Research on Longitudinal Thermoelastic Waves in an Orthotropic Anisotropic Hollow Cylinder Based on the Thermoelastic Theory of Green–Naghdi

Abstract

At present, many high-temperature pipelines need to carry out non-stop detection under high-temperature conditions, and an ultrasonic guided wave is undoubtedly one of the solutions with the highest potential to solve the problem. However, there is a lack of research on the propagation characteristics of longitudinal guided wave modes in high-temperature pipelines. Based on the Green–Naghdi (GN) generalized thermoelastic theory, a theoretical model of thermoelastic guided waves in an orthotropic hollow cylinder with a temperature field is established by using the Legendre polynomial series expansion method. Firstly, based on the GN thermoelastic theory, the coupling equations expressed by displacement and temperature are established by introducing the rectangular window function. The curves of dispersion, displacement, and temperature of the guided wave are numerically solved by using this equation. Subsequently, the influence of the diameter-to-thickness ratio on the dispersion of the longitudinal thermoelastic guided wave is analyzed at the same temperature. Finally, the effect of temperature field variation on the phase velocity dispersion is discussed, which provides a theoretical basis for the study of the dispersion characteristics of hollow cylindrical pipes containing temperature fields.

1. Introduction

With the continuous development of modern industry, pipes with anisotropic materials are widely used in the aerospace and transportation fields by virtue of their strong comprehensive performance, which often requires service under high-temperature conditions. For example, in the aerospace industry, these pipes are subjected to extreme temperatures in the engine. The probability of thermal stress, corrosion, and cracking in a high-temperature environment is much higher than that in a normal-temperature environment. These phenomena seriously affect the function and life of the equipment. Ultrasonic guided waves can achieve rapid non-destructive testing of pipeline defects with local removal of insulation and anti-corrosion layers. A previous study has shown that the coupling of the temperature and strain fields have an effect on the wave propagation speed and waveform [1]. It can be seen that in order to realize the online detection of guided waves in high-temperature pipes, accurately obtaining the propagation characteristics of guided waves under the temperature field is an important prerequisite for realizing the characterization of defects in the structures.

Currently, there are some methods to calculate the dispersion characteristics of guided waves in hollow cylinders. Among them, the transfer matrix and global matrix methods [2], which are based on the sub-wave expansion method, have been widely used in many materials as well as computational cases. However, the transfer matrix has the problem of numerical instability when solving large frequency–thickness products. Although the global matrix method solves the problem of numerical instability, it may take a long time to compute when the matrix is too large. To overcome the above problems, Lefebvre et al. [3] investigated the propagation characteristics of guided waves in multilayer piezoelectric material plates using Legendre’s orthogonal polynomial expansion method, which converts the traditional problem of solving transcendental equations into an eigenvalue problem by expanding the displacements into a polynomial series. After that, Yu et al. [4,5] solved the propagation characteristics of circumferential guided waves in a multilayer piezoelectric cylindrical plate and longitudinal guided waves in a multilayer hollow cylinder by using improved Legendre polynomials. Zhang et al. [6] investigated the axial guided wave characteristics of piezoelectric cylinders using Legendre’s series method. Liu et al. [7] obtained complete dispersion curves in complex domains based on Legendre’s polynomial method with stress and potential shift expansion. Zheng et al. [8,9] solved longitudinally guided waves in anisotropic hollow cylinders as well as circumferentially guided waves in multilayer composite hollow cylinders based on state vectors and Legendre orthogonal polynomials.

