Elastic wave propagation and scattering in heterogeneous ...

University of Nebraska - Lincoln

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Faculty Publications from Nebraska Center for Materials and Nanoscience, Nebraska Center for
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August 1999

Elastic wave propagation and scattering in
heterogeneous, anisotropic media: Textured
polycrystalline materials

Joseph A. Turner

University of Nebraska - Lincoln, [email protected]

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Turner, Joseph A., "Elastic wave propagation and scattering in heterogeneous, anisotropic media: Textured polycrystalline materials"
(1999). Faculty Publications from Nebraska Center for Materials and Nanoscience. Paper 60.
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Elastic wave propagation and scattering in heterogeneous,
anisotropic media: Textured polycrystalline materials

Joseph A. Turner
Department of Engineering Mechanics, W317.4 Nebraska Hall, University of Nebraska-Lincoln,
Lincoln, Nebraska 68588-0526

͑Received 23 September 1998; accepted for publication 3 May 1999͒

The propagation of elastic waves through heterogeneous, anisotropic media is considered.
Appropriate ensemble averaging of the elastic wave equation leads to the Dyson equation which
governs the mean response of the field. The Dyson equation is given here in terms of anisotropic
elastic Green’s dyadics for the medium with and without heterogeneities. The solution of the Dyson
equation for the mean response is given for heterogeneities that are weak. The formalism is further
specified for the case of equiaxed cubic polycrystalline metals with a single aligned axis. The
Green’s dyadics in this case are those for a transversely isotropic medium. Simple expressions for
the attenuations of the shear horizontal, quasicompressional, and quasishear waves are given in
terms of integrations on the unit circle. The derived expressions are limited to frequencies below the
geometric optics limit, but give the attenuations in a direct manner. Comparisons with previous
results are also discussed. It is anticipated that a similar approach is necessary for the study of wave
propagation in complex anisotropic materials such as fiber-reinforced composites. In addition, the
results are applicable to diffuse ultrasonic inspection of textured polycrystalline media. © 1999
Acoustical Society of America. ͓S0001-4966͑99͒04008-4͔

PACS numbers: 43.20.Bi, 43.20.Gp, 43.35.Cg ͓DEC͔

INTRODUCTION mined the phase velocity and scattering attenuation. Their
results were applicable for all frequencies from the Rayleigh
The study of wave propagation and scattering of elastic limit to the geometric optics limit due to their use of the
waves in heterogeneous, anisotropic media is related to non- Keller approximation alone without additional approxima-
destructive testing, materials characterization and acoustic tions. They have also examined correlations defined by both
emission of many important materials. Examples include equiaxed grains and grains with elongation.5,6 They further
polycrystalline media with texture, fiber-reinforced compos- noted the relations between the diffuse backscatter and the
ites, and extruded metal-matrix composites. Elastic waves presence of texture.7
which propagate through such media lose energy due to scat-
tering from the heterogeneous structure of the material. This The general scattering problem as discussed by these
scattering may be characterized by the attenuation of the me- authors and others involves a scattering integral with a
dium. If the medium is statistically isotropic, the attenuation Green’s function as its kernel. If the medium is statistically
is independent of propagation direction. In an anisotropic isotropic, this Green’s function clearly takes the form of the
medium the scattering attenuation is a function of propaga- Green’s function for the isotropic medium. In the case of
tion direction. The analysis of this scattering attenuation is, statistically anisotropic media this choice is less clear. Stanke
therefore, more complicated than that of the isotropic case. and Kino argue that the isotropic Green’s function may be
The study of statistically isotropic media and the correspond- used in the analysis for polycrystalline materials with
ing scattering attenuation has received considerable atten- texture.1 This argument was the basis of the work by Ahmed
tion. The problem of wave propagation in textured polycrys- and Thompson.5 They used the isotropic Green’s function as
talline materials has, however, received lesser attention. given by Lifshitz and Parkhamovski8 to describe the scatter-
Stanke and Kino1 briefly discuss the applicability of the ing in textured media. In addition, they simplified the analy-
Keller approximation2 to the case of polycrystalline media sis by using the polarization directions for the isotropic
with texture, although they provide no specific results. waves. Examination of the Keller approximation indicates
Hirsekorn was one of the first to carefully examine the scat- that the choice of Green’s function is not so clear. In the
tering in textured polycrystals as a function of frequency.3 original discussion,2 a small variable ␧ was defined as a mea-
The use of a single-sized, spherical grain resulted in unphysi- sure of the departure of the medium from homogeneity. The
cal oscillations at higher frequencies. She then extended her Green’s function within the scattering integral corresponded
theory, using the same perturbation approach, to determine to that for the medium for which the heterogeneity was zero.
the directional dependence of the phase velocity and attenu- For textured polycrystalline materials, the homogeneous me-
ation of the three wave types.4 The problem was considered dium is anisotropic. Thus, an anisotropic Green’s function
more recently by Ahmed and Thompson.5 They also exam- may be more appropriate for this and other similar problems.
ined the case of polycrystalline grains with an aligned axis Comparisons between these different solution methods ͑e.g.,
and developed algebraic equations governing the wave num- isotropic Green’s function with and without anisotropic po-
ber. The roots of these equations, found numerically, deter- larizations and anisotropic Green’s function͒ in terms of the

541 J. Acoust. Soc. Am. 106 (2), August 1999 0001-4966/99/106(2)/541/12/$15.00 © 1999 Acoustical Society of America 541

strength of the anisotropy will not be addressed here. pulse at location xЈ in the ␣th direction. The moduli are
The use of an anisotropic Green’s function for modeling
spatially variable and density is assumed uniform through-
the scattering in statistically anisotropic media is the subject
of this article. Here, the problem is formulated in terms of out. The units in Eq. ͑1͒ have been chosen such that the
the Dyson equation as discussed by Frisch9 and Weaver.10
The Dyson equation is easily solved in the spatial Fourier density is unity. The moduli are assumed to be spatially het-
transform domain within the limits of the first-order smooth-
ing approximation ͑FOSA͒, or Keller2 approximation. A fur- erogeneous and of the form C i j k l ( x) ϭ C 0 kl ϩ ␦ C i j k l ( x) .
ther approximation is also made which restricts the results to ij
frequencies below the high-frequency geometric optics limit. 0
The problem is further specified for the case of cubic poly- Thus, the moduli have the form of average moduli, C ij k l
crystalline grains with one aligned axis and with the other
two axes randomly oriented. In this case, the anisotropy re- ϭ͗Ci jkl(x)͘, plus a fluctuation about this mean, ␦Ci jkl(x).
duces to that of transverse isotropy. The result is the attenu-
ation as a function of direction and frequency for the shear The fluctuations are assumed to have zero mean,
horizontal, quasicompressional, and quasishear waves. In
particular, the angular dependence of the attenuations in the ͗␦Cijkl(x)͘ϭ0. The mean moduli are not necessarily
Rayleigh limit is obtained explicitly. Outside the Rayleigh isotropic—the material may have global anisotropy. The co-
limit, simple expressions for the attenuations of the shear
horizontal, quasicompressional, and quasishear waves are de- variance of the moduli is represented by an eighth-rank ten-
rived in terms of integrations on the unit circle. The results
are quantitatively similar to those of Ahmed and Thompson,5 sor
but are given here in a more direct manner. Differences in
the angular dependence are also seen due to the use of the ⌳ ͑ xϪ y͒ ijkl ϭ ͗ ␦ C i jk l͑ x͒ ␦ C ␣ ␤ ␥␦ ͑ y͒ ͘ . ͑2͒
anisotropic Green’s function here. ␣␤␥␦

