Preamble

Vibration of Functionally Graded Rectangular Plates with Internal Porosity in Fluid

Authors: Huang Xiaolin, Hao Xiqi, Li Liangjie, Xiao Weiwei
(School of Architecture and Transportation Engineering, Guilin University of Electronic Technology)

Abstract

To investigate the free vibration and dynamic response of metal-ceramic functionally graded plates with internal porosity submerged in fluid, this study calculates the fluid velocity potential function and hydrodynamic pressure based on interaction boundary conditions. A computational model for the physical property parameters of functionally graded materials (FGM) containing internal pores is established using the ceramic mass fraction as the fundamental parameter. Based on thin plate theory, the vibration governing equations for a functionally graded rectangular plate with four simply supported edges in a fluid medium are derived. The natural frequencies and dynamic responses are solved using the harmonic balance method. The research results indicate that the natural frequencies and dynamic responses of plates submerged within or at the bottom of the fluid decrease as the fluid depth increases. Conversely, for plates floating on the fluid surface, the natural frequencies and dynamic responses increase with fluid depth. The influence of porosity on the plate's natural frequency is not only related to the magnitude and distribution pattern of the pore volume fraction but is also influenced by factors such as the ceramic mass ratio and the fluid environment.

1. Introduction

Functionally graded materials (FGM) are advanced composite materials characterized by a continuous variation in composition and structure, which results in a seamless transition of mechanical properties. Due to their excellent thermal resistance and structural integrity, FGM plates are widely used in aerospace, marine engineering, and chemical equipment. During the fabrication process of these materials, micro-voids or porosities often inevitably form. These internal pores significantly affect the overall stiffness and density of the structure, thereby influencing its vibrational characteristics.

When an FGM plate operates in a fluid environment—such as a ship hull or a storage tank component—the fluid-structure interaction (FSI) becomes a critical factor. The fluid adds an "added mass" effect to the plate, which alters its dynamic behavior. While previous studies have extensively covered the vibration of solid FGM plates in a vacuum, research addressing the combined effects of internal porosity and fluid loading remains relatively limited.

2. Material Properties and Theoretical Model

2.1 Functionally Graded Material with Porosity

In this study, we consider a metal-ceramic FGM rectangular plate. The material properties are assumed to vary through the thickness direction according to the ceramic mass fraction. To account for internal defects, we introduce a porosity model. The effective material properties $P$ (such as Young's modulus $E$ and mass density $\rho$) of the porous FGM plate can be expressed as:

$$\begin{aligned} P(z) = P_m + (P_c - P_m) V_c(z) - \frac{\alpha}{2}(P_c + P_m) \Phi(z) \end{aligned}$$

where $P_c$ and $P_m$ represent the properties of the ceramic and metal phases, respectively, $V_c(z)$ is the volume fraction of the ceramic, and $\alpha$ denotes the porosity volume fraction. The function $\Phi(z)$ defines the distribution pattern of the pores (e.g., uniform or non-uniform distributions).

2.2 Fluid-Structure Interaction

The fluid is assumed to be inviscid, incompressible, and irrotational. The interaction between the plate and the fluid is governed by the velocity potential function $\phi$. Based on the boundary conditions at the interface, the hydrodynamic pressure $p$ exerted by the fluid on the plate surface is derived from the linearized Bernoulli equation:

$$\begin{aligned} p = -\rho_f \frac{\partial \phi}{\partial t} \end{aligned}$$

where $\rho_f$ is the fluid density. By solving the Laplace equation for the fluid domain with appropriate boundary conditions (submerged, bottom-mounted, or floating), the added mass effect is quantified.

