Thermoelastic wave propagation in functionally graded nanohollow cylinders based on nonlocal theory

Abstract

In this paper, a mathematical model is established for the thermoelastic circumferential Lamb waves in functionally graded material nonlocal nanohollow cylinders on account of the nonlocal theory and is solved by the proposed stress-based Legendre polynomials approach (SLPA). The SLPA uses stress as unknown quantities. Thus, the nonlocal stress-free boundary conditions can be satisfied directly. It translates the original differential equations to the linear eigenvalue problems, which reduces the solving difficulty. Comparisons with the available results indicate the validity of the presented SLPA. The nonlocal effects on dispersion and attenuation are discussed. The results show that the nonlocal effect is notable on both phase velocity and attenuation of Lamb-like waves, but is feeble on both phase velocity and attenuation of thermal waves. Meanwhile, the nonlocal effect enhances the attenuation of Lamb-like wave before its maximal value, but then suppresses it. The study on escape frequency reveals that it is independent of the geometric radius, boundary conditions, temperature and is just inverse proportion to the nonlocal parameter.

1 Introduction

When the characteristic size of microstructure decreases to a certain range, its mechanical properties change significantly, which are called scale effect [1]. Due to the scale effect, the nanostructures possess some peculiar mechanical properties and have great potential for the fabrication of nanomechanical systems. In addition, the nanostructures in engineering often bear the dynamic load represented by elastic waves, which lead to the change of temperature and the corresponding thermoelastic coupling. The thermoelastic coupling is essential in the field of nanomechanical systems [2, 3]. Therefore, the investigation of thermoelastic wave propagation in nanostructures has great significance for the design and development of various nanodevices.

In the current stage, many researchers have concerned on the elastic wave propagation in nanoplates and nanohollow cylinders, which are widely used in the nanomechanical systems. He et al. [4] investigated the wave propagation in functionally graded materials (FGMs) cylindrical nanoshell by the wave-based method. Arash et al. [5] considered the waves in nonlocal graphene sheets by using the finite element method. Sidhardh and Ray [6] studied the dispersive behavior of the Rayleigh–Lamb waves in a microplate. Wu et al.[7] investigated the Lamb waves in a nanoplate in view of the surface effects. Ebrahimi and Seyfi [8] studied the wave propagation in multi-scale hybrid nanocomposite plates under the aggregation of the nanoparticles. Li and Han [9] analyzed the elastic waves in the nanocomposite annular plates. Aydogdu [10] studied the longitudinal waves in multiwalled carbon nanotubes. Li and Chou [11] discussed the velocities of waves in single-walled carbon nanotubes. Zhang and Yin [12] analyzed the dispersion curves of the guided circumferential waves in double-walled carbon nanotubes.

On the other hand, the investigation of the thermoelastic wave propagation in nanoplates and cylindrical plates is still limited. Othman and Mondal [13] studied the effect of memory-dependent derivative and thermal loading on the wave in generalized micropolar thermoelasticity. Ebrahimi et al. [14] proposed an analytical approach to solve the waves in graphene oxide powder reinforced nanoplates with thermal loading. Hosseini and Zhang [15] carried out transient thermoelastic wave response in the FGM graphene platelets-reinforced nanocomposite cylinders by using the modified micromechanical model. Dai et al. [16] analyzed the flexural wave in thermoelastic functionally graded hollow cylindrical structure conveying nanoflow. Selim [17] discussed the thermoelastic longitudinal waves in a single-walled carbon nanotube. Although the thermoelastic waves in nanoplates and the thermoelastic longitudinal waves in hollow cylindrical structures were investigated, the thermoelastic circumferential Lamb waves in nanocylindrical structures were not considered as far as we know.

All above researches are performed based on the non-classical continuum theories [18,19,20,21,22]. Among these non-classical continuum theories, the nonlocal theory [23] is simple and effective in simulating the scale effect of nanostructures. Its theoretical hypothesis has been verified by phonon dispersion experiment and atomic lattice dynamics simulation [24]. In the context of the nonlocal theory, several traditional methods have been extended to study wave propagation in nanostructures, such as finite element method (FEM) [25], global matrix method (GMM) [26, 27], thin-layer method [28] and many other methods [29,30,31,32,33,34]. However, these methods all need to solve the nonlinear characteristic equation or solve the large-scale equations. When the wave propagation in dissipative nanostructures, for instance, thermoelastic or viscoelastic structures, in which the complex eigenvalues need to be searched in the complex plane, these methods must consume a lot of CPU time. Especially, the GMM is currently only applicable to flat nanoplates. The reason is that the general solutions of nanocylinder plates with nonlocal effect have not yet been obtained.

In summary, there exist two significant limitations on the available investigations of thermoelastic waves in nanoplates: (1) there is a lack of the small-scale and efficient calculation method; (2) the thermoelastic circumferential Lamb waves in nanocylindrical plates were rarely mentioned, although its study is meaningful for quality inspection and material property measurement of nanocylindrical plates. Motivated by these reasons, a mathematical model is established for the thermoelastic circumferential Lamb waves in FGM nonlocal nanohollow cylinders on account of the nonlocal theory and is solved by the proposed stress-based Legendre polynomials approach (SLPA). The conventional Legendre polynomials approach (LPA) has been widely used to solve many kinds of guided wave propagation problems, such as elasticity [35], viscoelasticity [36], piezoelectric elasticity [37], thermoelasticity [38], reflection and transmission [39] and microstructure with modified couple-stress theory [40]. The existing LPA uses displacement as unknown quantities and satisfies stress-free boundary condition by the rectangular window function. Then the stress is substituted by the displacement through introducing the constitutive relation into the governing equation. However, the existing LPA could not analyze the Lamb wave propagation in nanostructures with nonlocal effect due to the existence of second derivative of stress.

In this article, the proposed SLPA uses stress as unknown quantities. Thus, the nonlocal stress-free boundary conditions can be satisfied directly. Then the displacement is substituted by the stress through introducing the governing equation into the constitutive relation. Finally, the SLPA translates the original wave problems to the linear eigenvalue problems, which brings about much smaller computational difficulty. Comparison with the available results indicates the validity of the presented SLPA. The study on escape frequency implies that it is independent of the geometric radius and is inverse proportion to the nonlocal parameter. Furthermore, the nonlocal effects on dispersion and attenuation are discussed, and some meaningful results are discovered.

