Floating Elastic Plate

Introduction

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of physical structures such as a floating break water, an ice floe or a VLFS). The equations of motion were formulated more than 100 years ago and a discussion of the problem appears in Stoker 1957. The problem can be divided into the two and three dimensional formulations which are closely related. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.

Two Dimensional Problem

Equations of Motion

When considering a two dimensional problem, the [math]\displaystyle{ y }[/math] variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the Timoshenko Beam theory and Reddy-Bickford Beam theory where shear deformation of higher order is considered.

For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given by the following

[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p }[/math]

where [math]\displaystyle{ D }[/math] is the flexural rigidity, [math]\displaystyle{ \rho_i }[/math] is the density of the beam, [math]\displaystyle{ h }[/math] is the thickness of the beam (assumed constant), [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \eta }[/math] is the beam vertical displacement.

The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).

[math]\displaystyle{ \frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0 }[/math]

at the edges of the plate.

The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero pressure at the surface), i.e.

[math]\displaystyle{ p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t} }[/math]

where [math]\displaystyle{ \rho }[/math] is the water density and [math]\displaystyle{ g }[/math] is gravity, and [math]\displaystyle{ \phi }[/math] is the velocity potential. The velocity potential is governed by Laplace's equation through out the fluid domain subject to the free surface condition and the condition of no flow through the bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by [math]\displaystyle{ P }[/math] and the free surface by [math]\displaystyle{ F }[/math] the equations of motion for the Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] for water of Finite Depth are the following. At the surface we have the dynamic condition

[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta = i\omega \rho \phi, \, z=0, \, x\in P }[/math]

[math]\displaystyle{ 0= \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F }[/math]

and the kinematic condition

[math]\displaystyle{ \frac{\partial\phi}{\partial z} = i\omega\eta }[/math]

The equation within the fluid is governed by Laplace's Equation

[math]\displaystyle{ \nabla^2\phi =0 }[/math]

and we have the no-flow condition through the bottom boundary

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-h }[/math]

(so we have a fluid of constant depth with the bottom surface at [math]\displaystyle{ z=-h }[/math] and the free surface or plate covered surface are at [math]\displaystyle{ z=0 }[/math]). [math]\displaystyle{ g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \rho_i }[/math] and [math]\displaystyle{ \rho }[/math] are the densities of the plate and the water respectively, [math]\displaystyle{ h }[/math] and [math]\displaystyle{ D }[/math] the thickness and flexural rigidity of the plate.

Solution Methods

There are many different methods to solve the corresponding equations ranging from highly analytic such as the Wiener-Hopf to very numerical based on Eigenfunction Matching Method which are applicable and have advantages in different situations. We describe here some of the solutions which have been developed grouped by problem

Single Crack

The simplest problem to consider is one where there are only two semi-infinite plates of identical properties separated by a crack. A related problem in acoustics was considered by Kouzov 1963 who used an integral representation of the problem to solve it explicitly using the Riemann-Hilbert technique. Recently the crack problem has been considered by Squire and Dixon 2000 and Williams and Squire 2002 using a Green function method applicable to infinitely deep water and they obtained simple expressions for the reflection and transmission coefficients. Squire and Dixon 2001 extended the single crack problem to a multiple crack problem in which the semi-infinite regions are separated by a region consisting of a finite number of plates of finite size with all plates having identical properties. Evans and Porter 2005 further considered the multiple crack problem for finitely deep water and provided an explicit solution.

We present here the solution of Evans and Porter 2005 for the simple case of a single crack with waves incident from normal (they also considered multiple cracks and waves incident from different angles). The solution of Evans and Porter 2005 expresses the potential [math]\displaystyle{ \phi }[/math] in terms of a linear combination of the incident wave and certain source functions located at the crack. Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across the crack. They first define [math]\displaystyle{ \chi(x,z) }[/math] to be the Two-Dimensional solution to the Free-Surface Green Function for a Floating Elastic Plate given by

[math]\displaystyle{ \chi(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k_n h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-k_n|x|},\,\,\,(1) }[/math]

where

[math]\displaystyle{ C_n=\frac{1}{2}\left(h + \frac{(5\beta k_n ^4 + 1 - \alpha\gamma)\sin^2{(k_n h)}}{\alpha}\right), }[/math]

and [math]\displaystyle{ k_n }[/math] are the solutions of the Dispersion Relation for a Floating Elastic Plate.

