A hierarchy of asymptotic models for a fluid-loaded elastic layer

Abstract

A hierarchy of asymptotic models governing long-wave low-frequency in-plane motion of a fluid-loaded elastic layer is established. In contrast to a layer with traction-free faces, modelled by Neumann boundary conditions, a fluid-loaded one assumes more involved conditions along the interfaces, dictating a special asymptotic scaling. The latter corresponds to a fluid-borne bending wave, controlled by elastic stiffness of the layer and fluid inertia. In this case, the transverse inertia of the layer and fluid compressibility do not appear at zero-order approximation. The first-order approximation is associated with a Kirchhoff plate, immersed into incompressible fluid. In the studied free vibration setup, the fluid compressibility has to be taken into account only at third order, along with elastic rotary inertia. Transverse shear deformation enters the second-order approximation along with a few other corrections. The conventional impenetrability condition has to be also refined at second order. Dispersion relations corresponding to the developed asymptotic models are compared with the polynomial expansions of the full dispersion relation, obtained from the plane-strain problem of linear elasticity.

Keywords

Fluid-loaded elastic layer fluid-borne bending wave asymptotic models transverse shear deformation rotary inertia fluid compressibility fluid inertia impenetrability condition dispersion

1. Introduction

Numerous problems involving fluid loading of vibrating structures have been extensively studied in the literature since long ago. Particular interest in this area was manifested by Lord Rayleigh, who in his “Theory of Sound” [1] analysed the “reaction of the air on a vibrating circular plate.” The majority of the theoretical considerations on the subject start from the classical two-dimensional (2D) structural theories, such as the Kirchhoff theory for thin elastic plates (e.g., see books [2, 3] and the Rayleigh lecture [4]). We also cite insightful papers [5 –9] reporting on explicit analytical results spanning over eight decades.

At the same time, not much has yet been done for the mathematical justification of the models in fluid–structure interaction using heuristic plate and shell theories. Among a very few publications attempting to address this key issue, we mention Johansson et al. [10] using polynomial expansions along the thickness of a fluid-loaded flat elastic layer. However, the derivation in this paper lacks of asymptotic consistency and hardly can be useful for further implementation. Interesting observations in Sorokin and Chubinskij [11] concerned with the validity of the Kirchhoff theory, along with some other ad hoc setups, mostly promote an engineering approach.

There were also efforts, see Belov et al. [12] and Kaplunov et al. [13, 14], to adapt asymptotic results earlier established for elastic structures not interacting with fluid; in doing so, fluid loading was considered as the prescribed forcing terms in the equations governing plates or shells. In the latter case, all approximate theories for thin structures naturally follow from asymptotic analysis of Neumann boundary value problems for the three-dimensional (3D) equations in elasticity (e.g., see books [14 –18]). To this end, the results in Belov et al. [12] and Kaplunov et al. [13, 14] need more verification and development. The point is that fluid loading of an elastic structure is governed by more involved boundary conditions, rather than Neumann ones, modelling given mechanical loads on its faces. This motivated full asymptotic analysis of the coupled fluid–structure interaction problems.

A similar problem arises in analysis of a plate resting on a Winkler foundation (e.g., see Frýba [19]). The conventional approximate statement was recently compared with the outcomes of asymptotic analysis, starting from the associated problems in plane and 3D elasticity (see Erbaş et al. [20, 21]). In this case, the contact of an elastic layer with a Winkler foundation was modelled by mixed boundary conditions instead of a Neumann one. In addition, the assumption of a relatively soft foundation was made. It was demonstrated that the popular engineering setup of a Kirchhoff plate on a Winkler foundation fails near the lowest cut-off frequency (see also Erbaş [22]).

In this paper, we consider the simplest geometric configuration of an immersed elastic layer. Free in-plane vibrations are studied over the long-wave low-frequency domain. The focus is on the fluid-borne bending wave primarily influenced by plate stiffness and fluid inertia (e.g., see Talmant et al. [23]), which certainly is not a feature of a layer not interacting with fluid. The associated scaling is inspired by a preliminary dispersion analysis in Kaplunov et al. [24]. The proposed asymptotic procedure naturally extends that for a plate with traction free faces (e.g., see Kaplunov et al. [14], Goldenveizer et al. [25], and references therein). The leading order approximation is derived, along with three higher-order ones. Comparisons of the obtained shortened equations with the asymptotic behaviours of the exact dispersion relation illustrate the consistency of the presented results. The latter are used for the critical assessment of the well-known formulations using the classical and refined plate equations. It is demonstrated that plate transverse inertia and fluid compressibility can be neglected at leading order. The former comes to play at next order, whereas compressibility has to be taken into account only at third order. The effects of shear deformation, rotary inertia, and other mechanical phenomena are evaluated. The asymptotic correction to the traditional kinematic impenetrability condition along the fluid–solid interface is also established.

This paper is organised as follows. The statement of the problem is given in section 2. Section 3 is concerned with the derivation of the various asymptotic approximations, with asymptotic models of all four orders being presented in section 4. Section 5 presents dispersion relations for all four approximations and the numerical results are given therein, with concluding remarks in section 6.

2. Statement of problem

Consider free vibrations of an isotropic elastic layer of thickness $2 h$ immersed in a non-viscous compressible fluid, specifying Cartesian coordinates $- \infty < x_{1}, x_{2} < \infty, - h \leq x_{3} \leq h$ (see Figure 1); the axis $x_{2}$ , perpendicular to the plane $(x_{1}, x_{3})$ , is not shown in the figure. Throughout this paper, we use the following notation: $ρ$ and $ρ_{0}$ are the solid and fluid densities, respectively; $E$ is Young’s modulus, $ν$ is Poisson’s ratio, $c_{2} = \sqrt{E / 2 ρ (1 + ν)}$ is the shear wave speed in solid, and $c_{0}$ is the wave speed in fluid.