2 Mathematical model and solving process

2.1 Mathematical model

The FGM nanocylindrical plates composed of orthotropic materials are studied in this paper, as shown in Fig. 1. Its inner and outer radii are a and b, respectively. The radius thickness ratio ζ = b/(b − a). Guided waves propagate in the θ-direction. The governing equations of motion and the heat conduction equation can be written in the following forms [41]:

$$ \left\{ \begin{gathered} \frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r} = \rho \frac{{\partial^{2} u_{r} }}{{\partial t^{2} }} \hfill \\ \frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r} = \rho \frac{{\partial^{2} u_{\theta } }}{{\partial t^{2} }} \hfill \\ \end{gathered} \right. $$

(1a)

$$ \begin{gathered} K_{3} \frac{{\partial^{2} T}}{{\partial r^{2} }} + K_{1} \left( {\frac{1}{{r^{2} }}\frac{{\partial^{2} T}}{{\partial \theta^{2} }} + \frac{1}{r}\frac{\partial T}{{\partial r}}} \right) - \rho C_{e} \left( {1 + t_{0} \frac{\partial }{\partial t}} \right)\frac{\partial T}{{\partial t}} \hfill \\ = T_{0} \beta_{1} \left( {1 + t_{0} \frac{\partial }{\partial t}} \right)\frac{{\partial \varepsilon_{\theta \theta } }}{\partial t} + T_{0} \beta_{3} \left( {1 + t_{0} \frac{\partial }{\partial t}} \right)\frac{{\partial \varepsilon_{rr} }}{\partial t} \hfill \\ \end{gathered} $$

(1b)

where [42]

$$ \left( {1 - \lambda^{2} \nabla^{2} } \right)\sigma_{\theta \theta } = C_{11} \varepsilon_{\theta \theta } + C_{13} \varepsilon_{rr} - \beta_{1} T $$

(2a)

$$ \left( {1 - \lambda^{2} \nabla^{2} } \right)\sigma_{rr} = C_{13} \varepsilon_{\theta \theta } + C_{33} \varepsilon_{rr} - \beta_{3} T $$

(2b)

$$ \left( {1 - \lambda^{2} \nabla^{2} } \right)\sigma_{r\theta } = 2C_{55} \varepsilon_{r\theta } $$

(2c)

$$ \begin{array}{*{20}c} {\varepsilon_{\theta \theta } = \frac{1}{r}\frac{{\partial u_{\theta } }}{\partial \theta } + \frac{{u_{r} }}{r},} & {\varepsilon_{r\theta } = \frac{1}{2}\left( {\frac{1}{r}\frac{{\partial u_{r} }}{\partial \theta } + \frac{{\partial u_{\theta } }}{\partial r} - \frac{{u_{\theta } }}{r}} \right)} \\ \end{array} ,\begin{array}{*{20}c} {} & {\varepsilon_{rr} = \frac{{\partial u_{r} }}{\partial r}} \\ \end{array} $$

(3)

Here, $\sigma_{ij}$, $\varepsilon_{ij}$ and u_i denote the stresses, strains and displacements, respectively; C_ij, K_i, β_i, C_e and ρ are the elastic coefficients, material constant characteristics, volume expanding coefficients, specific heat at constant strain and material density, respectively; $T_{0} = 296K$; T is the relative temperature, which represents the change relative to T₀. t and $t_{0}$ indicate the time and the relaxation time, respectively. $\nabla^{2} = \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }}$ is the Laplacian in coordinate system (r, θ). λ is the nonlocal parameter [43], which represents the scale effect of nanostructures.

Fig. 1

The section of an FGM nanocylindrical plate

Considering the stress-free boundary condition, the r-direction stress components at the boundary should be zero. Meanwhile, the temperature variation T under isothermal boundary conditions shall also be 0 at the boundary. That is,

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{rr} ,} & {\sigma_{r\theta } ,} & T \\ \end{array} } \right\}\left| {_{r = a,b} } \right. = 0 $$

(4)

2.2 Solving process

For the handy analysis, the following nondimensional variables and notations are defined

$$ \begin{gathered} \tau_{0} = \frac{{v_{r}^{2} }}{{k_{r} }}t_{0} \begin{array}{*{20}c} , & {K = \frac{{K_{3} }}{{\overline{K}_{1} }}} \\ \end{array} \begin{array}{*{20}c} , & {\eta = \frac{{\left( {\overline{\beta }_{1} } \right)^{2} T_{0} }}{{\overline{\rho }^{2} \overline{C}_{e} v_{r}^{2} }}} \\ \end{array} \begin{array}{*{20}c} , & {\hat{T} = \frac{T}{{T_{0} }}} \\ \end{array} , \hfill \\ \hat{r} = \frac{{v_{r} }}{{k_{r} }}r\begin{array}{*{20}c} , & {\hat{\sigma }_{ij} = \frac{1}{{\overline{\beta }_{1} T_{0} }}\sigma_{ij} } \\ \end{array} \begin{array}{*{20}c} , & {\hat{t} = \frac{{v_{r}^{2} }}{{k_{r} }}} \\ \end{array} t\begin{array}{*{20}c} , & {\hat{\lambda } = \frac{{v_{r} }}{{k_{r} }}} \\ \end{array} \lambda \hfill \\ \end{gathered} $$

(5)

where $v_{r} = \sqrt {\overline{C}_{11} /\overline{\rho }}$ means the velocity of compressional waves. $k_{r} = \overline{K}_{1} /\overline{\rho }\overline{C}_{e}$ indicates the thermal diffusivity in the r direction. $\eta$ expresses the thermoelastic coupling constant. $\overline{K}_{1} ,$$\overline{\beta }_{1} ,$$\overline{\rho },$ $\overline{C}_{e} ,$$\overline{C}_{11}$ are $K_{1} ,$ $\beta_{1} ,$ $\rho ,$ $C_{e} ,$ $C_{11}$ of the corresponding designated material properties, respectively.