Simultaneously, it has been found that changes in the temperature field also have an effect on the propagation characteristics of sound waves. While there is an assumption of infinite velocity in the classical theory of heat conduction in solids, which is contrary to physical phenomena, various generalized thermoelastic theories have been developed. Among them, the LS model, the GL model, and the GN model [10] are the more classical models, and based on these thermoelastic theories, many researchers have investigated the propagation characteristics of thermoelastic waves. Verma et al. [11] investigated the dispersion characteristics of guided waves in isotropic plates based on GL theory and GN generalized thermoelastic theory. Al-Qahtani et al. [10] investigated the thermoelastic guided waves in transverse isotropic plates based on LS theory by using the semi-analytical finite element method. Based on GN theory, Li et al. [12] uses the finite element method to improve the formula and numerically analyzes the dispersion characteristics of guided waves in hollow cylinder structures. Yang et al. [13] proposed a method combining thermoelastic theory and the semi-analytical finite element method to study the effects of uniform and non-uniform thermal effects on phase velocity, group velocity, etc. Based on the Fourier expansion approach of Sububi’s generalized theory, Venkatesan et al. [14] examined the wave propagation properties of a generalized thermoelastic solid cylinder of any cross-section. The propagation of thermoelastic waves in an infinite-length hollow cylinder was studied by Erbay et al. [15]. In the meantime, Bao et al. [16] used Newman’s boundary integral approach, which is based on the first kind of Fredholm integral equations, to study the three-dimensional dispersion problem of thermoelastic waves. Based on experimental data, Dodson et al. [17] examined the temperature sensitivity of group velocity in aluminum alloy plates and examined how various temperatures affected Lamb waves. Zeng et al. [18] investigated the dispersion characteristics of guided waves in multilayer composite cylinders as well as thermoacoustic–elastic effects. Using Legendre’s gradient and the generalized thermoelastic theory of GN, Yu et al. [19,20,21] numerically solved the thermoelastic wave propagation characteristics of a functional gradient plate. They then expanded the application of this method to a curved spherical plate and hollow cylinders for the circumferential thermoelastic guided wave propagation problems. Following that, to investigate the guided wave propagation problem in fractional-order thermoelastic plates, Wang et al. [22] developed a modified Legendre polynomial approach based on the fractional-order thermoelastic theory. Wang et al. [23] investigated thermoelastic Lamb waves in functional gradient nanoplates using the integral form of the modified nonlocal theory. Yu et al. [24] and Zhang et al. [25] solved and analyzed the propagation characteristics of circumferential guided waves in a Lamb wave in a fractionally thermoelastic multilayer plate and in a fractionally viscoelastic hollow cylinder, respectively, based on the thermoelastic theoretical model of fractional order. Zhang et al. [26] investigated thermoelastic waves in rods with rectangular cross-sections based on the GL thermoelastic theory using modified bi-orthogonal Legendre polynomials. Ravi Chitikireddy et al. [27] studied the propagation characteristics of transient ultrasonic guided waves generated by concentrated heating on the outer surface of an infinitely anisotropic hollow cylinder. Based on the LS generalized thermoelastic theory, Wang et al. [28] studied the longitudinal guided waves of fractional order thermoelastic inhomogeneous hollow cylinders based on adiabatic boundary conditions by using the analytical Legendre polynomial series method.

Most of the above studies involve plate structures, and most of the studies on thermoelastic waves in hollow cylinders focus on the propagation characteristics of circumferential guided waves, with fewer studies on the propagation characteristics of thermoelastic guided waves in the longitudinal direction. Longitudinal guided waves have a high potential for use in pipeline inspection due to their propagation properties, which are particularly sensitive to circumferential faults. This study investigates the propagation characteristics of longitudinal thermoelastic waves in an orthotropic anisotropic hollow cylinder using the GN generalized thermoelastic model. In the first part, the thermoelastic guided wave dispersion model with temperature field is established by coupled thermodynamic equations. In the second part, the phase velocity dispersion curves of the longitudinal guided wave in an orthotropic anisotropic hollow cylinder as well as the displacements of the plasmas and the temperature variations are numerically computed, and the influences of the diameter-to-thickness ratio as well as the reference temperature on the dispersion characteristics are analyzed.

2. Theoretical Analyses

As is shown in Figure 1, a hollow cylinder is represented in the column coordinate system, with the axial direction extending indefinitely along the z-axis and the inner and outer surfaces satisfying the stress-free boundary condition. It is assumed that the inner radius of the cylinder is a, the outer radius is b, and the outer-radius-to-wall-thickness ratio is η = b/(b − a). The column coordinate system uses r, θ, and z to indicate the cylinder’s three directions: radial, circumferential, and axial.

The thermoelastic constitutive equation for orthotropic anisotropic materials in the column coordinate system can be expressed as:

$$\left\{\begin{array}{l}{\sigma }_{\theta \theta }\\ {\sigma }_{zz}\\ {\sigma }_{rr}\\ {\sigma }_{rz}\\ {\sigma }_{r\theta }\\ {\sigma }_{z\theta }\end{array}\right\}=\left[\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& 0& 0& 0\\ & c_{22}& c_{23}& 0& 0& 0\\ & & c_{33}& 0& 0& 0\\ & & & c_{44}& 0& 0\\ & symmetry& & & c_{55}& 0\\ & & & & & c_{66}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_{\theta \theta }\\ {\epsilon }_{zz}\\ {\epsilon }_{rr}\\ 2{\epsilon }_{rz}\\ 2{\epsilon }_{r\theta }\\ 2{\epsilon }_{z\theta }\end{array}\right\}-\left\{\begin{array}{l}{\beta }_1\\ {\beta }_2\\ {\beta }_3\\ 0\\ 0\\ 0\end{array}\right\}\left\{T\right\}$$

(1)

where $T$ is the temperature change above the uniform reference temperature $T_0$.

The geometric relations under the column coordinate system in the small deformation assumption are:

$$\begin{array}{l}{\epsilon }_{\theta \theta }=\frac{1}{r}\frac{\partial u_{\theta }}{\partial \theta }+\frac{u_r}{r},{\epsilon }_{zz}=\frac{\partial u_z}{\partial z},{\epsilon }_{rr}=\frac{\partial u_r}{\partial r},\\ {\epsilon }_{r\theta }=\frac{1}{2}\left(\frac{1}{r}\frac{\partial u_r}{\partial \theta }+\frac{\partial u_{\theta }}{\partial r}-\frac{u_{\theta }}{r}\right),{\epsilon }_{rz}=\frac{1}{2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right),\\ {\epsilon }_{\theta z}=\frac{1}{2}\left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)\end{array}$$

