Scattering of elastic waves by a sphere with orthorhombic anisotropy and application to polycrystalline material characterization

Scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry inside an isotropic medium is studied and applied to characterization of polycrystalline materials with anisotropic grains. For a single sphere the waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the relation among the scattered wave expansion coefficients and those of the incident wave, expressed as the transition ( 𝐓 ) matrix. For low frequencies the elements of the 𝐓 matrix are obtained in explicit form. According to the theory of Foldy the 𝐓 matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials, explicitly for low frequencies. Comparisons of the present method with previously published results and recent FEM results show a good correspondence for low frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.


Introduction
The study of wave propagation and scattering in inhomogeneous media have received extensive attention due to the need for the characterization of such media.The scattering of elastic waves by anisotropic grains in polycrystals is a classical problem that has important practical applications in various fields, such as seismology and non-destructive evaluation.Polycrystalline materials consist of numerous small crystals, also known as grains, which are typically anisotropic.Although the elastic constants are identical for all grains within a single phase polycrystalline material, the orientation is different for each grain.If the grains are randomly oriented and equiaxed, the overall properties of the material become isotropic.However, the presence of these grains leads to wave scattering and hence attenuation.To estimate the attenuation and effective phase velocity in polycrystals, different models such as individual particles, a regular array of particles, or a stochastic process can be used [1].In their work Stanke and Kino [1] address different aspects of each geometrical model and utilize a stochastic process to estimate the attenuation and phase velocity of polycrystalline materials with equiaxed anisotropic grains of cubic symmetry.In this approach, the micro-inhomogeneous elastic polycrystal is commonly simplified by replacing it with a mean isotropic medium, usually through a Voigt average of the properties of the crystal, and a continuous random fluctuation to account for the inhomogeneity of the medium.This approximation is referred to as the second order approximation (SOA), because it assumes that the second order degree of inhomogeneity is small.This methodology is further developed to compare attenuation in two and three dimensional polycrystals [2] and investigates the effect of various geometrical properties [3][4][5][6][7] as well as the anisotropy symmetries [6,[8][9][10] of the crystals.These advancements in the SOA method have recently been employed for non-destructive ultrasonic characterization of polycrystals [11,12].Another method is the self-consistent approach (SCA), proposed by Kube and Turner [13], where the mean elastic medium is determined through a self-consistent average instead of the Voigt average.The experimental wave velocity data evaluated against the SOA and SCA reveal a better agreement when the Voigt reference is used in second order models [14].A more extensive evaluation of the SOA method and its development are presented in the literature [6,15].
Development of the finite element method for investigation of phase velocity and attenuation in polycrystalline materials [15][16][17][18][19] suggests that the SOA models are not accurate when the grains are highly anisotropic and the degree of inhomogeneity increases, particularly in the low frequency limit [15,19,20].To address this issue, Huang et al. [15,19] develop approximation methods that consider the quadratic order of the degree of inhomogeneity in the SOA method.While these approximation methods are effective in estimating phase velocities accurately, there is still room for improvement in the estimation of attenuation [15].
Polycrystals can also be regarded as a special case of a medium with a distribution of inclusions.The attenuation and effective phase velocity in polycrystals can then be estimated by examining how waves scatter from individual grains.Foldy [21] develops a methodology to estimate the effective wave number of a medium with a distribution of inclusions (regarded as point scatterers) for scalar waves.This approach is further developed regarding correlations among scatterers positions, [22] elastic waves scattering and volume scatterers [23,24].Kanaun and Levin [25] referred to this neglection as the one particle solution.Based on the definition of the surrounding medium of the particle and the field acting on it, they provided a comprehensive overview of different approaches, including Foldy theory, for studying the dynamic response of inhomogeneous mediums with bounded inhomogeneities.In the case of polycrystalline materials, where grains possess anisotropic properties, it is necessary to consider the one particle solution of an anisotropic grain.For this purpose, Kanaun and Levin [25] assumed that the surrounding medium of the single particle is equivalent to the effective medium of the polycrystal.By using an integral representation method, the solution of the one particle problem is related to the Green function of the surrounding medium and the average of the elasticity tensor of the anisotropic particle over all orientations.
Boström [26] further investigated the one particle problem with orthorhombic symmetry of a spherical particle using series expansions of the fields inside and outside of the particle and evaluated effective wave numbers of polycrystals with anisotropic grains in 2D [27].Similarly, Jafarzadeh et al. [28,29] extend this study to 3D problems by calculating the transition () matrix of a single transversely anisotropic or cubic sphere, respectively, and estimate wave numbers in polycrystals with these crystal symmetries.Although these studies are limited to low and intermediate frequencies, they show good correspondence with FEM in estimating wave numbers in polycrystals with strongly anisotropic crystals.
The aim of this work is to characterize polycrystalline materials by finding the  matrix elements for a spherical obstacle with orthorhombic symmetry in an isotropic surrounding.The same approach as for a spherical obstacle with transverse isotropy [28] or cubic anisotropy [29] is taken.Expressing the constitutive equations for a material with orthorhombic symmetry in spherical coordinates reveals the complexity of the governing equations, involving explicit trigonometric functions in both the polar and azimuthal coordinates.By using expansions in vector spherical harmonics and power series in the angular and radial coordinates, respectively, the equations of motion result in recursion relations for the coefficients in the power series.The boundary conditions are applied and the  matrix elements for a single spherical obstacle are derived.These elements are employed along with Foldy theory to calculate the attenuation and phase velocity of polycrystalline materials explicitly for low frequencies.In addition to the explicit expression, a numerical calculation is performed to evaluate the attenuation and phase velocities for both low and intermediate frequencies.Overall, the paper contributes to a better understanding of wave propagation in polycrystalline materials with orthorhombic anisotropy.

