# Ultrasonic Attenuation in Polycrystalline Materials in 2D

## Abstract

Grains in a polycrystalline material, typically a metal, act as scatterers of ultrasonic waves and thus give rise to attenuation of the waves. Grains have anisotropic stiffness properties, typically orthotropic or cubic. A new approach is proposed to calculate attenuation in a 2D setting starting from the scattering by an anisotropic circle in an isotropic surrounding. This problem has recently been solved, giving explicit, simple expressions for the elements of the transition (T) matrix (which gives the relation between the the incoming and scattered fields) when the circle is small compared to the ultrasonic wavelengths. The T matrix can be used to calculate the total scattering cross section, which in turn can be used to estimate the attenuation in the material. Explicit expressions for the attenuation coefficient for longitudinal and transverse waves are obtained for a cubic material, and contrary to results in the literature these expressions are valid also for strong anisotropy. For the longitudinal attenuation coefficient a comparison with recent FEM results for Inconel 600 gives excellent agreement.

## Keywords

Ultrasonic attenuation 2D Cubic material## 1 Introduction

Grains in a polycrystalline material, typically a metal, act as scatterers of ultrasonic waves and thus give rise to attenuation of the waves. A model of the attenuation is useful to characterize polycrystalline materials by inversion from attenuation measurements. In ultrasonic nondestructive testing the scattering induced grain noise is also of importance. To estimate the attenuation and effective wave speed in polycrystals various approximate methods have been used. As examples Stanke and Kino [1] calculate the wave speed and attenuation using a perturbation method with weak anisotropy and give an excellent review of work prior to 1984, Hirsekorn [2] performs a similar analysis for textured materials (giving an anisotropic effective material), Thompson et al. [3] give an overview of scattering of elastic waves in simple and complex polycrystals, and Li and Rokhlin [4] study the scattering in general random anisotropic solids. These studies all use volume integral equation methods combined with some perturbation method, most often the Born approximation (which means a restriction to weak anisotropy). Recently finite element methods (FEM) have been used to compute the attenuation and effective wave speed in polycrystalline materials, see Van Pamel et al. [5, 6, 7] and Ryzy et al. [8]. For excellent and more extensive introductions (and many further references) these papers [1, 5, 6, 7, 8] can be consulted.

Here a different approach is proposed, where first the scattering by a single anisotropic grain in an effective isotropic surrounding is studied. The effective stiffnesses of the surrounding material are taken as the Voigt averages of the stiffnesses of the grain, see Van Pamel et al. [7] for a discussion of the appropriateness of this. Only the 2D problem for in-plane (P and SV) waves is treated and the grains are assumed to be cubic, this being the simplest and most common case. The grains are assumed small compared to the wavelength, usually called the Rayleigh regime, so it should be enough to assume them to be circular (the scattering at low frequencies is predominantly a volume effect, see Yang et al. [9] and Ryzy et al. [8]), and this case has recently been solved by Boström [10], giving simple explicit expressions. The simplest possible approach to estimate the attenuation is investigated and this yields a simple expression for the attenuation coefficient. It is, in particular, noted that the present approach does not assume weak anisotropy, a restriction that is made by other analytical approaches.

## 2 Scattering by an Anisotropic Circle

*a*residing in an isotropic medium. This problem has recently been solved by Boström [10] and here only the pertinent results are repeated. The material inside the circle is taken as cubic with stiffness constants \(c_1\), \(c_3\), and \(c_4\), so that the constitutive equations are

*x*and

*y*axes are along the principal directions. The density inside the circle is \(\rho _0\). The effective material surrounding the circle has the same density \(\rho _0\) and Lamé parameters \(\lambda _0\) and \(\mu _0\). These are taken as the Voigt averages, thus

*t*is time, is suppressed throughout. The longitudinal (pressure, P) wave number is \(k_p=\omega \sqrt{\rho _0/(\lambda _0+2\mu _0)}\) and the transverse (shear, S) is \(k_s=\omega \sqrt{\rho _0/\mu _0}\).

*r*,\(\varphi \) are employed and the following wavefunctions are introduced

*o*(odd) corresponds to the upper or lower row, respectively, of the trigonometric functions and gives waves which are symmetric or antisymmetric, respectively, with respect to \(\varphi =0\). The Neumann factor is \(\epsilon _0=1\) and \(\epsilon _m=2\) for \(m=1,2,\ldots \). For \(m=0\) there are just two wavefunctions, one odd for \(\tau =1\) and one even for \(\tau =2\). The upper index 0 on the wave functions denotes that they are regular, containing a Bessel function \(\hbox {J}_m\), the corresponding outgoing wavefunctions contain a Hankel function \(\hbox {H}_m^{(1)}\) and are denoted by an upper index +. The unit vectors are denoted \(\varvec{e}_r\), \(\varvec{e}_\varphi \), and \(\varvec{e}_z\).

*m*values) in steps of 4, in the case of 2D orthotropy there is a further coupling in steps of 2, see Boström [10].