Figure 1.

Fluid-loaded elastic layer.

We restrict ourselves to a plane strain problem in the coordinates $(x_{1}, x_{3})$ . Then, the equations of motion in linear elasticity may be written as:

\begin{matrix} \frac{\partial σ_{11}}{\partial x_{1}} + \frac{\partial σ_{31}}{\partial x_{3}} - ρ \frac{\partial^{2} v_{1}}{\partial t^{2}} = 0, \\ \frac{\partial σ_{13}}{\partial x_{1}} + \frac{\partial σ_{33}}{\partial x_{3}} - ρ \frac{\partial^{2} v_{3}}{\partial t^{2}} = 0 . \end{matrix}

(1)

Here and below, $σ_{kl} (k, l = 1, 2, 3)$ are the stresses, $v_{m} (m = 1, 3)$ are the displacements, and $t$ is the time. In these equations, the stresses and displacements satisfy the relations:

\begin{matrix} σ_{11} = \frac{E}{1 - ν^{2}} \frac{\partial v_{1}}{\partial x_{1}} + \frac{ν}{1 - ν} σ_{33}, \\ σ_{22} = \frac{E ν}{1 - ν^{2}} \frac{\partial v_{1}}{\partial x_{1}} + \frac{ν}{1 - ν} σ_{33}, \\ \frac{\partial v_{3}}{\partial x_{3}} = \frac{1}{E} (σ_{33} - ν (σ_{11} + σ_{22})), \\ \frac{\partial v_{1}}{\partial x_{3}} = - \frac{\partial v_{3}}{\partial x_{1}} + \frac{2 (1 + ν)}{E} σ_{31}, \end{matrix}

(2)

oriented to the forthcoming asymptotic analysis (e.g., see Kaplunov et al. [14]).

In addition, we have the wave equation for the fluid displacement potential $φ (x_{1}, x_{3}, t)$ :

\frac{\partial^{2} φ}{\partial x_{1}^{2}} + \frac{\partial^{2} φ}{\partial x_{3}^{2}} - \frac{1}{c_{0}^{2}} \frac{\partial^{2} φ}{\partial t^{2}} = 0 .

(3)

The interfacial conditions at $x_{3} = \pm h$ are:

σ_{31} = 0, σ_{33} = ρ_{0} \frac{\partial^{2} φ}{\partial t^{2}}, v_{3} = \frac{\partial φ}{\partial x_{3}} .

(4)

The aim of this paper is to derive asymptotic models of the formulated problem over long-wave low-frequency region, where $h << L$ and $h / c_{2} << T$ with $L$ and $T$ denoting a typical wavelength and time scale, respectively. The presentation below is not restricted to the leading order, also focusing on the peculiarities of higher-order ones.

3. Asymptotic derivation

3.1. Scaling and dimensionless equations

Let us scale the independent variables specified in the previous section by:

\begin{matrix} t = T τ, x_{1} = L ξ, x_{3} = {\begin{matrix} L η ζ, & if | x_{3} | < h, \\ L γ, & otherwise, \end{matrix} \end{matrix}

(5)

where $η = h / L << 1$ is a small geometric parameter. In addition, we assume that:

T = \frac{L}{η^{3 / 2} c_{2}},

(6)

which is motivated by the relation $ω h / c_{2} ~ {(kh)}^{5 / 2}$ between angular frequency $ω$ and wave number $k$ discovered for a fluid-borne bending wave from the asymptotic analysis of the dispersion relation for an immersed layer (see Kaplunov et al. [24]).

Now, we introduce the dimensionless stresses, displacements, and fluid potential, setting:

\begin{matrix} σ_{11} = E η σ_{11}^{*}, σ_{22} = E η σ_{22}^{*}, σ_{31} = E η^{2} σ_{31}^{*}, σ_{33} = E η^{3} σ_{33}^{*}, \\ v_{1} = L η v_{1}^{*}, v_{3} = {Lv}_{3}^{*}, φ = L^{2} φ^{*} . \end{matrix}

(7)

Here, all the starred quantities are assumed to be the same asymptotic order, i.e., of order unity. The asymptotic orders of the displacements and stresses in equation (7) are the same as for a bending wave on an elastic plate not contacting fluid, elastic plate (e.g., see Kaplunov et al. [14]). At the same time, for the latter, a typical time scale is $T = \frac{L}{η^{2} c_{2}}$ instead of equation (6).

Thus, equations (1) and (2) in the last section, taking into account relations (5) and (7), can be written in dimensionless form as:

\begin{matrix} \frac{\partial σ_{31}^{*}}{\partial ζ} = - \frac{\partial σ_{11}^{*}}{\partial ξ} + \frac{1}{2 (1 + ν)} η^{3} \frac{\partial^{2} v_{1}^{*}}{\partial τ^{2}}, \\ \frac{\partial σ_{33}^{*}}{\partial ζ} = - \frac{\partial σ_{31}^{*}}{\partial ξ} + \frac{1}{2 (1 + ν)} η \frac{\partial^{2} v_{3}^{*}}{\partial τ^{2}}, \end{matrix}

(8)

and

\begin{matrix} σ_{11}^{*} = \frac{1}{1 - ν^{2}} \frac{\partial v_{1}^{*}}{\partial ξ} + \frac{ν}{1 - ν} η^{2} σ_{33}^{*}, \\ σ_{22}^{*} = \frac{ν}{1 - ν^{2}} \frac{\partial v_{1}^{*}}{\partial ξ} + \frac{ν}{1 - ν} η^{2} σ_{33}^{*}, \\ \frac{\partial v_{3}^{*}}{\partial ζ} = η^{4} σ_{33}^{*} - ν η^{2} (σ_{11}^{*} + σ_{22}^{*}), \\ \frac{\partial v_{1}^{*}}{\partial ζ} = - \frac{\partial v_{3}^{*}}{\partial ξ} + 2 (1 + ν) η^{2} σ_{31}^{*} . \end{matrix}