(2)

When the volume force and heat source are both zero, the wave governing equation can be calculated using the GN generalized thermoelastic theory without energy dissipation combined with the equation of motion:

$$\left\{\begin{array}{l}\frac{\partial {\sigma }_{rr}}{\partial r}+\frac{1}{r}\frac{\partial {\sigma }_{r\theta }}{\partial \theta }+\frac{\partial {\sigma }_{rz}}{\partial z}+\frac{{\sigma }_{rr}-{\sigma }_{\theta \theta }}{r}=\rho \frac{{\partial }^2{u}_r}{\partial t^2}\\ \frac{\partial {\sigma }_{r\theta }}{\partial r}+\frac{1}{r}\frac{\partial {\sigma }_{\theta \theta }}{\partial \theta }+\frac{\partial {\sigma }_{\theta z}}{\partial z}+\frac{2{\sigma }_{r\theta }}{r}=\rho \frac{{\partial }^2{u}_{\theta }}{\partial t^2}\\ \frac{\partial {\sigma }_{rz}}{\partial r}+\frac{1}{r}\frac{\partial {\sigma }_{\theta z}}{\partial \theta }+\frac{\partial {\sigma }_{zz}}{\partial z}+\frac{{\sigma }_{rz}}{r}=\rho \frac{{\partial }^2{u}_z}{\partial t^2}\\ K_3\frac{{\partial }^{2}T}{\partial r^2}+K_1\left(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}T}{\partial {\theta }^2}\right)+K_2\frac{{\partial }^{2}T}{\partial z^2}-T_0\frac{{\partial }^2}{\partial t^2}\left({\beta }_2{\epsilon }_{zz}+{\beta }_3{\epsilon }_{rr}+{\beta }_1{\epsilon }_{\theta \theta }\right)=\rho C_e\frac{{\partial }^{2}T}{\partial t^2}\end{array}\right.$$

(3)

In the above Equations (1)–(3), ${\sigma }_{ij}$, $u_i$, and ${\epsilon }_{ij}$ represent stress, displacement, and strain, respectively; $c_{ij}$ represents the elastic constant of the material; $K_i$ is the constant property of the theoretical material; ${\beta }_i$ represents the coefficient of volumetric expansion; and $C_e$ represents the specific heat at a constant strain.

In order to satisfy the stress boundary conditions for the hollow cylinder, a rectangular window function is introduced:

$$\pi \left(r\right)=\left\{\begin{array}{l}1,a\le r\le b\\ 0 elsewhere\end{array}\right.$$

(4)

Introducing this into the constitutive equations yields stress-free boundary conditions at $r=a$ and $r=b$ with ${\sigma }_{rr}={\sigma }_{rz}={\sigma }_{r\theta }=0$:

$$\left\{\begin{array}{l}{\sigma }_{\theta \theta }\\ {\sigma }_{zz}\\ {\sigma }_{rr}\\ {\sigma }_{rz}\\ {\sigma }_{r\theta }\\ {\sigma }_{z\theta }\end{array}\right\}=\left[\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& 0& 0& 0\\ & c_{22}& c_{23}& 0& 0& 0\\ & & c_{33}& 0& 0& 0\\ & & & c_{44}& 0& 0\\ & symmetry& & & c_{55}& 0\\ & & & & & c_{66}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_{\theta \theta }\\ {\epsilon }_{zz}\\ {\epsilon }_{rr}\\ 2{\epsilon }_{rz}\\ 2{\epsilon }_{r\theta }\\ 2{\epsilon }_{z\theta }\end{array}\right\}\pi \left(r\right)-\left\{\begin{array}{l}{\beta }_1\\ {\beta }_2\\ {\beta }_3\\ 0\\ 0\\ 0\end{array}\right\}\left\{T\right\}\pi \left(r\right)$$

(5)

For a free harmonic wave propagating along the longitudinal direction in an infinitely long hollow cylinder, let its displacement and the component of temperature change be:

$$\begin{array}{l}{u}_r\left(r,\theta ,z,t\right)=\exp \left(ikz+in\theta -i\omega t\right)U\left(r\right)\\ u_{\theta }\left(r,\theta ,z,t\right)=\exp \left(ikz+in\theta -i\omega t\right)V\left(r\right)\\ u_z\left(r,\theta ,z,t\right)=\exp \left(ikz+in\theta -i\omega t\right)W\left(r\right)\\ T\left(r,\theta ,z,t\right)=\exp \left(ikz+in\theta -i\omega t\right)X\left(r\right)\end{array}$$

(6)

where $U\left(r\right)$, $V\left(r\right)$, and $W\left(r\right)$ denote the displacement amplitude along the three directions $r$, $\theta$, and $z$, respectively, $X\left(r\right)$ denotes the temperature amplitude along the $r$ direction, $k$ denotes the wave number, $\omega$ is the angular frequency, and $n$ denotes the number of circumferential orders.

