Floating Elastic Plate - Revision history

Meylan at 23:56, 15 June 2006

2006-06-15T23:56:56Z

← Older revision		Revision as of 23:56, 15 June 2006
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	~~= Introduction =~~		This page has moved to [[:Category:Floating Elastic Plate\|Floating Elastic Plate]].

	~~The floating elastic plate is one of~~ the ~~best studied problems in hydroelasticity. It can be used~~ to ~~model a range of~~		Please change the link to the new page
	~~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.~~

	~~The solution methods are divided up into those for [[Two-Dimensional Floating Elastic Plate]] and those for~~
	~~a [[Three-Dimensional Floating Elastic Plate]]~~


	~~[[Category:Linear Hydroelasticity]]~~

Meylan at 23:38, 15 June 2006

2006-06-15T23:38:07Z

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Hyuck: /* Equations of Motion */

2006-06-09T03:38:37Z

Equations of Motion

← Older revision		Revision as of 03:38, 9 June 2006
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	s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}w}{		s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}w}{
	\partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}w}{\partial		\partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}w}{\partial
	n\partial s^{2}}=0, ~~(boundaryplate)~~		n\partial s^{2}}=0,
	</math></center>		</math></center>
	[[Hildebrand65]] where <math>n</math> and <math>s</math> denote the normal and tangential		[[Hildebrand65]] where <math>n</math> and <math>s</math> denote the normal and tangential
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	water surface,		water surface,
	<center><math>		<center><math>
	p=-\rho \frac{\partial \phi }{\partial t}-\rho gW ~~(pressure)~~		p=-\rho \frac{\partial \phi }{\partial t}-\rho gW
	</math></center>		</math></center>
	where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density		where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
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	where		where
	<center><math>		<center><math>
	\beta =\frac{D}{g\rho a^{4}}\;\;\{~~mathrm~~ and}\ ~~\ =~~ \gamma =\frac{\rho _{i}h}{\rho		\beta =\frac{D}{g\rho a^{4}}, \; \mathbf{and} \; \gamma =\frac{\rho _{i}h}{\rho
	a}.		a}.
	</math></center>		</math></center>
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	frequency dependence. Therefore they can be expressed as the real part of a		frequency dependence. Therefore they can be expressed as the real part of a
	complex quantity whose time dependence is <math>e^{-i\sqrt{\alpha }t}</math> where <math>		complex quantity whose time dependence is <math>e^{-i\sqrt{\alpha }t}</math> where <math>
	\alpha <math> is the non-dimensional wavenumber and we write </math>\bar{W}(\bar{x},		\alpha </math> 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{		\bar{y},\bar{t})={Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{
	\alpha }\bar{t}}\right] \ <math>and</math>\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})=		\alpha }\bar{t}}\right] \ </math> 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		{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		}\bar{t}}\right] .</math> In the complex variables the equation of motion of the

Meylan: /* Introduction */

2006-06-08T03:54:28Z

Introduction

← Older revision		Revision as of 03:54, 8 June 2006
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	physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated		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		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.		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.

			The solution methods are divided up into those for [[Two-Dimensional Floating Elastic Plate]] and those for
			a [[Three-Dimensional Floating Elastic Plate]]

	= Two Dimensional Problem =		= Two Dimensional Problem =

Meylan: /* Equations of Motion */

2006-06-08T03:52:44Z

Equations of Motion

← Older revision		Revision as of 03:52, 8 June 2006
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	For a classical thin plate, the equation of motion is given by		For a classical thin plate, the equation of motion is given by
	<center><math>		<center><math>
	~~<math>~~D\nabla ^4 w + \rho _i h w = p~~</math>~~		D\nabla ^4 w + \rho _i h w = p
	</math></center>		</math></center>
	Equation ((plate)) is subject to the free edge boundary		Equation ((plate)) is subject to the free edge boundary