(9)

Similarly, the dimensionless form of equations (3) and (4) can be obtained using equations (5)–(7), which gives the following:

\frac{\partial^{2} φ^{*}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{*}}{\partial γ^{2}} - η^{3} δ^{2} \frac{\partial^{2} φ^{*}}{\partial τ^{2}} = 0,

(10)

and

\begin{matrix} {σ_{31}^{*} |}_{ζ = \pm 1} = 0, {σ_{33}^{*} |}_{ζ = \pm 1} = {\frac{r}{2 (1 + ν)} \frac{\partial^{2} φ^{*}}{\partial τ^{2}} |}_{γ = \pm η}, \\ {\frac{\partial φ^{*}}{\partial γ} |}_{γ = \pm η} = {v_{3}^{*} |}_{ζ = \pm 1}, \end{matrix}

(11)

where:

r = \frac{ρ_{0}}{ρ}, δ = \frac{c_{2}}{c_{0}} .

(12)

Now, expand the starred stress and displacement components in an asymptotic series as:

\begin{matrix} v_{1}^{*} = v_{1}^{(0)} + η v_{1}^{(1)} + η^{2} v_{1}^{(2)} + . . . \\ v_{3}^{*} = v_{3}^{(0)} + η v_{3}^{(1)} + η^{2} v_{3}^{(2)} + . . . \\ σ_{11}^{*} = σ_{11}^{(0)} + η σ_{11}^{(1)} + η^{2} σ_{11}^{(2)} + . . . \\ σ_{22}^{*} = σ_{22}^{(0)} + η σ_{22}^{(1)} + η^{2} σ_{22}^{(2)} + . . . \\ σ_{33}^{*} = σ_{33}^{(0)} + η σ_{33}^{(1)} + η^{2} σ_{33}^{(2)} + . . . \\ σ_{31}^{*} = σ_{31}^{(0)} + η σ_{31}^{(1)} + η^{2} σ_{31}^{(2)} + . . . \\ φ^{*} = φ^{(0)} + η φ^{(1)} + η^{2} φ^{(2)} + . . . \end{matrix}

(13)

leading to shortened forms of the original plane strain problem in hydro-elasticity.

In what follows, we derive four asymptotic approximations, considering antisymmetric motion about the midplane $x_{3} = 0$ . Also, due to the symmetry of the problem, we satisfy the interfacial conditions along the upper face $x_{3} = h$ only.

3.2. Leading order approximation

We consider the leading order approximation of the problem formulated in the previous section, keeping only the terms with the suffix $(0)$ in the asymptotic series (13). First, integrating equations (9)₃ and (9)₄ along the thickness variable $ζ$ , we obtain, respectively:

v_{3}^{(0)} = V_{3}^{(0)} (ξ, τ) and v_{1}^{(0)} = - ζ \frac{\partial V_{3}^{(0)}}{\partial ξ} .

(14)

Substituting the displacements (14) into equations (9)₁ and (9)₂, we obtain:

σ_{11}^{(0)} = - ζ \frac{1}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(0)}}{\partial ξ^{2}} and σ_{22}^{(0)} = - ζ \frac{ν}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(0)}}{\partial ξ^{2}} .

(15)

Next, integrating equation (8)₁ along $ζ$ , taking into account equations (14) and (15), gives:

σ_{31}^{(0)} = ζ^{2} \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(0)}}{\partial ξ^{3}} + A^{(0)} (ξ, τ),

(16)

where $A^{(0)}$ is an arbitrary function. Finally, inserting equations (14)–(16) into equation (8)₂ and integrating along $ζ$ , we obtain:

σ_{33}^{(0)} = - ζ^{3} \frac{1}{6 (1 - ν^{2})} \frac{\partial^{4} V_{3}^{(0)}}{\partial ξ^{4}} - ζ \frac{\partial A^{(0)}}{\partial ξ} .

(17)

Now, the fluid potential at leading order can be determined by inserting equation (13)₇ into equation (10), giving the Laplace equation:

\frac{\partial^{2} φ^{(0)}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{(0)}}{\partial γ^{2}} = 0,

(18)

governing incompressible fluid. Furthermore, the boundary conditions, obtained by inserting equation (13) into equation (11), are:

{σ_{31}^{(0)} |}_{ζ = 1} = 0, {σ_{33}^{(0)} |}_{ζ = 1} = {\frac{r}{2 (1 + ν)} \frac{\partial^{2} φ^{(0)}}{\partial τ^{2}} |}_{γ = η}, {\frac{\partial φ^{(0)}}{\partial γ} |}_{γ = η} = v_{3}^{(0)} .

(19)

By applying equations (19)₁ and (19)₂ into equations (16) and (17), respectively, we obtain:

A^{(0)} = - \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(0)}}{\partial ξ^{3}},

(20)

and

{\frac{2}{3 (1 - ν)} \frac{\partial^{4} V_{3}^{(0)}}{\partial ξ^{4}} - r \frac{\partial^{2} φ^{(0)}}{\partial τ^{2}} |}_{γ = η} = 0 .

(21)

The last equation governs bending of a thin elastic plate with the inertial term originating from fluid motion. Below, we discuss the related asymptotic model in detail.

