1 Introduction

Ultrasonic guided waves have already been demonstrated by various research works to have great potential and a good prospect in online nondestructive evaluation (NDE) and long-term structural health monitoring (SHM) of pipelines of various industries (Rose, 2014) due to its various corresponding advantages, such as single-point excitation, long detection range, high inspection efficiency, and 100% cross-sectional detectability. The three main types of transducers that are currently in service for guided wave inspection of pipelines are piezoelectric transducers (Alleyne and Cawley, 1996; Marty, 2002), magnetostrictive transducers (Kwun and Bartels, 1998; Kim Y.Y. et al., 2005; Cho et al., 2006; Turcu, 2008; Lee et al., 2009; Kim Y.G. et al., 2011), and electromagnetic acoustic transducers (Ribichini, 2011). Magnetostrictive transducers are more cost-effective, involve relatively simpler fabrication process, and have higher transduction efficiency than other types of transducers. However, it is highly important to develop a theoretical approach to consider the forced response and perturbation analysis of elastic pipes under magnetostrictive loadings, and the excitation of axisymmetric and non-axisymmetric guided waves in elastic hollow cylinders by finite size magnetostrictive transducers, with the unique feature of magnetostriction taken into consideration. These topics are rarely reported in the literature, but further investigation can provide guidance for the further development of magnetostrictive guided wave transducers.

Problems involving the forced response of a structure are typically solved using one of the following two approaches (Rose, 2014): integral transform technique or normal mode expansion (NME) method. The integral transform technique solves the guided wave excitations through transforms, such as Laplace, Fourier, Hankel, and Mellin. The NME method, on the other hand, analyzes the wave fields in the cylinder in the form of a sum of an infinite number of normal modes. Gazis (1959) showed that there exist an infinite number of normal modes in an elastic hollow cylinder, each with its own characteristics, such as phase velocity, group velocity, and wave structure profile. He obtained the general solution of harmonic waves propagating in an infinite long hollow cylinder, which has been very beneficial for long-range guided wave inspection on widely distributed pipelines. The forced response problem in a hollow cylinder problem was first studied by Ditri and Rose (1992) using the NME method to obtain the amplitude factors of different guided wave modes. Li and Rose (2001a) investigated the field distribution of non-axisymmetric longitudinal waves. The angular profile was calculated by taking into account the amplitude factors of every excited mode. The NME method was also used by Luo (2005), Zhang (2005), Mu (2008), and Mu and Rose (2008) in their studies. Several results have been reported on the source influence analysis of hollow cylinders with respect to piezoelectric loadings. However, a comprehensive perturbation analysis on pipes with respect to magnetostrictive loadings, considering unique features of magnetostriction, is essential for further development of guided wave magnetostrictive transducers and transducer arrays.

In this studies, the classical NME method is adopted to study the excitation of axisymmetric and non-axisymmetric guided wave modes in linear elastic hollow cylinders by magnetostrictive actuation. The phase velocity and frequency spectra of axisymmetric and non-axisymmetric guided wave modes excited by magnetostrictive transducers are modeled with a large static bias magnetic field assumption. The influence of load parameters is analyzed. Several novel results have been found about the source influence of different magnetostrictive loadings. The prediction of guided wave excitation by the NME method is verified by finite element numerical simulations and experiments.

2 NME method

2.1 Normal modes of a hollow cylinder

The NME method solving the forced loading problem is analogous to the eigenfunction expansion method in the field of mathematics, and the normal modes of cylinders serve as the eigenfunctions. The main idea of the NME method is to assume that the sought function can be written in the form of a series of known functions, i.e., the normal modes, each with an unknown amplitude. The goal is then to find a general expression for the unknown amplitudes or a numerical estimate of them. Note that the efficiency of NME method depends on two main considerations: completeness and orthogonality of the normal modes (Rose, 2014).

Consider a hollow cylinder in a cylindrical coordinate system as shown in Fig. 1. There exists an infinite number of propagating modes in the hollow cylinder. The velocity field due to a normal mode with circumferential order N in the mth family can be written as (Ditri and Rose, 1992; Li and Rose, 2001a):

$$\begin{array}{*{20}c}{v_m^N(r,\theta){{\rm{e}}^{{\rm{i}}(\omega t - k_m^Nz)}} = \left\{ {R_{mr}^N(r)\phi _r^N(N\theta){e_r}} \right.\quad \quad \quad \quad } \\ {\quad \quad \quad + R_{m\theta }^N(r)\phi _\theta ^N(N\theta){e_\theta }} \\ {\left. {\quad \quad \quad \;\;\quad \quad \quad + R_{mz}^N(r)\phi _z^N(N\theta){e_z}} \right\}{{\rm{e}}^{{\rm{i}}(\omega t - k_m^Nz)}},} \\ \end{array}$$
(1)

where ω and k are the angular frequency and wave number, respectively. Functions R(r) and ϕ(Nθ) denote the radical and angular field distributions of the velocity component of the normal mode with circumferential order N in the mth family, respectively. The phase velocity dispersion curves of an elastic pipe with the parameters given in Table 1 solved using the semi-analytical finite element method (Marzani, 2008; Zhang et al., 2014) for the axisymmetric and non-axisymmetric modes are shown in Fig. 2. Note that the curves in different colors denote modes with different circumferential orders.