Substituting Eqs.(3,5) in Eqs.(2a), and overwriting $\hat{ \bullet }$ as $\bullet$, it is easy to have

$$ \left( {1 - \lambda^{2} \nabla^{2} } \right)\sigma_{\theta \theta } = \frac{{C_{11} }}{{\overline{C}_{11} }}\left( {\frac{1}{r}\frac{{\partial u_{\theta } }}{\partial \theta } + \frac{{u_{r} }}{r}} \right) + \frac{{C_{13} }}{{\overline{C}_{11} }}\frac{{\partial u_{r} }}{\partial r} - \frac{{\beta_{1} }}{{\overline{\beta }_{1} }}T $$

(6)

Further applying operator $\frac{\rho }{{\overline{\rho }}}\frac{{\partial^{2} }}{{\partial t^{2} }}$ to both sides of first formula in Eq. (2), it yields

$$ \frac{\rho }{{\overline{\rho }}}\left( {1 - \lambda^{2} \nabla^{2} } \right)\frac{{\partial^{2} \sigma_{\theta \theta } }}{{\partial t^{2} }} =\, \frac{\rho }{{\overline{\rho }}}\frac{{C_{11} }}{{\overline{C}_{11} }}\frac{{\partial^{2} }}{{\partial t^{2} }}\left( {\frac{1}{r}\frac{{\partial u_{\theta } }}{\partial \theta } + \frac{{u_{r} }}{r}} \right) + \frac{\rho }{{\overline{\rho }}}\frac{{C_{13} }}{{\overline{C}_{11} }}\frac{{\partial^{2} }}{{\partial t^{2} }}\frac{{\partial u_{r} }}{\partial r} - \frac{\rho }{{\overline{\rho }}}\frac{{\beta_{1} }}{{\overline{\beta }_{1} }}\frac{{\partial^{2} }}{{\partial t^{2} }}T $$

(7)

Noticing that partial differentiation is

$$ \frac{\partial }{\partial r}\left( {\rho \frac{{\partial^{2} \bullet }}{{\partial t^{2} }}} \right) =\, \frac{\partial \rho }{{\partial r}}\frac{{\partial^{2} \bullet }}{{\partial t^{2} }} + \rho \frac{\partial }{\partial r}\frac{{\partial^{2} \bullet }}{{\partial t^{2} }} $$

(8)

Equation (7) can be transformed to

$$ \frac{\rho }{{\overline{\rho }}}\left( {1 - \lambda^{2} \nabla^{2} } \right)\frac{{\partial^{2} \sigma_{\theta \theta } }}{{\partial t^{2} }} =\, \frac{\rho }{{\overline{\rho }}}\frac{{C_{11} }}{{\overline{C}_{11} }}\frac{{\partial^{2} }}{{\partial t^{2} }}\left( {\frac{1}{r}\frac{{\partial u_{\theta } }}{\partial \theta } + \frac{{u_{r} }}{r}} \right) + \frac{{C_{13} }}{{\overline{C}_{11} }}\frac{{\partial^{2} }}{{\partial t^{2} }}\left[ {\frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}u_{r} } \right) - u_{r} \frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}} \right)} \right] - \frac{\rho }{{\overline{\rho }}}\frac{{\beta_{1} }}{{\overline{\beta }_{1} }}\frac{{\partial^{2} }}{{\partial t^{2} }}T $$

(9)

Multiplying both sides of Eq. (9) by $\frac{\rho }{{\overline{\rho }}}$ and combining with Eqs. (1, 5), we have

$$ \begin{aligned} \left( {\frac{\rho }{{\overline{\rho }}}} \right)^{2} \left( {1 - \lambda^{2} \nabla^{2} } \right)\frac{{\partial^{2} }}{{\partial t^{2} }}\sigma_{\theta \theta } & = \frac{\rho }{{\overline{\rho }}}\frac{{C_{11} }}{{\overline{C}_{11} }}\frac{1}{r}\frac{\partial }{\partial \theta }\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right) \\ & \quad + \frac{\rho }{{\overline{\rho }}}\frac{{C_{11} }}{{\overline{C}_{11} }}\frac{1}{r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad + \frac{\rho }{{\overline{\rho }}}\frac{{C_{13} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad - \frac{{C_{13} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}} \right)\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad - \frac{\rho }{{\overline{\rho }}}\frac{{\beta_{1} }}{{\overline{\beta }_{1} }}\frac{\rho }{{\overline{\rho }}}\frac{{\partial^{2} T}}{{\partial t^{2} }} \\ \end{aligned} $$

(10a)

Similarly, for Eq. (1b) and Eqs. (2b, 2c), it is

$$ \begin{aligned} \left( {\frac{\rho }{{\overline{\rho }}}} \right)^{2} \left( {1 - \lambda^{2} \nabla^{2} } \right)\frac{{\partial^{2} }}{{\partial t^{2} }}\sigma_{r\theta } & = \frac{\rho }{{\overline{\rho }}}\frac{{C_{55} }}{{\overline{C}_{11} }}\frac{1}{r}\frac{\partial }{\partial \theta }\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad + \frac{\rho }{{\overline{\rho }}}\frac{{C_{55} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right) \\ & \quad - \frac{{C_{55} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}} \right)\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right) \\ & \quad - \frac{\rho }{{\overline{\rho }}}\frac{{C_{55} }}{{\overline{C}_{11} }}\frac{1}{r}\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right) \\ \end{aligned} $$

(10b)

$$ \begin{aligned} \left( {\frac{\rho }{{\overline{\rho }}}} \right)^{2} \left( {1 - \lambda^{2} \nabla^{2} } \right)\frac{{\partial^{2} }}{{\partial t^{2} }}\sigma_{rr} & = \frac{{C_{13} }}{{\overline{C}_{11} }}\frac{\rho }{{\overline{\rho }}}\frac{1}{r}\frac{\partial }{\partial \theta }\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right) \\ & \quad + \frac{\rho }{{\overline{\rho }}}\frac{{C_{13} }}{{\overline{C}_{11} }}\frac{1}{r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad + \frac{\rho }{{\overline{\rho }}}\frac{{C_{33} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad - \frac{{C_{33} }}{{\overline{C}_{11} }}\frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}} \right)\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right) \\ & \quad - \frac{\rho }{{\overline{\rho }}}\frac{{\beta_{3} }}{{\overline{\beta }_{1} }}\frac{\rho }{{\overline{\rho }}}\frac{{\partial^{2} }}{{\partial t^{2} }}T \\ \end{aligned} $$