3. Governing Equations and Solution Method

Based on the classical thin plate theory, the governing equation for the transverse vibration of the FGM plate in fluid is established as:

$$\begin{aligned} D \nabla^4 w + \rho h \frac{\partial^2 w}{\partial t^2} = q_{fluid} \end{aligned}$$

where $w$ is the transverse displacement, $D$ is the bending stiffness, $\rho h$ is the mass per unit area, and

关键词

Vibration of Porous Functionally Graded Rectangular Plates in Fluid

HUANG Xiaolin, HAO Xiqi, LI Liangjie, XIAO Weiwei
School of Architecture and Transportation Engineering, Guilin University of Electronic Technology

Abstract: This study investigates the free vibration characteristics of porous functionally graded material (FGM) rectangular plates immersed in a fluid medium. By accounting for the influence of internal pores on material properties and the fluid-structure interaction effects, a comprehensive mathematical model is developed. The harmonic balance method is employed to analyze the vibration response.

Keywords: Functionally graded materials; Porosity; Hydroelasticity; Plates; Free vibration; Harmonic balance method

1. Introduction

Functionally graded materials (FGMs) are advanced composites characterized by a continuous variation of material properties along a specific direction, typically the thickness. This gradient is achieved by varying the volume fractions of the constituent materials, such as ceramics and metals. Due to their superior thermal resistance and structural integrity, FGM plates are widely utilized in aerospace, marine, and nuclear engineering. During the fabrication process of FGMs, micro-voids or pores can inevitably occur due to the difference in sintering temperatures between the constituents. These porosities significantly affect the mechanical behavior, including the stiffness and mass distribution of the structure.

When these structures operate in a fluid environment, such as in offshore platforms or underwater vehicles, the fluid-structure interaction (FSI) becomes a critical factor. The presence of fluid introduces added mass and damping effects, which shift the natural frequencies and alter the vibration modes of the plates. Therefore, understanding the vibration characteristics of porous FGM plates in fluid is essential for their safe and efficient design.

2. Material Properties of Porous FGM Plates

Consider a rectangular FGM plate with length $a$, width $b$, and thickness $h$. The material properties, such as Young's modulus $E$ and mass density $\rho$, are assumed to vary through the thickness $z$ according to a power-law distribution modified to account for porosity.

Two common porosity distributions are typically considered: even (uniform) and uneven (non-uniform) distributions. For a plate with even porosity, the effective material properties $P(z)$ can be expressed as:

$$P(z) = (P_c - P_m) \left( \frac{z}{h} + \frac{1}{2} \right)^k +$$

541004 Guilin

China

Abstract

The vibration and dynamic responses for porous metal/ ceramic functionally graded material plates in fluid were studied. Considering three types of plate-fluid interaction boundaries the potential function of speed and the hydrodynamic pressure were calculated. The effective material properties of the plates were built with the mass volume of ceramic. Based on the thin plate theory the governing equations of the simply supported plates were established and the natural frequencies and dynamic responses were calculated by using the harmonic balance method. The results show that the frequencies and responses of the plates submerged in fluid and at the bottom decrease with the increase of fluid depth. The frequencies and responses of the plate on fluid surface are raised as the water depth increases. The effect of pores on the natural frequencies is related to pores the mass volume of ceramic fluid and other factors.

Key words functionally graded material hydroelasticity plate free vibration harmonic balance

method

Engineering structures in fields such as transportation and naval architecture are frequently in contact with fluids, forming fluid-structure interaction (FSI) systems. Research has demonstrated that due to these coupling effects, the vibrational characteristics of structures immersed in fluid differ significantly from those in a vacuum. Currently, most studies on fluid-structure interaction problems focus on structures made of isotropic materials.

There is relatively little research concerning functionally graded materials (FGM). Among existing studies, Khorshidi \cite{ref1} investigated the hydroelastic vibration of vertical FGM plates partially in contact with fluid, discovering that the added mass of the fluid reduces the natural frequencies of the plate, with these frequencies decreasing as the depth of the fluid contact increases. Xu Yangjian et al. \cite{ref2} analyzed the influence of fluid temperature on the hydroelastic vibration characteristics of FGM plates, finding that as the temperature of the fluid medium increases, the overall temperature within the plate rises significantly, leading to a subsequent decrease in the plate's natural frequencies.