Fig. 1
figure 1

Stress-free hollow cylinder in a cylindrical coordi-nate system

Ri and Ro are the inner and outer radii, and ∂1D and ∂2D are the inner and outer boundaries, respectively

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Fig. 2
figure 2

Phase velocity dispersion curves for L(0, m), T(0, m), and F(N, m) modes in a steel pipe. Note: for interpre-tation of the references to color in this figure, the reader is referred to the web version of this article

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Table 1 Dimensions and material properties of a steel pipe
Full size table

According to the NME method, the generated particle velocity can then be expanded as

$$v{{\rm{e}}^{{\rm{i}}\omega t}} = \sum\limits_{N,{\rm{m}}} {A_m^Nv_m^N} {{\rm{e}}^{{\rm{i}}\omega t}},$$
(2)

where \(A_m^N\) is the NME amplitude of the mode with circumferential order N in the mth family containing wave component \({{\rm{e}}^{ - {\rm{i}}k_m^Nz}}\).

2.2 Orthogonality of normal modes

Before calculating the amplitude factor \(A_m^N\) by using the NME method, the orthogonality relation between the normal modes should be established. We start with the complex reciprocity relation (Auld, 1990), which is described as

$$\nabla \cdot (v_2^\ast \cdot {T_1} + {v_1} \cdot T_2^\ast) = 0,$$
(3)

where v and T represent the particle velocity and stress field of a solution to the linear elastic wave propagation equation, respectively. Let v1 and T1, v2 and T2 denote different normal modes of the cylinder:

$$\left\{ {\begin{array}{*{20}c}{{v_1} = v_m^N{{\rm{e}}^{ - {\rm{i}}k_m^Nz}},\;} & {{T_1} = T_m^N{{\rm{e}}^{ - {\rm{i}}k_m^Nz}},} \\ {{v_2} = v_n^M{{\rm{e}}^{ - {\rm{i}}k_n^Mz}},} & {{T_2} = T_n^M{{\rm{e}}^{ - {\rm{i}}k_n^Mz}}{.}} \\ \end{array} } \right.$$
(4)

After substituting Eq. (4) into Eq. (3), we obtain

$$ - 4{\rm{i}}(k_m^N - k_n^{M\ast })P_{nm}^{MN} = \int {\int\nolimits_D {{\nabla _{r\theta }} \cdot (v_n^{M\ast } \cdot T_m^N + v_m^N \cdot T_n^{M\ast })} } {\rm{d}}\sigma,$$
(5)

where D is the cross section and \(P_{nm}^{MN}\) is defined as

$$P_{nm}^{MN} = - {1 \over 4}\int {\int\nolimits_D {\left({v_n^{M\ast } \cdot T_m^N + v_m^N \cdot T_n^{M\ast }} \right)} } \cdot {e_z}{\rm{d}}\sigma{.}$$
(6)

For two different waveguide modes, the orthogonality relation can then be written as

$$P_{nm}^{MN} = 0,\quad {\rm{unless}}\;\;M = N\;\;{\rm{and}}\;\;k_n^M = k_m^N{.}$$
(7)

2.3 NME amplitude

In the complex reciprocity relation equation (3), let v1 represents the actual particle velocity field in the cylinder and v2 denotes the particle velocity field of a normal mode, which is given as

$$\left\{ {\begin{array}{*{20}c}{{v_1} = v = \sum\limits_{N = 0}^\infty {\sum\limits_m {A_m^N(z)v_m^N(r,\;\theta)},} } \\ {{v_2} = v_n^M(r,\;\theta){{\rm{e}}^{ - {\rm{i}}k_n^Mz}}{.}\quad \quad \quad \quad } \\ \end{array} } \right.$$
(8)

Using Eqs. (3), (7), and (8), after some intermediary transformations, one obtains

$$\begin{array}{*{20}c}{4P_{mm}^{NN}\left({{{\rm{d}} \over {{\rm{d}}z}} + {\rm{i}}k_m^N} \right)A_m^N(z)\quad \quad \quad \quad \quad \quad \quad } \\ { = \oint_{{\partial _1}D} {v_m^{N\ast } \cdot } (T \cdot {n_1}){\rm{d}}s + \oint_{{\partial _2}D} {v_m^{N\ast } \cdot (T \cdot {n_2})} {\rm{d}}s,} \\ \end{array}$$
(9)

where T·n1 and T·n2 are the loading conditions on the inner and outer boundaries, ∂1D and ∂2D, respectively, of the cylinder. Considering the traction-free condition on the inner and outer boundaries of the hollow cylinder (Fig. 1), Eq. (9) can be integrated with the result, that is, the general form of the forward normal mode amplitude:

