Introduction
We begin by presenting the solution for the case of a single crack with waves incident from normal. The solution method is derived from Squire and Dixon 2001 and Evans and Porter 2005.
We consider the entire free surface to be occupied by a Floating Elastic Plate with a single discontinuity at [math]\displaystyle{ x=x^\prime }[/math] (Fig. 1).
Image:GreenFunct.jpg
The governing equations are
[math]\displaystyle{
\nabla^2 \phi(x,z) = 0, -H\lt z\lt 0,
}[/math]
[math]\displaystyle{
\frac{\partial \phi(x,z)}{\partial z} =0, z=-H,
}[/math]
[math]\displaystyle{
{\left( \beta \frac{\partial^4}{\partial x^4} -
\gamma\alpha + 1\right)\frac{\partial \phi(x,z)}{\partial z} - \alpha \phi(x,z)}
= 0, z=0, x\neq x^\prime.
}[/math]
The Free-Surface Green Function for a Floating Elastic Plate 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-x^{\prime}), z=0,
}[/math]
where
[math]\displaystyle{
G(x,x^{\prime},z) = -i\sum_{n=-2}^\infty\frac{\sin{(k_n H)}\cos{(k_n(z+H))}}{2\alpha C(k_n)}e^{-k_n|x-x^{\prime}|},
}[/math]
[math]\displaystyle{
C(k_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_n^5 \sin(k_nH) - k_n \left(1 - \alpha \gamma \right) \sin(k_nH) =
-\alpha \cos(k_nH) \,
}[/math]
with [math]\displaystyle{ e^{i\omega t} }[/math] so that [math]\displaystyle{ n=-1,-2 }[/math] corresponding to the complex solutions with positive real part,
[math]\displaystyle{ n=0 }[/math] corresponding to the imaginary solution with negative imaginary part and
[math]\displaystyle{ n\gt 0 }[/math] corresponding to the real solutions with positive real part.
Green's Second Identity
Since φ and G are both twice continuously differentiable on U, where U represents the area bounded by the contour, S (Fig 1), the Green's second identity can be applied and gives
[math]\displaystyle{ \int_U \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV =
\oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS
}[/math]
where n repressents the plane normal to the boundary, S.
Our governing equations for G and [math]\displaystyle{ \phi }[/math] imply that the L.H.S of Green's second identity is zero so that
[math]\displaystyle{
0 = \oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS
}[/math]
expanding gives
[math]\displaystyle{
0 = -\int_{-\infty}^\infty \left( G {\partial \phi \over \partial z}|_{z=0} - \phi {\partial G \over \partial z}|_{z=0}\right)\, dx
+\int_{-h}^0 \left( G {\partial \phi \over \partial x}|_{x=N} - \phi {\partial G \over \partial x}|_{x=N}\right)\, dz
}[/math]
[math]\displaystyle{
+\int_{-\infty}^\infty \left( G {\partial \phi \over \partial z}|_{z=-h} - \phi {\partial G \over \partial z}|_{z=-h}\right)\, dx
-\int_{-h}^0 \left( G {\partial \phi \over \partial x}|_{x=-N} - \phi {\partial G \over \partial x}|_{x=-N}\right)\, dz
}[/math]
where we take the limit as N goes to infinity.
We now introduce the incident potential so that for x small
[math]\displaystyle{
\phi = A(e^{-k_0x} + R e^{k_0x})\cos{(k_n(z+H))}
}[/math]
[math]\displaystyle{
\phi_x = Ak_0(-e^{-k_0x} + R e^{k_0x})\cos{(k_n(z+H))}
}[/math]
for large x
[math]\displaystyle{
\phi = AT e^{-k_0x}\cos{(k_n(z+H))}
}[/math]
[math]\displaystyle{
\phi_x = -Ak_0T e^{-k_0x}\cos{(k_n(z+H))}
}[/math]
where we choose A to normalise and assume k_0 is positive imaginary.