The present formulation also allows the extension to the The covariance, ⌳, is given as a function of the differ-
full multiple scattering problem to be made in a straightfor- ence between two vectors, xϪy. This assumption implies
ward manner. Such an extension in terms of radiative trans- that the medium is statistically homogeneous. However, the
fer and diffusion has been discussed previously for the sta- additional assumption of statistical isotropy ͑that ⌳ is a func-
tistically isotropic case using similar methods.10–13 Although tion of ͉xϪy͉) is not made here. The power spectral density
the present application is for textured metals, the formalism of the moduli fluctuations, ⌳˜ , is given by the Fourier trans-
is sufficiently general to be applied to other heterogeneous form of the covariance
media with statistical anisotropy such as fiber-reinforced
composites and extruded metal-matrix composites. ͵⌳˜ ͑ p͒ ijkl ␦ϭ ⌳ ͑ r͒ ijkl ␦ e Ϫir–p d3r. ͑3͒
␣␤␥ ␣␤␥
In the next section, the Dyson equation is discussed in
terms of the appropriate Green’s dyadics. The formalism is Defining a temporal Fourier transform pair for the func-
developed for a general anisotropic material without refer- tions f (t) and ˜f (␻) by
ence to a particular symmetry class. The solution of the mean
response is further reduced in the succeeding section for the ͵˜f ͑␻ ͒ϭ f ͑t ͒ei␻t dt, ͑4͒
case of a transversely isotropic medium. The elastic modulus
tensor is specified for this case and expressions for the at- ͵1 ˜f ͑ ␻ ͒eϪi␻t d␻, ͑5͒
tenuation of each wave type are given. Finally, the covari-
ance tensor of the elastic moduli fluctuations is further speci- f͑t͒ϭ 2␲
fied for that of cubic polycrystals with texture and solutions
for the case of stainless steel with aligned ͓001͔ axes are allows Eq. ͑1͒ to be transformed as
given.
ͭ ͮ␻ 2 0 ‫ץ‬ ‫ץ‬ ‫ץ‬ ‫ץ‬
␦ j kϩ C ij ‫ץ‬x ‫ץ‬xl ϩ ‫ץ‬xi ␦ C i j kl͑ x͒ ‫ץ‬xl Gk␣͑ x,xЈ,␻ ͒
k l
i

ϭ␦ j␣␦3͑ xϪxЈ͒. ͑6͒

I. PRELIMINARY ELASTODYNAMICS The random nature of the medium suggests that the

The equation of motion for the elastodynamic response Green’s function, G, is of little value as it will also be a

of an infinite, linear-elastic material to deformation is given random function. The interesting quantities are instead those

in terms of the Green’s dyadic by related to the statistics of the response. These statistics in-
clude the mean response, ͗G͘, and the covariance of the
ͭ ͮ‫ץ‬2 ‫ץ‬ response, ͗GG*͘, with the * denoting a complex conjugate.
This article is devoted to examination of the mean response

with corresponding phase velocities and attenuations.

Wave propagation and scattering problems of this sort

do not lend themselves to solution by perturbation methods.
As Frisch points out, these solutions do not converge.9 In-

stead Frisch used diagrammatic methods for solution of the
mean response.9 The mean response, ͗G͘, is governed by the
Dyson equation which is given by9,10

‫ץ‬ ͵ ͵͗
Ϫ␦ jk ‫ץ‬t2 ϩ ‫ץ‬xi Ci jkl͑ x͒ ‫ץ‬xl Gk␣͑ x,xЈ;t ͒ 0 Gi0␤͑ x,y͒M ␤ j͑ y,z͒
G i ␣͑ x, xЈ ͒ ͘ ϭ G i␣ ͑ x, xЈ ͒ϩ

ϭ␦ j␣␦3͑ xϪxЈ͒␦͑ t ͒. ͑1͒ ϫ͗G j␣͑ z,xЈ͒͘ d3yd3z. ͑7͒

The second-rank Green’s dyadic, Gk␣(x,xЈ;t), defines In Eq. ͑7͒, the quantity G0 is the bare Green’s dyadic. It
the response at location x in the kth direction to a unit im- defines the response of the medium without

542 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 542

heterogeneities—the solution of Eq. ͑6͒ with ␦Cijkl(x)ϭ0. identical to that given by Weaver for a statistically isotropic
The second-rank tensor M is the mass or self-energy
operator.9 The Dyson equation, Eq. ͑7͒, is easily solved in medium. However, the use of an anisotropic Green’s dyadic
for representing G0, which is discussed below, is the main
Fourier transform space under the assumption of statistical
homogeneity. The spatial Fourier transform pair for G0 is new result here. The abbreviated presentation given in this

given by section serves as a reminder of the general procedure for the

͵ ͵Gi0␣͑p͒␦ 3͑ pϪ q͒ ϭ ͑ 2 1 ͒3 Gi0␣͑ x,xЈ͒ description of the mean response. The Dyson equation, Eq.
␲ ͑10͒, and the expression for the self-energy, Eq. ͑12͒, will be
used below for the derivation of the attenuations. Readers
ϫeϪip–xeiq–xЈ d3xd3xЈ, ͑8͒
interested in further details of the scattering theory are re-
ferred to the articles of Karal and Keller,2 Frisch,9 Stanke
and Kino,1 and Weaver.10

͵ ͵Gi0␣͑x,xЈ ͒ ϭ ͑ 2 1 ͒ 3 Gi0␣͑ p͒␦3͑ pϪq͒ II. GREEN’S DYADIC FOR TRANSVERSELY
␲ ISOTROPIC MEDIA

ϫeip–xeϪiq–xЈ d3pd3q. ͑9͒ The solution of the Dyson equation, Eq. ͑10͒, for the
mean response requires the Green’s dyadic for the bare me-
The Fourier transforms which define ͗G(p)͘ and M˜ (p) dium. The bare Green’s dyadic, G0, is defined as the solution
are given by expressions similar to that defining G0(p). The of the equation of motion, Eq. ͑6͒, without heterogeneities
assumption of statistical homogeneity ensures that they are ͓␦Ci jkl(x)ϭ0͔. The emphasis here is on anisotropic media
functions of a single wave vector in Fourier space. The with heterogeneities. Thus, the G0 required is that for an
Dyson equation can then be spatially Fourier transformed anisotropic medium. The lowest possible anisotropic symme-
and solved for ͗G(p)͘. The result is try class to be considered is that of a medium with a single
symmetry axis. Although this is the simplest case of global
͗G͑ p͒͘ϭ͓G0͑ p͒Ϫ1ϪM˜ ͑ p͔͒Ϫ1, ͑10͒ anisotropy, relevant transversely isotropic or uniaxial mate-
rials include fiber-reinforced composites and polycrystalline
where M˜ is the spatial transform of the self-energy. The media with fiber textures. These types of media have a single
symmetry axis defined here by the unit vector, nˆ . This direc-
Dyson equation is exact and describes the mean response of tion is termed the ‘‘fiber’’ direction although the medium
may not be composed of any fibers. The fourth-rank elastic
the medium. The main difficulty in the solution of Eq. ͑10͒ is moduli tensor, C, in a transversely isotropic medium is a
function of nˆ and is written
the representation of M. Approximations of M are often nec-

essary to obtain closed-form solutions of Eq. ͑10͒. The self-

energy, M, can be written as an expansion in powers of

moduli fluctuations. Approximation of M can then be made

to first order using the first term in such an expansion. Frisch Ci jklϭ␭Ќ␦i j␦klϩ␮Ќ͑ ␦ik␦ jlϩ␦il␦ jk͒ϩA͑ ␦i jnˆ knˆ l

discusses the equivalence of this technique, which he called ϩ␦klnˆ inˆ j͒ϩB͑ ␦iknˆ jnˆ lϩ␦ilnˆ jnˆ kϩ␦ jknˆ inˆ l
the first-order smoothing approximation ͑FOSA͒,9 and the
Keller approximation.2 The FOSA expression for M is given ϩ␦ jlnˆ inˆ k͒ϩDnˆ inˆ jnˆ knˆ l . ͑13͒
by10