$$\begin{array}{*{20}c}{A_{ + m}^N(z) = {{{{\rm{e}}^{ - {\rm{i}}k_m^Nz}}} \over {4P_{mm}^{NN}}}\int\nolimits_c^z {{{\rm{e}}^{{\rm{i}}k_m^N\xi }}} \quad \quad \quad \quad \quad } \\ { \cdot \left[ {\oint_{{\partial _1}D} {v_m^{N\ast } \cdot (T \cdot {n_1})} {\rm{d}}s} \right.} \\ {\quad \;\left. { + \oint_{{\partial _2}D} {v_m^{N\ast } \cdot } (T \cdot {n_2}){\rm{d}}s} \right]{\rm{d}}\xi {.}} \\ \end{array}$$
(10)

Only source condition on the outer boundary will be considered in this study.

3 Modeling of magnetostrictive transducers

Typical magnetostrictive transducers are modeled in this study (Hirao and Ogi, 2003; Ribichini et al., 2010; 2011; 2012). Ferromagnetic materials have the property that when placed in a magnetic field, they are mechanically deformed. This property is called the magnetostrictive effect. The reverse phenomenon in which the magnetic induction of the material changes when it is mechanically deformed is called inverse magnetostrictive effect. The magnetostrictive phenomenon and its inverse effect can be utilized for the generation and detection of ultrasonic guided waves. Magnetostrictive guided wave transducers, including longitudinal wave transducer based on Joule effect and its inverse effect and torsional wave transducer based on Wiedemann effect and its inverse effect, can produce axial and circumferential loading, respectively.

The magnetostrictive effect relating the magnetic and mechanical material states can be assumed to take a similar form as that for the piezoelectricity (Hirao and Ogi, 2003; Ribichini et al., 2010):

$$\left\{ {\begin{array}{*{20}c}{S = sT + dH,\;\;} \\ {B = {d^{\rm{T}}}T + \mu H,} \\ \end{array} } \right.$$
(11)

where S and T are strain and stress, while H and B are magnetic field and magnetic induction, respectively, s is the elastic compliance matrix, μ is the magnetic permeability matrix, and d=∂S/∂H is the piezomagnetic coupling matrix.

Eq. (11) assumes a linear S–H relation, but it is experimentally known that ferromagnetic materials exhibit a highly nonlinear magnetostriction curve. In the case of magnetostrictive guided wave transducers, a small dynamic magnetic field \(\tilde H\) is superimposed on a much larger static bias magnetic field \(\bar H\); therefore, the linear assumption is locally valid. When the small dynamic magnetic field \(\tilde H\) is superimposed on a larger static bias magnetic field \(\bar H\), the resulting strain can be decomposed into a static component \(\bar S\) and a dynamic component \(\tilde S\) (Ribichini et al., 2010):

$$\left\{ {\begin{array}{*{20}c}{H = \bar H + \tilde H,} \\ {S = \bar S + \tilde S{.}\;\;} \\ \end{array} } \right.$$
(12)

Here only the dynamic components of the constitutive equations are considered:

$$\left\{ {\begin{array}{*{20}c}{\tilde S = s\tilde T + d\tilde H,\;\;} \\ {B = {d^{\rm{T}}}\tilde T + \mu \tilde H,} \\ \end{array} } \right.$$
(13)

where the dynamic magnetostriction matrix d depends on the direction of the magnetization. Note that the linearized equations are valid in the large static bias magnetic field assumption, even when static component of linear constitutive equations is invalid. The first equation of Eq. (13) accounts for the direct magnetostrictive effect used in the generation of elastic waves, while the second equation describes the inverse magnetostriction used in the detection process.

Both the static bias magnetic field H0z and small dynamic magnetic field H z of longitudinal wave transducers are applied along the z direction, as is shown in Fig. 3. When H z is much smaller than H0z, the coupling matrix can be approximated as

$$d = {\left[ {\begin{array}{*{20}c}0 & 0 & 0 & 0 & {{{3{S_{\rm{t}}}} \over {{H_{0z}}}}} & 0 \\ 0 & 0 & 0 & {{{3{S_{\rm{t}}}} \over {{H_{0z}}}}} & 0 & 0 \\ { - {1 \over 2}\gamma } & { - {1 \over 2}\gamma } & \gamma & 0 & 0 & 0 \\ \end{array} } \right]^{\rm{T}}},$$
(14)

where γ=∂S3/∂H z is the slope of magnetostriction curve and St is the total magnetostrictive strain. Note that the S3 strain component is solely responsible for the generation of longitudinal waves, so the constitutive equations for longitudinal wave transducers can be simplified and given as

$$\left\{ {\begin{array}{*{20}c}{{{\tilde S}_3} = {s_{33}}{{\tilde T}_3} + {d_{33}}{{\tilde H}_z},} \\ {{B_z} = {d_{33}}{{\tilde \sigma }_3} + {\mu _3}{{\tilde H}_z}{.}} \\ \end{array} } \right.$$
(15)
Fig. 3
figure 3

