Introduction

Non-destructive and structural health monitoring systems (SHM) have been widely used in various fields for the ability to increase reliability and safety [1,2,3,4,5,6,7]. Lamb wave-based non-destructive and SHM techniques attract increasing attention and have been investigated in many aspects [8,9,10,11,12,13]. The analytical prediction is a simple and effective method for the propagation of Lamb wave. Characteristics like phase velocity, group velocity and dispersion can be obtained through the theoretical method [14], which play important roles in the application of different situations [15,16,17,18] of SHM.

The propagation of Lamb wave in plate-like structures has been discussed since the end of the nineteenth century, Rayleigh [19] and Lamb [20] first established the frequency equations, i.e., the well-known Rayleigh-Lamb equation. The plane Lamb wave is considered with the plane strain assumption in their derivation. As a consequence, the wavefront is assumed as an infinite plane normal to the propagation direction. Such results of the Lamb wave have been extended to anisotropic and viscoelastic media in the following investigation [21,22,23].

The theory of wave propagation in media including arbitrary amounts of parallel plates with different materials and thicknesses was proposed by Thomson in 1950 [24]. The stress of each layer is derived from the coupling conditions at interfaces and free boundary at the top as well as the bottom surface. The matrix method is adopted to produce frequency equations in a form suitable for understanding and computing. Following Thomson’s work, some researchers investigated the Lamb wave in multilayered media [25,26,27].

The above research work is based on the assumption of the plane wavefront, in which the size of excitation loading must be infinite. The reason is that the vibration normal to the propagation direction (but not through the plate thickness) is neglected in the plane wavefront assumption [28]. For this reason, the plane wave model (with plane wavefront assumption) is not exact in practice. In fact, almost all Lamb waves that existed in plates are circular crested waves [29, 30] whose wavefront travel as a circular from the origin. Goodman derived the circular crested Lamb wave in isotropic plates excited by axisymmetric loading [31]. The axisymmetric case is considered which implies with the same propagation distance, the vibration is constant along different propagation directions. Similarities and discrepancies between circular crested Lamb wave and plane Lamb wave are revealed. The frequency equation and across thickness displacements are proved to be the same as that of plane Lamb wave in the same plates [32, 33].

Besides the axisymmetric case of circular crested Lamb wave, the non-axisymmetric solutions are also of great significance for most of the excitation in practice is not axisymmetric. In this work, the complete solutions of both axisymmetric and non-axisymmetric Lamb wave in isotropic plates are derived and presented with the cylindrical coordinate system. The solution of the circular crested Lamb wave based on the governing equation is demonstrated in detail. The Helmholtz conversion concerning displacements is adopted for clarity of presentation. The displacement expressions and frequency equations are obtained by satisfying free boundary conditions at the surfaces of the plate, and characteristics of circular crested Lamb wave displacements are discussed. Simulations as well as experiments are conducted on plates to verify the results predicted by theoretical formulations.

The Fundamental Problem

Consider an isotropic plate in Fig. 1. A cylindrical coordinate system (r, θ, z) is adopted to describe the field of wave motion. The plate is infinite in the r–θ plane and has a finite thickness 2h in the longitudinal (z) direction. The upper and lower surfaces of the plate are located at z = h and z = − h, respectively.

Fig. 1
figure 1

An isotropic plate in a cylindrical coordinate system

Full size image

The time-harmonic waves are considered by means of a time dependence factor e−iωt, which is omitted for brevity in the following derivation, where ω = 2πf denotes the angular frequency, and f denotes the frequency. Based on the continuity conditions g(θ0) = g(θ0 + 2π) for an arbitrary field variable g (displacement, strain, stress et al.) with an arbitrary θ0, the angle dependence factor can be described as einθ (n = 0, 1, 2, …). The governing equations are expressed in an invariant form

$$(\lambda + \mu )\nabla \nabla \cdot {\varvec{u}} + \mu \nabla^{2} {\varvec{u}} = \rho \frac{{\partial^{2} {\varvec{u}}}}{{\partial t^{2} }},$$
(1)

where λ and μ are Lame’s constants, ρ is the mass density, u = (u, v, w)T is the displacement vector in (r, θ, z) direction. Equation (1) includes three partial differential equations with difficult to solve. A convenient method of dealing with this problem is through Helmholtz conversion [34]. The displacement vector is expressed in terms of a dilatational scalar potential φ and an equivoluminal vector potential ψ

$${\varvec{u}} = \nabla \varphi + \nabla \times {\varvec{\psi}},$$
(2)

with

$$\nabla \cdot {\varvec{\psi}} = 0.$$
(3)

In a cylindrical coordinate system, the displacement can be expressed as

$$\left\{ {\begin{array}{*{20}l} {u = \frac{\partial \varphi }{{\partial r}} + \frac{1}{r}\frac{{\partial \psi_{3} }}{\partial \theta } - \frac{{\partial \psi_{2} }}{\partial z}} \hfill \\ {v = \frac{1}{r}\frac{\partial \varphi }{{\partial \theta }} + \frac{{\partial \psi_{1} }}{\partial z} - \frac{{\partial \psi_{3} }}{\partial r}} \hfill \\ {w = \frac{\partial \varphi }{{\partial z}} + \frac{{\partial \psi_{2} }}{\partial r} + \frac{1}{r}\psi_{2} - \frac{1}{r}\frac{{\partial \psi_{1} }}{\partial \theta }} \hfill \\ \end{array} } \right.$$
(4)

where ψ = (ψ1, ψ2, ψ3)T. And the potentials φ and ψ satisfy the wave equations

$$\begin{aligned} c_{l}^{2} \nabla^{2} \varphi & = \frac{{\partial^{2} \varphi }}{{\partial t^{2} }} \\ c_{s}^{2} \nabla^{2} {\varvec{\psi}} & = \frac{{\partial^{2} {\varvec{\psi}}}}{{\partial t^{2} }}, \\ \end{aligned}$$
(5)

where

$$\begin{aligned} c_{l}^{2} & = (\lambda + 2\mu )/\rho \\ c_{s}^{2} & = \mu /\rho . \\ \end{aligned}$$
(6)

