An Innovated Vibration Equation for Longitudinal Plate by Using the Symmetric and Asymmetric Spectral Decomposition

Abstract

Thick wall structures involving longitudinal wave are typically utilized in aerospace engineering, nuclear power engineering, precision transmission device design, and pressure vessels design. Consequently, developing sophisticated dynamic models for thick plates is of paramount importance. However, the commonly used longitudinal vibration equation is of the second order, which is regarded as a plane stress problem. Its dispersion curve is a straight line, which cannot describe the actual dispersion in the plate. In this paper, the spectral analysis of Navier equation describing three-dimensional elasto-dynamics is carried out by using the symmetric and asymmetric spectral decomposition theory of differential operators and introducing the concept of virtual differential operators. The infinite product operator series describing longitudinal vibration are truncated into fourth order. The governing equation of longitudinal vibration consists of a fourth-order wave equation and a second-order wave equation. Owing to the fact that no a priori assumptions were introduced during the derivation of its dynamic equations, the proposed plate dynamic model boasts higher precision and is applicable across a broader frequency spectrum and for plates with greater thicknesses. This is a breakthrough in the longitudinal vibration equation of plates.

1. Introduction

Thick plates are widely used under dynamic loads in aerospace, nuclear power engineering, ocean engineering, civil engineering, and mechanical engineering [1,2,3,4,5]. Elastic wave techniques offer a means to characterize and simulate the stress–strain conditions generated by various dynamic loads in solid media or structures [6,7]. The propagation and scattering of elastic waves, dynamic stress concentration, and vibration localization in thick plates stand as significant frontier topics in the field of mechanics. Research into these areas can drive advancements and innovations in classical structural dynamics and problem-solving methodologies.

The classical bending vibration equation of thin plate is fourth-order [8]. The bending vibration governing equation of Mindlin thick plate commonly used in engineering is composed of a fourth-order wave equation and a second-order wave equation [9,10,11]. This set of partial differential equations has great advantages over other theories. However, for the engineering design of longitudinal vibration plate considering stress concentration, the theoretical results of plane stress problem in elasticity are mostly used [12]. The solution of plane stress problem uses the average stress and average displacement in the thickness direction of the plate in the process of constructing the solution, rather than the corresponding values at each point in the field [13,14,15]. The commonly used longitudinal vibration equation is of the second order, which is regarded as a plane stress problem. Its dispersion curve is a straight line, which cannot describe the actual dispersion in the plate. Therefore, when the displacement in the thickness direction of the plate changes sharply, the solution of plane stress may lead to wrong results, and there will be large errors in the analysis of high-frequency vibration stress of the plate in the plane. When studying the propagation of tension compression elastic wave, it is found that the plane stress approximation can be used to deal with the longitudinal vibration problem of plates only when the frequency is small and the wavelength is large.

Historically, in engineering design, the solution to the plane elasticity problem has frequently been employed to examine stress concentrations, rather than the solution for stretching plates. Examples include the Kirsch solution [12] for the elasticity problem and the plane stress problem in elastic dynamics. Nevertheless, the plane stress solution is defined in terms of average stresses and average displacements through the thickness of the plate, not in terms of the specific values at each individual point within the field [16]. This can lead to incorrect results when there are significant variations in displacements across the thickness. Such a situation arises when the plate undergoes longitudinal vibration at high frequencies. A comparison [17] with results based on the three-dimensional theory of elasticity indicates that the generalized plane stress approximation for a plate is only valid when the frequency is low and the wavelength is long. It is clear that the two-dimensional model differs significantly from the actual structure when addressing stress concentration problems. For these reasons, it is essential to establish accurate thick-walled stretching plate models to solve stress concentration issues in structures. Many researchers have put forward refined theories [18,19,20,21] to minimize engineering errors.

Hu Chao et al. [13,17] established an accurate dynamic equation for the bending vibration of a thick plate based on the Boussinesq-Galerkin general solution form of elastic dynamics and the theory of partial differential operators. His thick plate theory abandoned the basic assumptions of classical thin plate theory and used other assumptions based on three-dimensional elastic theory to study the mechanical problems of thick plates. However, the influence of lateral loads on the vibration of thick plate structures is not considered in this paper. Existing literature indicates that for low dimensional structures some engineering assumptions need to be adopted to establish vibration equations for thick plate structures [22,23,24]. For example, assuming the mechanical quantity as a polynomial of spatial coordinates, the deflection remains unchanged along the thickness direction and does not consider the effects of lateral compression deformation. However, research has shown that using polynomial assumptions can lead to incompatibility between the fundamental equations.