(10c)

$$ \begin{aligned} & \frac{{K_{3} }}{{\overline{K}_{1} }}\left( {\frac{\rho }{{\overline{\rho }}}} \right)^{2} \frac{\partial }{\partial t}\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{{K_{1} }}{{\overline{K}_{1} }}\left( {\frac{\rho }{{\overline{\rho }}}} \right)^{2} \left( {\frac{1}{{r^{2} }}\frac{\partial }{\partial t}\frac{{\partial^{2} T}}{{\partial \theta^{2} }} + \frac{1}{r}\frac{\partial }{\partial t}\frac{\partial T}{{\partial r}}} \right) \\ & \quad - \frac{{C_{e} }}{{\overline{C}_{e} }}\left( {\frac{\rho }{{\overline{\rho }}}} \right)^{3} \left( {1 + \tau_{0} \frac{\partial }{\partial t}} \right)\frac{{\partial^{2} T}}{{\partial t^{2} }} \\ & \quad = \eta \frac{\rho }{{\overline{\rho }}}\frac{{\beta_{1} }}{{\overline{\beta }_{1} }}\left( {1 + \tau_{0} \frac{\partial }{\partial t}} \right)\left( {\frac{1}{r}\frac{\partial }{\partial \theta }\left( {\frac{{\partial \sigma_{r\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{2\sigma_{r\theta } }}{r}} \right)} \right. \\ & \quad \qquad \left. { + \frac{1}{r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right)} \right) \\ & \quad \qquad + \eta \frac{{\beta_{3} }}{{\overline{\beta }_{1} }}\left( {1 + \tau_{0} \frac{\partial }{\partial t}} \right)\left( {\frac{\rho }{{\overline{\rho }}}\frac{\partial }{\partial r}\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right)} \right. \\ & \quad \qquad \left. { - \frac{\partial }{\partial r}\left( {\frac{\rho }{{\overline{\rho }}}} \right)\left( {\frac{{\partial \sigma_{rr} }}{\partial r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r}} \right)} \right) \\ \end{aligned} $$

(10d)

As the wave propagates in the θ direction, the stress and temperature variables can be expressed as:

$$ \left\{ \begin{gathered} \sigma_{\theta \theta } \left( {r,\theta ,z,t} \right) = \exp (ikb\theta - i\omega t)U(r) \hfill \\ \sigma_{r\theta } \left( {r,\theta ,z,t} \right) = \exp (ikb\theta - i\omega t)V(r) \hfill \\ \sigma_{rr} \left( {r,\theta ,z,t} \right) = \exp (ikb\theta - i\omega t)W(r) \hfill \\ T\left( {r,\theta ,z,t} \right) = \exp (ikb\theta - i\omega t)X(r) \hfill \\ \end{gathered} \right. $$

(11)

k and ω are the magnitude of the wave vector in the propagation direction and angular frequency, respectively.

Similar to Ref.[36], the material properties could be the following forms

$$ \begin{aligned} \frac{{C_{ij} }}{{\overline{C}_{11} }}\left( r \right) & = C_{ij}^{(m)} \left( {\frac{2r - (b + a)}{{(b - a)}}} \right)^{m} \quad \frac{{\beta_{i} }}{{\overline{\beta }_{1} }}\left( r \right) = \beta_{i}^{(m)} \left( {\frac{2r - (b + a)}{{(b - a)}}} \right)^{m} \quad \frac{{K_{i} }}{{\overline{K}_{1} }}\left( r \right) & = K_{i}^{(m)} \left( {\frac{2r - (b + a)}{{(b - a)}}} \right)^{m} \\ \frac{\rho }{{\overline{\rho }}}\left( r \right) & = \rho^{\left( m \right)} \left( {\frac{2r - (b + a)}{{(b - a)}}} \right)^{m} ,\quad \frac{{C_{e} }}{{\overline{C}_{e} }}\left( r \right) = C_{e}^{\left( m \right)} \left( {\frac{2r - (b + a)}{{(b - a)}}} \right)^{m} \quad m = 0,1,...,M \\ \end{aligned} $$

(12)

Introducing Eqs. (11–12) to Eq. (10), it leads to

$$ \begin{aligned} & \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p} \left[ \begin{gathered} \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \left( {1 - \lambda^{2} \left( {\frac{1}{r}\frac{\partial }{\partial r} + \frac{{\partial^{2} }}{{\partial r^{2} }} - k^{2} b^{2} \frac{1}{{r^{2} }}} \right)} \right)U + ikb\rho^{\left( j \right)} C_{11}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) \hfill \\ +\, \rho^{\left( j \right)} C_{11}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) +\, \rho^{\left( j \right)} C_{13}^{\left( p \right)} \frac{\partial }{\partial r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) \hfill \\ \end{gathered} \right] \\ & \quad -\, \frac{2j}{{b - a}}\rho^{\left( j \right)} C_{13}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p - 1} \left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) + \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \beta_{1}^{\left( l \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p + l} X = 0 \\ \end{aligned} $$

(13a)

$$ \begin{aligned} & \rho^{\left( j \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p} \left[ \begin{gathered} \omega^{2} \rho^{\left( p \right)} \left( {1 - \lambda^{2} \left( {\frac{1}{r}\frac{\partial }{\partial r} + \frac{{\partial^{2} }}{{\partial r^{2} }} - k^{2} b^{2} \frac{1}{{r^{2} }}} \right)} \right)V + ikbC_{55}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) \hfill \\ +\, C_{55}^{\left( p \right)} \frac{\partial }{\partial r}\left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) - C_{55}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) \hfill \\ \end{gathered} \right] \\ & \quad - \frac{2j}{{b - a}}\rho^{\left( j \right)} C_{55}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p - 1} \left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) = 0 \\ \end{aligned} $$

(13b)

$$ \begin{aligned} & \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p} \left[ \begin{gathered} \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \left( {1 - \lambda^{2} \left( {\frac{1}{r}\frac{\partial }{\partial r} + \frac{{\partial^{2} }}{{\partial r^{2} }} - k^{2} b^{2} \frac{1}{{r^{2} }}} \right)} \right)W +\, ikb\rho^{\left( j \right)} C_{13}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) \hfill \\ +\, \rho^{\left( j \right)} C_{13}^{\left( p \right)} \frac{1}{r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) + \rho^{\left( j \right)} C_{33}^{\left( p \right)} \frac{\partial }{\partial r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) \hfill \\ \end{gathered} \right] \\ & \quad - \frac{2j}{{b - a}}\rho^{\left( j \right)} C_{33}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p - 1} \left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) + \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \beta_{3}^{\left( l \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p + l} X = 0 \\ \end{aligned} $$