The scalar potential φ describes a longitudinal wave with the velocity \({{c}}_{{l}}\), while the vector potential ψ describes a shear wave with the velocity \({{c}}_{{s}}\). Equation (5) can be rewritten as

$$\left\{ {\begin{array}{*{20}l} {c_{l}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }}} \right)\varphi - \frac{{\partial^{2} \varphi }}{{\partial t^{2} }} = 0} \hfill \\ {c_{s}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }} - \frac{1}{{r^{2} }}} \right)\psi_{1} - c_{s}^{2} \frac{2in}{{r^{2} }}\psi_{2} - \frac{{\partial^{2} \psi_{1} }}{{\partial t^{2} }} = 0} \hfill \\ {c_{s}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }} - \frac{1}{{r^{2} }}} \right)\psi_{2} + c_{s}^{2} \frac{2in}{{r^{2} }}\psi_{1} - \frac{{\partial^{2} \psi_{2} }}{{\partial t^{2} }} = 0} \hfill \\ {c_{s}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }}} \right)\psi_{3} - \frac{{\partial^{2} \psi_{3} }}{{\partial t^{2} }} = 0} \hfill \\ \end{array} } \right.$$
(7)

It should be noticed that the second and third equations in Eq. (7) include both the potentials ψ1 and ψ2. Therefore, we introduce the transformation of variables

$$\left\{ {\begin{array}{*{20}l} {\tilde{\psi }_{1} = \psi_{1} + i\psi_{2} } \hfill \\ {\tilde{\psi }_{2} = \psi_{1} - i\psi_{2} } \hfill \\ \end{array} } \right..$$
(8)

The corresponding inverse transformation is

$$\left\{ {\begin{array}{*{20}l} {\psi_{1} = \frac{1}{2}\left( {\tilde{\psi }_{1} + \tilde{\psi }_{2} } \right)} \hfill \\ {\psi_{2} = - \frac{1}{2}i\left( {\tilde{\psi }_{1} - \tilde{\psi }_{2} } \right)} \hfill \\ \end{array} .} \right.$$
(9)

Equation (7) can be rewritten as

$$\left\{ {\begin{array}{*{20}l} {c_{l}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }}} \right)\varphi + c_{l}^{2} \frac{{\partial^{2} }}{{\partial z^{2} }}\varphi + \omega^{2} \varphi = 0} \hfill \\ {c_{s}^{2} \left[ {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{(n + 1)^{2} }}{{r^{2} }}} \right]\tilde{\psi }_{1} + c_{s}^{2} \frac{{\partial^{2} }}{{\partial z^{2} }}\tilde{\psi }_{1} + \omega^{2} \tilde{\psi }_{1} = 0} \hfill \\ {c_{s}^{2} \left[ {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{(n - 1)^{2} }}{{r^{2} }}} \right]\tilde{\psi }_{2} + c_{s}^{2} \frac{{\partial^{2} }}{{\partial z^{2} }}\tilde{\psi }_{2} + \omega^{2} \tilde{\psi }_{2} = 0} \hfill \\ {c_{s}^{2} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} - \frac{{n^{2} }}{{r^{2} }}} \right)\psi_{3} + c_{s}^{2} \frac{{\partial^{2} }}{{\partial z^{2} }}\psi_{3} + \omega^{2} \psi_{3} = 0} \hfill \\ \end{array} } \right..$$
(10)

Equation (10) includes four partial differential equations, each of which only contains one unknown function. The part concerning variable r in Eq. (10) is the Bessel equation, the solution of which can be expressed by the Bessel function.

The Fundamental Solutions

The solution of Eq. (10) can be expressed as

$$\left\{ {\begin{array}{*{20}l} {\varphi \left( {r,\theta ,z,t} \right) = J_{n} \left( {kr} \right)f_{0} \left( z \right)e^{in\theta } } \hfill \\ {\tilde{\psi }_{1} \left( {r,\theta ,z,t} \right) = J_{n + 1} \left( {kr} \right)f_{1} \left( z \right)e^{in\theta } } \hfill \\ {\tilde{\psi }_{2} \left( {r,\theta ,z,t} \right) = J_{n - 1} \left( {kr} \right)f_{2} \left( z \right)e^{in\theta } } \hfill \\ {\psi_{3} \left( {r,\theta ,z,t} \right) = J_{n} \left( {kr} \right)f_{3} \left( z \right)e^{in\theta } } \hfill \\ \end{array} } \right.,$$
(11)

where Jn is the Bessel function of order n satisfying

$$\left[ {\frac{\partial }{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r} + \left( {k^{2} - \frac{{n^{2} }}{{r^{2} }}} \right)} \right]J_{n} \left( {kr} \right) = 0.$$
(12)

Substitution of Eq. (11) into Eq. (10) yields

$$\left\{ {\begin{array}{*{20}l} {c_{l}^{2} f^{\prime\prime}_{0} \left( z \right) + \left( {\omega^{2} - c_{l}^{2} k^{2} } \right)f_{0} \left( z \right) = 0} \hfill \\ {c_{s}^{2} f^{\prime\prime}_{1} \left( z \right) + \left( {\omega^{2} - c_{s}^{2} k^{2} } \right)f_{1} \left( z \right) = 0} \hfill \\ {c_{s}^{2} f^{\prime\prime}_{2} \left( z \right) + \left( {\omega^{2} - c_{s}^{2} k^{2} } \right)f_{2} \left( z \right) = 0} \hfill \\ {c_{s}^{2} f^{\prime\prime}_{3} \left( z \right) + \left( {\omega^{2} - c_{s}^{2} k^{2} } \right)f_{3} \left( z \right) = 0} \hfill \\ \end{array} } \right.$$
(13)

The general solution of Eq. (13) can be written as

$$\left\{ {\begin{array}{*{20}l} {f_{0} \left( z \right) = A_{1} \cos \alpha z + A_{2} \sin\alpha z} \hfill \\ {f_{1} \left( z \right) = A_{3} \cos \beta z + A_{4} \sin\beta z} \hfill \\ {f_{2} \left( z \right) = A_{5} \cos \beta z + A_{6} \sin\beta z} \hfill \\ {f_{3} \left( z \right) = A_{7} \cos \beta z + A_{8} \sin\beta z} \hfill \\ \end{array} } \right.$$
(14)

where Ai (i = 1, 2, …, 8) is arbitrary coefficients, and

$$\alpha = \sqrt {\frac{{\omega^{2} }}{{c_{l}^{2} }} - k^{2} } ,\quad \beta = \sqrt {\frac{{\omega^{2} }}{{c_{s}^{2} }} - k^{2} }$$
(15)