Schematic of longitudinal wave transducers on the outer surface of a pipe

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For torsional wave transducers, the static bias magnetic field is applied along the θ direction, while the small dynamic magnetic field is applied along the z direction, as is shown in Fig. 4. Under the large static bias magnetic field assumption, the coupling matrix for torsional wave transducers can also be approximated as

$$d = {\left[ {\begin{array}{*{20}c}0 & 0 & 0 & 0 & 0 & {{{3{S_{\rm{t}}}} \over {{H_{0\theta }}}}} \\ { - {1 \over 2}\gamma } & \gamma & { - {1 \over 2}\gamma } & 0 & 0 & 0 \\ 0 & 0 & 0 & {{{3{S_{\rm{t}}}} \over {{H_{0\theta }}}}} & 0 & 0 \\ \end{array} } \right]^{\rm{T}}}{.}$$
(16)
Fig. 4
figure 4

Schematic of torsional wave transducers on the outer surface of a pipe

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Since only the S4 shear strain component is responsible for generation of torsional waves, the constitutive equations for torsional wave transducers can also be simplified as

$$\left\{ {\begin{array}{*{20}c}{{{\tilde S}_4} = {s_{44}}{{\tilde T}_4} + {d_{43}}{{\tilde H}_z},} \\ {{B_z} = {d_{43}}{{\tilde T}_3} + {\mu _3}{{\tilde H}_z}{.}} \\ \end{array} } \right.$$
(17)

Note that thin magnetostrictive patches such as nickel and ferrocobalt strips are often used to generate and measure elastic waves. The magnetostrictive longitudinal and torsional wave transducers described above can be modeled using commercial finite element software packages, such as COMSOL Multiphysics.

4 Perturbation analysis of cylinders under magnetostrictive loadings

As described in Section 3, magnetostrictive transducers are capable of providing circumferential and axial loadings (T=s−1(S−dH)) on the surface of a hollow cylinder. Therefore, only circumferential and axial magnetostrictive loadings on the outer boundary of the cylinder are considered in this study, which can be written as

$$T \cdot {n_1} = 0,$$
(18)
$$T \cdot {n_2} = \left\{ {\begin{array}{*{20}c}{ - {p_1}(\theta){p_2}(z){e_{\theta/z}},} & {\left\vert z \right\vert \leq L,\;\left\vert \theta \right\vert \leq \alpha, \;r = {R_{\rm{o}}},} \\ {0, \quad \quad \quad \quad \quad \;\;} & {\left\vert z \right\vert > L,\;\left\vert \theta \right\vert > \alpha, \;r = {R_{\rm{o}}},} \\ \end{array} } \right.$$
(19)

where p1(θ) and p2(z) are circumferential and axial loading functions, and L and α are axial and circumferential extent of the loading area, respectively.

Substituting Eqs. (18) and (19) into Eq. (10) gives

$$\begin{array}{*{20}c}{A_{ + m}^N(z) = {{R_{m\gamma }^N(b){{\rm{e}}^{ - {\rm{i}}k_m^Nz}}} \over {4P_{mm}^{NN}}}\left\langle {\Theta _\gamma ^N,\;{p_1}(\theta)} \right\rangle \cdot \left\langle {{p_2}(z),\;{{\rm{e}}^{{\rm{i}}k_m^Nz}}} \right\rangle,} \\ {z \geq L \;(\gamma = \theta, \;z),} \\ \end{array}$$
(20)

where \(\left\langle {\Theta _\gamma ^N,{p_1}(\theta)} \right\rangle\) and \(\left\langle {{p_2}(z),{{\rm{e}}^{{\rm{i}}k_m^Nz}}} \right\rangle\) are the circumferential and axial amplitude factors, respectively, which are defined as

$$\left\langle {\Theta _\gamma ^N,{p_1}(\theta)} \right\rangle = {H_\theta } = b\int\nolimits_\beta ^{\beta + 2\pi } {\Theta _\gamma ^N(N\theta)} {p_1}(\theta){\rm{d}}\theta,$$
(21)
$$\left\langle {{p_2}(z),\;{{\rm{e}}^{{\rm{i}}k_m^Nz}}} \right\rangle = {H_z} = \int\nolimits_{ - \infty }^\infty {{p_2}(z)} {{\rm{e}}^{{\rm{i}}k_m^Nz}}{\rm{d}}z{.}$$
(22)