Although these refined theories have revised and improved plane stress theories, they are essentially based on geometric approaches. The models are relatively simplistic because engineering assumptions were still utilized during their derivation, which has led to various limitations in their application to thick-walled structures, particularly when plates vibrate at higher frequencies. In this paper, departing from the geometric perspective and adopting an algebraic viewpoint, we propose a new and accurate elasto-dynamics theory for stretching plates. In the derivation process, we make use of the general solution proposed by Boussinesq-Galerkin and the operator theory of partial differential equations. By using the symmetric and asymmetric spectral decomposition theory of differential operators and introducing the concept of virtual differential operators, we obtain the refined elasto-dynamics equations for stretching plates. Since the dynamic equation is derived without any prior assumptions, the proposed dynamic equation for the plate is more accurate and can be applied over a broader frequency range and for greater thicknesses.

2. Derivation of Exact Wave Motion Equation for Stretching Thick Wall Structure

Based on the exact plate theory for the bending plate in reference [15], an exact elasto-dynamics theory without assumption for bending vibrations is presented by using the formal solution proposed by Boussinesq-Galerkin (B-G solution), and its dynamic equations are obtained under appropriate gauge conditions. The key point in Reference [15] is that the displacement at any point in the thick wall structure is derived using the Taylor series expansion of the exponential operator function. Nevertheless, the exact theory study is only for the flexural wave equation of bending plate. The longitudinal waves closely related to underwater targets have not been studied. In this paper, we focus on the exact dynamic equation of longitudinal wave.

According to the three-dimensional elasto-dynamics theory, the governing equation of the spatial displacement field is the Navier equation as follows:

$$\mu {\nabla }^{2}u+\left(\lambda +\mu \right)\nabla \left(\nabla \cdot u\right)=\rho {\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2}},$$

(1)

where $\mu , \lambda$ are the Lame constants, $\nabla =i\mathsf{\partial }/\mathsf{\partial }x+j\mathsf{\partial }/\mathsf{\partial }y+k\mathsf{\partial }/\mathsf{\partial }z$, and $\rho$ is the density.

In Equation (1), based on the Boussinesq-Galerkin solution (B-G solution), the solution is given as

$$u=2\left(1-\nu \right)\left({\nabla }^2-{\displaystyle \frac{1}{c_1^2}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}^2}}\right)G-\nabla \left(\nabla \cdot G\right),$$

(2)

where $c_1, c_2$ are longitudinal wave velocity and transverse wave velocity, $\nu$ is the Poisson ratio, and $G=\left(G_1,G_2,G_3\right)$ is the Somigliana vector potential function which satisfies the following relation as

$$\left({\nabla }^2-{\displaystyle \frac{1}{c_1^2}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}^2}}\right)\left({\nabla }^2-{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}^2}}\right)G=0.$$

(3)

Using the Taylor series expansion of the exponential operator function, the displacement at any point in the plate can be written as

$$u_x\left(x,y,z\right)=\exp \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_x\left(x,y,0\right),$$

(4a)

$$u_y\left(x,y,z\right)=\exp \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_y\left(x,y,0\right),$$

(4b)

$$u_z\left(x,y,z\right)=\exp \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_z\left(x,y,0\right).$$

(4c)

The fluctuation of plate stretching is a case of symmetric motion, and Equation (4) can be written as

$$u_x\left(x,y,z\right)=\sinh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_x\left(x,y,0\right),$$

(5a)

$$u_y\left(x,y,z\right)=\sinh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_y\left(x,y,0\right),$$

(5b)

$$u_z\left(x,y,z\right)=\cosh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)u_z\left(x,y,0\right),$$

(5c)

where $\sinh \left(\cdot \right)$ is hyperbolic sine function and $\cosh \left(\cdot \right)$ is hyperbolic cosine function.

The B-G solution can be written as

$$G_j\left(x,y,z\right)=\sinh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)G_j\left(x,y,0\right)=\cosh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right){\displaystyle \sum _{i=1}^2{G}_j^i\left(x,y,0\right)},$$

(6a)

$$G_3\left(x,y,z\right)=\cosh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right)G_3\left(x,y,0\right)=\cosh \left(z{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\right){\displaystyle \sum _{i=1}^2{G}_3^i\left(x,y,0\right)},$$

(6b)

where $G=G^1+G^2$, and $\left({\square }_j^2+{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{z}^2}}\right)G^j=0$; here, ${\square }_j^2={\nabla }^2-{\displaystyle \frac{1}{c_j^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}, \left(j=1,2\right)$ is the Lorentz operator.