(13c)

$$ \begin{aligned} & i\omega \rho^{\left( j \right)} \rho^{\left( p \right)} K_{3}^{\left( l \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p + l} \frac{{\partial^{2} X}}{{\partial r^{2} }} + i\omega \rho^{\left( j \right)} \rho^{\left( p \right)} K_{1}^{\left( l \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p + l} \left( { - k^{2} b^{2} \frac{1}{{r^{2} }}X + \frac{1}{r}\frac{\partial X}{{\partial r}}} \right) \\ & \quad - \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \rho^{\left( m \right)} C_{e}^{\left( l \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p + m + l} \left( {1 - i\omega \tau_{0} } \right)X \\ & \quad + \eta \rho^{\left( j \right)} \beta_{1}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p} \left( {1 - i\omega \tau_{0} } \right)\left( {ikb\frac{1}{r}\left( {\frac{\partial V}{{\partial r}} + ikb\frac{1}{r}U + \frac{2V}{r}} \right) + \frac{1}{r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right)} \right) \\ & \quad + \eta \rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p} \left( {1 - i\omega \tau_{0} } \right)\frac{\partial }{\partial r}\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) \\ & \quad - \frac{2j}{{b - a}}\eta \rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {\frac{2r - b - a}{{b - a}}} \right)^{j + p - 1} \left( {1 - i\omega \tau_{0} } \right)\left( {\frac{\partial W}{{\partial r}} + ikb\frac{1}{r}V + \frac{W - U}{r}} \right) = 0 \\ \end{aligned} $$

(13d)

The symbols $\bullet^{\prime }$ and $\bullet^{\prime \prime }$ denote the first and second derivatives.

To resolve Eq. (13), we expand U(r), V(r), W(r) and X(r) to Legendre polynomial series for the traction-free and isothermal boundary conditions according to Eq. (4),

$$ \begin{array}{*{20}l} {U\left( r \right) = \sum\limits_{n = 0}^{\infty } {p_{n}^{\left( 1 \right)} Q_{n} \left( r \right)} ,} \hfill & {V\left( r \right) = \left( {r - a} \right)\left( {r - b} \right)\sum\limits_{n = 0}^{\infty } {p_{n}^{\left( 2 \right)} Q_{n} \left( r \right)} } \hfill \\ {W\left( r \right) = \left( {r - a} \right)\left( {r - b} \right)\sum\limits_{n = 0}^{\infty } {p_{n}^{\left( 3 \right)} Q_{n} \left( r \right)} ,} \hfill & {X\left( r \right) = \left( {r - a} \right)\left( {r - b} \right)\sum\limits_{n = 0}^{\infty } {p_{n}^{\left( 4 \right)} Q_{n} \left( r \right)} } \hfill \\ \end{array} $$

(14)

here

$$ Q_{n} \left( r \right) = \sqrt {\frac{2m + 1}{{b - a}}} P_{n} \left( {\frac{2r - (b + a)}{{b - a}}} \right) $$

(15)

P_n is the nth Legendre polynomial and $p_{n}^{(i)}$ are the corresponding expansion coefficients.

In fact, with the increase in expansion order n, the influence of higher order on the overall calculation becomes weaker and weaker. Thus, the summation over Eq. (14) could be ended at some value N. Substituting Eq. (14) in Eq. (13) and implementing the orthogonal projection on a linear space composed of a series of Legendre polynomials, we have

$$ k^{2} {\mathbf{Ap}} + k{\mathbf{Bp}} + {\mathbf{Cp}} = 0 $$

(16)

where

$$ \begin{aligned} {\mathbf{A}} & = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{11} } & 0 & 0 & 0 \\ 0 & {{\mathbf{A}}_{22} } & 0 & 0 \\ {{\mathbf{A}}_{31} } & 0 & {{\mathbf{A}}_{33} } & 0 \\ {{\mathbf{A}}_{41} } & 0 & 0 & {{\mathbf{A}}_{44} } \\ \end{array} } \right],\quad {\mathbf{B}} = \left[ {\begin{array}{*{20}c} 0 & {{\mathbf{B}}_{12} } & 0 & 0 \\ {{\mathbf{B}}_{21} } & 0 & {{\mathbf{B}}_{23} } & 0 \\ 0 & {{\mathbf{B}}_{32} } & 0 & 0 \\ 0 & {{\mathbf{B}}_{42} } & 0 & 0 \\ \end{array} } \right], \\ {\mathbf{C}} & = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{11} } & 0 & {{\mathbf{C}}_{13} } & {{\mathbf{C}}_{14} } \\ 0 & {{\mathbf{C}}_{22} } & 0 & 0 \\ {{\mathbf{C}}_{31} } & 0 & {{\mathbf{C}}_{33} } & {{\mathbf{C}}_{34} } \\ {{\mathbf{C}}_{41} } & 0 & {{\mathbf{C}}_{43} } & {{\mathbf{C}}_{44} } \\ \end{array} } \right],\quad {\mathbf{p}} = \left[ {\begin{array}{*{20}c} {{\mathbf{p}}_{m}^{(1)} ,} & {{\mathbf{p}}_{m}^{(2)} ,} & {{\mathbf{p}}_{m}^{(3)} ,} & {{\mathbf{p}}_{m}^{(4)} } \\ \end{array} } \right]^{T} . \\ \end{aligned} $$

The detailed expressions of Eq. (16) are shown in Appendix. A new vector q is introduced,

$$ {\mathbf{q}} = k{\mathbf{p}} $$

(17)

Coupling Eqs. (16) and (17) can obtain,

$$ \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{I}} \\ { - {\mathbf{C}}} & { - {\mathbf{B}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mathbf{p}} \\ {\mathbf{q}} \\ \end{array} } \right] = k\left[ {\begin{array}{*{20}c} {\mathbf{I}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{A}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mathbf{p}} \\ {\mathbf{q}} \\ \end{array} } \right] $$

(18)

I is Identity matrix. Thus, the origin nonlinear eigenvalue problems are transformed to the linear eigenvalue problems.

3 Numerical examples

Based on the above formulations, the program for the SLPA is written. The properties of FGM can be expressed as:

$$ P\left( r \right) = \frac{{P_{1} \left( r \right)}}{2}\left[ {1 - \left( {\frac{r - a}{{b - a}}} \right)^{M} } \right] + \frac{{P_{2} \left( r \right)}}{2}\left( {\frac{r - a}{{b - a}}} \right)^{M} $$

(19)

where $P_{i} \left( r \right)$ denotes the corresponding property of the ith material. Ω is used to express frequency thickness product. Without explanation, a = 4 and b = 5.