4.1 Magnetostrictive partial loading

A magnetostrictive partial loading is non-axisymmetric and only covers a portion of the pipe surface over some circumferential angle, as is shown in Fig. 5. The loading distribution functions p1(θ) and p2(z) are given as (Shin and Rose, 1999)

$${p_1}(\theta) = \left\{ {\begin{array}{*{20}c}{{P_1}, } & {\left\vert \theta \right\vert \leq \alpha,} \\ {0, } & {\left\vert \theta \right\vert > \alpha,} \\ \end{array} } \right.$$
(23)
$${p_2}(z) = \left\{ {\begin{array}{*{20}c}{{P_2},} & {\left\vert z \right\vert \leq L,} \\ {0,} & {\left\vert z \right\vert > L{.} } \\ \end{array} } \right.$$
(24)
Fig. 5
figure 5

A hollow cylinder with partial loading of a trans-ducer with 2 L width and 2 α circumferential coverage angle

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Substituting Eqs. (23) and (24) into Eqs. (21) and (22), the source functions can be obtained:

$${H_\theta } = \left\{ {\begin{array}{*{20}c}{b{P_1}(2\alpha),\quad \quad } & {N = 0,} \\ {2b{P_1}{{\sin (N\alpha)} \over N},} & {N \geq 1,} \\ \end{array} } \right.$$
(25)
$${H_z} = {{2{P_2}\sin (k_m^NL)} \over {k_m^N}}{.}$$
(26)

According to Eq. (25), the circumferential amplitude factor H θ is a function of circumferential order N and loading angle α, but is independent of phase velocity and frequency. There is a linear relation between the axisymmetric circumferential amplitude factor and loading angle α, while the non-axisymmetric circumferential amplitude factor follows a sinusoidal trend as a function of the loading angle α for a certain circumferential order N. Also, H θ =0 (N≥1) when 2α=2π, which indicates that only axisymmetric modes will be excited if the magnetostrictive load is axisymmetric. If Nα=pπ, p=0, 1, 2, …, then H θ =0 (N≥1), which implies that normal modes with circumferential order N will not be excited. The relative circumferential amplitude factors for different loading angles of 45°, 90°, 180°, and 360° are shown in Fig. 6. It is evident that the amplitude of non-axisymmetric flexural modes decreases, while the loading angles of magnetostrictive partial loading increase. The flexural modes are completely suppressed when the loading angle approaches 360°.

Fig. 6
figure 6

Relative circumferential amplitude factors for axisymmetric and non-axisymmetric modes at loading angles of 2 α =45° (a), 90° (b), 180° (c), and 360° (d)

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According to Eq. (26), the axial amplitude factor H z is a function of loading length 2L and wave-number \(k_m^N\) of mode with circumferential order N in the mth family. Furthermore, H z will reach the maximum when \(k_m^NL = (2p + 1)\pi/2,\), p=0, 1, 2, …; thus, \(2L = (2p + 1)\lambda _m^N/2\), p=0, 1, 2, …, are efficient loading lengths for a mode at a particular frequency. The axial amplitude factor can be rewritten as a function of frequency f, phase velocity \(Cp_m^N\) and loading length 2L such that

$${H_z} = {{2{P_2}\sin \left[ {\left({2\pi f/(Cp_m^N)} \right)L} \right]} \over {2\pi f/(Cp_m^N)}}{.}$$
(27)

The phase velocity spectra (f=32 kHz) of H z with different loading lengths 2L are shown in Fig. 7. It can be observed that H z has a very broad phase velocity bandwidth, which means it has poor phase velocity selectivity. The frequency spectra of H z for axisymmetric modes T(0, 1) and L(0, 2) are shown in Fig. 8, which indicates that the axial amplitude factor has different frequency sensitivities for different loading lengths with the extreme point corresponding to the condition \(2L = (2p + 1)\lambda _m^N/2\), p=0, 1, 2, …

Fig. 7
figure 7

Phase velocity spectra of the axial amplitude fac-tor at different parameters: (a) 2 L =12.7 mm; (b) 2 L = 25.4 mm; (c) 2 L =50.8 mm; (d) 2 L =101.6 mm

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Fig. 8
figure 8

Frequency spectra of the axial amplitude factor at different parameters: (a) T (0, 1), 2 L =25.4 mm; (b) T (0, 1), 2 L =50.8 mm; (c) L (0, 2), 2 L =25.4 mm; (d) L (0, 2), 2 L =50.8 mm

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Substituting Eqs. (25) and (26) into Eq. (20), the amplitude of normal mode with circumferential order N in the mth family of an elastic hollow cylinder under magnetostrictive partial loading is obtained. A sample calculation of amplitude factor is carried out and shown in Fig. 9 for T(0, 1) and F(N, 2) modes generated by a 45° circumferential loading at 128 kHz in an elastic pipe with the parameters given in Table 1.