ͳ‫ץ‬ The above elastic constants are defined as Aϭ␯Ϫ␭Ќ , B
ϭ␮ʈϪ␮Ќ , and Dϭ␭Ќϩ2␮Ќϩ␭ʈϩ2␮ʈϪ2(␯ϩ2␮ʈ). The
M ␤ j͑ y,z͒Ϸ ‫ץ‬y ␣ ␦C␣␤␥␦ ͑ y͒ elastic moduli, ␭ʈϩ2␮ʈ , ␭Ќϩ2␮Ќ , ␮Ќ , ␮ʈ , and ␯ can be
defined alternatively as C11 , C33 , C44 , C66 , and C13 ,
ʹ‫ץ‬ 0 ‫ץ‬ ‫ץ‬ respectively.14
␥ ‫ץ‬zi ‫ץ‬zl
ϫ‫ץ‬y␦ Substitution of this form for C into the equation of mo-
G k͑ y, z͒ ␦ C i j kl͑ z͒ . ͑11͒
tion, Eq. ͑6͒, gives in direct notation
Such an approximation is assumed valid as long as the fluc-
tuations, ␦C, are not too large. The spatial Fourier transform, ͕␻2IϪ p2͓͑ ␮ЌϩB͑ pˆ •nˆ ͒2͒Iϩpˆ pˆ ͑ ␭Ќϩ␮Ќ͒
as defined by Eq. ͑8͒, of the self-energy, M, is then formu- ϩnˆ nˆ ͑ BϩD͑ pˆ •nˆ ͒2͒ϩ͑ AϩB ͒͑ pˆ •nˆ ͒

lated. Manipulation of this integration allows it to be reduced
to10

͵M˜ ␤j͑p͒ϭ d 3s G 0 ͑ s͒ p ␣ p l s ␦ s i ⌳˜ ͑ pϪ s͒ ijkl ␦ . ͑12͒ ϫ͑pˆ nˆ ϩnˆ pˆ ͔͖͒•G0͑p,nˆ ͒ϭI, ͑14͒
␥k ␣␤␥

Thus, the transform of the self-energy can be written as where p is the wave vector with magnitude, p, and direction
a convolution between the bare Green’s dyadic and the Fou-
rier transform of the covariance of the moduli fluctuations. pˆ .
This expression, Eq. ͑12͒, and the Dyson equation, Eq. ͑10͒,
are the primary results of this section which are used in the The above equation can be written in terms of the wave
remainder of the article. The components of M˜ , as discussed matrix, N, by14
in Sec. III, are used to calculate the phase velocity and at-
tenuation of the various propagation modes. Equation ͑12͒ is N jkGk␣͑ p,␻ ͒ϭ␦ j␣ . ͑15͒

The eigenvectors of N define the polarization directions
for the propagating waves.14 These eigenvectors can be

found directly from the wave matrix, N. Explicit expressions

543 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 543

of the directions of these eigenvectors are required for the Ϫ E sin2 ⌰ ͑24͒
results which follow. For this reason, an alternative method tan2␺ϭ PϩEcos2⌰ .
of determining the eigenvectors is given here.
The directions of uˆ 2 and uˆ 3 are defined in terms of ␺ for
The shear horizontal wave in a transversely isotropic later convenience. The above equation for ␺, Eq. ͑24͒, must
medium is known to be polarized perpendicular to the plane be used with care when numerical methods are employed.
defined by pˆ and nˆ . This vector, uˆ 1, is given by The value of ␺ which satisfies Eq. ͑24͒ may correspond to
either the quasi-P or quasi-SV wave. It should also be kept in
uˆ 1 ϭ ͉ pˆ ϫ nˆ ϭ pˆ ϫnˆ , ͑16͒
pˆ ϫ nˆ ͉ sin⌰ mind that the vectors uˆ 2 and uˆ 3 are functions of the direction
of propagation, pˆ , relative to the fiber direction, nˆ . This de-
where ⌰ defines the angle between the pˆ and nˆ . The identity pendence, ␺ϭ␺(⌰), which is evident in Eq. ͑24͒, will re-
dyadic can then be expanded as Iϭuˆ 1uˆ 1ϩ(IϪuˆ 1uˆ 1). Use of main implicit throughout.
the identity15
The bare Green’s dyadic may now be written

͑pˆ •nˆ ͒͑pˆ nˆ ϩnˆ pˆ ͒ G0͑ p͒ϭgS0H͑ p͒uˆ 1uˆ 1ϩgq0P͑ p͒uˆ 2uˆ 2ϩgq0SV͑ p͒uˆ 3uˆ 3 . ͑25͒

ϭuˆ 1uˆ 1͑ 1Ϫ͑ pˆ •nˆ ͒2͒ϪI͑ 1Ϫ͑ pˆ •nˆ ͒2͒ϩnˆ nˆ ϩpˆ pˆ , ͑17͒ The dispersion relations for the bare response of the SH,
qP, and qSV waves are given by
allows the last term in Eq. ͑14͒ to be expanded. The result is

͕uˆ 1uˆ 1͓ gS0H͑ p͔͒Ϫ1ϩ͑ IϪuˆ 1uˆ 1͒␻2 gS0H͑ p͒ϭ͓ ␻2Ϫ p2͑ ␮ЌϩBcos2⌰ ͔͒Ϫ1ϭ͓ ␻2Ϫ p2cS2H͔Ϫ1,
Ϫ p2͓͑ IϪuˆ 1uˆ 1͒Qϩ Ppˆ pˆ ϩEnˆ nˆ ͔͖•G0͑ p,nˆ ͒ϭI,
͑18͒ gq0P͑ p͒ϭ͓␻2Ϫ p2͑ Qϩ Pcos2␺ϩEcos2͑ ⌰ϩ␺ ͔͒͒Ϫ1
where the shear horizontal dispersion relation is
ϭ͓ ␻2Ϫ p2cq2P͔Ϫ1, ͑26͒

͓ gS0H͑ p͔͒Ϫ1ϭ␻2Ϫ p2͑ ␮ЌϩB͑ pˆ •nˆ ͒2͒. ͑19͒ gq0SV͑ p͒ϭ͓ ␻2Ϫ p2͑ Qϩ Psin2␺ϩEsin2͑ ⌰ϩ␺ ͔͒͒Ϫ1

The quantities Q, P, and E in Eq. ͑18͒ are defined by ϭ͓ ␻2Ϫ p2cq2SV͔Ϫ1,

Qϭ␮ЌϩB͑ pˆ •nˆ ͒2Ϫ͑ AϩB ͒͑ 1Ϫ͑ pˆ •nˆ ͒2͒, where the angle ␺ is defined by Eq. ͑24͒ and Q, P, and E are
defined in Eq. ͑20͒. The above expressions also describe the
Pϭ␭Ќϩ␮ЌϩAϩBϭ␯ϩ␮ʈ , ͑20͒ phase velocity, c␤ , as a function of propagation direction for
each wave type, ␤. The imaginary parts of these expressions
EϭAϩ2BϩD͑ pˆ •nˆ ͒2. will be used below and are given by

The quasi P and SV waves are polarized in directions Img 0 ͑ p͒ϭ Ϫ ␲ sgn͑ ␻ ͒ ␦ ͑ ␻ 2 Ϫ p 2 c 2 ͒. ͑27͒
␤ ␤
defined by uˆ 2 and uˆ 3 both of which lie in the pˆ -nˆ plane. They
form an orthonormal basis with uˆ 1 such that uˆ 3ϭuˆ 1ϫuˆ 2. The mean response of the heterogeneous medium is now
The vector uˆ 2 is directed at an angle ␺ from the propagation given by solution of the Dyson equation.
direction pˆ . Use of the directions uˆ 2 and uˆ 3 allows G0 to be
diagonalized. Thus IϪuˆ 1uˆ 1ϭuˆ 2uˆ 2ϩuˆ 3uˆ 3. The vectors pˆ and III. MEAN RESPONSE
nˆ are then written in terms of uˆ 2 and uˆ 3 as

pˆ ϭuˆ 2cos␺ϩuˆ 3sin␺, ͑21͒ The mean response, ͗G(p)͘, is given by solution of the
Dyson equation, Eq. ͑10͒, above. The solution of ͗G(p)͘ is
nˆ ϭuˆ 2cos͑ ␺ϩ⌰ ͒ϩuˆ 3sin͑ ␺ϩ⌰ ͒. ͑22͒ expressed in terms of G0(p) and M˜ „p). Like G0, the mean
Substitution into Eq. ͑18͒ gives
response and self-energy may be expanded in terms of the

orthonormal basis defined by uˆ 1 , uˆ 2, and uˆ 3. Thus,

͕uˆ 1uˆ 1͓ gS0H͑ p͔͒Ϫ1ϩuˆ 2uˆ 2͓ ␻2Ϫ p2͑ Qϩ Pcos2␺ ͗G͑ p͒͘ϭgSH͑ p͒uˆ 1uˆ 1ϩgqP͑ p͒uˆ 2uˆ 2ϩgqSV͑ p͒uˆ 3uˆ 3 ,
M˜ ͑ p͒ϭmSH͑ p͒uˆ 1uˆ 1ϩmqP͑ p͒uˆ 2uˆ 2ϩmqSV͑ p͒uˆ 3uˆ 3 ,
ϩEcos2͑ ⌰ϩ␺ ͔͒͒ϩuˆ 3uˆ 3͓␻2Ϫ p2͑ Qϩ Psin2␺ϩEsin2 ͑28͒