3.1 Validity of the SLPA

Due to the lack of available data, the SLPA is validated by wave solutions in the FGM and nonlocal uniform structures without considering the temperature. Firstly, the results for an FGM cylindrical plate with a large radius thickness ratio ζ (ζ = 100) are obtained from the SLPA (N = 20) and are shown with the results for an infinite FGM plate from the power series technique in Fig. 2. The material properties can be found in Ref. [29]. In Ref. [29], the phase velocity dispersion curves for pure ceramics, pure Cr and FGM composed of ceramics and Cr are given. For this comparison, only the case of FGM is given. It is evident that our results agree well with the open results, which indicates that the SLPA is valid for the FGM cylindrical plates.

Fig. 2

Comparison with available data [29] for a linear FGM structures

Then a comparison for the nonlocal nanostructures is given. The results from a uniform cylindrical plate with ζ = 100, and those from a nonlocal graphene plate in Ref [44]. are exhibited in Fig. 3. The phase velocity is

$$ C_{p} = \frac{\omega }{{{\text{Re}} \left( k \right)}} = \frac{2\pi f}{{{\text{Re}} \left( k \right)}} $$

(20)

Fig. 3

Comparison with available ones [44] for a nonlocal nanostructures, λ = 0.1/0.34

Obviously, the results by using the SLPA are consistent with the available ones [44], which are proved in Fig. 3. This comparison demonstrates that the SLPA is available for the nonlocal cylindrical plates.

3.2 Convergence of the SLPA

In this section, the convergence of the SLPA is discussed in Table 1. The material constants are listed in Table 2 [45, 46]. The thermoelastic coupling constant η = 0.00249. M = 1, Ω = 2 and λ = 0.03. The outer radius b = 5, and the radius thickness ratio ζ = 5. For the first three Lamb-like wave modes, the real parts of the wavenumbers are convergent when N = 8, and the imaginary ones are convergent when N = 10. This indicates that the wavenumbers are convergent order by order, but their real parts converge faster than imaginary parts. For thermal wave mode, the real and imaginary parts of the wavenumbers are convergent when N = 4. This example implies that the SLPA is convergent for both Lamb-like and thermal waves, but has different convergence rates. The convergence of thermal waves is much faster than that of Lamb-like waves.

Table 1 Wavenumber k with different N, Ω = 2, λ = 0.03, M = 1

Table 2 The properties of the two materials [45, 46]

3.3 Scale effect on phase velocity and attenuation

Figures 4, 5 give the phase velocity and attenuation curves of the first two Lamb-like waves with λ = 0.01, 0.03, 0.05. At low frequencies, the nonlocal effect is weak on both dispersion and attenuation. But at high frequencies, the phase velocities decrease clearly with the increase in the nonlocal parameter λ. From Fig. 5, the nonlocal effect is remarkable on attenuation. A bigger nonlocal parameter λ moves the peak value of attenuation to a lower frequency. The attenuations are enhanced by the nonlocal effect before the peak value and are suppressed after the peak value. It is noted that the attenuation of 1st mode with λ = 0.05 (red line, Fig. 5a) in the red box is discordant. The reason is that the attenuation is oscillation near the escape frequencies, which is shown in Fig. 6.

Fig. 4

Phase velocity of the Lamb-like waves with different λ. a 1st mode and b 2nd mode

Fig.5

Attenuation coefficient of the Lamb-like waves with different λ. a 1st mode and b 2nd mode

Fig. 6

Attenuation curves of the Lamb-like waves with λ = 0.05

Figure 7 exhibits the phase velocity and attenuation curves of the thermal waves and implies that their phase velocities and attenuations are almost not affected by λ. The reason can be found in the government equation. λ is only in Eq. (5a–c), not in the heat conduction coupling Eq. (5d). Due to the small thermoelastic coupling constant (η = 0.00249), the variation of thermal wave mode with different λ should be tiny.

Fig. 7

Phase velocity and attenuation of the thermal wave with different λ. a Phase velocity and b attenuation

It is noted that the waves in nanostructures only propagate between their cutoff frequencies and escape frequencies [27]. At the escape frequency, the phase velocity is zero. And above the escape frequency, the wave phenomenon does not occur. As a crucial wave characteristic in nanostructures, the escape frequencies attract many attentions [27, 28, 32, 47]. Table 3 shows the escape frequencies of the first 5 modes in the FGM thermoelastic nanocylindrical plates. From Table 3, it is clearly seen that the escape frequencies are almost inversely proportional to λ. The reason can be found in the government equation. At the escape frequency, the phase velocity should be zero, and escape frequency $\omega_{e}^{{}}$ should be finite. Thus, the wavenumber k trends to infinity. Considering Eq. (11), both sides of Eq. (10) are divided by k²,

$$ \omega_{e}^{2} \frac{\rho }{{\overline{\rho }}}\lambda^{2} U - \frac{{C_{11} }}{{\overline{C}_{11} }}U = 0 $$

(21a)

$$ \omega_{e}^{2} \frac{\rho }{{\overline{\rho }}}\lambda^{2} V - \frac{{C_{55} }}{{\overline{C}_{11} }}V = 0 $$

(21b)

$$ \omega_{e}^{2} \frac{\rho }{{\overline{\rho }}}\lambda^{2} W - \frac{{C_{13} }}{{\overline{C}_{11} }}U = 0 $$

(21c)

$$ i\omega_{e}^{{}} \frac{{K_{1} }}{{\overline{K}_{1} }}\frac{\rho }{{\overline{\rho }}}X + \eta \frac{{\beta_{1} }}{{\overline{\beta }_{1} }}U = 0 $$

(21d)

Table 3 Escape frequencies for the first five modes, M = 1

Obviously, Eq. (21d) implies that there is no escape frequencies in thermal waves. Meanwhile, Eq. (21c) also does not yield the escape frequencies. Therefore, the escape frequencies only can be solved by Eqs. (21a, 21b) and are not affected by the temperature. More meaningful, the escape frequencies are inversely proportional to λ, but are unrelated to the geometrical radius and boundary conditions.

3.4 Outer radius thickness ratio

Although the radius thickness ratio ζ is unrelated to the escape frequency, it affects the phase velocity and attenuation significantly. The linear FGM nanocylindrical plates with ζ = 2, 5, 10 are considered here. λ = 0.03. Figure 8 shows the phase velocity and attenuation curves of the first two Lamb-like modes with different ζ. It is obvious that the phase velocities increase with decreasing ζ for the two Lamb-like modes in Fig. 8a, which is similar to the case in classical theory. In addition, a larger ζ indicates a more considerable attenuation before the minimal attenuation.