Substituting Equations (2), (5) and (6) into Equation (3) to eliminate the strain, the control equation can be obtained as follows:

$$\begin{array}{l}\{inr{c}_{31}{V}^{\prime }+ik{r}^2{c}_{32}{W}^{\prime }+r^2{c}_{33}{U}^{″}-r^2{\beta }_3{X}^{\prime }-n^2{c}_{55}U+inr{c}_{55}{V}^{\prime }-in{c}_{55}V-k^2{r}^2{c}_{44}U\\ +ik{r}^2{c}_{44}{W}^{\prime }+rik{c}_{32}W+r{c}_{33}{U}^{\prime }-r{\beta }_{3}X-in{c}_{11}V-c_{11}U-ikr{c}_{12}W+r{\beta }_{1}X\}\pi \left(r\right)\\ +\{inr{c}_{31}V+r{c}_{31}U+ik{r}^2{c}_{32}W+r^2{c}_{33}{U}^{\prime }-r^2{\beta }_{3}X\}\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]=-r^2\rho {\omega }^{2}U\end{array}$$

(7a)

$$\begin{array}{l}\{inr{c}_{55}{U}^{\prime }+r^2{c}_{55}{V}^{″}-n^2{c}_{11}V+in{c}_{11}U-nkr{c}_{12}W+inr{c}_{13}{U}^{\prime }-inr{\beta }_{1}X\\ -k^2{r}^2{c}_{66}V-nkr{c}_{66}W+in{c}_{55}U+r{c}_{55}{V}^{\prime }-c_{55}V\}\pi \left(r\right)+\{rin{c}_{55}U\\ +r^2{c}_{55}{V}^{\prime }-r{c}_{55}V\}\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]=-r^2\rho {\omega }^{2}V\end{array}$$

(7b)

$$\begin{array}{l}\{ik{r}^2{c}_{44}{U}^{\prime }+r^2{c}_{44}{W}^{″}-nkr{c}_{66}V-n^2{c}_{66}W-nkr{c}_{21}V+ikr{c}_{21}U\\ -k^2{r}^2{c}_{22}W+ik{r}^2{c}_{23}{U}^{\prime }-ik{r}^2{\beta }_{2}X+ikr{c}_{44}U+r{c}_{44}{W}^{\prime }\}\pi \left(r\right)+\{ik{r}^2{c}_{44}U\\ +r^2{c}_{44}{W}^{\prime }\}\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]=-r^2\rho {\omega }^{2}W\end{array}$$

(7c)

$$\begin{array}{l}{r}^2{K}_3{X}^{″}+r{K}_1{X}^{\prime }-n^2{K}_{1}X-k^2{r}^2{K}_{2}X+ik{r}^2{T}_0{\omega }^2{\beta }_{2}W+r^2{T}_0{\omega }^2{\beta }_3{U}^{\prime }\\ +inr{T}_0{\omega }^2{\beta }_{1}V+r{T}_0{\omega }^2{\beta }_{1}U=-r^2\rho C_e{\omega }^{2}X\end{array}$$

(7d)

Since $r^2{T}_0{\omega }^2{\beta }_3{U}^{\prime }$ exists in Equation (7d), we must now address this equation:

$$\begin{array}{l}\rho r\left(r^2{K}_3{X}^{″}+r{K}_1{X}^{\prime }-n^2{K}_{1}X-k^2{r}^2{K}_{2}X\right)+ik{T}_0{\beta }_{2}r\cdot \rho r^2{\omega }^{2}W+T_0{\beta }_{3}r\cdot \rho r^2{\omega }^2{U}^{\prime }\\ +in{\beta }_1{T}_0\rho r^2{\omega }^{2}V+T_0{\beta }_1\rho r^2{\omega }^{2}U=-r^3{\rho }^2{C}_e{\omega }^{2}X\end{array}$$

(8)

Since $\rho r^2{\omega }^2{U}^{\prime }={\left(\rho r^2{\omega }^{2}U\right)}^{\prime }-2\rho r{\omega }^{2}U$, substituting in Equation (8) gives:

$$\begin{array}{l}\rho r\left(r^2{K}_3{X}^{″}+r{K}_1{X}^{\prime }-n^2{K}_{1}X-k^2{r}^2{K}_{2}X\right)+ik{T}_0{\beta }_{2}r\cdot \rho r^2{\omega }^{2}W+T_0{\beta }_{3}r\cdot {\left(\rho r^2{\omega }^{2}U\right)}^{\prime }\\ +in{\beta }_1{T}_0\rho r^2{\omega }^{2}V+T_0\left({\beta }_1-2{\beta }_3\right)\rho r^2{\omega }^{2}U=-r^3{\rho }^2{C}_e{\omega }^{2}X\end{array}$$

(9)