Thinh \cite{ref3} investigated the effects of parameters such as the Young's modulus of the ceramic phase, the material volume fraction index, fluid density, plate-fluid interaction boundary conditions, and geometric dimensions on the hydroelastic vibration frequencies of FGM plates. Gu Sen \cite{ref4} utilized Euler-Bernoulli beam theory to discuss the vibrational characteristics of functionally graded beams under photothermal excitation in a fluid; the results indicated that under the same excitation load, the vibration amplitude of simply supported beams is higher than that of clamped beams. These studies collectively find that when a structure vibrates in a fluid, its vibration frequency and dynamic response decrease significantly, and the magnitude of this reduction is related to factors such as fluid density, depth, and the interaction boundary conditions of the fluid.

Due to limitations in matrix materials or manufacturing techniques, internal pores are inevitably present in FGM components. Consequently, the dynamic characteristics of FGM components containing internal pores in a fluid environment are also a matter of significant concern. When studying the vibration of FGM pipes in fluid, Zhou Jie et al. \cite{ref5} found that when the fluid density is low, the natural frequency increases with porosity; however, as the fluid density increases, the natural frequency decreases as porosity increases.

By studying the free vibration of vertical porous FGM plates in contact with fluid, Farsani \cite{ref6} discovered that the pore distribution pattern also affects the natural frequency; specifically, FGM plates with a non-uniform symmetric distribution across the thickness exhibit the highest fundamental frequency, while those with a uniform distribution exhibit the lowest. Currently, there are few studies on the vibrational characteristics of porous FGM rectangular plates in fluid. Furthermore, when using the rule of mixtures to determine effective physical property parameters, most studies assume the pore volume is negligible and ignore its impact on the total volume calculation of the plate. This study accounts for the total volume of the pores and utilizes an improved rule of mixtures model to calculate the physical property parameters of the FGM. Based on thin plate theory, the vibration equations for a horizontally placed FGM plate in fluid are established. This work systematically analyzes the influence of factors such as plate-fluid interaction boundary conditions, porosity, and water depth on the natural frequencies and dynamic response, providing a theoretical reference for the design of FGM components containing internal pores in fluid environments.

1 功能梯度矩形板模型

As shown in [FIGURE:1], consider a rectangular ceramic-metal functionally graded material (FGM) plate with length $a$ and width $b$. The origin of the Cartesian coordinate system is located at one corner of the plate's geometric mid-plane, with the $z$-axis oriented vertically upward along the thickness direction. The material composition of the plate varies gradiently, transitioning from a metal-rich/ceramic-poor composition at the bottom to a metal-poor/ceramic-rich composition at the top.

It is assumed that the internal pores within the plate are micropores with low volume fractions; therefore, there is no connectivity or fracture between individual pores. Two types of internal pore distributions are considered in this study. The first type occurs within the metallic phase, where the distribution follows the metallic material gradient, resulting in a higher concentration at the bottom and a lower concentration at the top (Distribution 1). The second type arises from manufacturing limitations, specifically the difficulty of injecting the ceramic reinforcing phase into the center of the plate, which leads to a higher concentration of pores in the middle and fewer at the top and bottom (Distribution 2).

[FIGURE:2] illustrates the cross-sectional views of these two pore distributions. Let the volume fraction of pores relative to the metallic material volume be denoted as $\alpha$.

W t + W b = 1 （ 1 ）

where $w_c$ and $w_m$ represent the mass fractions of ceramic and metal, respectively, and $V_c$ and $V_m$ denote their corresponding volume fractions. The volume fraction of the ceramic phase can be calculated as:

$$V_c = \frac{w_c / \rho_c}{w_c / \rho_c + w_m / \rho_m}$$

where $\rho_c$ and $\rho_m$ are the mass densities of the ceramic and metal materials, respectively. Assuming the ceramic reinforcement follows a power-law distribution along the thickness direction, the volume fraction $V_c$ is defined by the material gradient index $p$. The ceramic distribution coefficient $C$ is determined by ensuring that the total mass of the ceramic remains constant across different distributions, such that:

V * t （ z ） = V t1 2 z + h 2 ( ) h

$N d z = \rho_t V_t a b h$ (5)

Assume the two types of pore distributions $\alpha^*(z)$ shown in [FIGURE:2] are:

z + h

$\alpha_{1}(z) = \alpha_{1} \left( \frac{1}{2} - z \right) h$ (Pore distribution $D_{1}$) (6)

The pore constants for different pore types are determined by ensuring that the total pore volume of the plate remains equivalent.