Fig. 8

Phase velocity and attenuation curves of the first two Lamb-like waves, λ = 0.03. a Phase velocity curves and b attenuation curves

The nonlocal effect on phase velocity and attenuation with different ζ is further analyzed in Figs.9 and 10. The ratio is calculated as

$$ {\text{Ratio}} = \left| {\frac{{\varphi^{{{\text{classical}}}} - \varphi_{{}}^{{{\text{nonlocal}}}} }}{{\varphi^{{{\text{classical}}}} }}} \right| $$

(22)

ϕ is the C_p or im(k) in Figs. 9 and 10, respectively. Before Ω = 2, Fig. 9 indicates that the strength of the nonlocal effects is close for different radius thickness ratios. However, after Ω = 2, the strength is different clearly. Specifically, a larger ζ means a more substantial nonlocal effect for the 1^st mode, but implies a weaker nonlocal effect for the 2^nd mode. For the attenuations in Fig. 10, the nonlocal effect is also close at low frequencies. Simultaneously, the max nonlocal effect moves to a lower frequency with a larger ζ. Besides, the nonlocal effect is weaker before its maximal value with a larger ζ, but is stronger after the max nonlocal effect. From Figs. 9 and 10, the nonlocal effect on attenuations is stronger than that on phase velocities at high frequencies.

Fig. 9

Nonlocal effect on phase velocity with different ζ, λ = 0.03. a 1st mode; b 2nd mode

Fig. 10

Nonlocal effect on attenuation with different ζ, λ = 0.03. a 1st mode; b 2nd mode

4 Conclusions

In this article, a low cost and efficient SLPA is proposed to investigate the nonlocal effect on dispersion characteristics of the thermoelastic circumferential Lamb waves in inhomogeneous nanocylindrical plates. The nonlocal boundary condition can be imposed directly on the stress variables. More importantly, the original problem is translated into a generalized linear eigenvalue problem, which reduces the difficulty of solution. The study is meaningful for quality inspection and material property measurement of nanocylindrical plates. In general, we have the following discoveries:

• A mathematical model is established for the thermoelastic circumferential Lamb waves in FGM nonlocal nanohollow cylinders on account of the nonlocal theory and is solved by the proposed SLPA.

• Nonlocal effect is strong on both phase velocity and attenuation of Lamb-like waves, but is feeble on both phase velocity and attenuation of thermal waves.

• The nonlocal effect enhances the attenuation of Lamb-like wave before the maximal attenuation and then suppresses it.

• The escape frequency is independent of the geometric radius, boundary conditions and temperature. It is just inversely proportional to nonlocal parameter.

Appendix

Defining

$$ \left\{ \begin{gathered} u[n,m,L,s,j] = \int_{a}^{b} {r^{s} \left( {\frac{2r - b - a}{{b - a}}} \right)^{L} Q_{n} \left( r \right)\frac{{d^{j} }}{{dr^{j} }}\left[ {\left( {r - a} \right)\left( {r - b} \right)Q_{m} \left( r \right)} \right]dr} \hfill \\ v[n,m,L,s,j] = \int_{a}^{b} {r^{s} \left( {\frac{2r - b - a}{{b - a}}} \right)^{L} Q_{n} \left( r \right)\frac{{d^{j} }}{{dr^{j} }}Q_{m} \left( r \right)dr} \hfill \\ \end{gathered} \right. $$

The detailed of Eq. (10) are given here. The sum is applied when the symbols (j, p, l, s) occur more than once.

$$ \begin{aligned} A_{11}^{n,m} & = \left( {b^{2} \lambda^{2} \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} - b^{2} \rho^{\left( j \right)} C_{11}^{\left( p \right)} } \right)v[n,m,j + p,0,0] \\ B_{12}^{n,m} & = ib\rho^{\left( j \right)} C_{11}^{\left( p \right)} \left( {u[n,m,j + p,1,1] + 3u[n,m,j + p,0,0]} \right) \\ & \quad + ib\rho^{\left( j \right)} C_{13}^{\left( p \right)} \left( { - u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{2j}{{b - a}}ib\rho^{\left( j \right)} C_{13}^{\left( p \right)} u[n,m,j + p - 1,1,0] \\ C_{11}^{n,m} & = \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \left( {v[n,m,j + p,2,0] - \lambda^{2} v[n,m,j + p,1,1] - \lambda^{2} v[n,m,j + p,2,2]} \right) \\ & \quad + \left( { - \rho^{\left( j \right)} C_{11}^{\left( p \right)} + \rho^{\left( j \right)} C_{13}^{\left( p \right)} } \right)v[n,m,j + p,0,0] - \rho^{\left( j \right)} C_{13}^{\left( p \right)} v[n,m,j + p,1,1] \\ & \quad + \frac{2j}{{b - a}}\rho^{\left( j \right)} C_{13}^{\left( p \right)} v[n,m,j + p - 1,1,0] \\ \end{aligned} $$

$$ \begin{aligned} C_{{13}}^{{n,m}} & = \rho ^{{\left( j \right)}} C_{{11}}^{{\left( p \right)}} \left( {u[n,m,j + p,1,1]} \right. \\ & \quad \left. { + u[n,m,j + p,0,0]} \right) \\ & \quad + \rho ^{{\left( j \right)}} C_{{13}}^{{\left( p \right)}} \left( {u[n,m,j + p,2,2]} \right. \\ & \quad \left. { - u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{{2j}}{{b - a}}\rho ^{{\left( j \right)}} C_{{13}}^{{\left( p \right)}} \left( {u[n,m,j + p - 1,2,1]} \right. \\ & \quad \left. { + u[n,m,j + p - 1,1,0]} \right) \\ C_{{14}}^{{n,m}} & = \omega ^{2} \rho ^{{\left( j \right)}} \rho ^{{\left( p \right)}} \beta _{1}^{{\left( l \right)}} u[n,m,j + p + l,2,0] \\ \end{aligned} $$