Fig. 9
figure 9

Amplitude factors for T (0, 1) and F (N, 2) modes generated by a 45° magnetostrictive circumferential load-ing at 128 kHz in an elastic pipe

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4.2 Magnetostrictive axial array loading

Consider the case of axial array loading in a comb transducer (Ditri et al., 1993; Rose et al., 1998; Hay and Rose, 2002; Philtron and Rose, 2014), as is shown in Fig. 10, where η magnetostrictive elements are equally spaced along the axial direction of a cylinder. Here, 2β is the element width and 2δ is the spacing. The loading distribution functions p1(θ) and p2(z) (without time delay) can be written as

$${p_{1}}(\theta) = {P_1}, \quad \theta \in [0,\;2\pi),$$
(28)
$${p_2}(z) = \left\{ {\begin{array}{*{20}c}{{P_2}, } & {z \in G,} \\ {0, } & {z \notin G,} \\ \end{array} } \right.$$
(29)

where

$$\begin{array}{*{20}c}{G = \bigcup\limits_{\xi = 1}^{\eta/2} {{g_\xi },} } \\ {{g_\xi } \equiv \left\{ {z:(2\xi - 1)\delta + 2(\xi - 1)\beta \leq \left\vert z \right\vert \leq (2\xi - 1)\delta + 2\xi \beta } \right\}{.}} \\ \end{array}$$
Fig. 10
figure 10

Schematic of elastic hollow cylinders under mag-netostrictive axial array loading

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Substituting Eqs. (21) and (22) into Eqs. (14) and (15), the source functions are obtained as:

$${H_\theta } = \left\{ {\begin{array}{*{20}c}{2{{\pi }}b{P_1}, } & {N = 0,} \\ {0, \quad \;\;} & {N \geq 1,\quad } \\ \end{array} } \right.$$
(30)
$$\begin{array}{*{20}c}{{H_z} = \sum\limits_{\xi = 1}^{\eta/2} {\int\nolimits_{{g_\xi }} {{P_2}{{\rm{e}}^{{\rm{i}}k_m^Nz}}{\rm{d}}z} } = {P_2}{{\sin (k_m^N\beta)} \over \beta }\quad \quad \quad \quad \quad \quad } \\ { \times {{\cos \left({k_m^N[(\eta - 1)D]} \right) - \cos \left({k_m^N[(\eta + 1)D]} \right)} \over {1 - \cos (2k_m^ND)}},} \\ \end{array}$$
(31)

where the element distance 2D= 2β+2δ.

According to Eq. (31), the axial amplitude factor depends on the element width 2β(Hβ), the element distance 2D, and the number of elements η(H D ). The extremum conditions of H β and H D are \(\tan (k_m^N\beta) = k_m^N\beta\) and \(\eta \cdot \tan (k_m^ND) = \tan (\eta \cdot k_m^ND)\), respectively, where \(k_m^N = p\pi/D\), p=0, 1, 2, …, and \(k_m^N = 2D/p\), p=0, 1, 2, ….

The phase velocity spectra (f=64 kHz) of axial amplitude factor at different values of 2β, 2D, and η are shown in Fig. 11. It is obvious that magnetostrictive axial array loading has good phase velocity selectivity. Furthermore, it can be seen from Figs. 11a and 11b that the center phase velocity is mainly dependent on element distance 2D; a larger element distance corresponds to a higher phase velocity. From Figs. 11a and 11c, a conclusion can be drawn that the phase velocity bandwidth is mainly dependent on the number of elements η. Moreover, it can be seen from Figs. 11b and 11d that the element width 2β makes a contribution to the amplitude of H z at the center phase velocity.

Fig. 11
figure 11

Phase velocity spectra of axial amplitude factor for axial array loading at different parameters: (a) η=4, 2ß=25.4 mm; (b) η=4, 2D=60 mm, 2ß= 25.4 mm; (c) η=8, 2D=40 mm, 2ß=25.4 mm; (d) η=4, 2D= 60 mm, 2ß=12.7 mm

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The frequency spectra of H z for axisymmetric modes T(0, 1) and L(0, 2) are shown in Fig. 12. The influence of array parameters 2β, 2D, and η on the frequency spectrum is similar to that on the phase velocity spectrum. Furthermore, if time delay e−iω(ξ·td) is applied to each element, the comb-type magnetostrictive transducer will provide even better phase velocity, frequency selectivity, and guided wave mode controllability.

Fig. 12
figure 12

Frequency spectra of axial amplitude factor for axial array loading at different parameters: (a) T (0, 1), 2 D =25.4 mm; (b) T (0, 1), 2 D =50.8 mm; (c) L (0, 2), 2 D = 25.4 mm; (d) L (0, 2), 2 D =50.8 mm

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4.3 Magnetostrictive circular array loading