-h/ 2 α 1 1 2 - z ( ) h d z = V b α abh （ 8 ）

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According to the modified rule of mixtures, the effective physical properties of functionally graded materials (FGMs), such as Young's modulus $E$ and mass density $\rho$, can be expressed as:

$$P(z) = (P_c - P_m) V_c(z) + P_m$$

In this expression, the subscripts $c$ and $m$ denote the corresponding material properties of the ceramic and metal phases, respectively. The term $V_c(z)$ represents the volume fraction of the ceramic phase, which varies along the thickness direction $z$.

2 流体作用

The fluid action on the plate during vibration is represented by the hydrodynamic pressure. This study considers the following scenarios for a plate submerged in a fluid with a free surface: (1) the plate is submerged at the bottom of a fluid with a free surface, where the fluid bottom is a rigid body; and (2) the plate is submerged in the middle of a fluid with a free surface, where the fluid bottom is also a rigid body. It is assumed that the fluid is incompressible, inviscid, and irrotational. Furthermore, the influence of the fluid beyond the extent of the plate surface is neglected.

Fluid Interaction Boundary Conditions: For the case where a flat plate is submerged at the bottom of a fluid with a free surface, the velocity potential function of the fluid can be calculated as follows:

φ 1 = e μ f z + c 1 e μ f （ 2 h 1 +h-z ）

respectively represent the plate thickness and the depth of the fluid on the plate; $k_n$ is the flexural wave wavenumber of the $n$-th mode; $w(x, y)$ is the deflection function of the plate; and $\Phi$ represents the fluid velocity potential coefficient, the specific expression of which is given by:

μ f = （ m π / a ） 2 + （ n π / b ）

c 1 = μ f g - ω 2

where $g$ is the acceleration due to gravity and $f$ is the vibration frequency. By substituting Eq. (1) into the Bernoulli equation, the hydrodynamic pressure acting on the upper surface of the plate ($z = h$) can be calculated as:

$$p = -\rho \frac{\partial \Phi}{\partial t}$$

where $\rho$ represents the density of the fluid. To avoid the complexities associated with nonlinear eigenvalue problems, the fluid velocity potential coefficients are typically expanded or linearized.

q w = - ρ w ∂ φ 1 ∂ t z = h 2 = - ρ w μ f

For the case of a flat plate floating on the free surface of a fluid, the velocity potential function of the fluid can be calculated as:

$$
\phi(x, y, z, t) = \text{Re} \left{ \phi(x, y, z) e^{-i\omega t} \right}
$$

where $\phi(x, y, z)$ represents the complex spatial velocity potential. Under the assumptions of an ideal, incompressible, and irrotational fluid, the velocity potential must satisfy the Laplace equation within the fluid domain:

$$
\nabla^2 \phi = 0
$$

The boundary conditions for this system are defined as follows:

1. Free Surface Condition: On the undisturbed free surface ($z=0$), excluding the area covered by the plate, the linearized combined kinematic and dynamic boundary condition is:
   $$
   K\phi - \frac{\partial \phi}{\partial z} = 0, \quad K = \frac{\omega^2}{g}
   $$
   where $g$ is the acceleration due to gravity and $\omega$ is the angular frequency of the wave.

2. Body Surface Condition: On the submerged surface of the plate ($S_B$), the normal velocity of the fluid must match the normal velocity of the plate:
   $$
   \frac{\partial \phi}{\partial n} = V_n
   $$
   where $n$ is the unit normal vector pointing into the fluid and $V_n$ is the velocity component of the plate in the normal direction.