The trigonometric function operator can be written as

$${\displaystyle \frac{\sin \left(z{\square }_j\right)}{{\square }_j}}={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{1}{\left(2n+1\right)!}{z}^{2n+1}}{\square }_j^{2n},$$

(7a)

$$\cos \left(z{\square }_j\right)={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{1}{\left(2n\right)!}{z}^{2n}}{\square }_j^{2n},$$

(7b)

here, $j=1,2$.

And we also can obtain the following relation as

$${\nabla }_{0}G={\displaystyle \frac{\mathsf{\partial }{G}_1}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }{G}_2}{\mathsf{\partial }y}}+{\displaystyle \frac{\mathsf{\partial }{G}_3}{\mathsf{\partial }z}}={\displaystyle \sum _{j=1}^2\frac{\sin \left(z{\square }_j\right)}{{\square }_j}}\left({\displaystyle \frac{\mathsf{\partial }{g}_1^j}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }{g}_2^j}{\mathsf{\partial }y}}-{\square }_j^2{g}_3^j\right).$$

(8)

For the sake of avoiding the non-uniqueness of unknown functions, two gauge conditions are adopted as follows:

$${\displaystyle \frac{\mathsf{\partial }{g}_1^j}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }{g}_2^j}{\mathsf{\partial }y}}=0,\left(j=1,2\right).$$

(9)

Equation (8) can be written as

$$\begin{array}{cc}\hfill {\nabla }_0\left({\nabla }_0\cdot G\right)=& -{\displaystyle \sum _{j=1}^2\left[\frac{\sin \left(z{\square }_j\right)}{{\square }_j}{\square }_j^2\frac{\mathsf{\partial }}{\mathsf{\partial }x}{g}_3^j\right]}i-{\displaystyle \sum _{j=1}^2\left[\frac{\sin \left(z{\square }_j\right)}{{\square }_j}{\square }_j^2\frac{\mathsf{\partial }}{\mathsf{\partial }y}{g}_3^j\right]}j\hfill \\ & -{\displaystyle \sum _{j=1}^2\left[\cos \left(z{\square }_j\right){\square }_j^2{g}_3^j\right]}k\hfill \end{array}.$$

(10)

The displacement in the plate can be expressed as

$$u=2\left(1-\nu \right)\left({\square }_1^2+{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{z}^2}}\right)G-{\nabla }_0\left({\nabla }_0\cdot G\right).$$

(11)

Its component-wise expressions can be written as

$$u_x={\displaystyle \frac{\sin (z{\square }_2)}{{\square }_2}}{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{g}_1^2+{\displaystyle \sum _{j=1}^2\frac{\sin (z{\square }_j)}{{\square }_j}}{\square }_j^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}{g}_3^j,$$

(12a)

$$u_y={\displaystyle \frac{\sin (z{\square }_2)}{{\square }_2}}{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{g}_2^2+{\displaystyle \sum _{j=1}^2\frac{\sin (z{\square }_j)}{{\square }_j}}{\square }_j^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}{g}_3^j,$$

(12b)

$$u_z=\cos (z{\square }_2){\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{g}_3^2+{\displaystyle \sum _{j=1}^2\cos (z{\square }_j){\square }_j^2{g}_3^j}.$$

(12c)

Considering the neutral surface displacement and the normal angle, the generalized displacement in the plate can be expressed as

$${\psi }_x={\left.-{\displaystyle \frac{\mathsf{\partial }{u}_x}{\mathsf{\partial }z}}\right|}_{z=0}=-{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{g}_1^2-{\displaystyle \sum _{j=1}^2{\square }_j^2}{\displaystyle \frac{{\mathsf{\partial }}^2{g}_3^j}{\mathsf{\partial }x}},$$

(13a)

$${\psi }_y={\left.-{\displaystyle \frac{\mathsf{\partial }{u}_y}{\mathsf{\partial }z}}\right|}_{z=0}=-{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}{g}_2^2-{\displaystyle \sum _{j=1}^2{\square }_j^2}{\displaystyle \frac{{\mathsf{\partial }}^2{g}_3^j}{\mathsf{\partial }y}},$$