and for both small and large x
[math]\displaystyle{
G = -i\frac{\sin{(k_0 H)}\cos{(k_0(z+H))}}{2\alpha C(k_0)}e^{-k_0|x-x^{\prime}|},
}[/math]
[math]\displaystyle{
G_x = -k_0(sgn(x-x^\prime))G
}[/math]
Now,
[math]\displaystyle{
\int_{-h}^0 \left( G {\partial \phi \over \partial x}|_{x=N} - \phi {\partial G \over \partial x}|_{x=N}\right)\, dz
}[/math]
[math]\displaystyle{
= \int_{-h}^0 \left( -Ak_0GT e^{-k_0N}\cos{(k_n(z+H))} + Ak_0GT e^{-k_0N}\cos{(k_n(z+H))} \right)\, dz
= 0
}[/math]
and
[math]\displaystyle{
-\int_{-h}^0 \left( G {\partial \phi \over \partial x}|_{x=-N} - \phi {\partial G \over \partial x}|_{x=-N}\right)\, dz
}[/math]
[math]\displaystyle{
=-\int_{-h}^0 \left( Ak_0G(-e^{k_0N} + R e^{-k_0N})\cos{(k_n(z+H))} - Ak_0G(e^{k_0N} + R e^{-k_0N})\cos{(k_n(z+H))} \right)\, dz
}[/math]
[math]\displaystyle{
=\int_{-h}^0 \left( 2Ak_0G\cos{(k_0(z+H))}e^{k_0N} \right)\, dz
}[/math]
[math]\displaystyle{
=\int_{-h}^0 \left( -i\frac{Ak_0}{\alpha C}\sin(k_0H)\cos^2{(k_0(z+H))} \right)\, dz
}[/math]
???????????? Hmmm is this right? I know A normalises, but should it be this complex. ?????????????
Also, 0ur governing equations imply [math]\displaystyle{ G {\partial \phi \over \partial z}|_{z=-h} = 0 }[/math] and [math]\displaystyle{ \phi {\partial G \over \partial z}|_{z=-h} = 0 }[/math] so that,
[math]\displaystyle{
\int_{-\infty}^\infty \left( G {\partial \phi \over \partial z}|_{z=-h} - \phi {\partial G \over \partial z}|_{z=-h}\right)\, dx =0
}[/math]
and we are left with
[math]\displaystyle{
0 = -\int_{-\infty}^\infty \left( G(x,x^\prime,z) \phi_z(x,z)|_{z=0} - \phi(x,z) G_z(x,x^\prime,z) |_{z=0}\right)\, dx
}[/math]
At z=0, the z variable disappears to give
[math]\displaystyle{
0 = -\int_{-\infty}^\infty \left( G(x,x^\prime) \phi_z(x) - \phi(x) G_z(x,x^\prime)\right)\, dx
}[/math]
We then substitute [math]\displaystyle{ G = \phi }[/math] to remove [math]\displaystyle{ \phi }[/math] and obtain
[math]\displaystyle{
\int_{-\infty}^{\infty}\left(
G_{z}\left( x,x^{\prime }\right) \frac{1}{\alpha}\left( \beta \partial_{x}^4 -\gamma\alpha + 1\right)\phi_{z}( x)
-G\left( x,x^{\prime }\right) \phi_{z}(x) \right) dx = 0
}[/math]
We now integrate by parts remembering that [math]\displaystyle{ \phi_z }[/math] is continuous everywhere except at [math]\displaystyle{ x = x^\prime }[/math] so that
[math]\displaystyle{
\int_{-\infty}^\infty(\partial_x^4\phi_z)G_z dx = \int_{-\infty}^{x^\prime}(\partial_x^4\phi_z)G_z dx + \int_{x^\prime}^\infty(\partial_x^4\phi_z)G_z dx
}[/math]
where
[math]\displaystyle{
\int_{-a}^b(\partial_x^4\phi_z)G_z dx = \int_a^b\phi(\partial_xG)dx - \phi(b)(\partial_x^3G(b) + \phi(a)(\partial_x^3G(a) + (\partial_x\phi(b))(\partial_x^2G(b))
}[/math]
[math]\displaystyle{
- (\partial_x\phi(a))(\partial_x^2G(a)) - (\partial_x^2\phi(b))(\partial_xG(b)) + (\partial_x^2\phi(a))(\partial_xG(a)) + (\partial_x^3\phi(b))G(b) - (\partial_x^3\phi(a))G(a)
}[/math]
and obtain
[math]\displaystyle{
\int_{-\infty}^{\infty}\left\{ \frac{1}{\alpha}\left( \beta \partial_{x}^4 - \gamma\alpha + 1\right)G_{z}\left( x,x^{\prime }\right)
- G( x,x^\prime)\right\} \phi_z(x)dx
}[/math]
[math]\displaystyle{
+ \frac{\beta}{\alpha}\left(\partial_{x}^3G_z(x,x^\prime)[\phi_z] - \partial_{x}^2 G_z(x,x^\prime)\partial_x[\phi_z] + \partial_{x} G_z(x,x^\prime)\partial_{x}^2[\phi_z] - G_z(x,x^\prime)\partial_{x}^3[\phi_z] \right)
=0
}[/math]
where [] denotes the jump in the function at [math]\displaystyle{ x = x^{\prime} }[/math].