The coefficients in Eq. (14) are dependently caused by Eq. (3) which can be expanded as

$$\frac{1}{r}\frac{\partial }{\partial r}(r\psi_{1} ) + \frac{in}{r}\psi_{2} + \frac{{\partial \psi_{3} }}{\partial z} = 0$$
(16)

Considering the transformation in Eq. (9) and using the properties of Bessel functions, the relation between different coefficients can be obtained

$$\left\{ {\begin{array}{*{20}l} {A_{7} = \frac{k}{2\beta }A_{4} - \frac{k}{2\beta }A_{6} } \hfill \\ {A_{8} = - \frac{k}{2\beta }A_{3} + \frac{k}{2\beta }A_{5} } \hfill \\ \end{array} } \right.$$
(17)

Substitution of Eq. (9) into Eq. (11) gives

$$\left\{ {\begin{array}{*{20}l} {\varphi \left( {r,\theta ,z,t} \right) = J_{n} \left( {kr} \right)f_{0} \left( z \right)e^{in\theta } } \hfill \\ {\psi_{1} \left( {r,\theta ,z,t} \right) = \frac{1}{2}\left[ {J_{n + 1} \left( {kr} \right)f_{1} \left( z \right) + J_{n - 1} \left( {kr} \right)f_{2} \left( z \right)} \right]e^{in\theta } } \hfill \\ {\psi_{2} \left( {r,\theta ,z,t} \right) = - \frac{1}{2}i\left[ {J_{n + 1} \left( {kr} \right)f_{1} \left( z \right) - J_{n - 1} \left( {kr} \right)f_{2} \left( z \right)} \right]e^{in\theta } } \hfill \\ {\psi_{3} \left( {r,\theta ,z,t} \right) = J_{n} \left( {kr} \right)f_{3} \left( z \right)e^{in\theta } } \hfill \\ \end{array} } \right..$$
(18)

The displacement in Eq. (4) can be expressed as

$$\begin{aligned} u & = \frac{\partial \varphi }{{\partial r}} + \frac{in}{r}\psi_{3} - \frac{{\partial \psi_{2} }}{\partial z} \\ & = \left\{ {kJ^{\prime}_{n} \left( {kr} \right)f_{0} \left( z \right) + \frac{in}{r}J_{n} \left( {kr} \right)f_{3} \left( z \right) + \frac{1}{2}i\left[ {J_{n + 1} \left( {kr} \right)f^{\prime}_{1} \left( z \right) - J_{n - 1} \left( {kr} \right)f^{\prime}_{2} \left( z \right)} \right]} \right\}e^{in\theta } \\ v & = \frac{in}{r}\varphi + \frac{{\partial \psi_{1} }}{\partial z} - \frac{{\partial \psi_{3} }}{\partial r} \\ & = \left\{ {\frac{in}{r}J_{n} \left( {kr} \right)f_{0} \left( z \right) + \frac{1}{2}\left[ {J_{n + 1} \left( {kr} \right)f^{\prime}_{1} \left( z \right) + J_{n - 1} \left( {kr} \right)f^{\prime}_{2} \left( z \right)} \right] - kJ^{\prime}_{n} \left( {kr} \right)f_{3} \left( z \right)} \right\}e^{in\theta } \\ w & = \frac{\partial \varphi }{{\partial z}} + \frac{{\partial \psi_{2} }}{\partial r} + \frac{1}{r}\psi_{2} - \frac{in}{r}\psi_{1} \\ & = e^{in\theta } \left\{ {J_{n} \left( {kr} \right)f^{\prime}_{0} \left( z \right) - \frac{1}{2}ik\left[ {J^{\prime}_{n + 1} \left( {kr} \right)f_{1} \left( z \right) - J^{\prime}_{n - 1} \left( {kr} \right)f_{2} \left( z \right)} \right]} \right. \\ & \quad \left. { - \frac{1}{2r}i\left[ {J_{n + 1} \left( {kr} \right)f_{1} \left( z \right) - J_{n - 1} \left( {kr} \right)f_{2} \left( z \right)} \right] - \frac{in}{{2r}}\left[ {J_{n + 1} \left( {kr} \right)f_{1} \left( z \right) + J_{n - 1} \left( {kr} \right)f_{2} \left( z \right)} \right]} \right\}. \\ \end{aligned}$$
(19)

Taking advantage of the properties existing in Bessel functions

$$\left\{ {\begin{array}{*{20}l} {J^{\prime}_{n} \left( {kr} \right) = \frac{1}{2}\left[ {J_{n - 1} \left( {kr} \right) - J_{n + 1} \left( {kr} \right)} \right]} \hfill \\ {\frac{2n}{x}J_{n} \left( {kr} \right) = J_{n - 1} \left( {kr} \right) + J_{n + 1} \left( {kr} \right)} \hfill \\ \end{array} } \right.,\quad n = 0, \pm 1, \pm 2, \ldots ,$$
(20)

the displacement can be expressed as

$$\begin{aligned} u & = \left\{ {J_{n - 1} \left( {kr} \right)\left[ {B_{1} \cos \alpha z + B_{2} \sin\alpha z + iB_{6} \cos \beta z + iB_{5} \sin\beta z} \right]} \right. \\ & \left. {\quad + J_{n + 1} \left( {kr} \right)\left[ { - B_{1} \cos \alpha z - B_{2} \sin\alpha z + iB_{4} \cos \beta z + iB_{3} } \right.\sin\beta z} \right\}e^{in\theta } \\ v & = \left\{ {J_{n + 1} \left( {kr} \right)\left[ {iB_{1} \cos \alpha z + iB_{2} \sin\alpha z + B_{3} \sin\beta z + B_{4} \cos \beta z} \right]} \right. \\ & \quad \left. { + J_{n - 1} \left( {kr} \right)\left[ {iB_{1} \cos \alpha z + iB_{2} \sin\alpha z - B_{5} \sin\beta z - B_{6} \cos \beta z} \right]} \right\}e^{in\theta } \\ w & = J_{n} \left( {kr} \right)\left\{ {\alpha \left( { - A_{1} \sin \alpha z + A_{2} \cos \alpha z} \right) - \frac{k}{2}i\left[ {\left( {A_{3} + A_{5} } \right)\cos \beta z + \left( {A_{4} + A_{6} } \right)\sin\beta z} \right]} \right\}e^{in\theta } , \\ \end{aligned}$$
(21)