Statement of the problem
Consider the scattering by a single spherical inclusion with radius  and orthorhombic material properties inside an isotropic full space, see Fig. 1.A known wave is incident on the inclusion and is assumed to be monochromatic with a fixed angular frequency  (the time factor exp(−) is suppressed).In this section wave propagation in the isotropic surrounding with density  and Lamé parameters  and  is reviewed, and in the following section the material properties and waves inside the anisotropic sphere are studied.
The wave numbers in the isotropic medium are  2  =  2 ∕( + 2) and  2  =  2 ∕ for longitudinal and transverse waves, respectively.For the spherical inclusion it is of course natural to use spherical coordinates (, , ).It is then convenient to use spherical vector wave functions   to express field quantities in the isotropic medium.These functions are defined as [30] The spherical vector wave functions are constructed from the derivatives of the spherical scalar wave functions, which are a product of a spherical Bessel function in the radial coordinate and a spherical harmonic in the polar and azimuthal coordinates.The spherical harmonics are defined as where,    (cos ) is an associated Legendre function of order  and degree .The Neumann factor is   with  0 = 1 and   = 2 for  = 1, 2, ….Also,  =  is for the upper row which is even with respect to  and  =  is for the lower row which is odd with respect to .It is then convenient to define vector spherical harmonics from the derivative of the spherical harmonics as ( The vector spherical harmonics constitute a complete orthonormal vector basis system on the unit sphere for vector valued functions.The spherical vector wave functions can then be expressed as The first index  = 1, 2, 3 specifies the type of the partial wave according to SH, SV, and P waves, respectively.The index  =  (even) and  (odd) indicates the parity of the wave with respect to the azimuthal coordinate, and the order is  = 0, 1, … and the degree is  = ,  + 1, … ( = 0 is only relevant for the P wave functions and is not defined for the others).The upper index ''0'' denotes the regular wave functions which contain spherical Bessel functions.The corresponding outgoing wave functions are denoted by the upper index ''+'' and contain spherical Hankel functions of the first kind to fulfill the radiation condition.
In the boundary conditions the radial traction is needed.This traction can be written The corresponding traction for each of the vector wave functions becomes () (  0 3 () The total wave outside the sphere is a sum of the incident and scattered waves () =  0 () +  + ().Since the source of the incident wave is assumed outside the sphere the incident wave can be expanded in terms of the regular wave functions and as the scattered wave must fulfill the radiation condition it can be expanded in the outgoing wave functions where the sums run through  = 1, 2, 3,  = , ,  = 0, 1, ..,  = ,  + 1, ….The expansion coefficients of the incident wave   are in principle known and the ones of the scattered wave ℎ  are to be determined.A general way to represent the scattering by an obstacle is to determine the transition matrix ( matrix) which gives the linear relation between the expansion coefficients of the incident and scattered waves This completes the necessary information in the surrounding medium to solve the scattering problem.In the following sections the anisotropic sphere is analyzed and the transition matrix is derived.