Substituting Equations (7a)–(7c) into Equation (9) above yields:

$$\begin{array}{l}\rho r\left(r^2{K}_3{X}^{″}+r{K}_1{X}^{\prime }-n^2{K}_{1}X-k^2{r}^2{K}_{2}X\right)\\ -ik{T}_0{\beta }_{2}r\left\{\begin{array}{l}\left(\begin{array}{l}ik{r}^2{c}_{44}{U}^{\prime }+r^2{c}_{44}{W}^{″}-nkr{c}_{66}V-n^2{c}_{66}W-nkr{c}_{21}V+ikr{c}_{21}U\\ -k^2{r}^2{c}_{22}W+ik{r}^2{c}_{23}{U}^{\prime }-ik{r}^2{\beta }_{2}X+ikr{c}_{44}U+r{c}_{44}{W}^{\prime }\end{array}\right)\pi \left(r\right)\\ +\left(ik{r}^2{c}_{44}U+r^2{c}_{44}{W}^{\prime }\right)\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]\end{array}\right\}\\ -T_0{\beta }_{3}r{\left\{\begin{array}{l}\left(\begin{array}{l}inr{c}_{31}{V}^{\prime }+ik{r}^2{c}_{32}{W}^{\prime }+r^2{c}_{33}{U}^{″}-r^2{\beta }_3{X}^{\prime }-n^2{c}_{55}U+inr{c}_{55}{V}^{\prime }-in{c}_{55}V-k^2{r}^2{c}_{44}U\\ +ik{r}^2{c}_{44}{W}^{\prime }+rik{c}_{32}W+r{c}_{33}{U}^{\prime }-r{\beta }_{3}X-in{c}_{11}V-c_{11}U-ikr{c}_{12}W+r{\beta }_{1}X\end{array}\right)\pi \left(r\right)\\ +\left(inr{c}_{31}V+r{c}_{31}U+ik{r}^2{c}_{32}W+r^2{c}_{33}{U}^{\prime }-r^2{\beta }_{3}X\right)\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]\end{array}\right\}}^{\prime }\\ -in{T}_0{\beta }_1\left\{\begin{array}{l}\left(\begin{array}{l}inr{c}_{55}{U}^{\prime }+r^2{c}_{55}{V}^{″}-n^2{c}_{11}V+in{c}_{11}U-nkr{c}_{12}W+inr{c}_{13}{U}^{\prime }-inr{\beta }_{1}X\\ -k^2{r}^2{c}_{66}V-nkr{c}_{66}W+in{c}_{55}U+r{c}_{55}{V}^{\prime }-c_{55}V\end{array}\right)\pi \left(r\right)\\ +\left(rin{c}_{55}U+r^2{c}_{55}{V}^{\prime }-r{c}_{55}V\right)\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]\end{array}\right\}\\ -T_0\left({\beta }_1-2{\beta }_3\right)\left\{\begin{array}{l}\left(\begin{array}{l}inr{c}_{31}{V}^{\prime }+ik{r}^2{c}_{32}{W}^{\prime }+r^2{c}_{33}{U}^{″}-r^2{\beta }_3{X}^{\prime }-n^2{c}_{55}U+inr{c}_{55}{V}^{\prime }-in{c}_{55}V-k^2{r}^2{c}_{44}U\\ +ik{r}^2{c}_{44}{W}^{\prime }+rik{c}_{32}W+r{c}_{33}{U}^{\prime }-r{\beta }_{3}X-in{c}_{11}V-c_{11}U-ikr{c}_{12}W+r{\beta }_{1}X\end{array}\right)\pi \left(r\right)\\ +\left(inr{c}_{31}V+r{c}_{31}U+ik{r}^2{c}_{32}W+r^2{c}_{33}{U}^{\prime }-r^2{\beta }_{3}X\right)\left[\delta \left(r-a\right)-\delta \left(r-b\right)\right]\end{array}\right\}\\ =-r^3{\rho }^2{C}_e{\omega }^{2}X\end{array}$$

(10)

The fluctuation equation above can be solved by expanding $U\left(r\right)$, $V\left(r\right)$, and $W\left(r\right)$ into Legendre orthogonal polynomials:

$$\begin{array}{l}U\left(r\right)={\displaystyle \sum _{m=0}^{\infty }{p}_m^1\cdot Q_m\left(r\right)}\\ V\left(r\right)={\displaystyle \sum _{m=0}^{\infty }{p}_m^2\cdot Q_m\left(r\right)}\\ W\left(r\right)={\displaystyle \sum _{m=0}^{\infty }{p}_m^3\cdot Q_m\left(r\right)}\end{array}$$

(11)

This is an isothermal boundary condition that satisfies $T=0$ at $r=a$ and $r=b$. Expand $X\left(r\right)$ as:

$$X\left(r\right)=\left(r-a\right)\left(r-b\right){\displaystyle \sum _{m=0}^{\infty }{p}_m^4\cdot Q_m\left(r\right)}$$

(12)

where

$$Q_m\left(r\right)=\sqrt{\frac{2m+1}{\left(b-a\right)}}{P}_m\left(\frac{2r-\left(b-a\right)}{b-a}\right)$$

where $P_m$ denotes the Legendre polynomial of order m. The value of m ranges from 0 to $\infty$, but in practice, m can only take a finite value of M. In this case, the higher order terms can be considered to be higher order minima and can be ignored.