The displacement field inside the circle is divided into four independent parts due to the symmetries. Thus the four parts are symmetric or antisymmetric with respect to the *x* and *y* axes. Here a symmetric displacement with respect to the *x* axis has an even *x* component and an odd *y* component and vice versa for symmetry with respect to the *y* axis. This also means that a symmetric displacement with respect to the *x* axis has an even *r* component and an odd \(\varphi \) component and vice versa for symmetry with respect to the *y* axis. The doubly symmetric wavefunctions have \(\sigma =e\) and \(m=0,2,4,\ldots \), the symmetric-antisymmetric (with respect to the *x* and *y* axes, respectively) ones have \(\sigma =e\) and \(m=1,3,\ldots \), the antisymmetric-symmetric ones have \(\sigma =o\) and \(m=1,3,\ldots \), and the doubly antisymmetric ones have \(\sigma =o\) and \(m=0,2,4,\ldots \). Thus, there is no coupling between these four groups, meaning that the corresponding T matrix elements vanish.

The displacement components inside the circle can be expanded in trigonometric series in the azimuthal coordinate appropriate to the symmetry of the component. A power series in the radial coordinate is then assumed and recursion relations can be set up for the expansion coefficients in these series, essentially solving the problem inside the circle.

## 3 The Attenuation Coefficient

*n*is the number density of grains, \(\gamma \) is the total scattering cross section, and

*c*is the relative density of grains. This formula for the attenuation is usually supposed to be valid only for dilute concentrations, typically \(c<0.05\) or less. As the grains fill the whole volume \(c=1\) is used. The rationale for this is that each grain scatters extremely little in the low frequency limit. This approach has been used also by others, see references in Stanke and Kino [1]. A schematic picture of the model is given in Fig. 1.

Materials | Copper | Inconel 600 | Iron | Aluminium |
---|---|---|---|---|

\(c_{1}\) (GPa) | 176.2 | 234.6 | 219.2 | 103.4 |

\(c_{3}\) (GPa) | 124.9 | 145.4 | 136.8 | 57.1 |

\(c_{4}\) (GPa) | 81.8 | 126.2 | 109.2 | 28.6 |

\(\rho \) (\(\hbox {kg/m}^3\)) | 8970 | 8260 | 7860 | 2760 |

\(\lambda _0\) (Gpa) | 96.8 | 104.6 | 102.0 | 54.4 |

\(\mu _0\) (Gpa) | 53.7 | 85.4 | 75.2 | 25.9 |

\(\beta \) (Gpa) | \(-28.1\) | \(-40.8\) | \(-34.0\) | \(-2.7\) |

\(\nu \) ( – ) | \(-1.37\) | \(-1.29\) | \(-1.24\) | \(-0.38\) |

Linear error (%) | 33 | 31 | 26 | 6.6 |

Other analytical approaches in the literature demand small anisotropy, either through a perturbation method in \(\nu \) or by using the Born approximation. In contrast the present method does not have such a limitation, instead the main limitation is that the Rayleigh regime is assumed. It can therefore be of interest to see the error induced by a linearization of the present result. This amounts to putting \(\beta =0\) in the denominator in Eqs. (19) and (20). The relative error \(\left| \alpha ^P_{\mathrm {lin}}-\alpha ^P\right| /\alpha ^P\), where \(\alpha ^P_{\mathrm {lin}}\) is the linearized form of \(\alpha ^P\), is therefore also given in Table 1. For the materials listed it is seen that this error ranges from \(6.6\%\) for aluminium, which is rather weakly anisotropic, to \(33\%\) for copper, which is rather strongly anisotropic. Except for aluminium this error is quite substantial.

Recently, FEM has been used to model attenuation due to grain scattering in both 2D and 3D [5, 6, 7, 8]. The only result that can be compared with the present one are the 2D longitudinal result for Inconel 600 of Van Pamel et al. [5]. Figure 2 shows the normalized longitudinal attenuation coefficient for Inconel 600 (material properties given in Table 1) as a function of normalized frequency in the Rayleigh regime for the present result and 2D FEM [5]. Also the 3D result of the Unified theory is shown. The grain radius is \(a=100\,\upmu \mathrm {m}\). The agreement between the present result and 2D FEM is excellent, the difference is about 5% at most, thus validating the present approach and the assumptions that are made (circular grains and neglect of multiple scattering). The unified theory is of the same order of magnitude, but as the frequency dependence is different the results must of course differ. It should be noted that the results are sensitive to the grain radius *a*. The present result assumes that all grains have the same radius and the 2D FEM result [5] seems to be for a case where the grain size is very uniform (more than would normally be the case in a real material).

## 4 Concluding Remarks

In the present paper simple expressions are derived for the longitudinal and transverse attenuation coefficients in 2D for a cubic polycrystalline material. The material may be strongly anisotorpic, instead the main limitation is that the result is only valid in the Rayleigh regime. It is also assumed that all the grains are circular with the same radius and that multiple scattering can be neglected, but these two limitations should be of minor importance as the comparison with FEM shows.

The main advantage of the present approach compared to previous results in the literature is that the anisotropy can be arbitrarily strong. A further advantage with the present approach is that it is straightforward to generalize it in various directions. Thus it is straightforward to consider a distribution of grain sizes, cf. Arguelles and Turner [13], or to consider grains of different types (as in a duplex material). It is also possible to include a distribution of inclusions of other types, like pores or cracks.

To be of real practical value the present approach should be generalized to 3D; such work is in progress.

## Notes

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