The orthorhombic sphere
In this section the material properties and governing equations of the orthorhombic sphere are described, then the waves are expanded using appropriate series.Starting with the material properties, the sphere has density  1 and material properties of orthorhombic symmetry in Cartesian coordinates.The constitutive relations can be expressed using reduced notation as where  11 ,  The governing elastodynamic equations are expressed in spherical coordinates For the present problem it is convenient to transform the constitutive relations in Eq. ( 15) from Cartesian to spherical coordinates (see Jafarzadeh et al. [28]).In spherical coordinates the stress-strain relations for a material with orthorhombic properties become A. Jafarzadeh et al.
where the strains in spherical coordinates are

𝜕𝑢 𝑟 𝜕𝜃
) . ( Here,  1 ,  ) . (26) In the isotropic limit the first two become the Lamé constants  1 and  1 of the sphere whereas the other seven vanish.Substituting ( 19)-( 25) into ( 16)-( 18), gives the governing equations of the displacement field; however, this leads to very large equations which are not given.Due to the trigonometric functions in  and  there are extensive couplings between different orders and degrees when expansions in spherical harmonics are later performed.
To solve the scattering problem the scattered wave outside the sphere and the interior wave inside the sphere are to be determined given the incident wave.Of course, expanding the displacement field inside the sphere in terms of the spherical vector wave functions does not satisfy the governing equations of an anisotropic medium.Instead, the vector spherical harmonics are appropriate for the expansion of the displacement inside the sphere since the displacement and stress fields outside the sphere are also in terms of these functions and this facilitates the application of the boundary conditions.Therefore, the displacement field inside the sphere is expanded as where  = 1, 2, … for  = 1, 2 and  = 0, 1, … for  = 3.The  dependent coefficients   () are expanded in power series in .By considering the regularity condition at the center of the sphere, these expansions are in which  30,−1 = 0. Here,  , are the unknown coefficients inside the sphere.The governing equations Eqs. ( 16)-( 18) provide recursion relations among these unknown coefficients inside the sphere.To derive these, the three governing equations Eqs. ( 16)-( 18) are considered as one vector valued function and are expanded in terms of the vector spherical harmonics  Here,   is the scalar product of a vector spherical harmonic and the governing vector equation.The orthogonality of the vector spherical harmonics means that all these coefficients, which are power series in , must vanish so that As the powers of  are linearly independent, the coefficient in front of every power of  must vanish, which provides the recursion relations among the unknown coefficients inside the sphere and leaves only one unknown for each partial wave.The general explicit expression of these recursion relations is complicated and is not given.The recursion relations also give couplings among the partial waves.Such coupling is discussed in detail for a sphere with transversely isotropic [28] and cubic [29] material properties.For the orthorhombic sphere, there is a similar coupling among the partial waves as for the cubic material, but with extra couplings of | ± 2| partial waves with  and | ± 4|.This is expected as there are trigonometric functions of order 2 in the strain-stress relations Eqs. ( 19)-( 24) (which are absent in the cubic case).
To find all the unknown expansion coefficients inside and outside the sphere there remains to apply the boundary conditions on the sphere.Assuming the sphere to be perfectly welded with the surrounding, the displacement and normal traction must be continuous at the surface of the sphere  = .The displacement field outside and inside of the sphere (Eqs.( 13) and ( 27)) are expanded in terms of the vector spherical harmonics and using the orthonormality of the vector spherical harmonics application of the boundary conditions is straightforward.For the traction boundary condition, the radial traction just inside the sphere is expanded in terms of the vector spherical harmonics as where   is the scalar product of the radial traction vector and vector spherical harmonics of order , ,  and .Such an expansion of the traction inside the sphere together with the traction outside the sphere (Eqs.( 9)-( 12)) makes it straightforward to apply the traction boundary condition.It is noted that   shows the same coupling among the coefficients  , as in   ().
The boundary conditions and the recursion relations give all the equations needed to determine all expansion coefficients and thereby the fields outside and inside the sphere.It is, in particular, possible to determine the  matrix elements.Due to their complexity, explicit expressions for all these equations cannot be stated, but for low frequencies it is possible to obtain explicit expressions for the leading order  matrix elements, and for higher frequencies it is possible to solve the equations numerically.
In the low frequency limit it is sufficient to expand the displacement field to power 3 in .With such a truncation in Eqs. ( 28)- (30), 64 partial waves remain for the displacement field inside the sphere, which are listed in Table 1.The continuity condition of displacement and traction for each vector spherical harmonic leads to a large system of equations to be solved.However, materials with orthorhombic symmetry have three orthogonal symmetry planes, which leads to a decoupling into eight independent subproblems.Table 1 presents the eight distinct sets of decoupled partial waves at low frequencies.In the low frequency limit, it has been established for cubic materials that the leading order  matrix elements for SH waves are of higher order than those for P and SV waves [28] and this is true also for orthorhombic materials.Therefore, it is adequate to only consider P and SV waves ( = 2 and 3).Additionally, only incident partial waves with orders  = 0, 1, and 2 are needed for determining the leading order  matrix elements at low frequencies.Thus all partial waves with  = 1 and  ≥ 3 inside the sphere (which are enclosed in parentheses in Table 1) are not of leading order at low frequencies.The subsequent section presents the leading order  matrix elements in the low frequency limit for each of the eight sets.