3. Bottom Condition: For a fluid of finite depth $h$, the vertical velocity at the seabed ($z=-h$) must vanish:
   $$
   \frac{\partial \phi}{\partial z} = 0 \quad \text{at } z = -h
   $$

4. Radiation Condition: At an infinite distance from the plate, the potential must represent outgoing waves, ensuring the uniqueness of the solution.

[FIGURE:1]

By solving the above boundary value problem, typically through the boundary element method (BEM) or analytical matching techniques, the pressure distribution on the bottom of the plate can be determined using the linearized Bernoulli equation:

$$
p = -\rho \frac{\partial \phi}{\partial t}
$$

This pressure distribution is then integrated over the wetted surface area

φ 2 = e μ f z + c 2 e - μ f z

Let $h$ represent the depth of the fluid beneath the plate. By substituting this into the Bernoulli equation, the hydrodynamic pressure acting on the plate can be calculated as:

q w = - ρ w ∂ φ 2 ∂ t z = - h 2 = - ρ w μ f

When a flat plate is submerged within a fluid—specifically considering the case where the plate is positioned in the middle of the fluid—let the water depth above the plate be denoted as $h_1$ and the water depth below the plate be denoted as $h_2$. The total hydrodynamic pressure acting on the plate is defined as the resultant force of the hydrodynamic pressures acting on both the upper and lower surfaces of the plate.

q w = - ρ w μ f 1 + c 1 e 2 μ f h 1

3 水弹性振动方程及求解

Based on composite thin plate theory, the governing equations for a composite plate containing internal porosity vibrating in a fluid can be derived as follows:

2 + q w = q （ 18 ）

$w$ is the deflection function of the rectangular plate; $F$ is the stress function, which relates to the in-plane forces as $\sigma_x = \frac{\partial^2 F}{\partial y^2}$, $\sigma_y = \frac{\partial^2 F}{\partial x^2}$, and $\tau_{xy} = -\frac{\partial^2 F}{\partial x \partial y}$. $q_d$ and $q$ represent the dynamic pressure of the fluid perpendicular to the plate surface and the external load, respectively. The fluid mass coefficient is denoted by $\gamma_w$. The linear differential operators $L(\cdot)$ and $\nabla^4(\cdot)$ are defined as:

$$\begin{aligned}
L(w, F) &= \frac{\partial^2 w}{\partial x^2} \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 w}{\partial y^2} \frac{\partial^2 F}{\partial x^2} - 2 \frac{\partial^2 w}{\partial x \partial y} \frac{\partial^2 F}{\partial x \partial y} \
\nabla^4 &= \frac{\partial^4}{\partial x^4} + 2 \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}
\end{aligned}$$

L 1 （） = D * 11 ∂ 4

L 2 （） = B * 21 ∂ 4

L 3 （） = A * 22 ∂ 4

L 4 （） = B * 21 ∂ 4

For equivalent stiffness elements, please refer to literature \cite{}. Assuming the boundaries of the plate are simply supported on all four sides, the deflection function and stress function that satisfy these boundary conditions can be defined as:

$$
\begin{aligned}
w(x, y) &= \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn} \sin \frac{m \pi x}{a} \sin \frac{n \pi y}{b} \
\phi(x, y) &= \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} B_{mn} \sin \frac{m \pi x}{a} \sin \frac{n \pi y}{b}
\end{aligned}
$$

w （ x ， y ， t ） = ∑

n = 1 w mn （ t ） sin m π x ( ) a sin n π y ( ) b （ 24 ）

m = 1 ∑

F （ x ， y ， t ） = ∑

Substituting into equation (1), we obtain:

m = 1 ∑

Using the harmonic balance method, we can derive the ordinary differential equations for the undetermined coefficients. Here, the matrices represent the mass and stiffness of the plate, respectively, while denotes the added mass matrix. Consequently, the equation (vibration_eq) represents the free vibration equation for a porous functionally graded rectangular plate immersed in a fluid. For a non-zero solution of (vibration_eq) to exist, the following condition must be satisfied:

$$\det ( \mathbf{K} - \omega^2 \mathbf{M} - \omega^2 \mathbf{M}_w ) = 0 \quad (27)$$

By solving this algebraic system of equations, the natural frequencies for each mode of free vibration can be obtained.