3.3. First-order approximation

In this section, we derive the first-order approximation of the problem formulated in section 3.1 but now keeping also the terms with the suffix (1) in the asymptotic series (13). Below, we omit some of the intermediate computations as the procedure is essentially the same as in the previous section. Now, we have:

\frac{\partial^{2} φ^{(1)}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{(1)}}{\partial γ^{2}} = 0,

(22)

and

\begin{matrix} v_{3}^{(1)} = V_{3}^{(1)} (ξ, τ), \\ v_{1}^{(1)} = - ζ \frac{\partial V_{3}^{(1)}}{\partial ξ}, \\ σ_{11}^{(1)} = - ζ \frac{1}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(1)}}{\partial ξ^{2}}, \\ σ_{22}^{(1)} = - ζ \frac{ν}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(1)}}{\partial ξ^{2}}, \\ σ_{31}^{(1)} = ζ^{2} \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(1)}}{\partial ξ^{3}} + A^{(1)} (ξ, τ), \\ σ_{33}^{(1)} = - ζ^{3} \frac{1}{6 (1 - ν^{2})} \frac{\partial^{4} V_{3}^{(1)}}{\partial ξ^{4}} - ζ \frac{\partial A^{(1)}}{\partial ξ} + ζ \frac{1}{2 (1 + ν)} \frac{\partial^{2} V_{3}^{(0)}}{\partial τ^{2}}, \end{matrix}

(23)

where $A^{(1)}$ is an arbitrary function, and the boundary conditions are:

{σ_{31}^{(1)} |}_{ζ = 1} = 0, {σ_{33}^{(1)} |}_{ζ = 1} = {\frac{r}{2 (1 + ν)} \frac{\partial^{2} φ^{(1)}}{\partial τ^{2}} |}_{γ = η}, {\frac{\partial φ^{(1)}}{\partial γ} |}_{γ = η} = v_{3}^{(1)} .

(24)

Note that the last term in the expression for $σ_{33}^{(1)}$ in equation (23) corresponds to elastic plate inertia which does not affect the leading order approximation.

Next, applying equations (24)₁ and (24)₂ into equations (23)₅ and (23)₆, respectively, we obtain:

A^{(1)} = - \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(1)}}{\partial ξ^{3}},

(25)

and

{\frac{2}{3 (1 - ν)} \frac{\partial^{4} V_{3}^{(1)}}{\partial ξ^{4}} + \frac{\partial^{2} V_{3}^{(0)}}{\partial τ^{2}} - r \frac{\partial^{2} φ^{(1)}}{\partial τ^{2}} |}_{γ = η} = 0 .

(26)

In what follows, we demonstrate that the above equation, combined with equation (21), corresponds to a fluid-loaded Kirchhoff plate.

3.4. Second-order approximation

Let us proceed towards deriving the second-order approximation. Analysis of relations (9) shows that at this order, the “Poisson effect” of the stress $σ_{33}$ on the stresses $σ_{11}$ and $σ_{22}$ as well as on the displacements $v_{3}$ , has to be accounted along with the transverse shear deformation (see also Kaplunov et al. [14] and Goldenveizer et al. [25]). In this case, we have as before:

\frac{\partial^{2} φ^{(2)}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{(2)}}{\partial γ^{2}} = 0,

(27)

and also:

\begin{matrix} v_{3}^{(2)} = ζ^{2} \frac{ν}{2 (1 - ν)} \frac{\partial^{2} V_{3}^{(0)}}{\partial ξ^{2}} + V_{3}^{(2)} (ξ, τ), \\ v_{1}^{(2)} = \frac{1}{6 (1 - ν)} ((2 - ν) ζ^{3} - 6 ζ) \frac{\partial^{3} V_{3}^{(0)}}{\partial ξ^{3}} - ζ \frac{\partial V_{3}^{(2)}}{\partial ξ}, \\ σ_{11}^{(2)} = \frac{1}{6 (1 - ν) (1 - ν^{2})} (2 (1 - ν) ζ^{3} + 3 (ν - 2) ζ) \frac{\partial^{4} V_{3}^{(0)}}{\partial ξ^{4}} \\ - ζ \frac{1}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(2)}}{\partial ξ^{2}}, \\ σ_{22}^{(2)} = \frac{ν}{6 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{3} - 3 ζ) \frac{\partial^{4} V_{3}^{(0)}}{\partial ξ^{4}} - ζ \frac{ν}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(2)}}{\partial ξ^{2}}, \\ σ_{31}^{(2)} = - \frac{1}{12 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{4} + 3 (ν - 2) ζ^{2}) \frac{\partial^{5} V_{3}^{(0)}}{\partial ξ^{5}} \\ + ζ^{2} \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(2)}}{\partial ξ^{3}} + A^{(2)} (ξ, τ), \\ σ_{33}^{(2)} = \frac{1}{60 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{5} + 5 (ν - 2) ζ^{3}) \frac{\partial^{6} V_{3}^{(0)}}{\partial ξ^{6}} \\ - ζ^{3} \frac{1}{6 (1 - ν^{2})} \frac{\partial^{4} V_{3}^{(2)}}{\partial ξ^{4}} - ζ \frac{\partial A^{(2)}}{\partial ξ} + ζ \frac{1}{2 (1 + ν)} \frac{\partial^{2} V_{3}^{(1)}}{\partial τ^{2}}, \end{matrix}

(28)

where $A^{(2)}$ is an arbitrary function, and the boundary conditions, similarly to equations (19) and (24), become:

{σ_{31}^{(2)} |}_{ζ = 1} = 0, {σ_{33}^{(2)} |}_{ζ = 1} = {\frac{r}{2 (1 + ν)} \frac{\partial^{2} φ^{(2)}}{\partial τ^{2}} |}_{γ = η}, {\frac{\partial φ^{(2)}}{\partial γ} |}_{γ = η} = {v_{3}^{(2)} |}_{ζ = 1} .