Low frequency T matrix elements
In this section all the leading order  matrix elements in the low frequency limit are explicitly stated.To keep the exposition brief, some details of the calculations are only explained for the first case with odd-odd-odd P-SV waves, and for the other cases only the truncated displacement expansion and the  matrix elements are presented.As shown in Table 1 for the odd-even-odd case there is no leading order  matrix element in the low frequency limit, so the other seven cases remain.For more calculational details see Jafarzadeh et al. [28,29] where the  matrix elements for a sphere with transversely isotropic and cubic material properties are derived, respectively.

Odd-odd-odd P-SV waves
For the case with odd-odd-odd P-SV waves,  = ,  = 1, and  = 1, the low frequency displacement field expansion is where A. Jafarzadeh et al.
As in Section 3, the equations of motion are projected onto  211 (, ) and  311 (, ) with only the two lowest order in  retained.The resulting two distinct equations are solved to give the two recursion relations Thus, two unknowns remain inside the sphere, and the unknowns outside the sphere are ℎ 201 and ℎ 301 .These are found by using the continuity of displacement and traction for  = 2 and  = 3 with  = 1 (again projecting), and the  matrix elements are calculated as ) . ( The  matrix elements only depend on the density of the sphere, and are independent of its elasticity constants.Thus, these elements are the same for any type of spherical inclusion.This is reasonable since the  = 1 case for low frequencies is related to the rigid body translation of the sphere.Consequently, all other  matrix elements of order  = 1 are the same regardless of their parity with respect to  and  and the following relations thus follows for the even-odd-odd and even-even-odd cases In the following subsections only the suitably truncated displacement field and the  matrix elements for the other cases are stated.

Even-odd-even P-SV waves
This case has  = ,  = 1, and  = 2 and the displacement expansion is The  matrix elements become ) . (42)