The dynamic response of each mode is solved using the Newmark-$\beta$ method. The total forced vibration response is then obtained by superimposing these individual modal responses.

4 算例比较与参数分析

In the following calculations, the dimensionless normalized frequency is defined as $\bar{\omega} = \omega a^2 \sqrt{\rho_c h / D_c}$, where the subscript $c$ denotes the physical property parameters of the ceramic phase.

[TABLE:1] presents the dimensionless fundamental frequencies of functionally graded rectangular plates under fluid-interaction boundary conditions, compared with results available in the existing literature. As shown in the table, the results of the present study are in close agreement with those reported in Ref. \cite{1}, with a maximum deviation of less than 0.5%.

Table 1: Comparison of dimensionless fundamental frequencies of hydroelastic free vibration for four-sided simply supported functionally graded rectangular plates.

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The following discussion examines the effects of fluid-structure interaction boundary conditions, porosity, ceramic mass fraction, material index, and fluid depth on the fundamental frequency of hydroelastic vibration and the dynamic response of functionally graded (FG) plates. The analysis considers various fluid interaction boundaries and different types of porosity distributions.

The functionally graded plate is composed of a ceramic reinforcement phase and a metallic matrix phase. The physical properties of the ceramic phase are given by $E_c = 380$ GPa.

The Poisson's ratio and density of the ceramic phase are $\nu_t = 0.3$ and $\rho_t = 3800 \text{ kg/m}^3$, respectively. The physical properties of the aluminum (Al) matrix are given by $E_b =$

70 GPa

$\nu_b = 0.3$ and $\rho_b = 2702 \text{ kg/m}^3$; the fluid density is $\rho_w = 1000 \text{ kg/m}^3$. Unless otherwise specified, the thickness of the plate is $h = 0.1 \text{ m}$, the time step is $\Delta t = 0.03 \text{ s}$, and the dynamic response refers to the dynamic deflection at the center point of the plate.

The influence on the fundamental frequency and dynamic response is analyzed. As shown in [FIGURE:N], the natural frequency decreases initially and then increases as the material gradient index $p$ increases, reaching a minimum value near $p = 1.5$. The dynamic response exhibits the opposite trend.

[FIGURE:N] illustrates the effects of porosity distribution types and porosity volume fractions on the dimensionless fundamental frequency and dynamic response. The results indicate that the dimensionless fundamental frequency for the first type of porosity distribution is greater than that of the second type. For all porosity distribution types, the dimensionless fundamental frequency of the functionally graded plate decreases as the porosity volume fraction $e_0$ increases. Under the same porosity type, the dimensionless fundamental frequency is highest under specific boundary conditions. Furthermore, it can be concluded that the dynamic response of the second porosity distribution is higher than that of the first, and the dynamic response increases with the increase of the porosity volume fraction.

[TABLE:N] lists the first several natural frequencies of a functionally graded plate with an aspect ratio of $a/b = 1.5$ under fluid-interaction boundary conditions, providing a comparison with the natural frequencies in a vacuum.

Under various fluid boundary conditions, the vibration frequencies of the plate are significantly lower than those in a vacuum, and this reduction becomes more pronounced for higher-order modes. Under specific boundary conditions, the frequency of the $(1,1)$ mode decreases to $41.56\%$ of its vacuum value, while the $(2,2)$ mode decreases to $22.08\%$. [TABLE:N] presents the dimensionless hydroelastic vibration frequencies of the functionally graded rectangular plate across different modes. As shown in [FIGURE:N], the natural frequency of the plate increases as the ceramic mass fraction increases, while the dynamic response follows the opposite trend. On average, for every increase in ceramic content, the normalized fundamental frequency increases accordingly.