The 4(M + 1) equations can be obtained by multiplying both sides of Equations (7a)–(7d) concurrently by $Q_j^{*}\left(r\right)$ picked from 0 to M. Then, using Legendre’s polynomials’ orthogonality feature, one can integrate the radius $r$ from $a$ to $b$:

$$\left\{\begin{array}{l}{A}_{11}^{j,m}{p}_m^1+A_{12}^{j,m}{p}_m^2+A_{13}^{j,m}{p}_m^3+A_{14}^{j,m}{p}_m^4=-{\omega }^2\rho \cdot M_m^j{p}_m^1\\ A_{21}^{j,m}{p}_m^1+A_{22}^{j,m}{p}_m^2+A_{23}^{j,m}{p}_m^3+A_{24}^{j,m}{p}_m^4=-{\omega }^2\rho \cdot M_m^j{p}_m^2\\ A_{31}^{j,m}{p}_m^1+A_{32}^{j,m}{p}_m^2+A_{33}^{j,m}{p}_m^3+A_{34}^{j,m}{p}_m^4=-{\omega }^2\rho \cdot M_m^j{p}_m^3\\ A_{41}^{j,m}{p}_m^1+A_{42}^{j,m}{p}_m^2+A_{43}^{j,m}{p}_m^3+A_{44}^{j,m}{p}_m^4=-{\omega }^2{\rho }^2{C}_e\cdot M{T}_m^j{p}_m^4\end{array}\right.$$

(13)

Equation (13) can now be stated as follows:

$$\left[\begin{array}{cccc}{A}_{11}^{j,m}& A_{12}^{j,m}& A_{13}^{j,m}& A_{14}^{j,m}\\ A_{21}^{j,m}& A_{22}^{j,m}& A_{23}^{j,m}& A_{24}^{j,m}\\ A_{31}^{j,m}& A_{32}^{j,m}& A_{33}^{j,m}& A_{34}^{j,m}\\ A_{41}^{j,m}& A_{42}^{j,m}& A_{43}^{j,m}& A_{44}^{j,m}\end{array}\right]\left\{\begin{array}{l}{p}_m^1\\ p_m^2\\ p_m^3\\ p_m^4\end{array}\right\}=-{\omega }^2\left[\begin{array}{cccc}\rho M_m^j& 0& 0& 0\\ 0& \rho M_m^j& 0& 0\\ 0& 0& \rho M_m^j& 0\\ 0& 0& 0& {\rho }^2{C}_{e}M{T}_m^j\end{array}\right]\left\{\begin{array}{l}{p}_m^1\\ p_m^2\\ p_m^3\\ p_m^4\end{array}\right\}$$

(14)

Equation (14) is an eigenvalue problem, where $-{\omega }^2$ denotes the eigenvalue, the phase velocity can be obtained from $V_{ph}=\omega /k$, and $p_m^i$ denotes the displacement as well as the distribution of the temperature field, by solving this eigenvalue problem, the eigenvalues as well as the eigenvectors of the equation can be obtained, and the acceptable solutions are those that tend to be numerically stable as the expansion term M increases.

For the longitudinal guided wave, its axisymmetric modes can be obtained when the circumferential order n = 0. At this point, Equation (7b) is independent of the other three equations, which represent the torsional modes. Equations (7a) and (7c), and the heat conduction Equation (7d) are coupled to represent the coupled thermoelastic wave. Thus, the dispersion characteristics of different modes can be obtained based on suitable fluctuation equations.

(1) When the circumferential order n = 0, there are symmetric modes.

Where the coupled thermoelastic wave longitudinal mode control equation is:

$$\left[\begin{array}{ccc}{A}_{11}^{j,m}& A_{13}^{j,m}& A_{14}^{j,m}\\ A_{31}^{j,m}& A_{33}^{j,m}& A_{34}^{j,m}\\ A_{41}^{j,m}& A_{43}^{j,m}& A_{44}^{j,m}\end{array}\right]\left\{\begin{array}{l}{p}_m^1\\ p_m^3\\ p_m^4\end{array}\right\}=-{\omega }^2\left[\begin{array}{ccc}\rho M_m^j& 0& 0\\ 0& \rho M_m^j& 0\\ 0& 0& {\rho }^2{C}_{e}M{T}_m^j\end{array}\right]\left\{\begin{array}{l}{p}_m^1\\ p_m^3\\ p_m^4\end{array}\right\}$$

Torsional modes:

$$\left[A_{22}^{j,m}\right]\left\{p_m^2\right\}=-{\omega }^2\left[\rho M_m^j\right]\left\{p_m^2\right\}$$

(2) When the circumferential order $n\ne 0$, this time indicates the bending mode, and the dispersion characteristic curve of the bending mode can be obtained according to control Equation (14).