Odd-even-even P-SV waves
This case has  = ,  = 2, and  = 2 and the displacement expansion is The  matrix elements become It is noticed that the three last cases, all with only  = 2 (and with no coupling to  = 0), are very similar, it is just to exchange the three shear moduli  44 ,  55 ,  66 .All these  matrix elements are independent of all the other stiffness constants.In the cubic and transversely isotropic cases, with  44 =  55 =  66 and  44 =  55 ,  66 = ( 11 −  12 )∕2, respectively, the same  matrix elements are obtained by Jafarzadeh et al. [28,29].
Looking at the  matrix elements, the leading order ones in the low frequency limit are all imaginary.The leading order real parts of the  matrix elements are of higher order and require higher order expansion of the displacement field.Using the ''hermitian'' property of the  matrix [32] the leading order real part of the  matrix elements is obtained from the leading order imaginary parts of them.
The derived  matrix elements are all that are of leading order at low frequencies.Low frequencies typically means    < 0.5, but this of course depends on the accuracy required.It must also be recognized that the wave numbers of the material inside the sphere play a role, so if one of the wave numbers inside the sphere is larger this may limit the range of frequencies further.There is, on the other hand, no limit on the strength of the anisotropy.This is in contrast with the perturbative Keller approximation [33] or the diagrammatic Bourret approximation [34], which are typically used in methods like SOA, and are only valid for small anisotropy (but has no limit in frequency).

Polycrystalline materials
Polycrystalline materials are composed of a collection of crystals with anisotropic properties.Due to the grainy structure, propagating waves in the medium scatter and attenuate.In the case of randomly oriented, equiaxed crystals, the macroscopic behavior of the polycrystals is isotropic and homogeneous.The medium is then characterized by the effective wave numbers   and   for the longitudinal and transverse waves, respectively.The effective wave numbers are complex numbers in which the real part gives the phase velocity and the imaginary part gives the attenuation of the wave.Since the scattering depends on the frequency, the effective phase velocity and attenuation are, in general, frequency dependent.However, for very low frequencies the state in the medium is quasi-static, attenuation is small, and the wave number provides a quasi-static velocity limit of the medium.Considering that a polycrystalline material is filled with crystals, the propagating wave constantly experiences multiple events of scattering.Modeling all these events of scattering is complicated, and different simplifying assumptions are considered in the literature to study wave propagation in these materials.
Here an approach where the scattering by each crystal is regarded as taking place in the effective and isotropic medium (called the matrix) of all the other grains is followed.Multiple scattering is neglected and it is assumed that each grain encounters the same incident wave throughout the whole medium.The matrix is then a homogeneous medium with no attenuation and constant phase velocities, which can be calculated by the Voigt average.Therefore, the method is called the Single Scattering Approximation in the Voigt homogenized medium (SSA-V).The matrix is defined by the longitudinal   = √ ∕( + 2) and transverse   = √ ∕ wave numbers.The density  for a single-phase polycrystalline material is the same as the single crystal density  1 .The Lamé parameters are the Voigt averages of a single crystal, which have the following relation to the stiffness constants of a material with orthorhombic symmetry [35]  = 1 15 ) . ( By substituting the surrounding medium of each crystal with a homogenized isotropic medium, each crystal contributes to a single scattering event which can be addressed by the calculated  matrix elements.With these assumptions, the theory of Foldy [21], neglecting correlations among the individual scatterers [24], is used to study the effective wave numbers of the polycrystal as [28,29] where   and   are the wave numbers of the matrix (which are the Voigt averages in the SSA-V method), and   and   are the polycrystalline effective wave numbers for longitudinal and transverse waves, respectively. = 4 3 ∕3 is the volume of a single grain and  is the relative density of the grains which for a polycrystalline materials is set to be  = 1 as the grains fill the volume.The grains in the polycrystalline material are considered to be spheres with the same radius  meaning that no distribution of grain size is considered.This is a reasonable approximation for low frequencies, for which the scattering is mainly a volume effect.The resulting complex effective wave number describes the attenuation and phase velocity of the composite medium as where  =  and  for longitudinal and transverse waves, respectively,   is the attenuation,   is the phase velocity in the matrix and   is the effective phase velocity.
Collecting everything, the leading order of the real and imaginary parts of the effective wavenumbers   becomes