(29)

Applying equations (29)₁ and (29)₂ into equations (28)₅ and (28)₆, respectively, we derive:

A^{(2)} = \frac{2 ν - 5}{12 (1 - ν) (1 - ν^{2})} \frac{\partial^{5} V_{3}^{(0)}}{\partial ξ^{5}} - \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(2)}}{\partial ξ^{3}},

(30)

and

{\frac{8 - 3 ν}{15 (1 - ν)^{2}} \frac{\partial^{6} V_{3}^{(0)}}{\partial ξ^{6}} + \frac{2}{3 (1 - ν)} \frac{\partial^{4} V_{3}^{(2)}}{\partial ξ^{4}} + \frac{\partial^{2} V_{3}^{(1)}}{\partial τ^{2}} - r \frac{\partial^{2} φ^{(2)}}{\partial τ^{2}} |}_{γ = η} = 0 .

(31)

Eliminating the sixth-order derivative in equation (31) using equation (21) (see Kaplunov et al. [14] and Goldenveizer et al. [25] for more detail), we arrive at:

\begin{matrix} {\frac{2}{3 (1 - ν)} \frac{\partial^{4} V_{3}^{(2)}}{\partial ξ^{4}} + \frac{\partial^{2} V_{3}^{(1)}}{\partial τ^{2}} + r \frac{\partial^{2}}{\partial τ^{2}} (\frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2} φ^{(0)}}{\partial ξ^{2}} - φ^{(2)}) |}_{γ = η} = 0 . \end{matrix}

(32)

This equation corresponds to a refined theory of plate bending under fluid loading, which, however, does not take into consideration rotary inertia. The latter appears only at the next order, in contrast to a free plate for which the effect of rotary inertia is of the same order as that of transverse shear deformation (see again Kaplunov et al. [14] and Goldenveizer et al. [25]).

3.5. Third-order approximation

Finally, we derive the third-order approximation of the original problem. This time, the inspection of dimensionless equations (8) and (10) indicates that rotary inertia and fluid compressibility have to be retained. Then, we obtain:

\frac{\partial^{2} φ^{(3)}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{(3)}}{\partial γ^{2}} - δ^{2} \frac{\partial^{2} φ^{(0)}}{\partial τ^{2}} = 0,

(33)

and

\begin{matrix} v_{3}^{(3)} = ζ^{2} \frac{ν}{2 (1 - ν)} \frac{\partial^{2} V_{3}^{(1)}}{\partial ξ^{2}} + V_{3}^{(3)} (ξ, τ), \\ v_{1}^{(3)} = \frac{1}{6 (1 - ν)} ((2 - ν) ζ^{3} - 6 ζ) \frac{\partial^{3} V_{3}^{(1)}}{\partial ξ^{3}} - ζ \frac{\partial V_{3}^{(3)}}{\partial ξ}, \\ σ_{11}^{(3)} = \frac{1}{6 (1 - ν) (1 - ν^{2})} (2 (1 - ν) ζ^{3} + 3 (ν - 2) ζ) \frac{\partial^{4} V_{3}^{(1)}}{\partial ξ^{4}} \\ - ζ \frac{1}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(3)}}{\partial ξ^{2}} + ζ \frac{ν}{2 (1 - ν^{2})} \frac{\partial^{2} V_{3}^{(0)}}{\partial τ^{2}}, \\ σ_{22}^{(3)} = \frac{ν}{6 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{3} - 3 ζ) \frac{\partial^{4} V_{3}^{(1)}}{\partial ξ^{4}} \\ - ζ \frac{ν}{1 - ν^{2}} \frac{\partial^{2} V_{3}^{(3)}}{\partial ξ^{2}} + ζ \frac{ν}{2 (1 - ν^{2})} \frac{\partial^{2} V_{3}^{(0)}}{\partial τ^{2}}, \\ σ_{31}^{(3)} = - \frac{1}{12 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{4} + 3 (ν - 2) ζ^{2}) \frac{\partial^{5} V_{3}^{(1)}}{\partial ξ^{5}} \end{matrix}

\begin{matrix} + ζ^{2} \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(3)}}{\partial ξ^{3}} - ζ^{2} \frac{1}{4 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(0)}}{\partial ξ \partial τ^{2}} + A^{(3)} (ξ, τ), \\ σ_{33}^{(3)} = \frac{1}{60 (1 - ν) (1 - ν^{2})} ((1 - ν) ζ^{5} + 5 (ν - 2) ζ^{3}) \frac{\partial^{6} V_{3}^{(1)}}{\partial ξ^{6}} \\ - ζ^{3} \frac{1}{6 (1 - ν^{2})} \frac{\partial^{4} V_{3}^{(3)}}{\partial ξ^{4}} + ζ^{3} \frac{1}{12 (1 - ν)} \frac{\partial^{4} V_{3}^{(0)}}{\partial ξ^{2} \partial τ^{2}} \\ - ζ \frac{\partial A^{(3)}}{\partial ξ} + ζ \frac{1}{2 (1 + ν)} \frac{\partial^{2} V_{3}^{(2)}}{\partial τ^{2}}, \end{matrix}

(34)

where $A^{(3)}$ is an arbitrary function, and the boundary conditions as usual take the form:

{σ_{31}^{(3)} |}_{ζ = 1} = 0, {σ_{33}^{(3)} |}_{ζ = 1} = {\frac{r}{2 (1 + ν)} \frac{\partial^{2} φ^{(3)}}{\partial τ^{2}} |}_{γ = η}, {\frac{\partial φ^{(3)}}{\partial γ} |}_{γ = η} = {v_{3}^{(3)} |}_{ζ = 1} .

(35)

Applying equations (35)₁ and (35)₂ into equations (34)₅ and (34)₆, respectively, we have:

A^{(3)} = \frac{2 ν - 5}{12 (1 - ν) (1 - ν^{2})} \frac{\partial^{5} V_{3}^{(1)}}{\partial ξ^{5}} - \frac{1}{2 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(3)}}{\partial ξ^{3}} + \frac{1}{4 (1 - ν^{2})} \frac{\partial^{3} V_{3}^{(0)}}{\partial ξ \partial τ^{2}} .