where

$$\begin{aligned} B_{1} & = \frac{k}{2}A_{1} ,\quad B_{2} = \frac{k}{2}A_{2} , \\ B_{3} & = - \frac{{k^{2} }}{4\beta }A_{3} + \frac{{k^{2} }}{4\beta }A_{5} - \frac{\beta }{2}A_{3} ,\quad B_{4} = \frac{{k^{2} }}{4\beta }A_{4} - \frac{{k^{2} }}{4\beta }A_{6} + \frac{\beta }{2}A_{4} \\ B_{5} & = - \frac{{k^{2} }}{4\beta }A_{3} + \frac{{k^{2} }}{4\beta }A_{5} + \frac{\beta }{2}A_{5} ,\quad B_{6} = \frac{{k^{2} }}{4\beta }A_{4} - \frac{{k^{2} }}{4\beta }A_{6} - \frac{\beta }{2}A_{6} . \\ \end{aligned}$$
(22)

Free Boundary Conditions

The expression of stresses can be derived from the stress and displacement relations

$$\left\{ {\begin{array}{*{20}l} {\sigma_{rz} = \mu \left( {\frac{\partial u}{{\partial z}} + \frac{\partial w}{{\partial r}}} \right)} \hfill \\ {\sigma_{\theta z} = \mu \left( {\frac{\partial v}{{\partial z}} + \frac{1}{r}\frac{\partial w}{{\partial \theta }}} \right)} \hfill \\ {\sigma_{zz} = \lambda \left( {\frac{\partial u}{{\partial r}} + \frac{u}{r} + \frac{1}{r}\frac{\partial v}{{\partial \theta }}} \right) + \left( {\lambda + 2\mu } \right)\frac{\partial w}{{\partial z}}} \hfill \\ \end{array} } \right.,$$
(23)

where σ represents the stress. The stresses can be described as

$$\begin{aligned} \sigma_{rz} & = \mu \left\{ {J_{n + 1} \left( {kr} \right)\left[ {C_{1} \sin \alpha z - C_{2} \cos \alpha z + iC_{6} \sin \beta z + iC_{5} \cos \beta z} \right]} \right. \\ & \quad \left. { + J_{n - 1} \left( {kr} \right)\left[ { - C_{1} \sin \alpha z + C_{2} \cos \alpha z + iC_{4} \sin \beta z + iC_{3} \cos \beta z} \right]} \right\}e^{in\theta } \\ \sigma_{\theta z} & = \mu \left\{ {J_{n + 1} \left( {kr} \right)\left[ { - iC_{1} \sin \alpha z + iC_{2} \cos \alpha z + C_{6} \sin \beta z + C_{5} \cos \beta z} \right]} \right. \\ & \quad \left. { + J_{n - 1} \left( {kr} \right)\left[ { - iC_{1} \sin \alpha z + iC_{2} \cos \alpha z - C_{4} \sin \beta z - C_{3} \cos \beta z} \right]} \right\}e^{in\theta } \\ \sigma_{zz} & = J_{n} \left( {kr} \right)\left[ { - \gamma A_{2} \sin \alpha z + i\chi \left( {A_{3} + A_{5} } \right)\sin \beta z} \right. \\ & \quad \left. { - \gamma A_{1} \cos \alpha z - i\chi \left( {A_{4} + A_{6} } \right)\cos \beta z} \right]e^{in\theta } , \\ \end{aligned}$$
(24)

where

$$\begin{aligned} C_{1} & = \alpha kA_{1} ,\quad C_{2} = \alpha kA_{2} , \\ C_{3} & = - \frac{{k^{2} }}{2}A_{3} + \frac{{\beta^{2} }}{2}A_{5} ,\quad C_{4} = - \frac{{k^{2} }}{2}A_{4} + \frac{{\beta^{2} }}{2}A_{6} \\ C_{5} & = - \frac{{\beta^{2} }}{2}A_{3} + \frac{{k^{2} }}{2}A_{5} ,\quad C_{6} = - \frac{{\beta^{2} }}{2}A_{4} + \frac{{k^{2} }}{2}A_{6} , \\ \end{aligned}$$
(25)

and

$$\gamma = \lambda k^{2} + \left( {\lambda + 2\mu } \right)\alpha^{2} ,\quad \chi = \mu \beta k.$$
(26)

It should be remarked that the stress components σrz and σθz include two parts described by different orders of Bessel functions Jn+1(kr) and Jn−1(kr), which affect the analysis and processing of boundary conditions. To avoid this problem, the conversion

$$\begin{aligned} \sigma_{1} & = \frac{{i\sigma_{rz} + \sigma_{\theta z} }}{2} = J_{n - 1} \left( {kr} \right)\left[ { - iC_{1} \sin \alpha z + iC_{2} \cos \alpha z - C_{4} \sin \beta z - C_{3} \cos \beta z} \right]e^{in\theta } \\ \sigma_{2} & = \frac{{i\sigma_{rz} - \sigma_{\theta z} }}{2} = J_{n + 1} \left( {kr} \right)\left[ {iC_{1} \sin \alpha z - iC_{2} \cos \alpha z - C_{6} \sin \beta z - C_{5} \cos \beta z} \right]e^{in\theta } \\ \end{aligned}$$
(27)

is introduced. The boundary conditions of free motion at the surfaces of the plate are

$$\left. {\sigma_{rz} } \right|_{z = \pm h} = \left. {\sigma_{\theta z} } \right|_{z = \pm h} = \left. {\sigma_{zz} } \right|_{z = \pm h} = 0.$$
(28)

The corresponding and equivalent conditions of Eq. (28) are

$$\left. {\sigma_{1} } \right|_{z = \pm h} = \left. {\sigma_{2} } \right|_{z = \pm h} = \left. {\sigma_{zz} } \right|_{z = \pm h} = 0.$$
(29)