Meylan: /* Equations of Motion */

2006-06-08T03:48:06Z

Equations of Motion

← Older revision		Revision as of 03:48, 8 June 2006
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	For a classical thin plate, the equation of motion is given by		For a classical thin plate, the equation of motion is given by
			<center><math>
	<math>D\nabla ^4 w + \rho _i h w = p</math>		<math>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~~
	~~<center><math>~~
	~~D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p,~~
	~~(plate)~~
	</math></center>		</math></center>
	~~where <math>W</math> is the floe displacement, <math>\rho _{i}</math> is the floe density, <math>h</math> is~~		Equation ((plate)) is subject to the free edge boundary
	~~the floe thickness, <math>p</math> is the pressure, and <math>D</math> is the modulus of rigidity~~
	~~of the floe (<math>D=Eh^{3}/12(1-\nu ^{2})</math> where <math>E</math> is the Young's modulus and <math>~~
	~~\nu <math> 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>W.</math>~~ Equation ((plate)) is subject to the free edge boundary
	conditions for a thin plate		conditions for a thin plate
	<center><math>		<center><math>
	\frac{\partial ^{2}W}{\partial n^{2}}+\nu \frac{\partial ^{2}W}{\partial		\frac{\partial ^{2}w}{\partial n^{2}}+\nu \frac{\partial ^{2}w}{\partial
	s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}W}{		s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}w}{
	\partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}W}{\partial		\partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}w}{\partial
	n\partial s^{2}}=0, (boundaryplate)		n\partial s^{2}}=0, (boundaryplate)
	</math></center>		</math></center>
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	water surface,		water surface,
	<center><math>		<center><math>
	p=-\rho \frac{\partial \~~Phi~~ }{\partial t}-\rho gW (pressure)		p=-\rho \frac{\partial \phi }{\partial t}-\rho gW (pressure)
	</math></center>		</math></center>
	where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density		where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
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	variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>		variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
	We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math> and		We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math> and
	the mass variables with respect to <math>\rho a^{3}</math>. ~~The non-dimensional~~		the mass variables with respect to <math>\rho a^{3}</math>.
	~~variables, denoted by an overbar, are~~
	~~<center><math>~~
	~~\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></center>~~
	In the non-dimensional variables equations ((plate)) and ((pressure)		In the non-dimensional variables equations ((plate)) and ((pressure)
	) become		) become
	<center><math>		<center><math>
	\beta \nabla ^{4}\bar{W}+\gamma \frac{\partial ^{2}\bar{W}}{\partial \bar{t}		\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)		^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{w}, (n-d_ice)
	</math></center>		</math></center>
	where		where
	<center><math>		<center><math>
	\beta =\frac{D}{g\rho a^{4}}\;\;=and\ \ = \gamma =\frac{\rho _{i}h}{\rho		\beta =\frac{D}{g\rho a^{4}}\;\;\{mathrm and}\ \ = \gamma =\frac{\rho _{i}h}{\rho
	a}.		a}.
	</math></center>		</math></center>

Meylan: /* Equations of Motion */

2006-06-08T03:45:06Z

Equations of Motion

Show changes

Meylan at 01:51, 2 June 2006

2006-06-02T01:51:46Z

← Older revision		Revision as of 01:51, 2 June 2006
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	=== Single Crack ===		=== 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_Porter_2005a\|~~Evans and Porter 2005]] further considered the multiple crack problem for finitely deep water and provided an explicit solution.		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		We present here the solution of [[Evans and Porter 2005]] for the simple
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	The solution to the problem of two semi-infinite plates with different properties can be		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_Davies_1968a\|~~Evans and Davies 1968]]		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.		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 ...		The explicit solution was not found until the work of ...
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	<math>D\nabla ^4 w + \rho _i h w = p</math>		<math>D\nabla ^4 w + \rho _i h w = p</math>

			[[Category:Linear Hydroelasticity]]

Meylan at 11:03, 1 June 2006

2006-06-01T11:03:39Z

Meylan: Revert

2006-05-31T20:19:52Z

Revert

Show changes

← Older revision		Revision as of 11:03, 1 June 2006
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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		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		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_1957a\|~~Stoker 1957]]. The problem can		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.		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.

Line 76:		Line 76:
	=== Single Crack ===		=== 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_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.		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_Porter_2005a\|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_Porter_2005a\|~~Evans and Porter 2005]] for the simple		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		case of a single crack with waves incident from normal (they also considered multiple cracks
	and waves incident from different angles).		and waves incident from different angles).
	The solution of [[~~Evans_Porter_2005a\|~~Evans and Porter 2005]] expresses the potential		The solution of [[Evans and Porter 2005]] expresses the potential
	<math>\phi</math> in terms of a linear combination of the incident wave and certain source functions located at the crack.		<math>\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.		Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across the crack.
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	====[[Eigenfunction Matching Method]]====		====[[Eigenfunction Matching Method]]====

	The eigenfunction matching solution was developed by [[~~Fox_Squire_1994a\|~~Fox and Squire 1994]].		The eigenfunction matching solution was developed by [[Fox and Squire 1994]].
	Essentially the solution is expanded on either side of the crack.		Essentially the solution is expanded on either side of the crack.