ϫ͑ ⌰ϩ␺ ͔͒͒Ϫ͑ uˆ 2uˆ 3ϩuˆ 3uˆ 2͓͒Esin͑ ⌰ϩ␺ ͒cos͑ ⌰ϩ␺ ͒ where it is again emphasized that the directions uˆ 1 , uˆ 2, and
uˆ 3 are dependent upon the propagation direction pˆ .
ϩ Psin␺cos␺͔͖•G0͑p͒ϭI. ͑23͒
The dispersion relations for the mean response are then
The vectors uˆ 2 and uˆ 3, which define the polarizations of
the quasi-P and quasi-SV waves, respectively, are eigenvec- given by the solution of the Dyson equation, Eq. ͑10͒, as
tors of N and diagonalize G0. Thus, the term in Eq. ͑23͒
g ␤͑ p͒ ϭ ͓ g 0 ͑ p͒Ϫ 1Ϫ m ␤͑ p͒ ͔ Ϫ 1
containing uˆ 2uˆ 3ϩuˆ 3uˆ 2 must be zero. This fact establishes a ␤
criterion for the angle ␺ given by
ϭ͓ ␻2Ϫ p2c␤2 Ϫm␤͑ p͔͒Ϫ1, ͑29͒

544 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 544

for each wave type, ␤. This expression for the dispersion from Eq. ͑30͒. The attenuation of each wave type is given by
relation of the mean response defines the phase velocity and
attenuation of each wave type. Solution of ͩ ͪ1 ␻ ͑31͒

␻2Ϫ p2c␤2 Ϫm␤͑ p͒ϭ0, ͑30͒ ␣␤͑ pˆ ͒ϭϪ 2␻c␤͑ pˆ ͒Imm␤ c␤ pˆ .

The final step in this derivation now lies in the expres-

for the wave vector p, is required given M˜ defined in Eq. sion for the imaginary part of the self-energy. The definition
͑12͒. The inverse Fourier transform of ͗G(p)͘ will be domi- of the self-energy was given by Eq. ͑12͒. The form of the
nated by the poles of the dispersion relations. The phase self-energy given by Eq. ͑28͒ is substituted into Eq. ͑12͒.

velocity is given by the real part of p and the attenuation by Appropriate inner products allow each component of the
the imaginary part. Such solutions of Eq. ͑30͒ are usually
done numerically using root finding techniques.5 However, self-energy to be determined independently. The wave num-
bers which appear in Eq. ͑12͒ are approximated to the same
explicit expressions for the attenuation can be determined
degree of the Born approximation discussed above. The in-
using an approximation valid below the high-frequency geo-
tegration over the magnitude of the wave vector is easily
metric optics limit. In this case, the wave vector p within the done due to the delta-function form of G0(s). The delta

self-energy is approximated as being equal to the bare wave functions are the result of consideration of the imaginary
vector. Such an approximation, m␤(p)Ϸm␤͓(␻/c␤) pˆ ͔, is parts of the dispersion relations given by Eq. ͑27͒. The at-
sometimes called a Born approximation.1,10 This approxima- tenuations for the three wave types, each defined by Eq. ͑31͒,

tion allows the imaginary part of p to be calculated directly are finally given by

ͭ ͵ ͩ ͪ ͵ ͩ ͪ␣ ••••uˆ 1pˆ sˆvˆ2
SH͑ pˆ ͒ϭ 1 ␲ d2sˆ cS5␻H͑4sˆ͒⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 1pˆ sˆvˆ1 ␲ d 2 sˆ ␻4 ⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 1pˆ sˆvˆ2
S3H͑ 4 SH͑ c SH͑ ϩ4 q5P͑ sˆ͒ SH͑ c qP͑
c pˆ ͒ c pˆ ͒ sˆ͒ c c pˆ ͒ sˆ͒
••••uˆ 1pˆ sˆvˆ1

͵ ͩ ͪ ͮ␲

ϩ4
d 2 sˆ ␻4 sˆ͒⌳˜ ␻ pˆ ؊ ␻ ••••uˆ 1pˆ sˆvˆ3 , ͑32͒
q5SV͑
cSH͑ pˆ ͒ sˆ
c c qSV͑ sˆ͒ ••••uˆ 1pˆ sˆvˆ3

ͭ ͵ ͩ ͪ ͵ ͩ ͪ␣
qP͑ pˆ ͒ ϭ 1 ␲ d 2 sˆ ␻4 ⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 2pˆ sˆvˆ1 ␲ d 2 sˆ ␻4 ⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 2pˆ sˆvˆ2
q3P͑ 4 S5H͑ sˆ͒ qP͑ c SH͑ ϩ q5P͑ sˆ͒ qP͑ pˆ ͒ c qP͑ ••••uˆ 2pˆ sˆvˆ2
c pˆ ͒ c c pˆ ͒ sˆ͒ 4 c c sˆ͒

••••uˆ 2pˆ sˆvˆ1

͵ ͩ ͪ ͮ␲

ϩ4
d 2sˆ ␻4 sˆ͒⌳˜ ␻ pˆ ؊ ␻ ••••uˆ 2pˆ sˆvˆ3 , ͑33͒

5 cqP͑ pˆ ͒ sˆ
c qS V͑ c qSV͑ sˆ͒ ••••uˆ 2pˆ sˆvˆ3

ͭ ͵ ͩ ͪ ͵ ͩ ͪ␣ ••••uˆ 3pˆ sˆvˆ2
qSV͑ pˆ ͒ϭ 1 ␲ d2sˆ cS5␻H͑4sˆ͒⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 3pˆ sˆvˆ1 ␲ d 2 sˆ ␻4 ⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 3pˆ sˆvˆ2
q3SV͑ 4 cqSV͑ pˆ ͒ cSH͑ sˆ͒ ϩ q5P͑ sˆ͒ cqSV͑ pˆ ͒ cqP͑ sˆ͒
c pˆ ͒ 4 c

••••uˆ 3pˆ sˆvˆ1

͵ ͩ ͪ ͮ␲ d 2 sˆ ␻4 sˆ͒⌳˜ ␻ pˆ ؊ ␻ sˆ ••••uˆ 3pˆ sˆvˆ3 . ͑34͒
q5SV͑ qSV͑ c qSV͑ ••••uˆ 3pˆ sˆvˆ3
ϩ4 c c pˆ ͒ sˆ͒

The integrals in the above equations are over the unit itly by ⌳˜ ( q) ••• • uuˆˆ ppˆˆ ssˆˆvvˆˆϭ ⌳˜ ( q) ijkl ␦uˆ ␤ uˆ k pˆ ␣ pˆ l sˆ isˆ ␦vˆ ␥ vˆ j .
sphere defined by sˆ. The direction pˆ defines the propagation ••• • ␣␤␥
direction, sˆ is the scattered direction, and the polarization
directions, uˆ and vˆ. In the above expressions the dependence The above expressions for the attenuation are more com-
of the vectors uˆ on pˆ and of vˆ on sˆ is implicit. The argument
of the autocorrelation is the difference between the incoming plicated than those for a statistically isotropic medium con-
and outgoing propagation directions. The inner products on sidered by Weaver.10 They reduce to the forms given there in
the autocorrelation of the moduli fluctuations are given in
terms of four unit vectors. This inner product is given explic- the case of statistical isotropy. The dependence of wave

speed and polarization direction on propagation direction

greatly complicates the integrations. In the next section, the

covariance is specified for equiaxed cubic polycrystals with

texture.