𝐷
) 2 ) In the low frequency limit the parameters   and   are small and the normalized attenuation and phase velocity can be approximated with good accuracy as These relations confirm the frequency dependence of attenuation and phase velocity at low frequencies, where the phase velocity is independent of the frequency and the attenuation depends on the fourth power of the frequency (or wave number).As mentioned, the frequency independent phase velocities at low frequencies provide macroscopic static properties of the polycrystal.
In the method just presented the matrix properties are determined by the Voigt average and this leads to a change in the phase velocities at low frequencies determined by   and   .Ideally this change should vanish (as it does in the case of isotropy), so nonzero   and   can be viewed as a measure of the degree of anisotropy, and they are henceforth used for this purpose.This also leads to another way to determine the Lamé parameters of the matrix by putting   =   = 0.These two equations can be rearranged to the following two nonlinear equations in  and It seems that this system only has a single solution where both  and  are positive and real (there are further solutions that are not physically meaningful).The method to calculate the Lamé parameters  and  by Eq. ( 56) is here called the static-consistent (SC) average.The static-consistent average shows an exact agreement with the self-consistent average calculated by the script provided by Kube and De Jong [35].(This article also give a comprehensive study of various anisotropy indices suggested in the literature and different averages and bounds of macroscopic homogenized static properties of polycrystals.)A modification of the SSA-V method is thus to define the matrix with the static-consistent elastic properties and this modified method is called the Single Scattering Approximation in the Static Consistent homogenized medium (SSA-SC).Such a consideration for the matrix properties is also suggested by Kube and Turner [13]).This method could be further generalized by iterating Eqs. ( 53) and (54) at finite frequencies, but this leads to attenuation and complex  and , so this is not further investigated.In summary the present approach makes the following assumptions and approximations: A. Jafarzadeh et al. 1.Each crystal is considered to be isolated from all other crystals, and they all experience the same incident wave propagating in the homogenized isotropic medium surrounding them.
2. Two assumptions are made regarding the surrounding medium: in the SSA-V approach the surrounding medium is considered to be the Voigt average of the properties of a single crystal, while in the self-consistent approach SSA-SC the surrounding medium is considered equal to the effective medium of the polycrystal in the static limit.
3. The scattered wave is studied in the far field of the surrounding homogenized medium.
4. The only statistical information of the geometrical properties of the grains considered is their effective volume, and the size distribution is neglected.
5. The explicit expressions are valid for the Rayleigh regime, where the size of the scatterers is much smaller than the wavelength of the incident wave.This is mainly due to the truncation of the  matrix elements to the dominant term in the low-frequency limit, which can be overcome by numerically evaluating higher orders.However, the validity of other assumptions, such as the neglect of the grain size distribution and multiple scattering, remains questionable for higher frequencies.