Consider the circular array loading case (Li and Rose, 2001b; 2002), as is shown in Fig. 13, where η equally sized (2L×2α) magnetostrictive elements are equally spaced along the circumferential direction of a cylinder with their circumferential position at θ= (2ξ−1)π/η, ξ=1, 2, …, η. The loading distribution functions p1(θ) and p2(z) can be written as

$${p_1}(\theta) = \left\{ {\begin{array}{*{20}c}{{P_1},} & {\theta \in G,} \\ {0,} & {\theta \notin G,} \\ \end{array} } \right.$$
(32)

where

$$\begin{array}{*{20}c}{G = \bigcup\limits_{\xi = 1}^\eta {{g_\xi },} } \\ {{g_\xi } \equiv \left\{ {\theta:{{(2\xi - 1)\pi } \over \eta } - \alpha < \theta < {{(2\xi - 1)\pi } \over \eta } + \alpha } \right\},} \\ {{p_2}(z) = \left\{ {\begin{array}{*{20}c}{{P_2},} & {\left\vert z \right\vert \leq L,\quad } \\ {0,} & {{\rm{otherwise}}{.}} \\ \end{array} } \right.} \\ \end{array}$$
(33)
Fig. 13
figure 13

Schematic of elastic hollow cylinders under mag-netostrictive circular array loading

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Substituting Eqs. (32) and (33) into Eqs. (21) and (22), the source functions are obtained as follows:

$$\begin{array}{*{20}c}{{H_\theta } = \left\{ {\begin{array}{*{20}c}{b{P_1}\eta (2\alpha),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } & {N = 0,} \\ {2b{P_1}{{\sin (N\alpha)} \over N}\sum\limits_{\xi = 1}^\eta {\cos } \left({{{N(2\xi - 1)\pi } \over \eta }} \right),} & {N \geq 1,} \\ \end{array} } \right.} \\ {\;\;\;\; = \left\{ {\begin{array}{*{20}c}{2b{P_1}{{\sin (p\eta \alpha)} \over p}{{(- 1)}^p},\quad N = p \eta, \quad \quad \quad \;\quad \;\;} \\ {0,\quad \quad \quad \quad \quad N \neq p\eta, \;{\rm{where}}\;p = 0,\;1,\;2,\; \cdots,} \\ \end{array} } \right.} \\ \end{array}$$
(34)
$${H_z} = {{2{P_2}\sin (k_m^NL)} \over {k_m^N}}{.}$$
(35)

Eq. (35) is the same as Eq. (26), and thus, the conclusions are not repeated here. According to Eq. (34), only modes with circumferential order N=pη, p=0, 1, 2, … can be excited in a hollow cylinder with a magnetostrictive circular array loading. The relative circumferential amplitude factor with respect to that of axisymmetric modes can be defined as

$$\left\vert {{{H_\theta ^N} \over {H_\theta ^0}}} \right\vert = {{\sin (p\eta \alpha)} \over {p\eta \alpha }} = {{\sin (p\Delta \pi)} \over {p\Delta \pi }},$$
(36)

where Δ=η(2α)/2π, Δ∈ [0, 1] is the loading area ratio.

As can be seen from Fig. 14, the relative amplitude of higher harmonics decreases as the loading area ratio Δ increases and the relative amplitude decreases at a faster rate for higher circumferential orders, which indicates that magnetostrictive circular array loading can be designed to generate a specific circumferential order of the guided wave modes. In addition, beam steering (Wilcox, 2003) and beam focusing (Hayashi et al., 2005; Sun et al., 2005) can be achieved when time delays are applied to the array elements.

Fig. 14
figure 14

Relative amplitudes of higher harmonics for cir-cular array loading

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5 Numerical simulation and experiment

In this section, summarized finite element numerical evaluations (Zhu, 2001; Drozdz, 2008; Moreau et al., 2012) and experiments demonstrate the generation of axisymmetric and non-axisymmetric guided wave modes in elastic hollow cylinders under magnetostrictive loadings. A steel pipe, with parameters given in Table 1, is considered in this study with the corresponding phase velocity dispersion curves shown in Fig. 2.

5.1 Magnetostrictive partial loading

5.1.1 Axial partial loading simulation

The first numerical evaluation involves the axial partial loading at f=128 kHz, with a loading area of axial length 50.8 mm and a circumferential angle 45°. According to the NME method, the angular profile for L(0, 2) and F(N, 3) modes at 1.5 m from the source was calculated and is shown in Fig. 15. Meanwhile, a model with the same parameters is established and calculated by using the finite element software Abaqus/Explicit. The axial displacements (U z ) of nodes at z=1.5 m are recorded, and their axial displacement angular distribution is plotted, which is shown in Fig. 15. The results demonstrated that these two models are in good agreement.

Fig. 15
figure 15

Angular profiles of L (0, 2) and F ( N , 3) modes generated by 45° magnetostrictive axial loading at 128 kHz in an elastic pipe: (a) theoretical prediction; (b) numerical simulation result

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5.1.2 Circumferential partial loading experiment

The experiment setup is shown in Fig. 16, where a small magnetostrictive transducer with a width of 50.8 mm is bonded on the surface of the pipe as a transmitter that covers an angle of 45° along the circumference of the pipe. The central transmitting frequency is f= 250 kHz. Thirty-two receivers were placed around the circumference of the pipe at a distance of 1.6 m from the transmitter. The theoretical and experimental results of angular profiles for the T(0, 1) and F(N, 2) modes at the frequency of 250 kHz are shown in Fig. 17, which are in good agreement. Since both theoretical and experimental results are in good agreement, the earlier described method involving magnetostrictive partial loading was verified by numerical evaluation and experiment.