545 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 545

IV. EQUIAXED POLYCRYSTALLINE MEDIA WITH ͗ C i j k l ͑ x͒ ͘ ϵ C 0 kl
TEXTURE ij

The above formalism is now specified for the particular ϭ␭Ќ␦i j␦klϩ␮Ќ͑ ␦ik␦ jlϩ␦il␦ jk͒
problem of equiaxed cubic polycrystalline media with one
aligned axis. This particular grain structure arises during ϩA͑ ␦i jnˆ knˆ lϩ␦klnˆ inˆ j͒
welding or solidification. In this case the transverse isotropy
is the result of alignment of one grain axis in all grains. The ϩB͑ ␦iknˆ jnˆ lϩ␦ilnˆ jnˆ kϩ␦ jknˆ inˆ lϩ␦ jlnˆ inˆ k͒
other axes are randomly oriented about that axis. Two as-
sumptions are made about the fluctuations of the elastic ϩDnˆ inˆ jnˆ knˆ l . ͑38͒
moduli. The tensorial and spatial components of covariance
are first assumed independent. This assumption implies The elastic modulus tensor for a single cubic crystal is

given by

3

͚Ci jkl͑ x͒ϭc12␦i j␦klϩc44͑ ␦ik␦ jlϩ␦il␦ jk͒ϩ␩ a ina nj a nk a n ,
nϭ1 l

⌳˜ ͑ p͒ ijkl ϭ ⌶ ijkl ␦W˜ ͑ p͒ , ͑35͒ ͑39͒
␣␤␥␦ ␣␤␥
where ␩ϭc11Ϫc12Ϫ2c44 is the single-crystal anisotropy.
n
where W˜ (p) is the Fourier transform of the spatial correla- The elements a i define the rotation matrix in terms of the

tion function. The grains are also assumed to be equiaxed Euler angles. Each grain is assumed to have a different ori-
such that W(r)ϭeϪr/L. The correlation length, L, is of the
order of the grain radius. The Fourier transform is then given entation such that the last term in Eq. ͑39͒ is dependent upon

by x. The ensemble average for this medium is defined by ro-

tation about the aligned axis of the grains. This average is

given by

W˜ ͑ q͒ ϭ L 3 . ͑36͒ ͵1 2␲ ͑40͒
ϩ
␲ 2͑ L 2 2 2 ͗f͘ϭ 2␲ 0 f͑␾͒d␾,

1 q ͒ where the dummy variable ␾ is defined in the plane perpen-
dicular to the direction of the aligned axis.
The forms of the attenuation given above contain the
difference of two vectors, W˜ (q)ϭW˜ (͓␻/c1(⌰)͔ pˆ Carrying out the average for Eq. ͑39͒ and equating like
Ϫ͓␻/c2(⌰Ј)͔ sˆ) as the argument for ⌳˜ (p). The form of terms with Eq. ͑38͒ gives
correlation function is dependent upon magnitude of this
␩␩ ͑41͒
vector. Trigonometry reduces the magnitude of this differ- ␭Ќϭc12ϩ 4 , ␮Ќϭc44ϩ 4 ,

ence to ␩ 7␩
AϭBϭϪ 4 , Dϭ 4 .
ͯ ͯq2ϭ
␻ ␻ 2 The average modulus tensor is then
1͑⌰ 2͑⌰
c ͒ pˆ ؊ Ј ͒ sˆ ͩ ͪ ͩ ͪ␩ ␩
c
͗Ci jkl͘ϭ c12ϩ 4 ␦i j␦klϩ c44ϩ 4 ͑ ␦ik␦ jlϩ␦il␦ jk͒
␻2 ␻2 2␻2
ϭ c 12͑ ⌰ ͒ ϩ c 22͑ ⌰ Ј ͒ Ϫ c ͑ ⌰͒c2͑ ⌰ Ј ͒ ͑ pˆ –sˆ͒. ͑37͒ Ϫ ␩ ͑ ␦ nˆ nˆ lϩ ␦ nˆ inˆ ͒Ϫ ␩ ͑ ␦ nˆ nˆ ϩ ␦ nˆ nˆ
4 4
1 i j k k l j ik j l i l j k

The form of the eighth-rank tensor, ⌶ ijkl ␦ , is now dis- nˆ nˆ nˆ inˆ 7␩ nˆ nˆ nˆ nˆ
␣␤␥ 4
ϩ ␦ lϩ ␦ ͒ ϩ . ͑42͒
cussed with regard to cubic crystallites. jk i j l k i j k l

A. Statistics of textured cubic polycrystalline media The eighth-rank covariance, ⌶ ijkl , is also a function of
with aligned †001‡ axis ␣␤␥␦

the single vector nˆ . It can therefore be constructed of Kro-

The average medium is characterized by the average necker deltas and pairs of nˆ ’s. The symmetry of the cubic
elastic modulus tensor, ͗Cijkl͘. For a statistically trans-
versely isotropic medium, the average modulus is a fourth- crystal also implies that this tensor must be invariant to per-
rank tensor which is a function of the unit vector nˆ . The most
mutation of Latin or Greek indices and exchange of all Latin
general form for this tensor is
for all Greek indices. Thus, the covariance of moduli fluc-

tuations may be written in terms of 14 independent tensors as

⌶ ijkl ␦ ϭ ͗ C i j k lC ␣␤␥ ␦͘ Ϫ ͗ C i jk l ͘ ͗ C ␣ ␤ ␥␦ ͘
␣␤␥

ͳͩ ͪͩ ͪʹ ͳ ʹͳ ʹϭ␩2 3 3 3 3

͚nϭ1 a ni a n a n a n ͚a n a n a ␥n a n ͚Ϫ ␩ 2 a ni a nj a nk a n ͚a ␣n a n a n a n
j k l ␣ ␤ ␦ l ␤ ␥ ␦
nϭ1 nϭ1 nϭ1

546 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 546

ϭb0͑ ␦i j␦klϩ␦ik␦ jlϩ␦il␦ jk͒͑ ␦␣␤␦␥␦ϩ␦␣␥␦␤␦ϩ␦␣␦␦␤␥͒ϩd0͑ ␦LL␦GG␦GL␦GLϪ72 terms͒

ϩh0͑ ␦GL␦GL␦GL␦GLϪ24 terms͒ϩb2͑ ␦LL␦LL␦GGnˆ Gnˆ Gϩ␦GG␦GG␦LLnˆ Lnˆ LϪ36 terms͒ϩc2͑ ␦GL␦GL␦LLnˆ Gnˆ G

ϩ␦GL␦GL␦GGnˆ Lnˆ LϪ144 terms)ϩd2͑ nˆ Gnˆ L␦GL␦GG␦LLϪ144 terms͒ϩh2͑ nˆ Gnˆ L␦GL␦GL␦GLϪ96 terms͒

ϩb4͓ nˆ inˆ jnˆ knˆ l͑ ␦␣␤␦␥␦ϩ␦␣␥␦␤␦ϩ␦␣␦␦␤␥͒ϩnˆ ␣nˆ ␤nˆ ␥nˆ ␦͑ ␦i j␦klϩ␦ik␦ jlϩ␦il␦ jk͔͒

ϩc4͑ ␦GG␦LLnˆ Gnˆ Gnˆ Lnˆ LϪ36 terms͒ϩd4͑ ␦LL␦GLnˆ Lnˆ Gnˆ Gnˆ Gϩ␦GG␦GLnˆ Lnˆ Lnˆ Lnˆ GϪ96 terms͒

ϩh4͑ ␦GL␦GLnˆ Lnˆ Lnˆ Gnˆ GϪ72 terms͒ϩb6͑ ␦GGnˆ Gnˆ Gnˆ Lnˆ Lnˆ Lnˆ Lϩ␦LLnˆ Lnˆ Lnˆ Gnˆ Gnˆ Gnˆ GϪ12 terms͒

ϩd6͑ ␦LGnˆ Lnˆ Lnˆ Lnˆ Gnˆ Gnˆ GϪ16 terms͒ϩd8nˆ inˆ jnˆ knˆ lnˆ ␣nˆ ␤nˆ ␥nˆ ␦ . ͑43͒