(36)

Also, after eliminating the term with a sixth-order derivative in the same manner as in the previous subsection, we arrive at:

\begin{matrix} \frac{2}{3 (1 - ν)} \frac{\partial^{4} V_{3}^{(3)}}{\partial ξ^{4}} + \frac{\partial^{2}}{\partial τ^{2}} (V_{3}^{(2)} + \frac{7 ν - 17}{15 (1 - ν)} \frac{\partial^{2} V_{3}^{(0)}}{\partial ξ^{2}}) \\ {+ r \frac{\partial^{2}}{\partial τ^{2}} (\frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2} φ^{(1)}}{\partial ξ^{2}} - φ^{(3)}) |}_{γ = η} = 0 . \end{matrix}

(37)

This equation is in agreement with the refined equation for a free plate, loaded along its faces, addressed in Kaplunov et al. [14] and Goldenveizer et al. [25].

4. Asymptotic models

In this section, we formulate asymptotic models, i.e., the shortened equations of motion for a thin elastic plate and fluid, as well as the impenetrability conditions along the interfaces, starting from the results established in the previous section. At leading order, from equations (18), (19)₃, and (21), we get:

{\frac{2}{3 (1 - ν)} \frac{\partial^{4} v_{3}^{*}}{\partial ξ^{4}} - r \frac{\partial^{2} φ^{*}}{\partial τ^{2}} |}_{γ = η} = 0,

(38)

\frac{\partial^{2} φ^{*}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{*}}{\partial γ^{2}} = 0,

(39)

and

{\frac{\partial φ^{*}}{\partial γ} |}_{γ = η} = v_{3}^{*},

(40)

where $v_{3}^{*} (ξ, 0, τ) = V_{3}^{(0)} (ξ, τ)$ and $φ^{*} (ξ, γ, τ) = φ^{(0)} (ξ, γ, τ)$ . In terms of original variables, these equations become:

{\frac{E h^{3}}{3 (1 - ν^{2})} \frac{\partial^{4} v_{3}}{\partial x_{1}^{4}} - ρ_{0} \frac{\partial^{2} φ}{\partial t^{2}} |}_{x_{3} = h} = 0,

(41)

\frac{\partial^{2} φ}{\partial x_{1}^{2}} + \frac{\partial^{2} φ}{\partial x_{3}^{2}} = 0,

(42)

and

{\frac{\partial φ}{\partial x_{3}} |}_{x_{3} = h} = v_{3} .

(43)

Here and below in this section, the transverse displacement $v_{3}$ is taken at the mid plane $x_{3} = 0$ , i.e., $v_{3} = v_{3} (x_{1}, 0, t)$ .

This is the simplest asymptotic model for a fluid-loaded elastic plate. It incorporates plate stiffness, fluid inertia, neglecting plate inertia, and fluid compressibility. In this case, the impenetrability condition takes a well-known form.

Now, we consider the sum of equations (21) and (26), multiplied by the small parameter $η$ , to obtain:

\begin{matrix} {\frac{2}{3 (1 - ν)} \frac{\partial^{4}}{\partial ξ^{4}} (V_{3}^{(0)} + η V_{3}^{(1)}) - r \frac{\partial^{2}}{\partial τ^{2}} (φ^{(0)} + η φ^{(1)}) |}_{γ = η} \\ + η \frac{\partial^{2}}{\partial τ^{2}} (V_{3}^{(0)} + η V_{3}^{(1)}) - η^{2} \frac{\partial^{2} V_{3}^{(1)}}{\partial τ^{2}} = 0 . \end{matrix}

(44)

Below, $η^{2}$ term in the last equation is neglected. Similarly, we have from equations (18), (22), (19)₃, and (24)₃, respectively:

\frac{\partial^{2}}{\partial ξ^{2}} (φ^{(0)} + η φ^{(1)}) + \frac{\partial^{2}}{\partial γ^{2}} (φ^{(0)} + η φ^{(1)}) = 0,

(45)

and

{\frac{\partial}{\partial γ} (φ^{(0)} + η φ^{(1)}) |}_{γ = η} = V_{3}^{(0)} + η V_{3}^{(1)} .

(46)

Formulae (44)–(46) can be re-written as:

{\frac{2}{3 (1 - ν)} \frac{\partial^{4} v_{3}^{*}}{\partial ξ^{4}} + η \frac{\partial^{2} v_{3}^{*}}{\partial τ^{2}} - r \frac{\partial^{2} φ^{*}}{\partial τ^{2}} |}_{γ = η} = 0,

(47)

where $v_{3}^{*} = V_{3}^{(0)} + η V_{3}^{(1)}$ and $φ^{*} = φ^{(0)} + η φ^{(1)}$ . In original variables, equation (47) becomes:

{\frac{E h^{3}}{3 (1 - ν^{2})} \frac{\partial^{4} v_{3}}{\partial x_{1}^{4}} + ρ h \frac{\partial^{2} v_{3}}{\partial t^{2}} - ρ_{0} \frac{\partial^{2} φ}{\partial t^{2}} |}_{x_{3} = h} = 0 .

(48)

As follows from equations (45) and (46), the last equation should be considered together with equation of fluid motion (42) and impenetrability condition (43). Thus, the plate inertia is the only correction coming at this order. As a result, relations (42), (43), and (48) govern vibrations of a Kirchhoff plate, immersed in incompressible fluid.

