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]

We model the ice floe as a thin plate of constant thickness and shallow draft following Wadhams1986 and Squire_Review. The thin plate equation 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, (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 ((plate)) is subject to the free edge boundary conditions for a thin plate

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

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 (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}},\;\;=and= \;\;\bar{\Phi}= \frac{\Phi }{a\sqrt{ag}}. }[/math]

In the non-dimensional variables equations ((plate)) and ((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}, (n-d_ice) }[/math]

where

[math]\displaystyle{ \beta =\frac{D}{g\rho a^{4}}\;\;=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})={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})= {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 ((n-d_ice)) is

[math]\displaystyle{ \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. (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{ (bvp) \left. \begin{matrix}{rr} \nabla ^{2}\phi =0, & -\infty \lt z\lt 0, \\ {\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\ {\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\; \mathbf{x}\in \Delta , \\ {\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{ x}\notin \Delta , \end{matrix} \right\} (bvp_nond) }[/math]

(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 (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.}}

(vibration)

\end{figure}

The boundary value problem ((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, (summerfield) }[/math]

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}, (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 ((bvp)) and ((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}}. (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]

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 abr_ste. A solution for water of finite depth could be found by simply using the depth dependent Green function Weh_Lait.

The integral equation ((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.

Solving for the Wave Induced Ice Floe Motion

To determine the ice floe motion we must solve equations ((plate2)) and ( (water)) simultaneously. We do this by expanding the floe motion in the free modes of vibration of a thin plate. The major difficulty with this method is that the free modes of vibration can be determined analytically only for very restrictive geometries, e.g. a circular thin plate. Even the free modes of vibration of a square plate with free edges must be determined numerically. This is the reason why the solution of jgrfloecirc was only for a circular floe.

Since the operator [math]\displaystyle{ \nabla ^{4}, }[/math] subject to the free edge boundary conditions, is self adjoint a thin plate must possess a set of modes [math]\displaystyle{ w_{i} }[/math] which satisfy the free boundary conditions and the following eigenvalue equation

[math]\displaystyle{ \nabla ^{4}w_{i}=\lambda _{i}w_{i}. }[/math]

The modes which correspond to different eigenvalues [math]\displaystyle{ \lambda _{i} }[/math] are orthogonal and the eigenvalues are positive and real. While the plate will always have repeated eigenvalues, orthogonal modes can still be found and the modes can be normalized. We therefore assume that the modes are orthonormal, i.e.

[math]\displaystyle{ \iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right) dS_{\mathbf{Q}}=\delta _{ij} }[/math]

where [math]\displaystyle{ \delta _{ij} }[/math] is the Kronecker delta. The eigenvalues [math]\displaystyle{ \lambda _{i} }[/math] have the property that [math]\displaystyle{ \lambda _{i}\rightarrow \infty }[/math] as [math]\displaystyle{ i\rightarrow \infty }[/math] and we order the modes by increasing eigenvalue. These modes can be used to expand any function over the wetted surface of the ice floe [math]\displaystyle{ \Delta }[/math] .

We expand the displacement of the floe in a finite number of modes [math]\displaystyle{ N, }[/math] i.e.

[math]\displaystyle{ w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) . (expansion) }[/math]

From the linearity of ((water)) the potential can be written in the following form

[math]\displaystyle{ \phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi) }[/math]

where [math]\displaystyle{ \phi _{0} }[/math] and [math]\displaystyle{ \phi _{i} }[/math] satisfy the integral equations

[math]\displaystyle{ \phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}} (phi0) }[/math]

and

[math]\displaystyle{ \phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y} )\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right) dS_{\mathbf{y}}. (phii) }[/math]

The potential [math]\displaystyle{ \phi _{0} }[/math] represents the potential due the incoming wave assuming that the displacement of the ice floe is zero. The potentials [math]\displaystyle{ \phi _{i} }[/math] represent the potential which is generated by the plate vibrating with the [math]\displaystyle{ i }[/math]th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into equation ((plate2)) to obtain

[math]\displaystyle{ \beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma \sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}. (expanded) }[/math]

To solve equation ((expanded)) we multiply by [math]\displaystyle{ w_{j} }[/math] and integrate over the plate (i.e. we take the inner product with respect to [math]\displaystyle{ w_{j}) }[/math] taking into account the orthogonality of the modes [math]\displaystyle{ w_{i} }[/math], and obtain

[math]\displaystyle{ \beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i \sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right) +\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left( \mathbf{Q}\right) dS_{\mathbf{Q}} (final) }[/math]

which is a matrix equation in [math]\displaystyle{ c_{i}. }[/math]

We cannot solve equation ((final)) without determining the modes of vibration of the thin plate [math]\displaystyle{ w_{i} }[/math] (along with the associated eigenvalues [math]\displaystyle{ \lambda _{i}) }[/math] and solving the integral equations ((phi0)) and (\ref {phii}). We use the finite element method to determine the modes of vibration Zienkiewicz and the integral equations ((phi0)) and ( (phii)) are solved by a constant panel method Sarp_Isa. The same set of nodes is used for the finite element method and to define the panels for the integral equation.