The above terms not explicitly given are written in terms Specific inner products required for the calculation of
of G and L. This notation implies the form of the terms given the attenuations are now determined. The attenuations will
by Greek or Latin indices. All possible permutations of the vary angularly only within the plane defined by the propaga-
particular form are implied and the number of terms of each tion direction and the fiber direction. Therefore, without loss
form is also given. For example, the factor c4 contains terms of generality, a reference plane is defined as the pˆ -nˆ plane.
such as ␦␣␤␦i jnˆ ␥nˆ ␦nˆ knˆ l and ␦␣␥␦iknˆ ␤nˆ ␦nˆ jnˆ l plus all other The following vectors are then defined with respect to a gen-
permutations of Greek and Latin indices for a total of 36 eral xyz coordinate system as
terms.
nˆ ϭzˆ,
The 764 terms in Eq. ͑43͒ completely define the covari-
ance in a coordinate-free manner. The averages given in pˆ ϭyˆsin⌰ϩzˆcos⌰, ͑45͒
terms of the Euler angles can be carried out and compared
with the results of the covariance given by the general form. sˆϭ xˆ sin⌰ Ј cos␾ Ј ϩ yˆ sin⌰ Ј sin␾ Ј ϩ zˆ cos⌰ Ј .
The coefficients are found to be

9␩2 The polarization vectors are then defined with respect to
d8ϭb6ϭϪd6ϭ 288 , these angles and ␺ ͓Eq. ͑24͔͒ as

uˆ 1ϭxˆ,

3␩2 uˆ 2ϭyˆsin␥ϩzˆcos␥, ͑46͒
b4ϭϪd4ϭϪh2ϭh0ϭ 288 ,
͑44͒ uˆ 3ϭϪyˆcos␥ϩzˆsin␥,
␩2 and
d2ϭc2ϭb0ϭϪd0ϭϪb2ϭϪc4ϭ288 ,

5␩2 vˆ 1ϭ xˆ sin␾ Ј Ϫ yˆ cos␾ Ј ,
h4ϭ 288 .
vˆ 2ϭ xˆ sin␥ Ј cos␾ Ј ϩ yˆ sin␥ Ј sin␾ Ј ϩ zˆ cos␥ Ј , ͑47͒

The forms of the attenuations given in Eqs. ͑32͒–͑34͒

above require various inner products on the covariance ten- vˆ 3ϭ Ϫ xˆ cos␥ Ј cos␾ Ј Ϫ yˆ cos␥ Ј sin␾ Ј ϩ zˆ sin␥ Ј ,
where the angles ␥ and ␥Ј used above are defined by
sor. These inner products have the general form of ⌶ • • • •qˆ pˆ sˆrˆ ,
• • • •qˆ pˆ sˆrˆ

where the vectors pˆ and sˆ represent the incoming and outgo-

ing and propagation directions, respectively, and the vectors ␥ϭ⌰ϩ␺͑⌰͒, ␥Јϭ⌰Јϩ␺͑⌰Ј͒. ͑48͒

qˆ and rˆ are vectors defining the polarization directions of the

particular wave. These vectors are perpendicular to the plane These angles, ␥ and ␥Ј, define the orientation angle of the
qP wave with respect to the nˆ direction in the pˆ -nˆ and sˆ-nˆ
defined by sˆ or pˆ and nˆ ͑for SH waves͒ or they lie in this planes, respectively.

plane ͑for qP and qSV͒. This general inner product can be Using these definitions of the relevant unit vectors, the
required inner products simplify considerably. The inner
written in terms of various combinations of inner products of products are for ␣SH ,

the vectors involved. The most general form is given explic-
itly elsewhere.16

547 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 547

⌶ • • • • uuˆˆ 11ppˆˆ ssˆˆvvˆˆ11ϭ ␩2 sin2 ⌰ Ј sin2 ⌰ , ͵␻4sin2␥sin2⌰ ␩2␲ 2␲
• • • • 32 ␣qP͑ ⌰ ͒ϭ cq3P͑ ⌰ ͒ 128 0 d␾Ј

⌶ • • • • uuˆˆ 11ppˆˆ ssˆˆvvˆˆ22ϭ ␩2 sin2 ⌰ Ј sin2 ␥ Ј sin2 ⌰ , ͑49͒ ͵ ͫϫ␲ W˜ qPϪSH͑ pˆ ,sˆ͒sin3⌰Ј
• • • • 32 0 cS5H͑ ⌰Ј͒

⌶ • • • • uuˆˆ 11ppˆˆ ssˆˆvvˆˆ33ϭ ␩2 sin2 ⌰ Ј cos2 ␥ Ј sin2 ⌰ , ϩ W˜ qPϪqP͑ pˆ ,sˆ͒sin3⌰ Јsin2␥ Ј
• • • • 32 cq5P͑ ⌰Ј͒

for ␣qP , ͬϩ
W˜ qSV͑ pˆ ,sˆ͒sin3⌰
qPϪ cq5SV͑ ⌰Ј͒ Ј cos2␥ Ј d⌰Ј, ͑53͒
␩2
⌶ • • • • uuˆˆ 22ppˆˆ ssˆˆvvˆˆ11ϭ 32 sin2 ⌰ Ј sin2 ␥ sin2 ⌰ ,
• • • •

⌶ • • • • uuˆˆ 22ppˆˆ ssˆˆvvˆˆ22ϭ ␩2 sin2 ⌰ Ј sin2 ␥ Ј sin2 ␥ sin2 ⌰ , ͑50͒ ͵␻4cos2␥sin2⌰ ␩2␲ 2␲
• • • • 32 ␣qSV͑ ⌰ ͒ϭ cq3SV͑ ⌰ ͒ 128 0 d␾Ј

⌶ • • • • uuˆˆ 22ppˆˆ ssˆˆvvˆˆ33ϭ ␩2 sin2 ⌰ Ј cos2 ␥ Ј sin2 ␥ sin2 ⌰ ; ͵ ͫϫ ␲ W˜ qSVϪSH͑ pˆ ,sˆ͒sin3⌰Ј
• • • • 32 0 cS5H͑ ⌰Ј͒

and, for ␣qSV , W˜ qSVϪqP͑ pˆ ,sˆ͒sin3⌰ Јsin2␥ Ј
cq5P͑ ⌰Ј͒
• • • • ␩2 ϩ
• • • • 32
⌶ uuˆˆ 33ppˆˆ ssˆˆvvˆˆ11ϭ sin2 ⌰ Ј cos2 ␥ sin2 ⌰ ,

ͬϩ
W˜ qSVϪqSV͑ pˆ ,sˆ͒sin3⌰ Ј cos2␥ Ј d⌰Ј. ͑54͒
cq5SV͑ ⌰Ј͒
⌶ • • • • uuˆˆ 33ppˆˆ ssˆˆvvˆˆ22ϭ ␩2 sin2 ⌰ Ј sin2 ␥ Ј cos2 ␥ sin2 ⌰ , ͑51͒
• • • • 32

⌶ • • • • uuˆˆ 33ppˆˆ ssˆˆvvˆˆ33ϭ ␩2 sin2 ⌰ Ј cos2 ␥ Ј cos2 ␥ sin2 ⌰ , These expressions are now nondimensionalized and sim-
• • • • 32 plified. Reference wave speeds are needed for the nondimen-
sionalization. Rather than using the Voigt average wave
where ␥ and ␥Ј are defined in Eq. ͑48͒. The expressions speeds, wave speeds characteristic of the anisotropic medium
given by Eqs. ͑49͒–͑51͒ are also directly related to the dif- are used. Average wave speeds are defined as
fuse propagation including backscatter.11–13
͵¯c␤ϭ 1 ␲ ͑55͒
Using the above inner products, the attenuations reduce 2
c␤͑ ⌰ ͒sin⌰ d⌰,
to
0

͵ ͵ ͫ␻4sin2⌰ ␩2␲ ␲ W˜ SHϪSH͑ pˆ ,sˆ͒sin3⌰Ј for each wave type, ␤. The three nondimensional frequen-
0 cS5H͑ ⌰Ј͒ cies are then defined as x␤ϭ␻L/¯c␤ . Finally, dimensionless
␣SH͑ ⌰ ͒ϭ cS3H͑ ⌰ ͒ 128 quantities which describe the slowness surface for each wave
2␲
type are defined by r␤(⌰)ϭ¯c␤ /c␤(⌰).
d␾Ј The expressions for the spatial correlation functions