(13b)

$$w={\left.u_z\right|}_{z=0}={\nabla }^2{g}_3^2+{\square }_1^2{g}_3^1.$$

(13c)

The rotational normal angle to the neutral surface can be expressed as

$${\psi }_x={\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }y}}, {\psi }_y={\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }y}}-{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }x}}.$$

(14)

The functions $g_1^2, g_2^2, g_3^2$ can be expressed by the neutral surface displacement and normal angle as

$$g_1^2=-T_2^{-2}\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }y}}+{\displaystyle \frac{\mathsf{\partial }E}{\mathsf{\partial }x}}\right),$$

(15a)

$$g_2^2=T_2^{-2}\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }E}{\mathsf{\partial }y}}\right),$$

(15b)

$$g_3^1=-T_2^{-2}{\square }_1^{-2}\left({\square }_1^{2}w+{\nabla }^{2}F\right),$$

(15c)

$$g_3^2=T_2^{-2}\left(F+w-E\right).$$

(15d)

where $T_j$ are time differential operators $T_j^2={\displaystyle \frac{1}{c_j^2}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}^2}},\left(j=1,2\right)$, ${\nabla }^{2}E=0$, $F=-{\square }_1^2{g}_3^1-{\square }_2^2{g}_3^2+E$.

In this way, the displacement can be derived as

$$\begin{array}{ll}\hfill u_x=& {\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin (z{\square }_j)}{{\square }_j}}{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}-{\displaystyle \frac{\sin (z{\square }_2)}{{\square }_2}}\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}\right)\hfill \\ & -T_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin (z{\square }_j)}{{\square }_j}}{\nabla }^2\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}\right)\hfill \end{array},$$

(16a)

$$\begin{array}{ll}\hfill u_y=& {\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin (z{\square }_j)}{{\square }_j}}{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }y}}-{\displaystyle \frac{\sin (z{\square }_2)}{{\square }_2}}\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }y}}-{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }x}}\right)\hfill \\ & -T_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin (z{\square }_j)}{{\square }_j}}{\nabla }^2\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }y}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }y}}\right)\hfill \end{array},$$

(16b)

$$u_z=\cos \left(z{\square }_1\right)w-T_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}\cos \left(z{\square }_2\right){\nabla }^2\left(F+w\right).$$

(16c)

In line with Hooke’s law, the stress components can be expressed as

$$\begin{array}{ll}\hfill {\tau }_{zx}=& 2{\mu }_M\cos \left(z{\square }_1\right){\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}-{\mu }_M\cos \left(z{\square }_2\right)\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }y}}\right)\hfill \\ & -2{\mu }_M{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}\cos \left(z{\square }_j\right){\nabla }^2\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}\right)\hfill \end{array},$$

(17a)

$$\begin{array}{ll}\hfill {\tau }_{zy}=& 2{\mu }_M\cos \left(z{\square }_1\right){\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }y}} -{\mu }_M\cos \left(z{\square }_2\right)\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }y}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }y}}-{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }x}}\right)\hfill \\ & -2{\mu }_M{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}\cos \left(z{\square }_j\right){\nabla }^2\left({\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }y}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }y}}\right) \hfill \end{array},$$

(17b)

$$\begin{array}{ll}\hfill {\sigma }_z=& 2{\mu }_M{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin \left(z{\square }_j\right)}{{\square }_j}}{\nabla }^2{\nabla }^2(\mathrm{F}+w)\hfill \\ & -({\lambda }_M+2{\mu }_M)T_1^{-2}{T}_2^{-2}{\displaystyle \frac{\sin \left(z{\square }_1\right)}{{\square }_1}}{\nabla }^2(\mathrm{F}+w)\hfill \\ & +({\lambda }_M+2{\mu }_M)T_1^2{\displaystyle \frac{\sin \left(z{\square }_1\right)}{{\square }_1}}w\hfill \\ & -2{\mu }_M{\displaystyle \frac{\sin \left(z{\square }_1\right)}{{\square }_1}}{\nabla }^{2}w+2{\mu }_M{\displaystyle \frac{\sin \left(z{\square }_2\right)}{{\square }_2}}{\nabla }^2(\mathrm{F}+w)\hfill \end{array}.$$

(17c)

Considering the normal stress boundary condition, the equalities can be given from Equation (17c) as follows:

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left[2\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(1)}+2{\square }_2^2{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}\cos \left({\displaystyle \frac{h}{2}}{\square }_j\right)(W+F^{(1)})\right.\\ \left.-\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)(W+F^{(1)})\right]\pm {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left[2{\square }_1\sin \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(2)}\right.+\\ 2T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}{\square }_j\sin \left(\frac{h}{2}{\square }_j\right)({\square }_1^2{F}^{(2)}+E)}\left.-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}({\square }_1^2{F}^{(2)}+E)\right]\\ ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left[\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(1)}\pm {\square }_2\sin \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(2)}\right]\end{array},$$

(18a)

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left[2\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(1)}+2{\square }_2^2{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right)(W+F^{(1)})}\right.\\ \left.-\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)(W+F^{(1)})\right]\pm {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left[2{\square }_1\sin \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(2)}+\right.\\ 2T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}{\square }_j\sin \left(\frac{h}{2}{\square }_j\right)({\square }_1^2{F}^{(2)}+E)}\left.-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}({\square }_1^2{F}^{(2)}+E)\right]\\ ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left[\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(1)}\pm {\square }_2\sin \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(2)}\right]\end{array},$$

(18b)

$$\begin{array}{l}\left(\lambda +2\mu \right)\left\{-\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)\left[\left({\square }_1^2{F}^{\left(2\right)}+E\right)+T_1^2{F}^{\left(2\right)}\right]\right.\\ \left.\pm {\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}\left[\kappa {\square }_2^2\left(W+F^{\left(1\right)}\right)+T_1^2{F}^{\left(1\right)}\right]\right\}\\ -2\mu {\nabla }^2\left[T_1^{-2}{\displaystyle \sum _{j=1}^2{\left(-1\right)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right)\left({\square }_1^2{F}^{\left(2\right)}+E\right)+\cos \left(\frac{h}{2}{\square }_1\right)F^{\left(2\right)}}\right]\\ \pm 2\mu {\nabla }^2\left[\sin \left({\displaystyle \frac{h}{2}}{\square }_1\right){\square }_1^{-1}{F}^{\left(1\right)}+{\square }_2^2{T}_2^{-2}{\displaystyle \sum _{j=1}^2{\left(-1\right)}^{j-1}\frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}\left(W+F^{\left(1\right)}\right)}\right]\\ =\pm {\displaystyle \frac{1}{2}}q+{\displaystyle \frac{1}{2}}q\end{array},$$

(18c)

where $h$ is the thickness of plates.

Based on the Riemann condition of complex variable function theory, Equation (18a) and Equation (18b) can be transformed into

$$\begin{array}{l}2\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(1)}+2{\square }_2^2{T}_2^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right)(W+F^{(1)})}\\ -\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)(W+F^{(1)})\pm \left[2{\square }_1\sin \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(2)}+2T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}\right.\\ \times \left.{\square }_j\sin \left({\displaystyle \frac{h}{2}}{\square }_j\right)({\square }_1^2{F}^{(2)}+E)-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}({\square }_1^2{F}^{(2)}+E)\right]=0\end{array},$$

(19a)

$$\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(1)}\pm {\square }_2\sin \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(2)}=0.$$

(19b)

The vibration of plate structure is decomposed into the symmetric and asymmetric antisymmetric motions. As we see, the bending vibration of plate is an asymmetric antisymmetric motion, and the stretching vibration of plate is the symmetric movement. The following equations can be derived by separating the symmetric and asymmetric antisymmetric functions as

$$\begin{array}{l}{\square }_1\sin \left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(2)}+T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}{\square }_j\sin \left(\frac{h}{2}{\square }_j\right)({\square }_1^2{F}^{(2)}+E)}\\ -{\displaystyle \frac{1}{2\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}({\square }_1^2{F}^{(2)}+E)=0\end{array}.$$

(20a)

$${\square }_2\sin \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{(2)}=0.$$

(20b)

According to the integral function theory, Equation (20b) can be expanded in series as

$${\square }_2\sin \left({\displaystyle \frac{h}{2}}{\square }_2\right)f^{\left(2\right)}={\square }_2^2{\displaystyle \prod _{m=1}^{\infty }\left[1-\frac{h^2{\square }_2^2}{4m^2{\pi }^2}\right]f^{\left(2\right)}}.$$

(21)

By means of truncating the infinite series, the wave motion equation to describe the shear field is given as

$${\nabla }^{2}f-{\displaystyle \frac{1}{c_2^2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}f=0.$$

(22)