3. Numerical Examples

Based on the above theory, the dispersion characteristics of longitudinal thermoelastic guided waves in hollow cylinders are calculated. In order to verify the correctness of the theoretical analysis and algorithm, the phase velocity dispersion curves of circumferential thermoelastic waves in an orthogonal anisotropic cylinder are first calculated [19] and compared with the results in this paper. Meanwhile, based on the above theory, the phase velocity dispersion curves of longitudinal thermoelastic waves as well as the displacement versus temperature curves in a hollow cylinder of silicon nitride with an inner radius of 9 mm and a thickness of 1 mm are calculated. Then, the effect of the diameter-to-thickness ratio on the longitudinal mode dispersion characteristics is discussed. Finally, the effect of temperature on the propagation characteristics of longitudinal thermoelastic waves was investigated. The material properties of silicon nitride are shown in

Table 1. Material properties of silicon nitride.

Symbol	Units	Quantity
c11	109 N·m−2	574
c12	109 N·m−2	127
c13	109 N·m−2	127
c23	109 N·m−2	195
c22	109 N·m−2	433
c33	109 N·m−2	433
c44	109 N·m−2	119
c55	109 N·m−2	108
c66	109 N·m−2	108
ρ	103 kg·m−3	3.2
Ce	J·kg·deg·m−1	670
β1	106 N·deg−1·m−2	3.22
β2	106 N·deg−1·m−2	2.71
β3	106 N·deg−1·m−2	2.71
K1	W/m·K	55.4
K2	W/m·K	43.5
K3	W/m·K	43.5

.

3.1. Numerical Validation

In order to compare the dispersion relation of circumferential thermoelastic waves with that of a hollow cylinder of silicon nitride in the literature [21], the same material as well as geometrical dimensions were used in this paper for the calculations. The geometrical dimensions in the literature are an inner radius of 9 m and a tube thickness of 1 m. Where Figure 2a,b show the result plots in the references, the result of this paper as a term is taken as M = 8, and the results are shown in Figure 2c,d. From Figure 2, it can be seen that the results of this paper are consistent with those in the literature, i.e., it proves that the theoretical analysis and calculation in this paper are correct.

3.2. Characterization of Longitudinal Thermoelastic Guided Wave

Longitudinal thermoelastic guided wave propagation characteristics in orthotropic anisotropic hollow cylinders are investigated using hollow cylinders of silicon nitride as an example. Based on the theoretical method in the first part, a theoretical model of the longitudinal thermoelastic behavior of silicon nitride is established. It should be pointed out that the cutoff term in the calculation is taken to be M = 10, and the inner radius, outer radius, and temperature of the temperature field in which the hollow cylinder is located are 293 K. Based on the above model, the longitudinal thermoelastic guided wave phase velocity dispersion curves of the hollow cylinder of silicon nitride, as well as the phase velocity dispersion curves when there is no coupling of the heat transfer equation, are calculated numerically, compared, and analyzed, as shown in Figure 3.

As can be seen from the above Figure 3, the dispersion curves of the longitudinal guided waves in the thermoelastic hollow cylinder are very similar to the dispersion curves in the elastic structure when there is no coupled temperature effect. And, due to the coupling of the temperature field, new modes, namely thermal modes, have appeared in the figure. These have been marked in the above figure, from which it can be seen that these thermal modes are very similar to the elastic modes and are very close to the torsional modes. Moreover, all these thermal modes have a cutoff frequency and dispersion behavior, and their phase velocity decreases with the increase in the frequency and finally reaches a steady state.

3.2.1. Longitudinal Modes

In this paper, we focus on the analysis of the dispersion characteristics of longitudinal modes when the circumferential order n = 0. Using the above equations, the longitudinal mode dispersion curves of the thermoelastic waveguide of the hollow cylinder of silicon nitride are solved, and the dispersion characteristics are discussed in comparison with those when there is no coupled temperature field.

In Figure 4b, the phase velocity curves without the coupled temperature term are shown in red, and the phase velocity with the coupled temperature term is shown in black. It can be seen by comparison that for the L(0,1) mode, the difference between the two cases is very small. There is a slight difference at lower frequencies, and the difference between the two curves gradually decreases and finally tends to coincide with the increase in frequency. It shows that the effect on the L(0,1) mode is small. For the L(0,2) mode, the difference is small at the cutoff frequency, and the difference gradually increases with the increase in frequency. The difference is largest in the 4–6 MHz interval, while it can be seen that the convergence frequency is higher when the temperature is not added. For the other modes, there are generally differences near the cutoff frequency, and the convergence frequency decreases after the temperature field is added.

3.2.2. Displacement Distribution and Temperature Variation Distribution of Longitudinal Modes

Figure 5 shows the displacement distribution curves of the first three longitudinal modes at k = 2 and the temperature change curves. From the figure, it can be seen that the magnitude of the displacement change in the mass point in the elastic mode is much larger than that in the thermal mode. Meanwhile, there is no circumferential component in the displacement of the mass in the longitudinal mode. For the L (0,1) mode, the radial component $u_r$ of the displacement is dominant in the wall thickness range, and the axial component $u_z$ has a weak influence. Moreover, the trend of change shows that the radial displacement varies relatively little over the entire range of wall thicknesses, whereas the axial displacement component shows a large amplitude variation. The magnitude of temperature variation is also larger for this mode. For the L (0,2) mode, the axial component of the mass displacement, $u_z$, varies in a smaller magnitude, and the radial displacement of the mass, $u_r$, varies in a larger range. For the thermal mode represented by mode 3, the magnitude of the displacement changes at the plasma point decreases sharply at this point compared to the previous two modes. The mass displacement of this mode is similar to that of the L (0,2) mode. The axial displacement component $u_z$ changes by a smaller magnitude, and the radial displacement component $u_r$ changes by a larger magnitude.

