nL5

cT Γ2

(

Bω2

1 + (ω/ωB)2

+4(3γ5 + 2)

15π3γ5

ρb2L

cT

ω4[1 − (ω/ωB)2]

[1 + (ω/ωB)2]2

)

. (52)

Note that, for low frequencies and low damping, ω ≪ ωB,both damping coefficients scale with frequency as

α ∼ C1(B)ω2 + C2ω4 (53)

This is a reflection of the fact that the damping of thecoherent wave has two origins: internal losses, scaling likefrequency squared, and originated by disorder, scalinglike frequency to the fourth.

Finally, for fixed total dislocation line density, that isfixed nL, measured in units of inverse surface, attenua-tion coefficients scale with segment length L as

α ∼ L4

and the velocity changes ∆v = c − v scale as

∆v

c∼ L2

just as in the GL theory [Granato & Lucke, 1966]. Moreon this below.

1. Comparison of current theory

with attenuation experiments

The results developed in the previous paragraphs canbe put to the test since measurements of longitudinaland transverse wave attenuation are available. Indeed,quality factors Q have been measured for copper singlecrystals [Ogi et al., 1999], copper polycrystals [Ledbet-ter, 1995] and LiNbO3 [Ogi et al., 2004] using resonantultrasound spectroscopy (RUS), and its electromagneticvariant (EMAR) for copper single crystals [Ogi et al.,1999]. The frequency ranges of these experiments fallinto what has been called “low frequencies” in the calcu-lations above, and it is a simple matter to find the ratioof the longitudinal to transverse quality factors:

Q−1T

Q−1L

≃ 3

4γ ∼ 1.30 − 1.56, (54)

QL/QT

Polycrystallyne copper using RUS (a) 1.6

LiNbO3 using RUS (b) 1.7

Copper single crystals using RUS (c) 1.4

Copper single crystals using EMAR (c) 2.1-2.3

Current theory 1.3-1.6

TABLE I: The ratio of longitudinal to transverse wave attenu-ation according to current theory (bottom line) and accordingto experiments: (a) Ledbetter & Fortunko [1995]; (b) Ogi et

al., [2004]; (c) Ogi et al., [1999]

Note that γ is just the ratio of longitudinal to trans-verse wave propagation velocity, a quantity determinedcompletely by the material at hand. In this ratio thereis no dependence on material density, elastic constants,or dislocation paramenters such as mass, line tension orattenuation coefficient. There are thus no adjustable pa-rameters. The range of values given in Eqn. (54) re-flects a range in Poisson’s ratio of 0.25 to 0.35. TableI compares the data with this value. The agreement issatisfactory.

2. Granato-Lucke theory as a special case of current theory

The theory of Granato & Lucke [1966] is a special caseof the theory developed in the present paragraph. Indeed,its starting point is the set of equations

ρ∂2

∂t2σ(~x, t) − µ

∂2

∂x22

σ(~x, t) = −µρ bΛ

L

∫

ds ξ(~x, s, t),

mξ + Bξ − Γξ′′ = µσ. (55)

where Λ is the total length of movable dislocation (Λ =nL in our notation), ξ is a scalar string (a simplification

of our ~X), and σ is a scalar stress (a simplification of ourσij). This system is solved looking for solutions of theform

σ(x2, t) = σ0e−αGLx2eiω(t−x2/vGL), (56)

that is, attenuated waves traveling along the x2 direction,where αGL and vGL are, respectively the attenuation andthe effective wave velocity in the GL model, and it isassumed that ξ does not depend on x3.

It is a straightforward exercise to check that Eqns.(55) are obtained within our multiple scattering formal-ism when looking at the system of Figure 10. That is,a purely transverse waves is incident on an ensemble ofdislocation segments randomly placed but not randomlyoriented: they must all be parallel, along the x3 direc-tion, according to Eqn. (55). Also, all the Burgers vec-tors must point along the same, x1, direction. In the longwavelength limit, attenuation length and effective wavevelocity obtained from this calculation coincides with our

12

kTvT

inc

x3

x2

x1

FIG. 10: Applying a multiple scattering formalism to ashear wave incident upon a medium filled with many, ran-domly placed but parallel, dislocation segments reproducesthe Granato-Lucke theory [Maurel et al., 2005b].

own, up to numerical coefficients. The distinguishing fea-ture of our theory is that it can clearly discriminate be-tween longitudinal and transverse polarizations. Also,the effective medium corresponding to the situation de-picted in Figure 10 is anisotropic so that an effective wavevector will depend on the direction of propagation.