Consequently, the source functions for a single crack at [math]\displaystyle{ x=0 }[/math] can be defined as

[math]\displaystyle{ \psi_s(x,z)= \beta\chi_{xx}(x,z),\,\,\, \psi_a(x,z)= \beta\chi_{xxx}(x,z),\,\,\,(2) }[/math]

It can easily be shown that [math]\displaystyle{ \psi_s }[/math] is symmetric about [math]\displaystyle{ x = 0 }[/math] and [math]\displaystyle{ \psi_a }[/math] is antisymmetric about [math]\displaystyle{ x = 0 }[/math].

Substituting (1) into (2) gives

[math]\displaystyle{ \psi_s(x,z)= { -\frac{\beta}{\alpha} \sum_{n=-2}^\infty \frac{g_n\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|} }, \psi_a(x,z)= { {\rm sgn}(x) i\frac{\beta}{\alpha}\sum_{n=-2}^\infty \frac{g_n'\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|}}, }[/math]

where

[math]\displaystyle{ g_n = ik_n^3 \sin{k_n h},\,\,\,\, g'_n= -k_n^4 \sin{k_n h}. }[/math]

We then express the solution to the problem as a linear combination of the incident wave and pairs of source functions at each crack,

[math]\displaystyle{ \phi(x,z) = e^{-k_0 x}\frac{\cos(k_0(z+h))}{\cos(k_0h)} + (P\psi_s(x,z) + Q\psi_a(x,z))\,\,\,(3) }[/math]

where [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] are coefficients to be solved which represent the jump in the gradient and elevation respectively of the plates across the crack [math]\displaystyle{ x = a_j }[/math]. The coefficients [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] are found by applying the edge conditions and to the [math]\displaystyle{ z }[/math] derivative of [math]\displaystyle{ \phi }[/math] at [math]\displaystyle{ z=0 }[/math],

[math]\displaystyle{ \frac{\partial^2}{\partial x^2}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0,\,\,\, {\rm and}\,\,\,\, \frac{\partial^3}{\partial x^3}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0. }[/math]

The reflection and transmission coefficients, [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] can be found from (3) by taking the limits as [math]\displaystyle{ x\rightarrow\pm\infty }[/math] to obtain

[math]\displaystyle{ R = {- \frac{\beta}{\alpha} (g'_0Q + ig_0P)} }[/math]

and

[math]\displaystyle{ T= 1 + {\frac{\beta}{\alpha}(g'_0Q - ig_0P)} }[/math]

Two Semi-Infinite Plates of Different Properties

The next most simple problem is two semi-infinite plates of different properties. Often one of the plates is taken to be open water which makes the problem simpler. In general, the solution method developed for open water can be extended to two plates of different properties, the exception to this is the Residue Calculus solution which applies only when one of the semi-infinite regions is water.

Wiener-Hopf

The solution to the problem of two semi-infinite plates with different properties can be solved by the Wiener-Hopf method. The first work on this problem was by Evans and Davies 1968 but they did not actually develop the method sufficiently to be able to calculate the solution. The explicit solution was not found until the work of ...

Eigenfunction Matching Method

The eigenfunction matching solution was developed by Fox and Squire 1994. Essentially the solution is expanded on either side of the crack.

Three Dimensional Problem

Equations of Motion

For a classical thin plate, the equation of motion is given by

[math]\displaystyle{ D\nabla ^4 w + \rho _i h w = p }[/math]

The Equation of Motion for the Ice Floe

Ice floes range in size from much smaller to much larger than the dominant

wavelength of the ocean waves. However there are two reasons why solutions

for ice floes of intermediate size (a size similar to the wavelength) are

the most important. The first is that at these intermediate sizes ice floes

scatter significant wave energy. The second is that, since it is wave

induced flexure which determines the size of ice floes in the MIZ, ice floes

tend to form most often at this intermediate length.

The theory for an ice floe of intermediate size which is developed in this

paper obviously also applies to small or large floes. However, if the

solution for a small or large floe is required then the appropriate simpler

theory should be used. Small ice floes (ice floes much small than the

wavelength) should be modelled as rigid \citep{Masson_Le,Massondrift}. Large

ice floes (ice floes much larger than the wavelength) should be modelled as

infinite and flexible \citep{FoxandSquire}. In the intermediate region,

where the size of the wavelength is similar to the size of the ice floe, the

ice floe must be modelled as finite and flexible.