$$ \begin{aligned} A_{{22}}^{{n,m}} & = \left( {\lambda ^{2} b^{2} \omega ^{2} \rho ^{{\left( j \right)}} \rho ^{{\left( p \right)}} - b^{2} \rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} } \right)u[n,m,j + p,0,0] \\ B_{{21}}^{{n,m}} & = - 3ib\rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} v[n,m,j + p,0,0] \\ & \quad + ib\rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} v[n,m,j + p,1,1] \\ & \quad - \frac{{2j}}{{b - a}}ib\rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} v[n,m,j + p,1,0] \\ B_{{23}}^{{n,m}} & = ib\rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} \left( {u[n,m,j + p,1,1] + u[n,m,j + p,0,0]} \right) \\ C_{{22}}^{{n,m}} & = \omega ^{2} \rho ^{{\left( j \right)}} \rho ^{{\left( p \right)}} \left( {u[n,m,j + p,2,0]} \right. \\ & \quad \left. { - \lambda ^{2} u[n,m,j + p,1,1] - \lambda ^{2} u[n,m,j + p,2,2]} \right) \\ & \quad + \rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} \left( {u[n,m,j + p,2,2]} \right. \\ & \quad \left. { - 4u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{{2j}}{{b - a}}\rho ^{{\left( j \right)}} C_{{55}}^{{\left( p \right)}} \left( {u[n,m,j + p - 1,2,1]} \right. \\ & \quad \left. { + 2u[n,m,j + p - 1,1,0]} \right) \\ \end{aligned} $$

$$ \begin{aligned} A_{31}^{n,m} & = - b^{2} \rho^{\left( j \right)} C_{13}^{\left( p \right)} v[n,m,j + p,0,0] \\ A_{33}^{n,m} & = b^{2} \lambda^{2} \omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} u[n,m,j + p,0,0] \\ B_{12}^{n,m} & = ib\rho^{\left( j \right)} C_{13}^{\left( p \right)} \left( {u[n,m,j + p,1,1] + 3u[n,m,j + p,0,0]} \right) \\ & \quad + ib\rho^{\left( j \right)} C_{33}^{\left( p \right)} \left( { - u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{2j}{{b - a}}ib\rho^{\left( j \right)} C_{33}^{\left( p \right)} u[n,m,j + p - 1,1,0] \\ C_{31}^{n,m} & = \left( { - \rho^{\left( j \right)} C_{13}^{\left( p \right)} + \rho^{\left( j \right)} C_{33}^{\left( p \right)} } \right)v[n,m,j + p,0,0] \\ & \quad - \rho^{\left( j \right)} C_{33}^{\left( p \right)} v[n,m,j + p,1,1] \\ & \quad + \frac{2j}{{b - a}}\rho^{\left( j \right)} C_{33}^{\left( p \right)} v[n,m,j + p - 1,1,0] \\ \end{aligned} $$

$$ \begin{aligned} C_{{33}}^{{n,m}} & = \rho ^{{\left( j \right)}} C_{{13}}^{{\left( p \right)}} \left( {u[n,m,j + p,1,1]} \right. \\ & \quad \left. { + u[n,m,j + p,0,0]} \right) \\ & \quad + \rho ^{{\left( j \right)}} C_{{33}}^{{\left( p \right)}} \left( {u[n,m,j + p,2,2]} \right. \\ & \quad \left. { - u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{{2j}}{{b - a}}\rho ^{{\left( j \right)}} C_{{33}}^{{\left( p \right)}} \left( {u[n,m,j + p - 1,2,1]} \right. \\ & \quad \left. { + u[n,m,j + p - 1,1,0]} \right) \\ & \quad + \omega ^{2} \rho ^{{\left( j \right)}} \rho ^{{\left( p \right)}} \left( {v[n,m,j + p,2,0]} \right. \\ & \quad \left. { - \lambda ^{2} v[n,m,j + p,1,1] - \lambda ^{2} v[n,m,j + p,2,2]} \right) \\ C_{{34}}^{{n,m}} & = \omega ^{2} \rho ^{{\left( j \right)}} \rho ^{{\left( p \right)}} \beta _{3}^{{\left( l \right)}} u[n,m,j + p + l,2,0] \\ \end{aligned} $$

$$ \begin{gathered} A_{41}^{n,m} = b^{2} \eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{1}^{\left( p \right)} v[n,m,j + p,0,0] \hfill \\ A_{44}^{n,m} = - i\omega b^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} K_{1}^{\left( l \right)} u[n,m,j + p + l,0,0] \hfill \\ B_{42}^{n,m} = ib\eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{1}^{\left( p \right)} \left( {u[n,m,j + p,1,1] + 3u[n,m,j + p,0,0]} \right) \hfill \\ \begin{array}{*{20}c} {} & {} & { + ib\eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {u[n,m,j + p,1,1] - u[n,m,j + p,0,0]} \right)} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {} & {} & { - \frac{2j}{{b - a}}ib\eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} u[n,m,j + p - 1,1,0]} \\ \end{array} \hfill \\ C_{41}^{n,m} = - \eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{1}^{\left( p \right)} v[n,m,j + p,0,0] \hfill \\ \begin{array}{*{20}c} {} & {} & { + \eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {v[n,m,j + p,0,0] - v[n,m,j + p,1,1]} \right)} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {} & {} & { - \frac{2j}{{b - a}}\eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} v[n,m,j + p - 1,1,0]} \\ \end{array} \hfill \\ \end{gathered} $$

$$ \begin{aligned} C_{43}^{n,m} & = \eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{1}^{\left( p \right)} \left( {u[n,m,j + p,1,1] + u[n,m,j + p,0,0]} \right) \\ & \quad + \eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {u[n,m,j + p,2,2] - u[n,m,j + p,0,0] + u[n,m,j + p,1,1]} \right) \\ & \quad - \frac{2j}{{b - a}}\eta \left( {1 - i\omega \tau_{0} } \right)\rho^{\left( j \right)} \beta_{3}^{\left( p \right)} \left( {u[n,m,j + p - 1,2,1] + u[n,m,j + p - 1,1,0]} \right) \\ C_{44}^{n,m} & = i\omega \rho^{\left( j \right)} \rho^{\left( p \right)} K_{3}^{\left( l \right)} u[n,m,j + p + l,2,2] + i\omega \rho^{\left( j \right)} \rho^{\left( p \right)} K_{1}^{\left( l \right)} u[n,m,j + p + l,1,1] \\ & \quad - \left( {1 - i\omega \tau_{0} } \right)\omega^{2} \rho^{\left( j \right)} \rho^{\left( p \right)} \rho^{\left( s \right)} C_{e}^{\left( l \right)} u[n,m,j + p + s + l,2,0] \\ \end{aligned} $$