Substitution of Eq. (29) into Eq. (27) yields

$$\begin{aligned} & \mp iC_{1} \sin \alpha h + iC_{2} \cos \alpha h \mp C_{4} \sin \beta h - C_{3} \cos \beta h = 0 \\ & \pm iC_{1} \sin \alpha h - iC_{2} \cos \alpha h \mp C_{6} \sin \beta h - C_{5} \cos \beta h = 0 \\ & \mp \gamma A_{2} \sin \alpha h \pm i\chi \left( {A_{3} + A_{5} } \right)\sin \beta h - \gamma A_{1} \cos \alpha h - i\chi \left( {A_{4} + A_{6} } \right)\cos \beta h = 0 \\ \end{aligned}$$
(30)

Equation (30) can be written as its equivalent expression

$$\left\{ {\begin{array}{*{20}l} {2\alpha kiA_{2} \cos \alpha h + \left( {k^{2} A_{3} - \beta^{2} A_{5} } \right)\cos \beta h = 0} \hfill \\ { - 2\alpha kiA_{1} \sin \alpha h + \left( {k^{2} A_{4} - \beta^{2} A_{6} } \right)\sin \beta h = 0} \hfill \\ { - 2\alpha kiA_{2} \cos \alpha h + \left( {\beta^{2} A_{3} - k^{2} A_{5} } \right)\cos \beta h = 0} \hfill \\ {2\alpha kiA_{1} \sin \alpha h + \left( {\beta^{2} A_{4} - k^{2} A_{6} } \right)\sin \beta h = 0} \hfill \\ {\left( {k^{2} - \beta^{2} } \right)iA_{1} \cos \alpha h + \beta k\left( {A_{4} + A_{6} } \right)\cos \beta h = 0} \hfill \\ {\left( {\beta^{2} - k^{2} } \right)iA_{2} \sin \alpha h + \beta k\left( {A_{3} + A_{5} } \right)\sin \beta h = 0} \hfill \\ \end{array} } \right..$$
(31)

Frequency Equations

Equation (31) can be divided into two independent parts as

$$\left\{ {\begin{array}{*{20}l} { - 2\alpha kiA_{1} \sin \alpha h + \left( {k^{2} A_{4} - \beta^{2} A_{6} } \right)\sin \beta h = 0} \hfill \\ {2\alpha kiA_{1} \sin \alpha h + \left( {\beta^{2} A_{4} - k^{2} A_{6} } \right)\sin \beta h = 0} \hfill \\ {\left( {k^{2} - \beta^{2} } \right)iA_{1} \cos \alpha h + \beta k\left( {A_{4} + A_{6} } \right)\cos \beta h = 0} \hfill \\ \end{array} } \right.$$
(32)

for symmetric (S) motion, and

$$\left\{ {\begin{array}{*{20}l} {2\alpha kiA_{2} \cos \alpha h + \left( {k^{2} A_{3} - \beta^{2} A_{5} } \right)\cos \beta h = 0} \hfill \\ { - 2\alpha kiA_{2} \cos \alpha h + \left( {\beta^{2} A_{3} - k^{2} A_{5} } \right)\cos \beta h = 0} \hfill \\ {\left( {\beta^{2} - k^{2} } \right)iA_{2} \sin \alpha h + \beta k\left( {A_{3} + A_{5} } \right)\sin \beta h = 0} \hfill \\ \end{array} } \right.$$
(33)

for anti-symmetric (A) motion. The frequency equations of both S motion and A motion can be derived from Eqs. (32) and (33) with the requirement of a nontrivial solution for each set of equations as

$$\sin \beta h\left| {\begin{array}{*{20}c} { - 2\alpha k\sin \alpha h} & {k^{2} \sin \beta h} & { - \beta^{2} \sin \beta h} \\ 0 & 1 & { - 1} \\ {\left( {k^{2} - \beta^{2} } \right)\cos \alpha h} & {\beta k\cos \beta h} & {\beta k\cos \beta h} \\ \end{array} } \right| = 0$$
(34)

for S motion, and

$$\cos \beta h\left| {\begin{array}{*{20}c} {2\alpha k\cos \alpha h} & {k^{2} \cos \beta h} & { - \beta^{2} \cos \beta h} \\ 0 & 1 & { - 1} \\ {\left( {\beta^{2} - k^{2} } \right)\sin \alpha h} & {\beta k\sin \beta h} & {\beta k\sin \beta h} \\ \end{array} } \right| = 0$$
(35)

for A motion. Both Eqs. (34) and (35) include shear horizontal (SH) wave and Lamb wave. For Lamb wave, frequency equations are expressed as

$$\frac{\tan \beta h}{{\tan \alpha h}} = - \left[ {\frac{{4\alpha \beta k^{2} }}{{\left( {\beta^{2} - k^{2} } \right)^{2} }}} \right]^{ \pm 1} ,$$
(36)

where + and − describe S motion and A motion, respectively. The frequency equations of circular crested Lamb wave are the same as those of plane Lamb wave [20, 35], and the frequency-dependent wavenumber k can be calculated from the corresponding frequency equation. The transcendental frequency equations provide an infinite number of solutions, leading to infinite series of S modes as well as A modes. The S modes are described as S(m, n), and the A modes are described as A(m, n), where m (m = 0, 1, 2, …) indicates the mth solution of the corresponding frequency equation (km) and n (n = 0, 1, 2, …) indicates the circumferential order.

The Displacement Expressions

The frequency equations have been divided into two parts: S wave and A wave. The two kinds of waves are discussed separately. For S motion, Eq. (32) can be written as

$$\left( {\begin{array}{*{20}c} { - 2\alpha k\sin \alpha h} & {k^{2} \sin \beta h} & { - \beta^{2} \sin \beta h} \\ {2\alpha k\sin \alpha h} & {\beta^{2} \sin \beta h} & { - k^{2} \sin \beta h} \\ {(k^{2} - \beta^{2} )\cos \alpha h} & {\beta k\cos \beta h} & {\beta k\cos \beta h} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {iA_{1} } \\ {A_{4} } \\ {A_{6} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right).$$
(37)

A fundamental solution of Eq. (37) is

$$\left\{ {\begin{array}{*{20}l} {A_{1} = 2i\beta k\cos \beta h} \hfill \\ {A_{4} = (k^{2} - \beta^{2} )\cos \alpha h} \hfill \\ {A_{6} = (k^{2} - \beta^{2} )\cos \alpha h} \hfill \\ \end{array} } \right..$$
(38)