The following equation can be obtained by separating the symmetric functions from the mid-plane of Equation (18c) as

$$\begin{array}{l}{\displaystyle \frac{1}{2}}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)\left[{\displaystyle \frac{1}{\kappa }}({\square }_1^2{F}^{(2)}+E)+T_2^2{F}^{(2)}\right]\\ +{\nabla }^2\left[T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right)({\square }_1^2{F}^{(2)}+E)+\cos }\left({\displaystyle \frac{h}{2}}{\square }_1\right)F^{(2)}\right]=-{\displaystyle \frac{1}{4\mu }}q\end{array}$$

(23)

According to Equations (20a) and (23), the governing equations of the transverse normal strain function $E$ and the generalized displacement potential function $F^{\left(2\right)}$ can be obtained as

$$\left[\begin{array}{l}{D}_{11}\begin{array}{cc}& D_{12}\end{array}\\ D_{21}\begin{array}{cc}& D_{22}\end{array}\end{array}\right]\left[\begin{array}{l}E\\ F^{\left(2\right)}\end{array}\right]=\left[\begin{array}{l}0\\ {\displaystyle \frac{q}{4\mu }}\end{array}\right],$$

(24)

where the expression of each operator is

$$\begin{array}{l}{D}_{11}=2T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{\nabla }^2-2{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}+{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}},\\ D_{12}=2T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{\nabla }^2{\square }_1^2+{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}{\square }_1^2,\\ D_{21}=T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right){\nabla }^2}-{\displaystyle \frac{1}{2\kappa }}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right),\\ D_{22}=T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}\cos \left(\frac{h}{2}{\square }_j\right){\nabla }^2{\nabla }^2}-{\displaystyle \frac{1}{2\kappa }}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right){\nabla }^2+\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right){\nabla }^2.\end{array}$$

According to the determinant of the operator matrix Equation (24), the equation of the transverse normal strain function $E$ can be obtained as

$$\begin{array}{l}\left[T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{\nabla }^2-2{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}+{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}\right]E\\ +\left[T_1^{-2}{\displaystyle \sum _{j=1}^2{(-1)}^{j-1}}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_j\right)}{{\square }_j}}{\nabla }^2{\square }_1^2+{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}{\square }_1^2\right]F=0\end{array}.$$

(25a)

$$\begin{array}{c}2T_1^{-2}\left[{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)-{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right)\right]{\nabla }^2{\nabla }^{2}E\\ -2\left[{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_1\right)}{{\square }_1}}\cos \left({\displaystyle \frac{h}{2}}{\square }_2\right)\right.\left.-{\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right){\nabla }^{2}E\right]\\ -{\displaystyle \frac{1}{2\kappa }}\cos \left({\displaystyle \frac{h}{2}}{\square }_1\right){\displaystyle \frac{\sin \left(\frac{h}{2}{\square }_2\right)}{{\square }_2}}{T}_2^{2}E=-{\displaystyle \frac{1}{4\mu }}{D}_{12}q\end{array}.$$

(25b)

After the truncation of the infinite order operator series, the fourth-order wave equation of the stretching vibration of the plate can be obtained as

$$\begin{array}{l}\left[\left(3-4\kappa \right){\nabla }^2-\left(1-2{\kappa }^2\right)T_2^2-{\displaystyle \frac{24}{h^2}}\left(1-2\kappa \right)\right]E\\ +\left[\left(3-2\kappa \right){\nabla }^2-T_2^2-{\displaystyle \frac{24}{h^2}}\right]\left({\nabla }^2-T_1^2\right)F^{\left(2\right)}=0\end{array},$$

(26a)

$$\begin{array}{l}{\nabla }^2{\nabla }^{2}E-12\left[{\displaystyle \frac{1}{h^2}}+{\displaystyle \frac{2-{\kappa }^2}{24\left(1-\kappa \right)}}{T}_2^2\right]{\nabla }^{2}E\\ +{\displaystyle \frac{3}{1-\kappa }}\left({\displaystyle \frac{1}{h^2}}+{\displaystyle \frac{1+3\kappa }{24}}{T}_2^2\right)T_2^{2}E\\ =-{\displaystyle \frac{1}{16\left(1-\kappa \right)\mu }}\left[2\left(1-\kappa \right){\nabla }^2+{\square }_2^2-{\displaystyle \frac{24}{h^2}}\right]{\square }_1^{2}q\end{array}.$$

(26b)

Without loss of generality, the solution of vibration harmonic of the problem is studied, set as

$$E=\tilde{E}{\mathrm{e}}^{-\mathrm{i}\omega t}, F^{\left(2\right)}={\tilde{F}}^{\left(2\right)}{\mathrm{e}}^{-\mathrm{i}\omega t}, f^{\left(2\right)}={\tilde{f}}^{\left(2\right)}{\mathrm{e}}^{-\mathrm{i}\omega t},$$

(27)

where $\omega$ is angular frequency of stretching plate.