The effects of fluid-interaction boundary conditions and fluid depth on the dimensionless fundamental frequency and dynamic response are investigated. As seen in [FIGURE:N], at a constant fluid depth, the dimensionless fundamental frequency varies significantly across different boundary conditions. Specifically, the dimensionless fundamental frequency decreases as the fluid depth increases under certain conditions, whereas it increases with depth under others. When the fluid depth exceeds a certain threshold, the dimensionless fundamental frequencies for various boundary conditions remain largely constant and tend to converge. Regarding the dynamic response, among the various boundary conditions, the response is minimized under specific constraints. [FIGURE:N] shows the dynamic response at different water depths under fluid-interaction conditions; the dynamic response of the plate in water is much smaller than that in a vacuum. When the water depth exceeds a specific limit, the dynamic response time-history curves become very similar.

1072 应用力学学报

Subsequently, the dynamic response time curves for various depths tend to converge. This indicates that the influence of the fluid on the vibration characteristics of the plate gradually stabilizes as the water depth increases. This phenomenon occurs because the velocity potential function progressively converges toward a consistent state as the fluid depth increases.

Fluid interaction boundary conditions and the effect of fluid depth on dynamic response

5 结

This study improves the calculation model for the effective physical property parameters of functionally graded material (FGM) plates containing pores. Based on thin plate theory, the equations of motion for the hydroelastic vibration of rectangular FGM plates with internal porosity are established and solved. The investigation discusses the effects of plate-fluid interaction boundary conditions, fluid depth, porosity, and the material gradient index on the fundamental frequency of free vibration and the resulting dynamic response.

The main conclusions are as follows: Compared to a vacuum environment, the presence of fluid not only reduces the natural frequencies but also suppresses the dynamic response under forced vibration. Under the same fluid depth, the natural vibration frequency of the plate varies with the boundary conditions, while the dynamic response is minimized under specific boundary constraints.

Under certain boundary conditions, the natural vibration frequency changes with fluid depth, whereas under other boundary conditions, the frequency exhibits the opposite trend. Furthermore, the influence of fluid depth on both the natural frequency and the dynamic response stabilizes and ceases to change once the depth reaches a certain threshold.

Among the various pore distributions studied, the distribution designated as $P_1$ results in the highest natural frequency and the minimum dynamic response. This indicates that the $P_1$ distribution has the most significant impact on the mechanical performance of the plate. The influence of internal pores on the fundamental frequency is related not only to the pore volume fraction and distribution pattern but also to the specific plate-fluid interaction boundary conditions. Additionally, the natural frequency of the plate increases with the ceramic mass ratio. As the material gradient index $p$ increases, the natural frequency initially decreases and then increases, with the dynamic response showing the opposite trend. The natural frequency reaches its minimum value—and the dynamic response its maximum—when the material gradient index is in the range of $1.5 < p < 2.0$. This suggests that this interval should be avoided during structural design.

References:
[1] Vibration and Acoustic Radiation of Thin Plate Structures in Fluids [Beijing: Science Press].
[2] XIAO Yihua, HAN Xu, HU De'an. Simulating fluid-structure interaction with FE-SPH method. Chinese Journal of Applied Mechanics.
[3] LIU Hui, DENG Xuhui, ZHAO Ke, et al. Effects of different constraints on the dynamics of deep-sea mining pipelines. Chinese Journal of Applied Mechanics.

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Influences of Fluid and Structural Parameters on Flow-Induced Vibrations of Plate-Fluid Structures

LU Li, YANG Yiren
(Southwest Jiaotong University)

Abstract

This paper investigates the influence of various fluid and structural parameters on the flow-induced vibration characteristics of a plate-fluid coupled system. By establishing a mathematical model for the interaction between a flexible plate and a surrounding fluid medium, we analyze how changes in physical properties—such as fluid density, flow velocity, and structural stiffness—affect the stability and oscillatory behavior of the system. The results provide theoretical insights into the design and safety assessment of engineering structures subjected to fluid-structure interaction (FSI).

1 Introduction

The coupling between fluid flow and structural deformation is a critical phenomenon in various engineering fields, including aerospace, civil engineering, and marine energy harvesting. When a flexible plate is immersed in or subjected to a fluid flow, the resulting fluid-structure interaction (FSI) can lead to complex vibrational behaviors, such as flutter or divergence. Understanding how specific parameters influence these vibrations is essential for optimizing structural performance and preventing catastrophic failure.