0 must now be discussed. Using the form of the spatial Fourier
transform of the correlation function, given by Eq. ͑36͒, the
ϩ W˜ SHϪ qP͑ pˆ ,sˆ͒sin3⌰ Ј sin2 ␥ Ј functions W˜ ␤Ϫ␥(pˆ ,sˆ) are determined. In terms of the above
cq5P͑ ⌰Ј͒ dimensionless quantities, we find

ͬϩ
W˜ SHϪqSV͑ pˆ ,sˆ͒sin3⌰ Јcos2 ␥ Ј d⌰Ј, ͑52͒
cq5SV͑ ⌰Ј͒

W˜ ␤Ϫ ␥͑ pˆ , sˆ͒ϭ ␲ 2͑ 1 ϩ x 2 r ␤2 ͑ ⌰ ͒ϩ x ␥2 r ␥2 ͑ ⌰ L3 2 x x ␥r ␤͑ ⌰ ͒ r ͑ ⌰ Ј ͒ pˆ –sˆ͒2 ͑56͒
␤ Ј͒Ϫ
␤ ␥

for the incoming wave type ␤ and outgoing wave type ␥. integration over ␾Ј in Eqs. ͑52͒–͑54͒ can be done in closed
The inner product, pˆ –sˆϭcos⌰cos⌰Јϩsin⌰sin⌰Јsin␾Ј. The form. The resulting dimensionless attenuations are then

548 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 548

␣ SH͑ ⌰ ͒ L ϭ x 4 ␩2 4 r S3H͑ ⌰ ͒ sin2⌰
SH 64␳ 2¯c SH

ͫ ͩ ͪ ͩ ͪ ͬϫ 5
¯c SH 5 ¯c SH
I SHϪSHϩ I SHϪqP ¯c qP ¯c qSV ,
ϩ I SHϪqSV

͑57͒

␣ qP͑ ⌰ ͒ L ϭ x 4 ␩2 4 r q3P͑ ⌰ ͒ sin2⌰ sin2 ␥
qP 64␳ 2¯c qP

ͫ ͩ ͪ ͩ ͪ ͬϫ
¯c qP 5 ¯c qP 5
¯c SH ¯c qSV
I qPϪSH ϩ I qPϪqPϩ I qPϪqSV ,

͑58͒ FIG. 1. Slowness surfaces for stainless steel with texture.

␣ qSV͑ ⌰ ͒ L ϭ x 4 ␩2 4 r q3SV͑ ⌰ ͒ sin2⌰ cos2␥ ͵␲
qSV 64␳ 2¯c qSV
I␤ϪqSVϭ rq5SV͑ ⌰Ј͒cos2͑ ⌰Јϩ␺͑ ⌰Ј͒͒sin3⌰Ј d⌰Ј,
ͫ ͩ ͪ ͩ ͪϫ 0
¯c qSV 5 ¯c qSV 5
¯c SH ¯c qP for all incoming wave types ␤. Thus, the angular depen-
I qSVϪSH ϩ I qSVϪqP dence of the attenuations in the Rayleigh limit is seen explic-
itly. The attenuations are dependent upon the cube of the
ͬϩIqSVϪqSV , ͑59͒ slowness multiplied by a factor related to the polarization
type. The attenuation ␣SH varies as rS3H(⌰)sin2⌰, ␣qP varies
with the density, ␳, now included. as rq3P(⌰)sin2(⌰ϩ␺(⌰))sin2⌰, and ␣qSV varies as
The above terms within the square brackets represent r q3SV( ⌰ ) cos2(⌰ϩ␺(⌰))sin2⌰.

integrals which are given by the general forms B. Results for stainless steel

͵I ␤ Ϫ SHϭ ␲ r S5H͑ ⌰Ј͒X␤ϪSHsin3⌰ Ј d ⌰ Ј, Numerical results are now presented for the specific case
0
͑ X 2 Ϫ SHϪ Y 2 Ϫ SH͒ 3/2 of stainless steel with aligned ͓001͔ axes. The material prop-
␤ ␤ erties used are5

͵I ␤ Ϫ qPϭ ␲ r q5P͑ ⌰ Ј ͒X ␤ϪqPsin2͑ ⌰ Јϩ␺͑⌰ Ј ͒ ͒ sin3⌰ Ј d ⌰ Ј , c11ϭ2.16ϫ1011 Pa, c12ϭ1.45ϫ1011 Pa,
0 c44ϭ1.29ϫ1011 Pa, ␳ϭ7860 kg/m3.
͑ X 2 qPϪ Y 2 qP͒ 3/2 ͑63͒
␤ ␤Ϫ
Ϫ ͑60͒

I ␤ϪqSV The slowness surfaces calculated using the dispersion

relations for the bare medium, Eqs. ͑26͒, are shown in Fig. 1.
These results agree well with those of previous authors.5

͵ϭ ␲ r q5SV͑ ⌰ Ј ͒X␤ϪqSVcos2͑ ⌰Јϩ␺͑ ⌰ Ј ͒ ͒sin3⌰ Ј d ⌰ Ј ,
0
͑ X 2 Ϫ qSVϪ Y 2 Ϫ qSV͒ 3/2
␤ ␤

with

X ␤Ϫ␥ϭ 1 ϩ x 2 r 2 ͑ ⌰ ͒ ϩ x 2 r 2 ͑ ⌰ Ј ͒
␤ ␤ ␥ ␥

Ϫ2x␤x␥r␤͑ ⌰ ͒r␥͑ ⌰Ј͒cos⌰cos⌰Ј,

Y ␤Ϫ␥ϭ2x␤x␥r␤͑ ⌰ ͒r␥͑ ⌰Ј͒sin⌰sin⌰Ј, ͑61͒

for the different wave types, ␤ and ␥.
In the Rayleigh limit, these integrals simplify consider-

ably. They become independent of incident direction and fre-
quency and are given by

͵␲

I␤ϪSHϭ rS5H͑ ⌰Ј͒sin3⌰Ј d⌰Ј,
0

͵␲ FIG. 2. Dimensionless attenuation in the Rayleigh limit as a function of

I␤ϪqPϭ rq5P͑ ⌰Ј͒sin2͑ ⌰Јϩ␺͑ ⌰Ј͒͒sin3⌰Ј d⌰Ј, direction for the SH, qP, and qSV waves. The dimensionless attenuation ␣L
0
͑62͒ has been normalized by the fourth power of the dimensionless frequency for

the respective wave type: ␣ SHL / x 4 , ␣ qPL / x 4 , and ␣ qSVL / x 4 .
SH qP qSV

549 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 549

FIG. 3. Angular dependence of the normalized SH attenuation, ␣SH /kSH , FIG. 4. Angular dependence of the normalized qP attenuation, ␣qP /kqP , for
for various frequencies, xSH . various frequencies, xSH .

Attenuation results, using Eqs. ͑57͒–͑59͒ are given in are similar to the SH attenuation. The cross-fiber attenuation
terms of the single dimensionless frequency xSH increases more rapidly than other directions as the frequency
ϭ␻L/¯cSH . The calculations are primarily limited to xSH increases. Similar results were seen by Hirsekorn4 and
Ͻ10 corresponding to the onset of the geometric optics limit Ahmed and Thompson.5
observed by Ahmed and Thompson.5 ͑Their dimensionless
frequency, ␹ϭkd, is based on the grain diameter. Here, this The normalized qSV attenuation, ␣qSV /kqSV , is shown
quantity contains the grain radius.͒ The required integrals in Fig. 5 at the same frequencies as Figs. 3 and 4. The at-
were calculated numerically using a Newton–Cotes quadra- tenuation in the fiber and cross-fiber directions is zero as
expected from the form of the modulus fluctuations. The
ture function quad8 available in the software package direction of maximum ␣qSV , as observed in Fig. 5, is seen to
Matlab.17 be a function of frequency. This result is shown more explic-
itly in Fig. 6 in which the direction of maximum ␣qSV is
In the Rayleigh limit, the integrals in Eqs. ͑60͒ reduce to plotted versus xSH . This direction is 47.1° in the Rayleigh
those given by Eqs. ͑62͒. For the case considered here, with limit as was seen in Fig. 2. The direction of maximum ␣qSV
parameters given by Eq. ͑63͒, these integrals are I␤ϪSH then increases with increasing frequency. The peak in this
ϭ1.623, I␤ϪqPϭ1.068, and I␤ϪqSVϭ0.6271. In the Ray- maximum occurs at a frequency of about xSHϭ1.8 at an
leigh regime the attenuation depends on the fourth power of angle of 51.8°. The direction of maximum ␣qSV then de-
creases for larger values of xSH . The result presented here
frequency. Thus, the angular dependence of the three attenu- differs from that presented by Ahmed and Thompson.5 Their
ations is described by the quantity ␣L/x4. This parameter for maximum ␣qSV occurs at 45° in the Rayleigh regime. At
each wave type is shown in Fig. 2. The SH and qP waves higher frequencies they noted an increase in the angle of
maximum ␣qSV , although the shift from 45° is not as dra-
have their maxima perpendicular to the fiber direction. All
FIG. 5. Angular dependence of the normalized qSV attenuation, ␣qSV /kqSV ,
wave types have zero attenuation in the fiber direction—the for various frequencies, xSH .