The displacements of this case can be obtained by substituting Eq. (38) into displacement expression in Eq. (21)

$$\begin{aligned} u & = \frac{1}{2}\left[ {J_{n + 1} \left( {kr} \right) - J_{n - 1} \left( {kr} \right)} \right]\left[ { - 2\beta k^{2} \cos \beta h\cos \alpha z + \beta \left( {k^{2} - \beta^{2} } \right)\cos \alpha h\cos \beta z} \right]e^{in\theta } \\ v & = \frac{1}{2i}\left[ {J_{n + 1} \left( {kr} \right) + J_{n - 1} \left( {kr} \right)} \right]\left[ { - 2\beta k^{2} \cos \beta h\cos \alpha z + \beta \left( {k^{2} - \beta^{2} } \right)\cos \alpha h\cos \beta z} \right]e^{in\theta } \\ w & = J_{n} \left( {kr} \right)\left[ { - 2\alpha \beta k\cos \beta h\sin \alpha z - k\left( {k^{2} - \beta^{2} } \right)\cos \alpha h\sin \beta z} \right]e^{in\theta } . \\ \end{aligned}$$
(39)

For A motion, Eq. (33) can be written as

$$\left( {\begin{array}{*{20}c} {2\alpha k\cos \alpha h} & {k^{2} \cos \beta h} & { - \beta^{2} \cos \beta h} \\ { - 2\alpha k\cos \alpha h} & {\beta^{2} \cos \beta h} & { - k^{2} \cos \beta h} \\ {\left( {\beta^{2} - k^{2} } \right)\sin \alpha h} & {\beta k\sin \beta h} & {\beta k\sin \beta h} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {iA_{2} } \\ {A_{3} } \\ {A_{5} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right).$$
(40)

A fundamental solution of Eq. (40) is

$$\left\{ {\begin{array}{*{20}l} {A_{2} = 2i\beta k\sin \beta h} \hfill \\ {A_{3} = \left( {\beta^{2} - k^{2} } \right)\sin \alpha h} \hfill \\ {A_{5} = \left( {\beta^{2} - k^{2} } \right)\sin \alpha h} \hfill \\ \end{array} } \right..$$
(41)

The displacements of this case are derived by substituting the coefficients into displacement expressions

$$\begin{aligned} u & = \frac{1}{2}\left[ {J_{n + 1} \left( {kr} \right) - J_{n - 1} \left( {kr} \right)} \right]\left[ { - 2\beta k^{2} \sin \beta h\sin \alpha z + \beta \left( {k^{2} - \beta^{2} } \right)\sin \alpha h\sin \beta z} \right]e^{in\theta } \\ v & = \frac{1}{2i}\left[ {J_{n + 1} \left( {kr} \right) + J_{n - 1} \left( {kr} \right)} \right]\left[ { - 2\beta k^{2} \sin \beta h\sin \alpha z + \beta \left( {k^{2} - \beta^{2} } \right)\sin \alpha h\sin \beta z} \right]e^{in\theta } \\ w & = J_{n} \left( {kr} \right)\left[ {2\alpha \beta k\sin \beta h\cos \alpha z + k\left( {k^{2} - \beta^{2} } \right)\sin \alpha h\cos \beta z} \right]e^{in\theta } . \\ \end{aligned}$$
(42)

The displacement field of circular crested Lamb wave has been exactly solved by using a theoretical method. Lamb wave travels along the radial direction from the origin and holds as a standing wave in circumferential and longitudinal directions. Two cases of solution are obtained for one S case as well as one A case with respect to the z-axis.

Results and Discussions

The displacements of Lamb wave can be described as

$${\varvec{u}} = \left( \begin{array}{l} u \hfill \\ v \hfill \\ w \hfill \\ \end{array} \right) = \left( \begin{array}{l} R_{u} \varTheta_{u} Z_{u} \hfill \\ R_{v} \varTheta_{v} Z_{v} \hfill \\ R_{w} \varTheta_{w} Z_{w} \hfill \\ \end{array} \right).$$
(43)

Lamb wave displacements depend on the radial distance, circumferential direction, and frequency et al. It is necessary to investigate the effect of these parameters on the Lamb wave displacement field.

Characteristics Lamb Wave Displacements in Radial Direction

The displacements of the Lamb wave are governed by the Bessel function in the radial direction. For a large value of r, the Bessel functions converge to the asymptotic expression

$$J_{s} \left( {kr} \right) \to \sqrt {\frac{2}{\pi kr}} \cos \left( {kr - \frac{s}{2}\pi - \frac{1}{4}\pi } \right)\quad \left( {kr \to \infty } \right).$$
(44)

Therefore, the radial components of displacements in the far field satisfy

$$\left\{ {\begin{array}{*{20}l} {R_{u}^{{\left( {m,n} \right)}} \left( r \right) = \frac{1}{2}\left[ {J_{n + 1} \left( {k_{m} r} \right) - J_{n - 1} \left( {k_{m} r} \right)} \right] \to \sqrt {\frac{2}{{\pi k_{m} r}}} \sin \left( {k_{m} r - \frac{n}{2}\pi - \frac{1}{4}\pi } \right)} \hfill \\ {R_{v}^{{\left( {m,n} \right)}} \left( r \right) = \frac{1}{2}\left[ {J_{n + 1} \left( {k_{m} r} \right) + J_{n - 1} \left( {k_{m} r} \right)} \right] \to 0} \hfill \\ {R_{w}^{{\left( {m,n} \right)}} \left( r \right) = J_{n} \left( {k_{m} r} \right) \to \sqrt {\frac{2}{{\pi k_{m} r}}} \cos \left( {k_{m} r - \frac{n}{2}\pi - \frac{1}{4}\pi } \right)} \hfill \\ \end{array} } \right.\quad (k_{m} r \to \infty ).$$
(45)

In the far field, only radial and longitudinal displacements exist in the Lamb wave. Caused by the sine and cosine functions, the radial and longitudinal displacements have phase differences in the time domain. The asymptotic functions can be divided into two parts, one of which describes the attenuation in amplitude and the other one describes the fluctuation in the time domain. The amplitudes of displacements decrease with the radial distance r following O(\(1/\sqrt r\)) power rule as they propagate away from the origin [36]. Caused by the trigonometric functions contained in asymptotic functions, the displacement patterns become periodic at large radial distances.