Previous studies have demonstrated that the dynamic response of a plate is highly sensitive to the properties of the fluid and the boundary conditions of the structure. This study builds upon existing research by systematically varying fluid and structural parameters to quantify their impact on the system's stability boundaries and vibration modes.

2 Mathematical Modeling

To analyze the coupled system, we consider a thin elastic plate of length $L$, width $W$, and thickness $h$. The plate is subjected to a fluid flow with velocity $U$ and density $\rho_f$. The governing equation for the transverse displacement $w(x, y, t)$ of the plate can be expressed as:

$$D \nabla^4 w + \rho_s h \frac{\partial^2 w}{\partial t^2} = \Delta P(x, y, t)$$

where $D = \frac{Eh^3}{12(1-\nu^2)}$ represents the flexural rigidity, $\rho_s$ is the structural density, and $\Delta P$ is the fluid pressure difference acting on the plate surfaces.

The fluid is assumed to be inviscid and incompressible, governed by the potential flow theory. The aerodynamic or hydrodynamic loads are coupled with the structural motion through the kinematic boundary condition at the interface.

[FIGURE:1]

3 Results and Discussion

3.

Southwest Jiaotong University ， 2009 ， 44 （ 3 ）： 370- 374 （ in Chi-

Dong Yu, Yang Yiren, Lu Li. Frequency analysis of rectangular plates vibrating in still water bounded by rigid walls. Applied Mathematics and Mechanics.

Chen Yong, Zhang Jun, Wang Yu, et al. Non-linear transient analysis of shock-loaded circular plate in damping medium. Journal of Vibration and Shock.

Kutlu A, Ugurlu B, Omurtag M H, et al. Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid. Ocean Engineering.

Khorshidi K, Bakhsheshy A. Free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid. Acta Mechanica.

Karimi M, Khorshidi K, Dimitri R, et al. Size-dependent hydroelastic vibration of FG microplates partially in contact with a fluid. Composite Structures.

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Introduction

The study of rectangular plates coupled with fluid remains a fundamental topic in applied mathematical modeling. Early foundational work, such as that by Lamb \cite{LAMB}, investigated the vibrations of an elastic plate in contact with water, providing critical insights into the interaction between structural dynamics and fluid mechanics. Building upon these classical theories, modern research has shifted toward advanced materials. For instance, Pang et al. \cite{PANG} explored the free vibration of functionally graded graphene-reinforced composite (FG-GRC) plates, highlighting the significant impact of material distribution on structural performance.

[FIGURE:1]

Free Vibration of Functionally Graded Graphene-Reinforced Composite Plates

Functionally graded materials (FGMs) offer a unique advantage by allowing for a tailored spatial distribution of material properties. In the context of graphene-reinforced composites, the reinforcement can be distributed according to specific patterns to optimize the stiffness-to-weight ratio. The free vibration analysis of such plates involves solving complex governing equations that account for the coupling between the elastic deformation of the composite and the surrounding environment.

[TABLE:1]

As shown in [TABLE:1], the mechanical properties of FG-GRC plates are highly sensitive to the weight fraction of graphene and the specific grading pattern employed. Mathematical models typically utilize higher-order shear deformation theories to accurately capture the kinematic behavior of these thin-walled structures. When these plates are coupled with a fluid medium, the added mass effect and fluid pressure must be integrated into the dynamic equations, often resulting in a shift in the natural frequencies compared to vibrations in a vacuum.

[FIGURE:2]

The integration of graphene as a reinforcement phase significantly enhances the fundamental frequencies of the plates. Research by Pang et al. \cite{PANG} demonstrates that the distribution of graphene—whether uniform or functionally graded—plays a decisive role in the structural response. By applying advanced numerical methods, researchers can predict the modal shapes and frequency parameters essential for the design of aerospace and marine components where fluid-structure interaction is a critical consideration.

nese journal of applied mechanics ， 2020 ， 37 （ 2 ）： 558-565 （ in Chi-