Numerical results for polycrystalline materials
This section presents numerical evaluation of the attenuation and phase velocity for the 50 different polycrystals with single-phased equiaxed crystals listed in Table 2, characterized by orthorhombic, tetragonal, hexagonal, and cubic symmetries.A brief overview is provided for the have a better monotonic relation with accurate SC results compared to   and   .This is partly due to the separation of the degree of anisotropy in the longitudinal and transverse directions, whereas anisotropy indices aim to indicate the anisotropy of a material with a single parameter, disregarding the direction.However, as shown in Table 2 and Fig. 2(a-b), a material may have quite different degrees of anisotropy in various directions.
In the following, longitudinal attenuation and phase velocity for the materials listed in Table 2 are evaluated using the SSA, SOA, and DAM methods, with a focus on strongly anisotropic materials in the longitudinal direction.
The longitudinal attenuation and phase velocity are calculated for the materials listed in Tables 2 and 3, ordered based on the longitudinal anisotropy   for each symmetry of the crystals.Considering the good correspondence of the present methods for estimation of the attenuation and phase velocity for weakly anisotropic materials, the study is focused on strongly anisotropic materials.However, normalized root-mean-square deviation (NRMSD) of the longitudinal attenuation with respect to FEM is presented in Table 3 for all materials.NRMSD quantifies the overall difference in the frequency range of    ≤ 0.5 and is calculated as root mean square of ( Model  ∕ FEM  − 1).In the following the normalized longitudinal attenuation    and normalized longitudinal phase velocity   ∕  are depicted versus normalized frequency    for different materials.The normalization parameters,   and   , are determined by the Voigt average.For each material, the quantities are calculated with the R-SSA-V method (dashed blue lines), SSA-V method (solid blue lines), R-SSA-SC method (dashed green lines), SSA-SC method (solid green lines), SOA method (dotted black lines), DAM method (solid red lines), and the FEM results (circles).
Fig. 3 shows (a) normalized longitudinal attenuation    and (b) normalized longitudinal phase velocity   ∕  versus normalized frequency    for the strongest anisotropic materials in the longitudinal (Sr 1 Mg 6 Ga 1 ) and transverse (Al 2 Cu 2 O 6 ) cases.Fig. 4 shows normalized longitudinal attenuation    versus normalized frequency    for the three polycrystals of, orthorhombic, tetragonal, hexagonal and cubic symmetries with the highest anisotropy factors   as stated in Table 2 (excluding the materials shown in Fig. 3).Finally, Fig. 5 shows normalized longitudinal phase velocity   ∕  versus normalized frequency    for the same polycrystals as in Fig. 4.
Combining the results of Figs.3-5 and Table 2, the following observations are of interest: The R-SSA-V and R-SSA-SC methods are valid only in the Rayleigh limit when the wavelength is much larger than the grain radius.In this frequency range they should be very attractive to use due to their simplicity and the fact that they are valid also for strong anisotropy.The range of validity of the Rayleigh results are somewhat hard to specify as it can vary significantly depending on the degree of anisotropy, ranging from    > 0.5 for weakly anisotropic materials to    < 0.1 for strongly anisotropic materials, such as Sr 1 Mg 6 Ga 1 .The Sr 1 Mg 6 Ga 1 crystals possess orthorhombic symmetry, with different longitudinal wavenumbers along each coordinate axis.The ratio of these wavenumbers to the Voigt wave number of the macroscopically isotropic medium is    ∕  = 0.82,    ∕  = 1.65, and    ∕  = 0.66 for longitudinal wave numbers in the , , and  directions, respectively.The significant variation in wave velocities in different directions inside the crystals impacts the range of validity of the Rayleigh results.
For the phase velocity evaluation, the SSA-V method estimation of phase velocity shows a good correspondence with FEM for   < 0.1, however, the difference with FE calculation increases for higher degrees of anisotropy.On the other hand, for the DAM and SSA-SC methods all studied materials (except Sr 1 Mg 6 Ga 1 ) show good agreement with FEM results.Theoretically, DAM quasi-static phase velocity should be equal to R-SSA-SC phase velocity and the small differences that appear for the strongly anisotropic materials seem to be due to the fewer iterations considered in the evaluation of the self-consistent phase velocity by Huang et al. [15].Fig. 3. Normalized attenuation    and normalized phase velocity   ∕  versus normalized frequency    for longitudinal waves evaluated by the Rayleigh asymptote of the SSA methods, R-SSA-V (dashed blue lines), R-SSA-SC (dashed green lines), numerical evaluation of the SSA methods, SSA-V (solid blue lines), SSA-SC (solid green lines), SOA (dotted black lines), DAM method (solid red lines) and numerical FEM results (circles), for Al 2 Cu 2 O 6 and Sr 1 Mg 6 Ga 1 with highest transverse and longitudinal anisotropy degree, respectively.
For the attenuation estimation, the correspondence of the SSA methods with FEM results has an almost monotonic correlation with the anisotropy degree   .The SSA-V method shows a good agreement with NRSMD less than 10% for materials with   < 0.1, while the SSA-SC method shows such correspondence for all materials except Sr 1 Mg 6 Ga 1 with   = 0.26.On the other hand, the correspondence of SOA and DAM methods with FEM are not directly related to the anisotropy degree   .Overall, both methods show better correspondence with FEM for weakly anisotropic materials, and the agreement of DAM is higher than the SOA method.However, there are plenty of exceptions for all types of crystal symmetries.As the accuracy of the FEM results depend on several factors discussed in the last section, the validity of the mentioned analytical approaches is best to be examined with experimental results.
For Sr 1 Mg 6 Ga 1 , with an extremely high degree of anisotropy, none of the studied methods has a good agreement with FEM.The lowest NRMSD in the range of 0.15 <    < 0.5 is for the SSA-SC method with 3% difference for the phase velocity and 60% difference for the attenuation.4. Normalized attenuation    versus normalized frequency    for longitudinal waves evaluated by the Rayleigh asymptote of the SSA methods, R-SSA-V (dashed blue lines), R-SSA-SC (dashed green lines), numerical evaluation of the SSA methods, SSA-V (solid blue lines), SSA-SC (solid green lines), DAM method (solid red lines), and numerical FEM results (circles), for three highly anisotropic polycrystals of cubic, hexagonal, tetragonal and orthorhombic symmetries.This appears to be due to the high degree of anisotropy and strong attenuation, even at low frequencies.As a result, the frequency range of 0.15 <    < 0.5 is beyond the Rayleigh limit for this material and is in the region where the geometrical properties of crystals are becoming significant.This confirms that the Rayleigh limit not only depends on the ratio of the grain size and wavelength, but also on the elasticity of the grains and the attenuation.