Next, similarly to the derivation above, we have from equations (21), (26), and (32) neglecting $O (η^{3})$ terms:

\begin{matrix} {\frac{2}{3 (1 - ν)} \frac{\partial^{4} v_{3}^{*}}{\partial ξ^{4}} - r (1 - η^{2} \frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2}}{\partial ξ^{2}}) \frac{\partial^{2} φ^{*}}{\partial τ^{2}} |}_{γ = η} + η \frac{\partial^{2} v_{3}^{*}}{\partial τ^{2}} = 0, \end{matrix}

(49)

where $v_{3}^{*} = V_{3}^{(0)} + η V_{3}^{(1)} + η^{2} V_{3}^{(2)}$ and $φ^{*} = φ^{(0)} + η φ^{(1)} + η^{2} φ^{(2)}$ . In this case, the impenetrability condition to within the same truncation error becomes (see equations (11), (23), and (28)):

\begin{matrix} \frac{\partial φ^{*}}{\partial γ} |_{γ = η} = (1 + η^{2} \frac{ν}{2 (1 - ν)} \frac{\partial^{2}}{\partial ξ^{2}}) v_{3}^{*} . \end{matrix}

(50)

In terms of original variables, equations (49) and (50) are given by:

{\frac{E h^{3}}{3 (1 - ν^{2})} \frac{\partial^{4} v_{3}}{\partial x_{1}^{4}} + ρ h \frac{\partial^{2} v_{3}}{\partial t^{2}} - ρ_{0} (1 - h^{2} \frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2}}{\partial x_{1}^{2}}) \frac{\partial^{2} φ}{\partial t^{2}} |}_{x_{3} = h} = 0,

(51)

and

{\frac{\partial φ}{\partial x_{3}} |}_{x_{3} = h} = (1 + \frac{ν h^{2}}{2 (1 - ν)} \frac{\partial^{2}}{\partial x_{1}^{2}}) v_{3} .

(52)

The last two relations have to be considered together with equation (42) for incompressible fluid. It is remarkable that at this order $O (η^{2})$ , corrections appear both in the equation of plate motion and in the impenetrability condition. However, as it has been already mentioned, see also equation (8)₁, the rotary inertia is still beyond the accuracy of the asymptotic model, given by equations (42), (51), and (52).

Finally, we deduce from equations (21), (26), (32), and (37), neglecting $O (η^{4})$ terms:

\begin{matrix} {\frac{2}{3 (1 - ν)} \frac{\partial^{4} v_{3}^{*}}{\partial ξ^{4}} - r (1 - η^{2} \frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2}}{\partial ξ^{2}}) \frac{\partial^{2} φ^{*}}{\partial τ^{2}} |}_{γ = η} \\ + η (1 + η^{2} \frac{7 ν - 17}{15 (1 - ν)} \frac{\partial^{2}}{\partial ξ^{2}}) \frac{\partial^{2} v_{3}^{*}}{\partial τ^{2}} = 0, \end{matrix}

(53)

where $v_{3}^{*} = V_{3}^{(0)} + η V_{3}^{(1)} + η^{2} V_{3}^{(2)} + η^{3} V_{3}^{(3)}$ and $φ^{*} = φ^{(0)} + η φ^{(1)} + η^{2} φ^{(2)} + η^{3} φ^{(3)}$ . Now, the equation governing fluid motion, taking into account formulae (18), (22), (27), and (33) and neglecting $O (η^{4})$ terms, can be written as:

\begin{matrix} \frac{\partial^{2} φ^{*}}{\partial ξ^{2}} + \frac{\partial^{2} φ^{*}}{\partial γ^{2}} - η^{3} δ^{2} \frac{\partial^{2} φ^{*}}{\partial τ^{2}} = 0 . \end{matrix}

(54)

In original variables, equation (53) takes the form:

\begin{matrix} \frac{E h^{3}}{3 (1 - ν^{2})} \frac{\partial^{4} v_{3}}{\partial x_{1}^{4}} + ρ h (1 + h^{2} \frac{7 ν - 17}{15 (1 - ν)} \frac{\partial^{2}}{\partial x_{1}^{2}}) \frac{\partial^{2} v_{3}}{\partial t^{2}} \\ {- ρ_{0} (1 - h^{2} \frac{8 - 3 ν}{10 (1 - ν)} \frac{\partial^{2}}{\partial x_{1}^{2}}) \frac{\partial^{2} φ}{\partial t^{2}} |}_{x_{3} = h} = 0, \end{matrix}

(55)

together with the equation governing the compressible fluid (3) (equation (54) rewritten in original variables) and impenetrability condition (52), this equation corresponds to the third-order asymptotic model. In comparison with second-order model above, the latter incorporates rotary inertia and fluid compressibility. It is worth noting that the aforementioned third-order model coincides with the refined asymptotic formulation for a free plate in Kaplunov et al. [14] and Goldenveizer et al. [25]; see also Belov et al. [12], where fluid is treated as a mechanical load, prescribed along the faces.

5. Dispersion relations for derived asymptotic models

In this section, we derive the dispersion relations corresponding to the approximate formulations established in section 4.

Let us begin with the travelling wave solution of the problem (41)–(43), setting:

\begin{matrix} v_{3} = e^{i (k x_{1} - ω t)}, \\ φ = - \frac{1}{k} e^{- k (x_{3} - h) + i (k x_{1} - ω t)}, \end{matrix}

(56)

where $k$ and $ω$ are the wavenumber and angular frequency, respectively. On substituting equation (56) into the aforementioned formulae, we arrive at the dispersion relation:

Ω^{2} = \frac{2}{3 r (1 - ν)} K^{5},

(57)

where $K$ and $Ω$ are the dimensionless wavenumber and frequency, respectively, with:

K = kh, Ω = \frac{ω h}{c_{2}} .

(58)

Next, substituting equation (56) into equations (42), (43), and (48), we arrive at the dispersion relation, corresponding to the first-order asymptotic model:

Ω^{2} = \frac{2}{3 (1 - ν)} \frac{K^{5}}{r + K} .

