# Modelling of Ultrasonic Bulk Wave Scattering by an Axial Crack in a Pipe

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## Abstract

Modelling of ultrasonic bulk wave scattering by an internal, infinitely long, axial crack in a thick-walled pipe is considered. The problem is formulated as a hypersingular integral equation for the crack-opening displacement (COD), the hypersingularity arises in the Green’s tensor. The COD is expanded in Chebyshev functions which have the correct square-root singularity along the crack edges, thereby regularizing the integral equation. To discretize the integral equation it is likewise projected onto the same Chebyshev functions. A model of an ultrasonic rectangular vertically polarized shear wave contact probe is developed, and the signal response is calculated using a reciprocity argument. Some numerical examples demonstrate the possible application of the method, in particular investigating the importance of the pipe curvature.

## Keywords

Non-destructive testing Wave propagation Cylindrical geometry Pipe Crack## 1 Introduction

Ultrasonic nondestructive testing (NDT) is routinely used in some branches of industry, examples being aerospace and nuclear power industries. For a few decades mathematical models of the testing have led to more or less refined theories for varying test situations. There are many benefits from a reliable and validated model. It is easy to perform parametric studies with a model, and it is thus a useful tool in the development of testing procedures. A model can also enhance the physical “feeling”, i.e. to know a priori when certain parameters are important, and be a useful educational tool. A good model is also a prerequisite when attempting to solve inverse problems.

One important area of application of ultrasonic NDT is the testing of pipes. This can involve the long-range investigation of corrosion and cracks in pipelines. Then guided waves of relatively low frequencies are used, and in this field a lot of work has been performed, see e.g. Bai et al. [3], Benmeddour et al. [4], Fletcher et al. [9] and Duan and Kirby [8]. It should be noted that the field of guided waves has been active for quite some time, i.e. Rose et. al [11]. Velichko and Wilcox [13] have investigated the relationship between guided wave solutions for plates and pipes. However, there is also an interest in testing of pipes at higher frequencies where guided waves are no longer a useful concept. This is, in particular, of importance in the nuclear power industry where a lot of thick-walled pipes are being tested. For modelling purposes this testing is often approximated as taking place in a plate, but this is not always a valid approximation. Thus there is an interest in modelling ultrasonic testing for defects in pipes when the frequency is in an intermediate range so that neither a guided wave approach nor a plate approximation is applicable. Very little work seems to have been done in this area, although Olsson [10] considers the scattering by a defect in a pipe at low and intermediate frequencies using what may be called a T matrix approach. This work considers the wave scattering inside a thick-walled cylinder by a spherical cavity, excited by a point force.

In the present paper a model of ultrasonic testing for an interior axial crack in a pipe is developed using an integral equation approach. The crack is for simplicity taken as infinitely long in the axial direction, but in practice it is enough if the crack is longer than the width of the lobe from the transmitting ultrasonic probe. A realistic model of an ordinary contact probe is used, both in transmission and reception, the latter through a reciprocity argument. The method starts from a hypersingular integral equation over the crack which contains the Green’s tensor of the pipe. The free (singular) part of this Green’s tensor is expanded in plane waves in the crack coordinate system whereas the part added to fulfill the boundary conditions on the pipe walls are expanded in cylindrical waves. This type of approach has large similarities with the work by Bövik and Boström [7] where the same approach is taken for a crack in a plate.

The method used in this article can be summarized as starting from an integral representation, Eq. (4), contains the Green’s tensor for the pipe, Eq. (16) and the crack opening displacement (COD). The traction operator is applied to the integral representation and the field point is taken to the crack surface, however, if the limit is taken inside the integral the integral become improper, Eq. (17). To regularize the integral, and allowing the limit to be taken inside the integral, the integral equation is projected on a set of Chebyshev functions and as a Fourier transform. The COD is expanded in a similar manner. The projection of the integral equation and expansion of the COD integration over the crack surface leads to an equation for the unknown coefficients for the COD, Eq. (29). The incoming field is modelled using a probe model, Eq. (32), derived by Boström and Wirdelius. When the COD and incoming field are fully known a reciprocal argument is used to calculate the signal response in Eq. (40).