Concluding remarks
This paper examines the scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry in an isotropic environment.To accomplish this spherical coordinates are utilized and the displacement field inside the sphere is expanded in terms of vector spherical harmonics in the angular coordinates and powers in the radial coordinate.This leads to recursion relations among the expansion coefficients.The boundary conditions on the sphere are then used to determine the elements of the transition matrix ().At low frequencies the leading order elements are expressed explicitly in simple form.The contributions of monopole, dipole, and quadrupole elements are all taken into account.The dipole elements are not coupled to the other elements and depend solely on the density of the sphere, and are, in fact, the same as for an isotropic sphere.The monopole elements, however, are coupled with the quadrupole elements for even waves of even order ( =  and  = 0, 2) in azimuthal coordinates.These elements depend on the elasticity of the sphere but not on the density or shear moduli.On the other hand, the remaining quadrupole elements are uncoupled and only depend on the shear moduli of the sphere.
Furthermore, the wave propagation in single phase polycrystalline materials with cubic, hexagonal, tetragonal, and orthorhombic symmetry of the crystals is analyzed using the Foldy theory.To assess the degree of anisotropy of the grains, two parameters for measuring degrees of anisotropy in longitudinal and transverse directions are introduced.The Voigt average has typically been used to calculate the unperturbed elasticity constants for determining attenuation and phase velocity in polycrystals.However, this method seems to be not suitable for highly anisotropic materials as it does not show a good agreement in quasi static phase velocity estimation with accurate SC results.To address this, a new methodology is proposed that calculates the average macroscopic homogenized and isotropic properties of polycrystals.This average is shown to be equivalent with self

Fig. 1 .
Fig. 1.The orthorhombic sphere with radius  and the incident wave   .

Fig. 2 .
Fig. 2.The ratio of quasi-static phase velocities to Voigt phase velocities for (a) longitudinal ( −  ∕  ) and (b) transverse ( −  ∕  ) waves, versus longitudinal anisotropy degree   and transverse anisotropy degree   , respectively.Quasi-static phase velocities are evaluated by SSA-V (blue dots), SC (green dots), and SOA (Black dots) for 50 polycrystals of Table2.The ratio of SC quasi-static phase velocities to Voigt phase velocities for longitudinal (   ∕  ) and transverse (   ∕  ) waves is depicted in Figure(c) and (d), respectively.The anisotropy indices   are shown on the bottom -axis as dark green dots, while   is represented on the top -axis as red dots.

Fig.
Fig.4.Normalized attenuation    versus normalized frequency    for longitudinal waves evaluated by the Rayleigh asymptote of the SSA methods, R-SSA-V (dashed blue lines), R-SSA-SC (dashed green lines), numerical evaluation of the SSA methods, SSA-V (solid blue lines), SSA-SC (solid green lines), DAM method (solid red lines), and numerical FEM results (circles), for three highly anisotropic polycrystals of cubic, hexagonal, tetragonal and orthorhombic symmetries.

Table 1
Table of couplings among the partial waves in the Rayleigh limit.

Table 2
Table of material properties, anisotropy degrees and anisotropy indices [15].

Table 3
[15]D with respect to FEM in longitudinal attenuation for SOA, DAM, SSA-V, and SSA-SC methods[15].