Fig. 16
figure 16

A setup of circumferential partial loading experiment

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Fig. 17
figure 17

Angular profiles of T (0, 1) and F ( N , 2) modes generated by 45° magnetostrictive circumferential load-ing at 250 kHz in an elastic pipe: (a) theoretical predic-tion; (b) experimental result

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5.2 Magnetostrictive axial array loading

Another numerical evaluation involved an axial array loading at f=128 kHz. Two sets of array parameters were designed for the purpose of L(0, 1) excitation and L(0, 2) excitation, respectively: (1) 2β=12 mm, 2δ=4 mm and (2) 2β=15 mm, 2δ=6 mm. Finite element models were established and calculated. Fig. 18 shows the snapshot for wave propagation of L(0, 1) mode and L(0, 2) mode. The L(0, 1) mode is dominant on the condition of parameter (1) and L(0, 2) mode is dominant on the condition of parameter (2), which demonstrates the mode controllability of magnetostrictive axial array load.

Fig. 18
figure 18

Snapshot of the wave propagation of L (0, 1) (a) and L (0, 2) (b) modes in the steel pipes

(a) Parameter (1): 2β=12 mm, 2δ=4 mm; (b) Parameter (2): 2β=15 mm, 2δ=6 mm

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The wave structures of L(0, 1) and L(0, 2) at f=128 kHz are shown in Fig. 19. It can be observed that radial displacement (U r ) plays a dominant role for L(0, 1) mode, whereas the axial displacement (U z ) for L(0, 2) mode at f=128 kHz. Both r-component and z-component of the displacement field on the external surface of the pipe at the distance of 0.7 m from the source are recorded and converted into video signals, which are shown in Fig. 20. The above observation is confirmed once again, but it also shows that the modes, especially L(0, 1), are not as pure as expected, as radial displacement is dominant for L(0, 1) while the external load is in the axial direction. Therefore, radial load, which the magnetostrictive transducer cannot provide, may have the best chance to excite pure L(0, 1) mode.

Fig. 19
figure 19

Wave structure of L (0, 1) (a) and L (0, 2) (b) modes at 128 kHz in the steel pipe

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Fig. 20
figure 20

Video signals extracted from the excitation of L (0, 1) and L (0, 2) modes

(a) Parameter (1): 2β=12 mm, 2δ=4 mm; (b) Parameter (2): 2β=15 mm, 2δ=6 mm

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5.3 Magnetostrictive circular array loading

5.3.1 Circular array axial load simulation

The last numerical evaluation involves circular array axial load at f=64 kHz, where the number of elements η=2 and the loading area ratio Δ=0.5. The finite element model is established and used for calculation. Displacements of all the nodes at the distance of 1.5 m from the source are recorded. The received signals form a 2D time-circumference data matrix. A Fourier transformation is performed on the data matrix in the circumferential direction, and the circumferential order of the excited modes is extracted. Fig. 21 shows that only modes with circumferential order N=2p, p=1, 2, … are excited, which is in accordance with the theoretical prediction.

Fig. 21
figure 21

Circumferential orders of excited modes for cir-cular array axial load numerical simulation

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5.3.2 Circular array circumferential load experiment

A two-element magnetostrictive transducer is bonded around the circumference of the pipe with the loading area ratio Δ=0.5 and the excitation frequency f=250 kHz. Thirty-two receivers are placed around the circumference of the pipe at a distance of 1.6 m from the transmitter. The received signals from 32 receivers form a 2D data matrix. A 2D Fourier transformation is performed on the data matrix, and the circumferential order of excited modes is extracted. Fig. 22 verifies the theoretical prediction once again.

Fig. 22
figure 22

Circumferential orders of excited modes for cir-cular array circumferential load experiment

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6 Conclusions

This paper has adopted and developed the classical NME method to analyze forced response and perturbation analysis of a hollow cylinder considering the finite size magnetostrictive loadings, including partial loading, axial array loading, and circular array loading, with the unique feature of magnetostriction taken into consideration. The phase velocity and frequency spectra of guided wave amplitude excited by magnetostrictive loading are analyzed in detail, and some basic conclusions have been reached.

Some novel results of a comprehensive perturbation analysis on elastic hollow cylinders under magnetostrictive loadings are summarized as follows. Magnetostrictive partial loading is able to excite guided wave modes of the same family, but has poor phase velocity and frequency selectivity. Axisymmetric loading excites axisymmetric modes only. Angular profiles in a hollow cylinder are sensitive to the circumferential loading length. Magnetostrictive axial array loading has good phase velocity and frequency selectivity and guided wave mode controllability. According to phase velocity and frequency spectra, center phase velocity and center frequency are mainly determined by element distance. Magnetostrictive circular array load only excites guided wave modes whose circumferential order is integer multiple of the number of elements.