material properties do not vary in that direction. The qSV

wave is seen to have zero attenuation in the fiber direction

and perpendicular to the fiber direction. This result is the
same as previous work.4,5 Here, however, the same symme-

try is not seen. The qSV attenuation actually peaks at 47.1°

rather than 45°. The qSV attenuation greater than the peak is

slightly higher than that for propagation directions below the

peak as well. Thus, the inclusion of the wave polarizations

through the anisotropic Green’s dyadic results in slight dif-

ferences from the case without.

Results outside the Rayleigh regime are calculated using
the complete integrals, Eqs. ͑60͒. The directional dependence
of the attenuation as a function of frequency is presented

first. Figure 3 shows the normalized SH attenuation,
␣SH /kSH , as a function of propagation direction for five dif-
ferent frequencies. These results may be contrasted with the

results in the Rayleigh limit. As the frequency increases we

see that the attenuation in the cross-fiber direction increases

more than in other directions. The results for the normalized
qP attenuation, ␣qP /kqP , are shown in Fig. 4. These results

550 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 550

FIG. 6. Direction of maximum qSV attenuation as a function of frequency, FIG. 7. Normalized SH attenuation, ␣SH /kSH , as a function of dimension-
less frequency, xSH , for propagation directions of 45°, 69.5°, and 90°.
xSH . This angle is about 47.1° in the low-frequency limit, peaks at about
51.8° at xSHϭ1.8 and then decreases for higher frequencies. V. DISCUSSION
The propagation and scattering of elastic waves in het-
matic as that seen here. This difference is also attributed to
erogeneous, anisotropic media has been examined. Appropri-
the inclusion of polarization direction in the present results. ate ensemble averaging of the elastic wave equation resulted
in the Dyson equation, governing the mean field. The prob-
The appearance of the additional peak in the SV attenu- lem was further specified for the case of transverse isotropy.
The Green’s dyadic for a transversely isotropic medium was
ation at about 69° is observed. The presence of such a peak used to derive expressions for the attenuation of the shear
horizontal, quasicompressional, and quasishear waves. The
at the same angle was also observed by Ahmed and Thomp- covariance of moduli fluctuations were determined in
coordinate-free form for cubic polycrystalline materials with
son. They suggested that this phenomenon was related to the aligned ͓001͔ axes. The final forms of the attenuations for the
three wave types were given by simple expressions involving
stochastic-geometric transition. However, the presence of integrations over the unit circle. The integrands are depen-
dent upon inner products on the convariance of modulus
this peak in the present results shows that this is not the fluctuations and factors of phase velocity. The simple form
of the results makes them particularly useful for nondestruc-
case—the above derivation is limited to frequencies exclud- tive testing and materials characterization research and for
inclusion of attenuation in numerical models of elastic wave
ing the high-frequency geometric limit. The above forms of propagation such as those by Spies.14,18,19 The results pre-

the attenuation allow this phenomenon to be examined more FIG. 8. Normalized qP attenuation, ␣qP /kqP , as a function of dimensionless
frequency, xSH , for propagation directions of 45°, 69.5°, and 90°.
closely. The integrals ISHϪqSV and IqSVϪSH in the equations
for attenuation, Eqs. ͑60͒, have peaks at about 69°. This peak Joseph A. Turner: Scattering in textured polycrystals 551

is the result of the form of the spatial correlation function

between the SH and qSV modes. The term

X SHϪ qSV( X 2 qSVϪ Y 2 qSV) Ϫ 3/2 determines the specific di-
SHϪ SHϪ

rection of this peak and is a function of the angular phase

velocities and propagation direction. Closer examination

shows that this peak occurs at the angle corresponding to the

intersection of the SH and qSV slowness surfaces of 69.3°.

The appearance of a similar peak is seen in the angular plot

of SH attenuation, Fig. 3, at xSHϭ10.
Finally, results are presented for the normalized attenu-

ations as a function of frequency for several propagation di-

rections. In Figs. 7 and 8 the normalized SH and qP attenu-

ations are plotted versus dimensionless frequency, xSH , for
propagation directions of 45°, 69.5°, and 90°. The SH attenu-

ation in the cross-fiber direction is seen to increase more

rapidly than other directions for increasing frequency. The

qP attenuation has a local maximum which is a function of

propagation direction. Similar results were given by Ahmed
and Thompson.5 The normalized qSV attenuation is plotted

versus frequency, xSH , for propagation directions of 45° and
69.5° in Fig. 9. The attenuation at 69.5° increases more rap-

idly than that at 45°. However, at higher frequencies the ratio

between the two appears to be constant. Thus, the peak at

69.5° observed in the angular plots, Fig. 5, is not expected to

become larger than that of 45°, within the frequency limits

used here.

551 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999

FIG. 9. Normalized qSV attenuation, ␣qSV /kqSV , as a function of dimen- 2 F. C. Karal and J. B Keller, ‘‘Elastic, electromagnetic, and other waves in
sionless frequency, xSH , for propagation directions of 45° and 69.5°. The
attenuation at 69.5° is initially much smaller than that at 45°. It then in- a random medium,’’ J. Math. Phys. 5, 537–547 ͑1964͒.
creases more quickly than that at 45°. The ratio between the two appears to 3 S. Hirsekorn, ‘‘The scattering of ultrasonic waves in polycrystalline ma-
then be constant at higher frequencies.
terials with texture,’’ J. Acoust. Soc. Am. 77, 832–843 ͑1985͒.
sented here are also directly related to diffuse field methods 4 S. Hirsekorn, ‘‘Directional dependence of ultrasonic propagation in tex-
such as backscatter techniques. In addition, the above for-
malism holds for other forms of global anisotropy. The bare tured polycrystals,’’ J. Acoust. Soc. Am. 79, 1269–1279 ͑1986͒.
Green’s dyadic may be expanded in a similar fashion. The 5 S. Ahmed and R. B. Thompson, ‘‘Propagation of elastic waves in equi-
main difficulty will then lie in the determination of the form
of the covariance of elastic moduli fluctuations. axed stainless-steel polycrystals with aligned ͓001͔ axes,’’ J. Acoust. Soc.
Am. 99, 2086–2096 ͑1996͒.
ACKNOWLEDGMENTS 6 S. Ahmed and R. B. Thompson, in Effects of Preferred Grain Orientation

The Alexander von Humboldt Foundation and the and Grain Elongation on Ultrasonic Wave Propagation in Stainless Steel,
Fraunhofer-Institut fu¨r zersto¨rungsfreie Pru¨fverfahren ͑IzfP͒,
in Saarbru¨cken, Germany are both gratefully acknowledged edited by D. O. Thompson and D. E. Chimenti ͑Plenum, New York,
for support during a portion of this work. The author also 1992͒, Vol. 11B, pp. 1999–2006.
thanks M. Spies for conversations regarding this work. 7 S. Ahmed and R. B. Thompson, ‘‘Influence of Columnar Microstructure

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552 J. Acoust. Soc. Am., Vol. 106, No. 2, August 1999 Joseph A. Turner: Scattering in textured polycrystals 552