D. Three dimensions, low angle grain boundaries

It is well known that low angle grain boundaries can bemodeled as arrays of dislocations [Bragg, 1940; Burgers,1940; Read & Schockley, 1950; Shockley & Read, 1949],see Figure 11.

θb

FIG. 11: Low angle grain boundaries can be considered asarrays of dislocations [Maurel et al., 2006].

On the other hand, Zhang et al. [2004] have measuredthe frequency dependence of acoustic attenuation in poly-crystalline copper with enough accuracy to find that itscales as a linear combination of quadratic and quartic

terms:

α ∼ c1ω2 + c2ω

4. (57)

While there appears to be general agreement that thequartic term arises from Rayleigh scattering by the grainboundaries, the quadratic term is a surprise. Is it possibleto rationalize these findings within a multiple scatteringframework?

We have already seen that this type of behavior is whathappens, for long wavelengths, when a coherent wavepropagates in a medium filled with many dislocations.The quadratic term is associated with internal losses, andthe quartic term is associated with losses due to the dis-order: The multiple scattering takes energy away fromthe forward-moving coherent wave.

To proceed, the many dislocations considered in Eqn.(28) are distributed in two classes: one class is groupedas a number of parallel segments, that mimic a low an-gle grain boundary. Then, each such class is randomlydistributed throughout space, see Figure 12. The detailsin two dimensions can be found in Maurel et al. [2006],and, in three dimensions, in Maurel et al. [2007b].

b

D

L

w

hy

FIG. 12: Dislocation segments grouped in two classes mimic,at long wavelengths, the response of randomly oriented grainboundaries to acoustic waves [Maurel et al., 2007b]. Here,w would be the grain size, D and L the rectangular grainlengths, and h the spacing between dislocations within a grainboundary.

This modeling does not capture the topology of thegrain boundaries, in the sense that such walls must meetat joints, to actually enclose grains. However, long wave-length propagation should be independent of these char-acteristics. The result is [Maurel et al., 2007b] thatthe prefactor of the quartic term in Eqn. (57) can beobtained with reasonable values for the material understudy, without adjustable parameters. The usual sourceof attenuation in polycrystals, changes in the elastic con-stants from grain to grain, can be incorporated as anadditive effect, in perturbation theory. The prefactor ofthe quadratic term can be fit assuming that the drag onthe dynamics of the dislocations making up the wall isone to two orders of magnitude smaller than the valueusually accepted for isolated dislocations. As mentioned

13

in Section II, the expressions (3) for mass and line tensionare valid for isolated dislocations. More precisely, theyinvolve a cut off that is the distance to the nearest defect.In the case at hand, this distance is h, small comparedto wavelength, and these values should be revisited for amore accurate assessment.

V. CONCLUSIONS AND OUTLOOK

We have provided a fairly complete theory of the inter-action of elastic waves with dislocations, both in isolationand in large numbers, leading to the understanding of anumber of quantitative experimental results, as well asof a number of visualizations. What next?

As stated in the Introduction, one motivation for thiswork has been to develop non intrusive probes of plasticbehavior of materials. The next logical step might be tovalidate the results obtained with the current theory withthose obtained using conventional techniques. For exam-ple two identical materials differing only in their densityof dislocations, a situation that appears not impossible toachieve, would give rise to different speeds of sound, lead-ing to different resonant frequencies in identically-shapedsamples. The dislocation density so inferred could thenbe compared with the density measured through Trans-mission Electron Microscopy.

Assuming that step to be successful, the next stepwould be to develop suitable, hopefully portable instru-mentation to probe dislocation density in any sample ofinterest. The obstacle to be overcome would be to differ-entiate the contribution of dislocations from the contri-bution of other scatterers such as inclusions, vacancies,grain boundaries and so on. This does not look an in-surmountable job given the specific nature, such as angu-lar dependence, of the radiation scattered by dislocationscompared to other defects.

Acknowledgments

J.-F. Mercier contributed significantly to the initialstages of this project, including the cases of single andmany dislocations in two dimensions. D. Boyer also con-tributed significantly to the work for model grain bound-aries in two dimensions. We are grateful to E. Bitzek,R. Espinoza, N. Mujica, A. Sepulveda, D. Shilo, andE. Zolotoyabko for useful discussions. The support ofECOS-CONICYT, CNRS-CONICYT, FONDAP Grant11980002, FONDECYT Grants 1030556, 1060820, andACT 15 (Anillo Ciencia y Tecnologıa) is gratefully ac-knowledged.

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