Figure 2 presents the radial components of Lamb wave displacements at various propagation distances. The axisymmetric case n = 0 and a non-axisymmetric case n = 2 are selected as examples, and λ′ is the wavelength. As predicted, circumferential displacement vanishes for case n = 0. For case n = 2, circumferential displacement exists in the near field and decreases rapidly with the propagation distance increases, which can be neglected after several wavelengths in comparison with the other two displacement components.

Fig. 2
figure 2

Radial components of displacements: a n = 0 and b n = 2

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Figure 3 presents the radial components of displacement and asymptotic amplitudes. The cases n = 0–3 are taken into consideration. And the asymptotic amplitude is described as

$$A\left( {kr} \right) = \sqrt {\frac{2}{\pi kr}} ,$$
(46)

where k is the wave number. In the far field, as expected, asymptotic amplitudes can accurately describe the amplitude decrease. Little discrepancy between the displacement components and the asymptotic amplitudes appears in the near field for the reason that the asymptotic amplitudes of corresponding Bessel functions are precise only in the far field.

Fig. 3
figure 3

Radial component of displacement and asymptotic amplitude: a the radial displacement u and b the longitudinal displacement w

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Characteristics of Lamb Wave Displacements in the Circumferential Direction

Lamb wave field versus circumferential component θ is also concerned for it affects the amplitude distribution in the circumferential direction. Ignoring the effect of the initial phase, the circumferential components of displacement can be expressed as

$$\begin{aligned} \varTheta_{u}^{{\left( {m,n} \right)}} \left( \theta \right) & = \varTheta_{w}^{{\left( {m,n} \right)}} \left( \theta \right) = {\text{Re}} \left( {e^{in\theta } } \right) = \cos \left( {n\theta } \right) = \varTheta_{1} \\ \varTheta_{v}^{{\left( {m,n} \right)}} \left( \theta \right) & = {\text{Re}} \left( {ie^{in\theta } } \right) = - \sin \left( {n\theta } \right) = \varTheta_{2} . \\ \end{aligned}$$
(47)

For the case n = 0, the amplitude distributions of displacement are uniform in the circumferential direction, representing an axisymmetric wave. For the case, n ≠ 0, several maxima as well as minima exist in different circumferential directions. The maxima of radial and longitudinal displacements occur at θ = Nπ/n (integer N = 0, 1, 2, …, 2n − 1), while the maxima of circumferential displacement appear at θ = (N + 1/2)π/n, as presented in Fig. 4.

Fig. 4
figure 4

Amplitude distributions of displacements in circumferential direction: a the radial and longitudinal displacements and b the circumferential displacement

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Characteristics of Lamb Wave Displacement Field

Figure 5 presents the amplitude distributions of the across-thickness displacements for Lamb wave at various radial distances and circumferential angles by assuming Zu = Zv = Zw = 1. As discussed above, only radial and longitudinal displacements exist in the Lamb wave for the case n = 0. For n = 1 and n = 2, circumferential displacements exist near the origin and almost vanish in the far field. There are two and four obvious maxima of amplitude for n = 1 and n = 2, respectively. The maxima of radial and longitudinal displacements appear at θ = 0°, 180° and θ = 0°, 90°, 180°, 270°, while those of circumferential displacement appear at θ = 90°, 270° and θ = 45°, 135°, 225°, 315°. For each case, the amplitude of displacement decreases with the propagation distance. The results in Fig. 5 show that both propagation distance and propagation direction have effects on the amplitudes of the Lamb wave displacements.

Fig. 5
figure 5

Displacement field of Lamb wave: a radial displacement u and b circumferential displacement v and c longitudinal displacement w

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The normalized displacements of A(0,2) mode (fh = 0.75 MHz-mm) at various propagation directions with different propagation distances are presented in Fig. 6. The displacements in Fig. 6b, d are normalized by dividing the same coefficients with Fig. 6a, c, respectively. Circumferential displacement vanishes at the direction θ = 0° and exists alone at the direction θ = 45°. All of these displacements decrease with the increasing propagation distance, which is caused by the amplitude attenuation of Bessel functions. In the far field, the circumferential displacement can be neglected in comparison with the other two displacements.

Fig. 6
figure 6

Displacements of A(0,2) mode (fh = 0.75 MHz-mm): a θ = 0° and r = λ′ and b θ = 0° and r = 10λ′ and c θ = 45° and r = λ′ and d θ = 45° and r = 10λ′

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Numerical and Experimental Verifications

To verify the theoretical prediction of frequency equations and displacement expressions, both the numerical simulation and experimental study are carried out here.

Numerical Verification

As presented in Fig. 7a, the simulation is conducted on a 200 × 200 × 3 mm plate using ABAQUS. As an example, the aluminum plate is selected. The elasticity modulus, Poisson’s ratio and mass density of the aluminum plate are adopted as 70 GPa, 0.33 and 2700 kg/m3. Absorbing regions are added at each edge of the plate [37, 38] to avoid the reflection waves from the edges of the plate. A non-axisymmetric loading is applied at the origin of the upper surface of the plate as an actuator to excite a circular crested Lamb wave, as presented in Fig. 7b. Two sets of sensing points are placed at two circular rings around the origin with the radius 10 mm and 80 mm, respectively. Each set has 36 sensing points with 10° increments shown in Fig. 7c. Moreover, a set of sensing points is placed 80 mm away from the origin through the longitudinal direction to receive the across thickness signals presented in Fig. 7d. To obtain the signal at various propagation distances, a set of sensing points is placed at a straight line through the origin from 10 to 80 mm with a 1 mm increment. A 5-cycle sinusoid tone-burst signal enclosed in a Hanning window with the center frequency fc at 200 kHz is chosen as an excitation signal. The N-cycle sinusoid tone-burst signal S(t) enclosed in the Hanning window with the center frequency fc can be described as

$$S(t) = \frac{1}{2}\left[ {1 - \cos \left( {\frac{{2\pi f_{c} t}}{N}} \right)} \right]\sin (2\pi f_{c} t).$$
(48)
Fig. 7
figure 7

Finite element model of aluminum plate: a the whole model and b the actuator and c sensing points at circular and d sensing points through thickness

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The mesh density and time step are sufficiently small to resolve the smallest wavelength and simultaneously capture the highest frequency response, respectively.

