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| − | = Introduction =
| + | This page has moved to [[:Category:Floating Elastic Plate|Floating Elastic Plate]]. |
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| − | The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
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| − | physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
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| − | more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
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| − | 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.
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| − | = Two Dimensional Problem =
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| − | == Equations of Motion ==
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| − | When considering a two dimensional problem, the <math>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.
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| − | For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
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| − | by the following
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| − | <math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>
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| − | where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
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| − | <math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
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| − | and <math>\eta</math> is the beam vertical displacement.
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| − | The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
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| − | <math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>
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| − | at the edges of the plate.
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| − | The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
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| − | pressure at the surface), i.e.
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| − | <math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>
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| − | where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
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| − | is the velocity potential. The velocity potential is governed by Laplace's equation through out
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| − | the fluid domain subject to the free surface condition and the condition of no flow through the
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| − | bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
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| − | [[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
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| − | [[Finite Depth]] are the following. At the surface
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| − | we have the dynamic condition
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| − | <math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
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| − | i\omega \rho \phi, \, z=0, \, x\in P</math>
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| − | <math>0=
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| − | \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
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| − | and the kinematic condition
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| − | <math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
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| − | The equation within the fluid is governed by [[Laplace's Equation]]
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| − | <math>\nabla^2\phi =0 </math>
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| − | and we have the no-flow condition through the bottom boundary
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| − | <math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
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| − | (so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
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| − | free surface or plate covered surface are at <math>z=0</math>).
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| − | <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
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| − | are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
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| − | the thickness and flexural rigidity of the plate.
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| − | == Solution Methods ==
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| − | There are many different methods to solve the corresponding equations ranging from highly analytic such
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| − | as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
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| − | applicable and have advantages in different situations. We describe here some of the solutions
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| − | which have been developed grouped by problem
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| − | === Single Crack ===
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| − | 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_1963a|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_Dixon_2000A|Squire and Dixon 2000]] and [[Williams_Squire_2002A|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_Dixon_2001a|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_Porter_2005a|Evans and Porter 2005]] further considered the multiple crack problem for finitely deep water and provided an explicit solution.
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| − | We present here the solution of [[Evans_Porter_2005a|Evans and Porter 2005]] for the simple
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| − | case of a single crack with waves incident from normal (they also considered multiple cracks
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| − | and waves incident from different angles).
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| − | The solution of [[Evans_Porter_2005a|Evans and Porter 2005]] expresses the potential
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| − | <math>\phi</math> in terms of a linear combination of the incident wave and certain source functions located at each of the cracks.
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| − | Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across each crack.
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| − | We will briefly present the solution of [[Evans_Porter_2005a|Evans and Porter 2005]]. They first define
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| − | <math>\chi(x,z)</math> to be the Two-Dimensional solution to the [[Free-Surface Green Function for a Floating Elastic Plate]]
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| − | given by
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| − | <math>
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| − | \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)
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| − | </math>
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| − | where
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| − | <math>
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| − | C_n=\frac{1}{2}\left(h + \frac{(5\beta k_n ^4 + 1 - \alpha\gamma)\sin^2{(k_n h)}}{\alpha}\right),
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| − | </math>
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| − | and <math>k_n</math> are the solutions of the [[Dispersion Relation for a Floating Elastic Plate]].
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| − | Consequently, the source functions for a single crack at <math>x=0</math> can be defined as
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| − | <math>
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| − | \psi_s(x,z)= \beta(\chi_{xx}(x,z),\,\,\,
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| − | \psi_a(x,z)= \beta(\chi_{xxx}(x,z),\,\,\,(2)
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| − | </math>
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| − | It can easily be shown that <math>\psi_s</math> is symmetric about <math>x = 0</math> and
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| − | <math>\psi_a</math> is antisymmetric about <math>x = 0</math>.
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| − | Substituting (1) into (2) gives
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| − | <math>
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| − | \psi_s(x,z)=
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| − | {
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| − | -\frac{\beta}{\alpha}
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| − | \sum_{n=-2}^\infty
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| − | \frac{g_n\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|} },
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| − | \psi_a(x,z)=
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| − | {
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| − | {\rm sgn}(x) i\frac{\beta}{\alpha}\sum_{n=-2}^\infty
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| − | \frac{g_n'\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|}},
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| − | </math>
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| − | where
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| − | <math>
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| − | g_n = ik_n^3(\sin{(k(n) h)},
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| − | g'_n= -k_n^4(\sin{(k_n h)}.
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| − | </math>
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| − | We then express the solution to the problem as a linear combination of the
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| − | incident wave and pairs of source functions at each crack,
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| − | <math>
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| − | \phi(x,z) =
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| − | { Ie^{-k_(0)(x-r_1)}\frac{\cos(k_(0)(z+h))}{\cos(k_(0)h)} }
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| − | + \(P_n\psi_s(x,z) + Q_n\psi_a(x,z))\,\,\,(3)
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| − | </math>
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| − | where <math>P_n</math> and <math>Q_n</math> are coefficients to be solved which represent the jump in the gradient
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| − | and elevation respectively of the plates across the crack <math>x = a_j</math>.
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| − | The coefficients <math>P_n</math> and <math>Q_n</math> are found by applying the edge conditions and to
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| − | the <math>z</math> derivative of <math>\phi</math> at <math>z=0</math>.
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| − | The reflection and transmission coefficients, <math>R_1(0)</math> and <math>T_\Lambda(0)</math> can be found from (3)
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| − | by taking the limits as <math>x\rightarrow\pm\infty</math> to obtain
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| − | <math>
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| − | R_1(0) e^{-\kappa_(0)r_1}= {- \frac{\beta}{\alpha}\sum_{n=1}^{\Lambda-1}
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| − | \frac{e^{\kappa(0)r_n}}{2k_0C_0}(g'_0Q_n + ig_0P_j)}
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| − | T_\Lambda(0) e^{\kappa(0)l_\Lambda}= 1 + {\frac{\beta}{\alpha}\sum_{n=1}^{\Lambda -1}\frac{e^{-\kappa(0)r_n}}{2k_0C_0}(g'_0Q_n - ig_0P_j)}
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| − | </math>
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| − | === Two Semi-Infinite Plates of Different Properties ===
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| − | The next most simple problem is two semi-infinite plates of different properties. Often one of
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| − | the plates is taken to be open water which makes the problem simpler. In general, the solution method
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| − | developed for open water can be extended to two plates of different properties, the exception to
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| − | this is the [[Residue Calculus]] solution which applies only when one of the semi-infinite regions
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| − | is water.
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| − | ====[[Wiener-Hopf]]====
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| − | The solution to the problem of two semi-infinite plates with different properties can be
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| − | solved by the Wiener-Hopf method. The first work on this problem was by [[Evans_Davies_1968a|Evans and Davies 1968]]
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| − | but they did not actually develop the method sufficiently to be able to calculate the solution.
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| − | The explicit solution was not found until the work of ...
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| − | ====[[Eigenfunction Matching Method]]====
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| − | ====[[Residue Calculus]]====
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| − | = Three Dimensional Problem =
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| − | == Equations of Motion ==
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| − | For a classical thin plate, the equation of motion is given by
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| − | <math>D\nabla ^4 w + \rho _i h w = p</math>
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