Scattering of elastic waves by an anisotropic sphere with application to polycrystalline materials

Scattering of a plane wave by a single spherical obstacle is the archetype of many scattering problems in various branches of physics. Spherical objects can provide a good approximation for many real objects, and the analytic formulation for a single sphere can be used to investigate wave propagation in more complex structures like particulate composites or grainy materials, which may have applications in non-destructive testing, material characterization, medical ultrasound, etc. The main objective of this thesis is to investigate an analytical solution for scattering of elastic waves by an anisotropic sphere with various types of anisotropy. Throughout the thesis a systematic series expansion approach is used to express displacement and traction fields outside and inside the sphere. For the surrounding isotropic medium such an expansion is made in terms of the traditional vector spherical wave functions. However, describing the fields inside the anisotropic sphere is more complicated since the classical methods are not applicable. The first step is to describe the anisotropy in spherical coordinates, then the expansion inside the sphere is made in the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere provide recurrence relations among the unknown expansion coefficients. The remaining expansion coefficients outside and inside the sphere can be found using the boundary conditions on the sphere. Thus, this gives the scattered wave coefficients from which the transition T matrix can be found. This is convenient as the T matrix fully describes the scattering by the sphere and is independent of the incident wave. The expressions of the general T matrix elements are complicated, but in the low frequency limit it is possible to obtain explicit expressions.The T matrices may be used to solve more complex problems like the wave propagation in polycrystalline materials. The attenuation and wave velocity in a polycrystalline material with randomly oriented anisotropic grains are thus investigated. These quantities are calculated analytically using the simple theory of Foldy and show a very good correspondence for low frequencies with previously published results and numerical computations with FEM. This approach is then utilized for an inhomogeneous medium with local anisotropy, incorporating various statistical information regarding the geometrical and elastic properties of the inhomogeneities

Scattering of elastic waves by an anisotropic sphere

Scattering of a plane wave by a single spherical obstacle is the archetype of many scattering problems in physics and geophysics. Spherical objects can provide a good approximation for many real objects, and the analytic formulation for a single sphere can be used to investigate wave propagation in more complicated structures like particle composites or grainy materials, which may have application in non-destructive testing, material characterization, medical ultrasound, etc. The main direction of this thesis is to investigate an analytical solution for scattering of elastic waves by an anisotropic sphere in the special case with transverse isotropy. Throughout the thesis a systematic series expansion approach is used to derive displacement and traction fields outside and inside the sphere. For the surrounding isotropic medium such an expansion is made conveniently in terms of the traditional vector spherical wave functions. However, describing the fields inside the anisotropic sphere is more complicated since the classical methods are not applicable anymore. The first step is to describe the anisotropy in spherical coordinates, then the expansion inside the sphere is made in the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere provide recurrence relations among the unknown expansion coefficients. The remaining expansion coefficients outside and inside the sphere can be found using the boundary conditions on the sphere. Thus, this gives the scattered wave coefficients from which the transition T matrix can be found. This is convenient as the T matrix fully describes the sphere and is independent of the incident wave. The expressions of the general T matrix elements are complicated, but in the low frequency limit it is possible to obtain explicit expressions.The T matrices may be used to solve more complicated problems like the wave propagation in polycrystalline materials. The attenuation and wave velocity in a polycrystalline material with randomly oriented transversely isotropic grains is thus investigated. These quantities are calculated analytically using the simple theory of Foldy and show a very good correspondence for low frequencies with previously published results and numerical computations with FEM

Scattering of elastic waves by a transversely isotropic sphere and ultrasonic attenuation in hexagonal polycrystalline materials

The scattering of elastic waves by a transversely isotropic sphere in an isotropic medium is considered. The elastodynamic equations inside the sphere are transformed to spherical coordinates and the displacement field is expanded in the vector spherical harmonics in the angular coordinates and powers in the radial coordinate. The governing equations inside the sphere then give recurrence relations among the expansion coefficients. Then all the remaining expansion coefficients for the fields outside and inside the sphere are found using the boundary conditions on the surface of the sphere. As a result, the transition (T) matrix elements are calculated and given explicitly for low frequencies. Using the T matrix and the theory of Foldy an explicit expression for the effective complex wave number of transversely isotropic (hexagonal) polycrystalline materials are presented for low frequencies. Numerical comparisons are made with previously published results and with recent FEM results and show a very good correspondence with FEM for low frequencies. As opposed to other published methods there is no limitation on the degree of anisotropy with the present approach

Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

Scattering of elastic waves by an anisotropic sphere with cubic symmetry inside an isotropic medium is studied. 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 a 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 expansion coefficients of the scattered wave. This determines the transition (T) matrix elements which are calculated explicitly to the leading order for low frequencies. Using the theory of Foldy, the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials with cubic symmetry, explicitly for low frequencies and numerically for intermediate frequencies. Numerical comparisons of the present method with previously published results and recent finite element method (FEM) results show a good correspondence for low and intermediate frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods

Scattering of elastic waves by a sphere with cubic anisotropy

Scattering of elastic waves in materials with inhomogeneities is a classical problem in physics and geophysics, and have applications in non-destructive testing, material characterization, medical ultrasound, etc. The classical analytical solution of the scattering by a single isotropic spherical obstacle provides a good approximation and a basis for more complicated problems and gives a deep understanding of the scattering phenomenon [1].However, plenty of natural and synthetic materials, specifically the grains in a metal, are known to be anisotropic. Recently, the scattering of elastic waves by a circle with cubic anisotropy is studied in 2D by Bostrom [2, 3], and Jafarzadeh et al. [4] use the same method to study the 3D scattering problem for a transversely isotropic sphere. The present work is a continuation of these studies and investigates the 3D scattering by a sphere withcubic anisotropy. Consider the scattering of a single spherical obstacle with cubic anisotropy contained in a three-dimensional, homogeneous, isotropic and infinite elastic medium. In the isotropic surrounding the classical approach is used with the displacement field constructed as a superposition of incident and scattered waves, which are expanded in spherical vector wave functions. Inside the sphere the stress-strain relations are given in Cartesian coordinates for the cubic material and these are first transformed to spherical coordinates. These relations then become inhomogeneous in that they contain factors with trigonometric functions in both angular coordinates and the same becomes true also for the equations of motion. To proceed it is useful to expand the displacement into a series of vector spherical harmonics where each coefficient in turn is expanded into a power series in the radialcoordinate. It follows from the equation of motion that the coefficients in the power series obey certain recursion relations, thus reducing the number of unknowns inside the sphere. Expressing also the stresses as a series in the vector spherical harmonics, the rest of the unknowns are determined by the continuity of the displacement and traction on the sphere boundary. As a result, the transition (T) matrix elements, relating the expansion coefficients of the scattered wave to those of the incident wave, are calculated. It is, in particular, possible to obtain explicit expressions for the leading order T matrix elements for low frequencies.References[1] V. Varadan, A. Lakhtakia, and V. Varadan, Field representations and introduction to scattering. North-Holland, Amsterdam, 1991.[2] A. Bostrom, "Scattering by an anisotropic circle," Wave Motion, vol. 57, pp. 239-244, 2015.[3] A. Bostrom, "Scattering of in-plane elastic waves by an anisotropic circle," The Quarterly Journal of Mechanics and Applied Mathematics, vol. 71, pp. 139-155, 2018.[4] A. Jafarzadeh, P. D. Folkow, and A. Bostrom, "Scattering of elastic sh waves by transversely isotropic sphere," in Proceedings of the International Conference on Structural Dynamic, EURODYN, vol. 2, pp. 2782-2797, 2020

Scattering of elastic SH waves by transversely isotropic sphere

The scattering by a transversely isotropic sphere in a three-dimensional, homoge-neous, isotropic and infinite elastic medium is considered. The problem is a scalar one andis phrased as a scattering problem for an incident SH elastic waves propagating in the direc-tion parallel to the axis of material symmetry. The elastodynamic equations inside the sphereare transformed to spherical coordinates and the displacement field is expanded in associatedLegendre functions in the polar coordinate and powers in the radial coordinate. This leadsto recursion relations for the expansion coefficients inside the sphere. Using the boundaryconditions on the surface of the sphere results in a system of equations for all the expansioncoefficients for the fields outside and inside the sphere. As a result, the transition (T) matrixelements are calculated and given explicitly for low frequencies

Calculation of the T matrix elements for one of the twelve cases from Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

Mathematica code for calculation of T matrix elements for m=2 even PS even l odd SH odd

Calculation of the explicite wave numbers for polycrystalline materials from Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

Mathematica code to calculate effective wave numbers of the polycrystalline materials given the T matrix element

Note on the supplementary materials from Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

A short discription of the provided supplementary material

Data for replication of the figures from Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

A set of XLS files that includes data for replicating the figure