We model the ice floe as a thin plate of constant thickness and shallow

draft following \citet{Wadhams1986} and \citet{Squire_Review}. The thin

plate equation \citep{Hildebrand65} gives the following equation of motion

for the ice floe

[math]\displaystyle{ D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p, \label{plate} }[/math]

where [math]\displaystyle{ W }[/math] is the floe displacement, [math]\displaystyle{ \rho _{i} }[/math] is the floe density, [math]\displaystyle{ h }[/math] is

the floe thickness, [math]\displaystyle{ p }[/math] is the pressure, and [math]\displaystyle{ D }[/math] is the modulus of rigidity

of the floe ([math]\displaystyle{ D=Eh^{3}/12(1-\nu ^{2}) }[/math] where [math]\displaystyle{ E }[/math] is the Young's modulus and [math]\displaystyle{ \nu \lt math\gt is Poisson's ratio). Visco-elastic effects can be included by making }[/math] D </math> have some imaginary (damping)\ component but this will not be done here

to keep the presented results simpler. We assume that the plate is in

contact with the water at all times so that the water surface displacement

is also [math]\displaystyle{ W. }[/math] Equation (\ref{plate}) is subject to the free edge boundary

conditions for a thin plate

[math]\displaystyle{ \dfrac{\partial ^{2}W}{\partial n^{2}}+\nu \dfrac{\partial ^{2}W}{\partial s^{2}}=0,\;\;\;\text{\textrm{and}}\mathrm{\;\;\;}\dfrac{\partial ^{3}W}{ \partial n^{3}}+\left( 2-\nu \right) \dfrac{\partial ^{3}W}{\partial n\partial s^{2}}=0, \label{boundaryplate} }[/math]

\citep{Hildebrand65} where [math]\displaystyle{ n }[/math] and [math]\displaystyle{ s }[/math] denote the normal and tangential

directions respectively.

The pressure, [math]\displaystyle{ p }[/math], is given by the linearized Bernoulli's equation at the

water surface,

[math]\displaystyle{ p=-\rho \frac{\partial \Phi }{\partial t}-\rho gW \label{pressure} }[/math]

where [math]\displaystyle{ \Phi }[/math] is the velocity potential of the water, [math]\displaystyle{ \rho }[/math] is the density

of the water, and [math]\displaystyle{ g }[/math] is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length

variables with respect to [math]\displaystyle{ a }[/math] where the surface area of the floe is [math]\displaystyle{ 4a^{2}. }[/math]

We non-dimensionalise the time variables with respect to [math]\displaystyle{ \sqrt{g/a} }[/math] and

the mass variables with respect to [math]\displaystyle{ \rho a^{3} }[/math]. The non-dimensional

variables, denoted by an overbar, are

[math]\displaystyle{ \bar{x}=\frac{x}{a},\;\;\bar{y}=\frac{y}{a},\;\;\bar{z}=\frac{z}{a},\;\;\bar{ W}=\frac{W}{a},\;\;\bar{t}=t\sqrt{\frac{g}{a}},\;\;\text{and}\;\;\bar{\Phi}= \frac{\Phi }{a\sqrt{ag}}. }[/math]

In the non-dimensional variables equations (\ref{plate}) and (\ref{pressure} ) become

[math]\displaystyle{ \beta \nabla ^{4}\bar{W}+\gamma \frac{\partial ^{2}\bar{W}}{\partial \bar{t} ^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{W}, \label{n-d_ice} }[/math]

where

[math]\displaystyle{ \beta =\frac{D}{g\rho a^{4}}\;\;\text{and\ \ }\gamma =\frac{\rho _{i}h}{\rho a}. }[/math]

We shall refer to [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] as the stiffness and mass

respectively.