## 2 Formulation of the Scattering Problem

*x*,

*y*,

*z*and cylindrical coordinates \(r,\varphi ,z\) are used. The open internal crack is infinitely long in the axial direction and is placed at \(\varphi = 0\) between \(r = a\) and \(r=b\), and the crack surface is denoted \(S_c\), see Fig. 1. The material of the pipe is assumed to be isotropic and homogeneous with Lamé parameters \(\lambda \) and \(\mu \) and density \(\rho \). The equation of motion for the displacement field, \(\mathbf {u}\), in an elastic solid is (Achenbach [1]),

*m*, this set of basis functions represents outwards traveling waves. There are also regular waves (\({\mathrm{Re}}{\mathbf {\chi }}_{\tau \sigma m}\)), which are obtained by replacing the Hankel functions with Bessel functions, \({\text {J}}_{m}\). The Neumann factor is defined as \(\epsilon _m = 2 - \delta _{m0}\),

*h*is the axial wave number and the radial wave numbers are defined as \(q_i = (k_i^2 - h^2)^{1/2}, i = p,s\). The radial wave numbers are chosen to have a non-negative imaginary part. The basis functions have several indices that represent the mode (\(\tau =1,2,3\)), parity (\(\sigma = o,e\)) and order (\(m=0,1,2\ldots \)), which are combined to a multi-index, \(\mathbf {\chi }_k \equiv {\mathbf {\chi }}_{\tau \sigma m}\). The first two modes are shear waves (\(\tau =1,2\)) and the third is a pressure wave (\(\tau = 3\)), the parity is determined by the parity of the trigonometric function. These basis functions can be divided into two groups \(\tau \sigma = 1o2e3e\) (symmetric in \(\varphi \)) and \(\tau \sigma = 1e2o3o\) (anti-symmetric in \(\varphi \)). The basis functions will be used to construct the Green’s tensor for the cylindrical pipe and to expand the incoming field produced by the probe.

## 3 Green’s Tensors

*x*-coordinate is aligned with the crack as seen in Fig. 1. The quantities \(f_{jn}\), \(f_{jn}^*\) are defined as

*x*and

*z*of the Green’s tensor.

## 4 The Integral Equation

*z*, in the following way

*g*(

*r*) takes care of the linear transformation from the interval \([-1,1]\) to the crack interval [

*a*,

*b*].

*z*is also taken to yield

## 5 Transmitting and receiving probes

The model of the transmitting probe is similar to the one used by Boström and Wirdelius [5] to model a probe on a planar surface. The action of the probe is thus modelled as an applied traction as the boundary condition where the probe is located. The field from the transmitting probe is obtained by determining the displacement field induced by this given set of boundary conditions, in the absence of a defect.

*m*is the Fourier series number, and

*h*is the Fourier transform variable. Twelve equations are thus obtained to solve for the twelve expansion coefficients \(\xi _k^1\) and \(\xi _k^2\).

*P*is the electric power exciting the elastodynamic field. Collapsing the surface to the crack and using the traction free boundary condition on the crack this is further simplified to

## 6 Numerical Examples

*q*integral in the first term in (30). Due to branch cuts the integral is extended into the complex plane, and the curve which is integrated along is chosen as

*r*and \(r'\) in (30) are evaluated using a Gauss-Legendre scheme with 50 points. The sums over

*m*that appear in some places are truncated at \(m_{\text {max}} = r_o\text {Re}(k_s) + 5\).