By applying non-axisymmetric surface loading, both axisymmetric and non-axisymmetric Lamb waves can be excited. At the excitation frequency, only S(0,n) and A(0,n) can propagate for a long distance. The displacements obtained from sensing points are a linear combination of both axisymmetric and non-axisymmetric modes, denoted as S(0,*) and A(0,*), where

$${\text{S}}(0,*) = \sum\limits_{n = 0}^{\infty } {c_{n} {\text{S}}(0,n)} ,\quad {\text{A}}(0,*) = \sum\limits_{n = 0}^{\infty } {d_{n} {\text{A}}(0,n)}$$
(49)

and cn and dn are coefficients.

The displacement field (longitudinal displacement w) of the circular crested Lamb wave is presented in Fig. 8. The Lamb wave displacement field extends as circumferences with the center at the actuator center. At any given moment, Lamb wave at different propagation directions travels for the same distance from the actuator, which is consistent with the theoretical formulations.

Fig. 8
figure 8

Displacement field of circular crested Lamb wave

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Circumferential distributions of the excited Lamb wave displacement are presented in Fig. 9. As predicted, the displacements of both S and A modes are non-axisymmetric which indicates that non-axisymmetric Lamb wave exists in plates. The circumferential displacement v can be neglected in comparison with other displacements after propagating 80 mm. This phenomenon is caused by the characteristic of Bessel functions

$$J_{n + 1} \left( {kr} \right) + J_{n - 1} \left( {kr} \right) \to 0,\quad kr \to \infty$$
(50)
Fig. 9
figure 9

Circumferential distributions of Lamb wave displacements: a S(0,*) mode after propagating 80 mm and b A(0,*) mode after propagating 80 mm

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Figure 10 shows the across-thickness displacements of Lamb waves after propagating 80 mm. Circumferential displacement is ignored for the amplitude is small enough in comparison to other displacements. The across-thickness displacements obtained from simulations compare very well with theory, providing verification to theoretical formulations.

Fig. 10
figure 10

Across thickness displacements of Lamb waves excited by non-axisymmetric loading in the far field: a S(0,*) mode and b A(0,*) mode

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The displacement amplitudes of A(0,*) mode varying with propagation distance are presented in Fig. 11. As predicted in theory, the amplitudes of both radial and longitudinal displacements decrease following \({\text{O}}(1/\sqrt r )\) in the far field [36] (after propagating 20 mm, the wavelength is about 10 mm) because the amplitude of all modes decreases following the same power rule. Circumferential displacement amplitude rapidly decays to negligible with propagation distance increasing.

Fig. 11
figure 11

Displacement amplitudes of A(0,*) mode varying with propagation distance

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Experimental Verification

The experiment is carried out on a 1250 × 1250 × 3 mm aluminum plate. Three 6 × 6 × 0.5 mm square transducers (American Piezo Ceramics, APC-851) are mounted on the top surface of the plate as an actuator. These thin transducers can reduce the effect of induced bending moment applied to the plate. Seven 8 mm(diameter) × 0.45 mm(thickness) circular transducers (American Piezo Ceramics, APC-851) are placed around the actuators at θ = 0°–90° with 15° increments numbered #1–#7 as sensors. The distance between the actuator and the sensors is 300 mm. The layout and location of the actuator and sensors are presented in Fig. 12. A 5-cycle Hanning window modulated sinusoidal tone burst waveforms with a center frequency at 150–250 kHz with 10 kHz increment is generated as the input signal to each actuator to excite Lamb waves. Signals are generated and received by a damage diagnostic system at a sampling rate of 12 MHz and a sampling point of 8000.

Fig. 12
figure 12

Layout and location of actuator and sensors

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Similar to the simulation, signals obtained from the experiment are also the linear combination of axisymmetric and non-axisymmetric modes as described in Eq. (49). Figure 13 shows the comparison of signals received by different sensors. The blue solid lines represent the Lamb wave signals received by different sensors, and the red dotted lines represent the start and end times of corresponding Lamb modes. The first arrival waveforms in Fig. 13 are cross talks, which are produced by the coupling effect of the circuit. As shown in Fig. 13, the arrival times of both S(0,*) and A(0,*) modes at different propagation directions with the same propagation distance are about the same, which demonstrates that Lamb wave propagates as a circle with the center at the actuator center.

Fig. 13
figure 13

Arrival times of S(0,*) mode and A(0,*) mode

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Figure 14 shows the circumferential distributions of the excited Lamb wave. The points represent the amplitudes of signals received by different sensors, and the lines represent the connection lines of points. Both S and A modes have non-uniform distributions at different propagation directions, which proves the existence of non-axisymmetric Lamb wave in plates.

Fig. 14
figure 14

Circumferential distributions of the excited Lamb wave modes: a S(0,*) mode at 150 kHz and b S(0,*) mode at 200 kHz and c A(0,*) mode at 150 kHz and d A(0,*) mode at 200 kHz

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A comparison of the arrival time for circular crested Lamb wave is presented in Fig. 15. The blue lines represent the Lamb wave signals received by sensor #1 at different frequencies in the experiment, and the red lines represent the theoretical arrival times of Lamb modes. The theoretical arrival times of S(0,*) and A(0,*) modes are calculated based on group velocity obtained from frequency equations in Eq. (36). Excellent agreements are observed, which proves the frequency equations of circular crested Lamb wave.

Fig. 15
figure 15

Comparison of the arrival time of experiment and theory

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Conclusions

The complete and exact solutions of circular crested Lamb waves in isotropic plates are obtained and presented with the cylindrical coordinate system. The solutions represent a traveling wave in the radial direction and a standing wave in the other two orthogonal coordinate directions. The displacement field of a circular crested Lamb wave is governed by the Bessel function in the radial direction and harmonic functions in the circumferential direction. Similar to plane Lamb wave, circular crested Lamb wave is divided into two motion types, i.e., S motion and A motion. Two families of circular crested Lamb modes are generated and identified by wavenumber in the radial direction (k) and circumferential direction (n). Characteristics of circular crested Lamb wave displacements are discussed in detail. Governed by Bessel functions, the radial component describes the amplitude attenuation of displacement. And the circumferential component affects the distribution of amplitude in different propagation directions. Some numerical and experimental results are presented to provide verification for the theoretical formulations. Characteristics of Lamb wave including circular crested wave field, non-axisymmetric property, across thickness displacements and group velocity obtained from simulation and experiment are compared with those of theory. The good agreement between numerical or experimental and theoretical results verifies the theoretical formulations.