(59)

Now, we set:

\begin{matrix} φ = - \frac{1}{k} (1 - \frac{ν h^{2} k^{2}}{2 (1 - ν)}) e^{- k (x_{3} - h) + i (k x_{1} - ω t)}, \end{matrix}

(60)

with $v_{3}$ , given by equation (56), in equations (42), (51), and (52), resulting in the dispersion relation for the second-order model, given by:

Ω^{2} = \frac{40 (1 - ν) K^{5}}{60 r {(1 - ν)}^{2} + 12 (4 r K^{2} + 5 K) (1 - ν)^{2} + 3 r ν (3 ν - 8) K^{4}} .

(61)

Similarly, insert $v_{3}$ from equation (56) and:

\begin{matrix} φ = - {(k^{2} - \frac{ω^{2}}{c_{0}^{2}})}^{- \frac{1}{2}} (1 - \frac{ν h^{2} k^{2}}{2 (1 - ν)}) e^{- {(k^{2} - \frac{ω^{2}}{c_{0}^{2}})}^{\frac{1}{2}} (x_{3} - h) + i (k x_{1} - ω t)} . \end{matrix}

(62)

into the equations of the third-order approximation, given by equations (3), (52), and (55). As a result, we have:

\begin{matrix} Ω^{2} (60 (1 - ν)^{2} H + 3 r (3 ν - 8) (2 (ν - 1) + ν K^{2}) K^{2} + 4 (1 - ν) (17 - 7 ν) K^{2} H \\ - 30 r (1 - ν) (2 (ν - 1) + ν K^{2})) = 40 (1 - ν) K^{4} H, \end{matrix}

(63)

where:

H = \sqrt{K^{2} - Ω^{2} δ^{2}} .

(64)

It can be easily verified that at $Ω ~ K^{\frac{5}{2}} (Ω << 1, K << 1)$ , the four-term polynomial asymptotic expansion (69) of the full dispersion relation (65), presented in Appendix 1, also follows from shortened dispersion relation (63). The similar observation is also true for the associated one-, two-, and three-term polynomial expansions in Appendix 1, which can be derived from the dispersion relations (57), (59), and (61), respectively.

Numerical results for an aluminium layer immersed in water are presented in Figures 2 –5. The problem parameters are $c_{0} = 1480 m s^{- 1}$ , $ν = 0.2$ , $c_{2} = 3156 m s^{- 1}$ , $ρ = 2710 kg m^{- 3}$ , and $ρ_{0} = 1000 kg m^{- 3}$ . The dispersion curves for the fluid-borne bending wave calculated from the full dispersion relation (65) and the leading (57), first (59), second (61), and third-order (63) dispersion relations, associated with the derived asymptotic models, are displayed in Figure 2. This figure shows that the dispersion curves (65) and (63) virtually coincide over the depicted frequency range.

Figure 2.

Comparison of the full dispersion relation (65) (solid black line) with the dispersion relations (57) (dashed orange line), (59) (dashed green line), (61) (dashed blue line), and (63) (dashed red line), corresponding to zero-, first-, second-, and third-order asymptotic models, respectively.

Figure 3.

Comparison of the dispersion relation (59) (dashed red line), corresponding to the first-order asymptotic model, with full dispersion relation (65) (solid black line) and two-term expansion in equation (69) (dotted blue line).

Figure 4.

Comparison of the dispersion relation (61) (dashed red line), corresponding to the second-order asymptotic model, with full dispersion relation (65) (solid black line) and the three-term asymptotic expansion (69) (dotted blue line).

Figure 5.

Comparison of the dispersion relation (63) (dashed red line), corresponding to the third-order asymptotic model, with full dispersion relation (65) (solid black line) and the four-term asymptotic expansion (69) (dotted blue line).

Figures 3 –5 show the comparison of the dispersion relations (59), (61) and (63) with two-, three-, and four-term polynomial expansions in Appendix 1 (see equation (69) together with equation (70)).

6. Conclusion

Asymptotic analysis of a plane problem for a fluid-loaded elastic layer is developed over a long-wave low-frequency domain. The chosen scaling (5), corresponding to a fluid-borne bending wave, is different to that for a free elastic layer. At the same time, the relative asymptotic orders of displacements and stresses (see equation (7)) are the same as for a ‘free layer (see Kaplunov et al. [14] and Goldenveizer et al. [25]).

Four asymptotic approximations are derived. At leading (zero) order, only elastic plate stiffness and fluid inertia have to be retained (see equation (41)). The plate transverse inertia appears at the next order. The associated fluid–structure interaction model (42), (43), and (48) involves a Kirchhoff plate immersed in incompressible fluid. Fluid compressibility has to be taken into consideration only at third order. Thus, the approximation of incompressible fluid often appears to be accurate enough to be adapted together with shortened low-dimensional equations governing free plate vibrations, in particular, the aforementioned Kirchhoff equation and some of its refinements. However, it might be expected that fluid compressibility is essential for forced vibrations analysis. In the latter case, asymptotic procedure will involve two independent parameters, corresponding to typical wavelength $L$ and time scale $T$ . For a free plate such two-parametric procedure is presented in Kaplunov et al. [14, 26].

In contrast to a free plate, the effects of transverse shear deformation arises at second order, whereas that of the rotary inertia has to be kept only at third order, see equations (51) and (55), respectively. It is remarkable that third-order equation of plate motion (55) also follows from the refined asymptotic theory for a free plate in Kaplunov et al. [14] and Goldenveizer et al. [25], in which fluid is treated as a prescribed mechanical load. It is also worth noting that the correction to the traditional impenetrability condition (43) comes at second order (see equation (52)).

The dispersion relations obtained from the derived asymptotic models (57), (59), (61), and (63) are compared with the expansions of the “exact” dispersion relation (65), demonstrated in Appendix 1. Numerical examples are also presented.

Footnotes

Appendix 1 Acknowledgements

Bursary for S.S’s PhD by Keele University is gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ludmila Prikachikova

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