The above analysis shows that the amplitude of the mass displacement of the thermal mode is much smaller than that of the first two modes. However, the amplitude of temperature change in these three modes is similar, and the boundary condition of isotherm is satisfied on both the inner and outer surfaces.

3.3. Effect of Diameter-to-Thickness Ratio on the Longitudinal Modes of Thermoelastic Guided Waves

In this section, the effect of different diameter-to-thickness ratios on the dispersion relation of thermoelastic guided waves is investigated. That is, the phase velocity dispersion curves of four longitudinal thermoelastic guided wave longitudinal modes at the same wall thickness and different diameter–thickness ratios are calculated. The thickness is taken as 1 mm, and the outer diameter is taken as 5 mm, 10 mm, 15 mm, and 20 mm, and their diameter-to-thickness ratios are 5, 10, 15, and 20.

The calculation is carried out with the up-to-term M = 10 and the circumferential order n = 0. The results are shown in Figure 6 below, where the diameter to thickness ratio is denoted by η.

In order to facilitate the analysis of the change rule of phase velocity under the above four diameter-to-thickness ratios, the first three modes in the above four diagrams are combined into the following Figure 7 for analysis. By analyzing the local enlargement of the three regions in Figure 7a–c, it can be seen that the effect of the diameter–thickness ratio on the dispersion of the longitudinal modes is as follows: In the dispersion curves of the first two modes, with the reduction in the diameter–thickness ratio, the cutoff frequency is gradually increasing. That is, in the figure, it is expressed as the curve shifted to the right. And, for the thermal modes, the cutoff frequency with the reduction in the diameter–thickness ratio is gradually reduced. The two modes show opposite trends.

3.4. Effect of Different Reference Temperatures on the Propagation Characteristics

To investigate the effect of temperature on the propagation properties of longitudinally guided waves in hollow cylinders, based on the above theory, the longitudinal mode dispersion curves at different temperatures are numerically solved by taking the temperature gradient as 50 K. In this section, the dispersion curves of the first three modes are selected for ease of analysis.

Figure 8a–c are the local magnification diagrams of the three modes. It can be seen that under the given temperature range, the phase velocities of the three modes show a corresponding change rule. That is, with the gradual increase in temperature, the phase velocity values of the three modes show a decreasing trend. Therefore, by studying the effect of temperature on the propagation characteristics of guided waves, it can provide a certain theoretical basis and guidance for the non-destructive testing of hollow cylindrical pipes at high temperatures.

4. Conclusions

In this paper, based on the GN generalized thermoelastic model, a theoretical model of guided wave propagation in hollow cylindrical pipes containing a temperature field is established. The longitudinal thermoelastic guided waves and the curves of displacement and temperature are solved. The effects of diameter-to-thickness ratio and temperature on the propagation characteristics of guided waves in hollow cylinders are investigated. The following conclusions can be drawn:

(1) Firstly, based on the GN generalized thermoelastic theoretical model, the governing equations of the thermoelastic guided wave are obtained by coupling the heat conduction equations. On the basis of satisfying the stress-free as well as isothermal boundary conditions, the acquisition of the dispersion curve of the thermoelastic guided wave is realized by using Legendre polynomials.

(2) It is analytically obtained that the coupled heat conduction equation leads to a thermal mode. The mode is similar to the elastic mode, with a cutoff frequency as well as a dispersion phenomenon. Also, by solving the displacement and temperature variation curves, it is determined that the amplitude of the mass displacement of the thermal mode is much smaller than that of the other modes, but the amplitude of their temperature variation is close to the same value.

(3) The effect of the diameter-to-thickness ratio on the propagation characteristics is obtained by solving the longitudinal guided wave dispersion curves at different diameter-to-thickness ratios. For L(0,1) and L(0,2), the cutoff frequency gradually increases with the decrease in the diameter–thickness ratio. For the thermal mode, the cutoff frequency of the mode decreases as the diameter–thickness ratio decreases.

(4) The phase velocity dispersion curve from 293 K to 593 K is calculated with a temperature gradient of 50 K. By analyzing the first three modes, the results show that with the gradual increase in temperature, the phase velocity shows a decreasing trend.

(5) In this paper, the propagation characteristics of longitudinal thermoelastic waves in hollow cylinders are studied by coupling GN generalized thermoelastic theory without energy dissipation. Although the linear and small deformation assumptions based on the model simplify the calculation model and make it more feasible in numerical calculation, these assumptions may be challenged under some high-frequency or large-deformation conditions, and there are some deviations. Therefore, in future studies, we plan to expand the existing model, consider the addition of a nonlinear strain–displacement relationship, and supplement the influence of other high-frequency effects to continue to improve the model.