Two Semi-Infinite Elastic Plates of Identical Properties

Introduction

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.

Governing Equations

We consider the entire free surface to be occupied by a Floating Elastic Plate with a single discontinuity at [math]\displaystyle{ x=0 }[/math]. The equations are the following

[math]\displaystyle{ \nabla^2 \phi = 0, -H\lt z\lt 0, }[/math] [math]\displaystyle{ \frac{\partial \phi}{\partial z} =0, z=-H, }[/math] [math]\displaystyle{ {\left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha \phi} = 0, z=0, x\neq 0. }[/math]

The Two-Dimensional solution to the Free-Surface Green Function for a Floating Elastic Plate given by

[math]\displaystyle{ G(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.

[math]\displaystyle{ \beta k^5 \sin(kH) + k \left(1 - \alpha \gamma \right) \sin(kH) = -\alpha \cos(kH). }[/math]

The Green function [math]\displaystyle{ G(x,z) }[/math] representing outgoing waves as [math]\displaystyle{ |x|\rightarrow \infty }[/math] satisfies

[math]\displaystyle{ \nabla^2 G = 0, -H\lt z\lt 0, }[/math] [math]\displaystyle{ \frac{\partial G}{\partial z} =0, z=-H, }[/math] [math]\displaystyle{ {\left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial G}{\partial z} - \alpha G} = \delta(x), z=0, }[/math]

We then use Green's second identity If φ and ψ are both twice continuously differentiable on U, then

[math]\displaystyle{ \int_U \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV = \oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS }[/math]

If we then substitiute the Free-Surface Green Function for a Floating Elastic Plate which satisfies the following equations (plus the Sommerfeld Radiation Condition far from the body)

[math]\displaystyle{ \nabla^2 G = 0, -H\lt z\lt 0, }[/math] [math]\displaystyle{ \frac{\partial G}{\partial z} =0, z=-H, }[/math] [math]\displaystyle{ {\left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial G}{\partial z} - \alpha G} = \delta(x), z=0, }[/math]

for ψ we obtain

[math]\displaystyle{ 0 = \int_{-\infty}^{\infty}\left( G_{n}\left( x,x^{\prime }\right) \phi \left( x ^{\prime }\right) -G\left( x,x^{\prime }\right) \phi _{n}\left( x^{\prime }\right) \right) dx^{\prime } }[/math]

We then substitute for [math]\displaystyle{ \phi }[/math] to obtain {\left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial G}{\partial z}

[math]\displaystyle{ \int_{-\infty}^{\infty}\left( G_{n}\left( x,x^{\prime }\right) \left( \frac{\beta}{\alpha} \partial_x^4 - \gamma\alpha + 1\right)\phi_{n}\left( x^{\prime }\right) -G\left( x,x^{\prime }\right) \phi _{n}\left( x^{\prime }\right) \right) dx^{\prime } = 0 }[/math]

We then consider the following integral

[math]\displaystyle{ \int_{-\infty}^{\infty} \left( \beta \partial_x^4 - \gamma\alpha + 1\right)\phi }[/math]

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]