The material of the pipe is kept fixed and is chosen as steel with the following material properties: \({\mu }^{\star }={{79.3}\mathrm{GPa}}\), \({\lambda }^{\star }={86.6}\mathrm{GPa}\) and \({\rho } = {7800}\mathrm{kgm}^{-3}\). As this is more realistic and as it increases the convergence damping is assumed to be present in the form \({\mu } ={{\mu }^{{\star }}} {\left( 1+{\mathrm{i}}{\epsilon }\right) }\) and \({\lambda } ={\lambda }^{\star }(1+{\mathrm{i}} {\epsilon })\), where \({\epsilon }=0.01\) is chosen. The probe is scanning in pulse-echo along a quarter circle on the outer surface of the cylinder at 700 positions. The probe is of shear wave type approximately \(t{10}\mathrm{mm}\) by \({10}\mathrm{mm}\) and is operating at the fixed frequency \({1}\mathrm{MHz}\). The angle of the probe, \({\gamma }\), is set to 30 \({{^{\circ }}}\), 45\({{^{\circ }}}\), or 60\({{^{\circ }}}\). The signal response is normalized with the largest value for each probe with the same angle, although strictly speaking the probes are not completely identical for different radii of the outer pipe wall. The crack width in the radial direction is chosen as \({5}\mathrm{mm}\) and it is placed \({1}\mathrm{mm}\) from the inner wall, so it is practically surface-breaking. To investigate the influence of the curvature of the wall the wall thickness is kept constant and the inner and outer radii are varied. This has been performed for two different wall thicknesses, \({10}\mathrm{mm}\) and \({20}\mathrm{mm}\), and several different inner and outer radii.

Minimum radii for the nominal probe ray (\(r_{min}\) in Fig. 2) for different probe angles and outer radii, all units are in millimeters

\(r_o\) | \(r_\text {min}\) for \(\gamma \) | \(30 {^{\circ }}\) | \(45 {^{\circ }}\) | \(60 {^{\circ }}\) |
---|---|---|---|---|

40 | 20.0 | 28.3 | 34.6 | |

50 | 25.0 | 35.3 | 43.3 | |

60 | 30.0 | 42.4 | 52.0 |

Figures 6, 7, and 8 show the same type of curves with the same outer radii but with the doubled wall thickness 20 mm. In these cases the crack is situated further from the scanning surface and the values therefore generally decrease (the normalization is still performed with largest value for a particular probe and these appear in Figs. 3, 4, 5). This decrease is largest in Fig. 8 where the probe angle is \({60}{{^{\circ }}}\). Particularly for the smallest outer radius 40 mm the drop in peak value is about 30 dB as compared with the corresponding curve in Fig. 5 for the smaller wall thickness. The reason for this is that the ray from the probe completely misses the crack in this case as it only goes down to the radius 34.6 mm whereas the crack only extends out to 26 mm.

The present results have not been validated with independent results with some other method. One possibility is to perform an experimental investigation and this is of course a very good way. Another way is to perform numerical work by some different method, the method that first comes to mind is the finite element method (FEM), but in the present fully 3D case this will be quite demanding, although it is feasible at least for lower frequencies.

## 7 Conclusions

This paper gives a model to determine the signal response from an infinite axial crack in a thick-walled pipe. A hypersingular integral equation for the COD is used and it is crucial that the COD is expanded in a system that has the right singularity along the crack edges. A model of an ultrasonic probe on a curved surface is used together with reciprocity to obtain the signal response. As compared to a plate the signals are generally more difficult to interpret, and it is also noted that there is only a finite penetration depth for angled probes in a pipe.

This work shows that there are some interesting effects when investigating the scattering of a crack in an pipe. Thus there is an interest in extending the present work to other crack types in a pipe, as a rectangular (as the present paper but finite in the axial direction), but still axial, crack, or a radial crack.

## Notes

### Acknowledgements

This work is supported by the Swedish Radiation Safety Authority (SSM) and this is gratefully acknowledged. The authors would like to thank Associated Professor Per-\(\AA \) ke Jansson for the help provided.

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