We will determine the response of the ice floe to wave forcing of a single

frequency (the response for more complex wave forcing can be found by

superposition of the single frequency solutions). Since the equations of

motion are linear the displacement and potential must have the same single

frequency dependence. Therefore they can be expressed as the real part of a

complex quantity whose time dependence is [math]\displaystyle{ e^{-i\sqrt{\alpha }t} }[/math] where [math]\displaystyle{ \alpha \lt math\gt is the non-dimensional wavenumber and we write }[/math]\bar{W}(\bar{x}, \bar{y},\bar{t})=\func{Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{ \alpha }\bar{t}}\right] \ [math]\displaystyle{ and }[/math]\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})= \func{Re}\left[ \phi \left( \bar{x},\bar{y},\bar{z}\right) e^{-i\sqrt{\alpha

}\bar{t}}\right] .</math> In the complex variables the equation of motion of the

ice floe (\ref{n-d_ice}) is

[math]\displaystyle{ \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. \label{plate2} }[/math]

From now on we will drop the overbar and assume all variables are

non-dimensional.

Equations of Motion for the Water

We require the equation of motion for the water to solve equation (\ref {plate2}). We begin with the non-dimensional equations of potential theory

which describe linear surface gravity waves

[math]\displaystyle{ \label{bvp} \left. \begin{array}{rr} \nabla ^{2}\phi =0, & -\infty \lt z\lt 0, \\ {\dfrac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\ {\dfrac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\; \mathbf{x}\in \Delta , \\ {\dfrac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{ x}\notin \Delta , \end{array} \right\} \label{bvp_nond} }[/math]

(\citet{Weh_Lait}). As before, [math]\displaystyle{ w }[/math] is the displacement of the floe and [math]\displaystyle{ p }[/math]

is the pressure at the water surface. The vector [math]\displaystyle{ \mathbf{x=(}x,y) }[/math] is a

point on the water surface and [math]\displaystyle{ \Delta }[/math] is the region of the water surface

occupied by the floe. The water is assumed infinitely deep. A schematic

diagram of this problem is shown in Figure \ref{vibration}.

\begin{figure}[tbp]

\begin{center}

\epsfbox{vibration.eps}

\end{center}

\caption{{The schematic diagram of the boundary value problem and the

coordinate system used in the solution.}}

\label{vibration}

\end{figure}

The boundary value problem (\ref{bvp}) is subject to an incident wave which

is imposed through a boundary condition as [math]\displaystyle{ \left| \mathbf{x}\right| \rightarrow \infty }[/math]. This boundary condition, which is called the

Sommerfeld radiation condition, is essentially that at large distances the

potential consists of a radial outgoing wave (the wave generated by the ice

floe motion) and the incident wave. It is expressed mathematically as

[math]\displaystyle{ \lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left( \frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{ \mathrm{In}})=0, \label{summerfield} }[/math]

\citep{Weh_Lait}. The incident potential (i.e. the incoming wave) [math]\displaystyle{ \phi ^{ \mathrm{In}} }[/math] is

[math]\displaystyle{ \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta +y\sin \theta )}e^{\alpha z}, \label{input} }[/math]

where [math]\displaystyle{ A }[/math] is the non-dimensional wave amplitude.

The standard solution method to the linear wave problem is to transform the

boundary value problem into an integral equation using a Green function

\citep{john1,

john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary

value problem (\ref{bvp}) and (\ref{summerfield}) becomes

[math]\displaystyle{ \phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x} ;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x} )\right) dS_{\mathbf{y}}. \label{water} }[/math]

The Green function [math]\displaystyle{ G_{\alpha } }[/math] is

[math]\displaystyle{ G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x} -\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y} |)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha | \mathbf{x}-\mathbf{y}|)\right) , }[/math]

\citep{Weh_Lait,jgrfloecirc}, where [math]\displaystyle{ J_{0} }[/math] and [math]\displaystyle{ Y_{0} }[/math] are respectively

Bessel functions of the first and second kind of order zero, and [math]\displaystyle{ \mathbf{ H_{0}} }[/math] is the Struve function of order zero \citep{abr_ste}. A solution for

water of finite depth could be found by simply using the depth dependent

Green function \citep{Weh_Lait}.

The integral equation (\ref{water}) will be solved using numerical

integration. The only difficulty arises from the non-trivial nature of the

kernel of the integral equation (the Green function). However, the Green

function has no [math]\displaystyle{ z }[/math] dependence due to the shallow draft approximation and

depends only on [math]\displaystyle{ |\mathbf{x}-\mathbf{y}|. }[/math] This means that the Green

function is one dimensional and the values which are required for a given

calculation can be looked up in a previously computed table.