In the following analysis, the time factor and the symbol ‘~’ in the generalized displacement functions are omitted. Taking Equation (27) into Equation (26b), the following equations can be expressed as

$$\underset{j=1}{\overset{2}{\Pi }}({\nabla }^2+{\alpha }_j^2)E=0,$$

(28a)

$${\nabla }^2{f}^{\left(2\right)}+k_2^2{f}^{\left(2\right)}=0,$$

(28b)

where ${\alpha }_j\left(j=1,2\right)$ are traveling wave numbers which satisfy the following expression

$${\alpha }^4+12\left[{\displaystyle \frac{1}{h^2}}-{\displaystyle \frac{2-{\kappa }^2}{24\left(1-\kappa \right)}}{k}_2^2\right]{\alpha }^2-{\displaystyle \frac{3}{1-\kappa }}\left({\displaystyle \frac{1}{h^2}}-{\displaystyle \frac{1+3\kappa }{24}}{k}_2^2\right)k_2^2=0,$$

(28c)

here, $k_2={\displaystyle \frac{\omega }{c_2}}$.

3. Different Dispersion Relations of Different Theories

The dispersion curve is an important verification tool for analytical or numerical solutions of vibration equations. If the dispersion curve derived from the theoretical vibration equation is consistent with the experimental measurement results, it can prove the correctness of the equation. For numerical methods, dispersion curves can be used to verify the accuracy of the model. When the wavelength is much larger than the plate thickness, the longitudinal wave in the plate can be considered as non-dispersive. The longitudinal wave dispersion equation based on the classical thin plate theory is presented as

$${\displaystyle \frac{c_p}{c_1}}=\sqrt{{\displaystyle \frac{E}{\rho \left(1-{\nu }^2\right)}}/{\displaystyle \frac{E\left(1-\nu \right)}{\rho \left(1+\nu \right)\left(1-2\nu \right)}}}={\displaystyle \frac{\sqrt{\left(1+\nu \right)\left(1-2\nu \right)}}{1-\nu }},$$

(29)

When the wavelength is smaller than the plate thickness, the plate needs to be considered as three-dimensional. Its longitudinal wave dispersion equation derived from Rayleigh–Lamb equation is presented as

$${\displaystyle \frac{\tanh \left(kh\sqrt{1-\frac{{c_p}^2}{{c_1}^2}}\right)}{\tanh \left(kh\sqrt{1-\frac{1}{\kappa }\frac{{c_p}^2}{{c_1}^2}}\right)}}={\displaystyle \frac{{\left(2-\frac{1}{\kappa }\frac{{c_p}^2}{{c_1}^2}\right)}^2}{4\sqrt{\left(1-\frac{{c_p}^2}{{c_1}^2}\right)\left(1-\frac{1}{\kappa }\frac{{c_p}^2}{{c_1}^2}\right)}}},$$

(30)

The dispersion equation based on the exact plate theory in this paper can be derived from Equation (28) as follows:

$$\begin{array}{l}{\displaystyle \frac{1+3\kappa }{8{\kappa }^2\left(1-\kappa \right)}}{\left(\alpha h\right)}^2{\left({\displaystyle \frac{c_p}{c_1}}\right)}^4-\left[{\displaystyle \frac{2-{\kappa }^2}{2\left(1-\kappa \right)\kappa }}{\left(\alpha h\right)}^2+{\displaystyle \frac{3}{\left(1-\kappa \right)\kappa }}\right]{\left({\displaystyle \frac{c_p}{c_1}}\right)}^2\\ +{\left(\alpha h\right)}^2+12=0\end{array}$$

(31)

Dispersion curve is the “visual language” of vibration equation in wave propagation problems. It not only reveals the basic characteristics of wave propagation speed, wave mode differentiation, and dispersion effect, but also provides the core basis for engineering applications such as medium characterization, damage detection, and structural design. In the next section, we will analyze the accuracy of different vibration equations by discussing and comparing the dispersion curves between the vibration equations in this paper and other vibration equations.

4. Comparison and Discussion of the Exact Plate Theory

In this paper, the fourth-order wave equation describing the longitudinal vibration of a plate is obtained by truncating the operator series in Equation (28). Figure 1, Figure 2 and Figure 3 describe the relationship curve between the ratio of phase velocity to longitudinal wave velocity and the product of wave number and thickness for different stretching plate dynamics theory.

Through the dispersion curve shown as in Figure 1, Figure 2 and Figure 3, we can see that the dispersion curve based on the vibration equation of the exact stretching plate theory is very close to the dispersion curve of longitudinal vibration equation in three-dimensional elasto-dynamics, while the dispersion curve of longitudinal vibration equation of plate based on plane stress problem is a straight line. Figure 1 shows that when the Poisson’s ratio is small ($\nu =0.1$), the dispersion curve based on the exact stretching plate theory and the dispersion curve based on the three-dimensional elasto-dynamics are almost coincident. Only when the dimensionless product of wave number and thickness $\alpha h$ < 2, the three dispersion curves are very adjoining. Figure 2 ($\nu =0.25$) and Figure 3 ($\nu =0.45$) show that when the Poisson’s ratio is larger, the dispersion curve based on the exact stretching plate theory and the dispersion curve based on the three-dimensional elasto-dynamics are almost coincident only if the dimensionless product of wave number and thickness is smaller. When the dimensionless product of wave number and thickness $\alpha h$ is larger, there is a certain difference between the two curves. Nevertheless, the difference is still tiny.

The analytical solution of the Navier equation of three-dimensional elasto-dynamics theory cannot be derived under general conditions but only under certain specific boundary conditions or initial conditions. In this paper, the spectral decomposition of Navier equations describing three-dimensional elasto-dynamics is carried out by using the spectral decomposition theory of differential operators and introducing the concept of virtual differential operators. Taking the plate with upper and lower boundaries as the research object (here, the plate can be either thin plate, thick plate or super thick plate), the longitudinal vibration and bending vibration are separated by Cauchy–Riemann condition, and the equation is expanded into infinite product operator series according to the integral function theory. Finally, the infinite product operator series is truncated to obtain the governing equation describing the vibration of plate with arbitrary thickness. There is no geometric assumption in the whole derivation process but only the truncation of operator product. From the dispersion curve, when the operator product is truncated into fourth-order partial differential equations, the dispersion relation is accurate enough, while the classic dispersion curve of longitudinal vibration is a straight line. Therefore, the proposed exact wave motion equation for stretching plate structure based on the operator spectral decomposition theory in this paper is reliable. And for the governing equation of longitudinal vibration, which consists of a fourth-order wave equation and a second-order wave equation, it is easy to solve.

5. Conclusions

In this paper, based on differential operator symmetric and asymmetric spectral decomposition, an exact dynamic modeling method is proposed to address the problem of insufficient accuracy in classical plane stress problem for longitudinal vibration modeling of thick-walled structures. In view of the three-dimensional elastic dynamics Navier equation, the concept of virtual differential operators is introduced, combined with the Boussinesq-Galerkin general solution and Taylor series expansion of exponential operator functions. Subsequently, the longitudinal vibration control equation composed of fourth-order wave equation and second-order wave equation is derived.

Compared with traditional theories, the core advantage of this model is that no geometric assumptions are introduced in the derivation process, and only simplification is achieved through operator product truncation, thus breaking through the limitations of the classical second-order longitudinal vibration equation (plane stress problem). Through comparative analysis of dispersion curves, it is shown that the dispersion curve corresponding to the precise theory proposed in this paper is highly consistent with the results of three-dimensional elastic dynamics theory: when the Poisson’s ratio is small, the dispersion curves of the two almost completely overlap. Even with an increase in Poisson’s ratio, there are only slight differences when the dimensionless product of wavenumber thickness is large, and the overall consistency is much better than the linear dispersion curve of classical plane stress theory. This proves that the model has higher accuracy in longitudinal vibration analysis of wide frequency range and thick plates and can effectively solve the error problem caused by the approximation of plane stress problem under high-frequency vibration.

Above all, the fourth-order and second-order coupled control equations derived in this article have both accuracy and solvability, providing a reliable theoretical tool for dynamic stress analysis, damage detection, and structural optimization design of thick-walled structures in aerospace, nuclear power engineering, pressure vessels, and other fields. It also provides a new algebraic perspective for modeling plate vibration problems in elastic dynamics.