The integral can be simplified using the delta function property of the Green function
to give us
[math]\displaystyle{
\phi_{z}\left( x\right)
= -\beta\left(\partial_{x}^3 G_z [\phi_z] - \partial_{x}^2 G_z [\partial_{x}\phi_z]
+ \partial_{x} G_z [\partial_{x}^2\phi_z] - G_z [\partial_{x}^3\phi_z]\right)
}[/math]
We can write the equation in terms of [math]\displaystyle{ \phi }[/math] as was done by Porter and Evans 2005 but there is
no real point because the boundary conditions are given in terms of [math]\displaystyle{ \phi_z }[/math] since this represents
the displacement.
We include the boundary conditions at infinity, which we omitted earlier, to give the full equation
[math]\displaystyle{
\phi_{z}(x,z)
= \phi_z^\mathrm{In} - \beta(\partial_x^3 G_z[\phi_z] - \partial_{x}^2 G_z[\partial_{x}\phi_z]
+ \partial_{x} G_z [\partial_{x}^2\phi_z] - G_z [\partial_{x}^3\phi_z])
}[/math]
which can be solved by applying the edge conditions at [math]\displaystyle{ x=x^\prime }[/math] and z = 0
[math]\displaystyle{
\partial_x^2\phi_z=0,\,\,\,
{\rm and}\,\,\,\,
\partial_x^3\phi_z=0.
}[/math]
Solution
We re-express [math]\displaystyle{ \phi_z }[/math] as
[math]\displaystyle{
\phi_{z} = \phi_{z}^{\mathrm{In}} - \psi_a [\phi_z] + \psi_s [\partial_{x}\phi_z]
- \chi_a[\partial_{x}^2\phi_z] + \chi_s [\partial_{x}^3\phi_z]
}[/math]
where
[math]\displaystyle{
\chi_s(x-x^\prime) = \beta G_z = \frac{i\beta}{2\alpha} \sum_{n=-2}^\infty\frac{k_n\sin^2{(k_n H)}}{C(k_n)}e^{-k_n|x-x^\prime|},
}[/math]
[math]\displaystyle{
\chi_a(x-x^\prime) = \beta\partial_x G_z = -sgn(x-x^\prime)\frac{i\beta}{2\alpha} \sum_{n=-2}^\infty\frac{k_n^2\sin^2{(k_n H)}}{C(k_n)}e^{-k_n|x-x^\prime|},
}[/math]
[math]\displaystyle{
\psi_s(x-x^\prime) = \beta\partial_x^2 G_z = \frac{i\beta}{2\alpha} \sum_{n=-2}^\infty\frac{k_n^3\sin^2{(k_n H)}}{C(k_n)}e^{-k_n|x-x^\prime|},
}[/math]
[math]\displaystyle{
\psi_a(x-x^\prime) =\beta\partial_x^3 G_z = -sgn(x-x^\prime)\frac{i\beta}{2\alpha} \sum_{n=-2}^\infty\frac{k_n^4\sin^2{(k_n H)}}{C(k_n)}e^{-k_n|x-x^\prime|}
}[/math]
and
[math]\displaystyle{
\phi_z^{In} = e^{-k_0x}\frac{\sin(k_0(z+H))}{\sin(k_0H)}
}[/math]
The edge conditions given above imply that [math]\displaystyle{ [\partial_{x}^2\phi_z] }[/math] and [math]\displaystyle{ [\partial_{x}^3\phi_z] }[/math] are zero so that [math]\displaystyle{ \phi_z }[/math] becomes
[math]\displaystyle{
\phi_{z}( x) = \phi_{z}^{\mathrm{In}}(x) -\psi_a [\phi_z] + \psi_s [\partial_{x}\phi_z]
}[/math]
We are now left with two unknowns which can be solved using the two edge conditions. To solve, we use
[math]\displaystyle{
\partial_x^2\phi_z = \partial_x^2\phi_z^{In} - \partial_x^2\psi_a [\phi_z] + \partial_x^2\psi_s [\partial_{x}\phi_z]
}[/math]
[math]\displaystyle{
= k_0^2e^{-k_0x} \frac{\sin(k_0z+H)}{\sin(k_0H)} + \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}e^{-k_n|x-x^\prime|}
\left[sgn(x-x^\prime)k_n^6[\phi_z] + k_n^5[\partial_{x}\phi_z] \right]
}[/math]
and
[math]\displaystyle{
\partial_x^3\phi_z = \partial_x^3\phi_z^{In} - \partial_x^3\psi_a [\phi_z] + \partial_x^3\psi_s [\partial_{x}\phi_z]
}[/math]
[math]\displaystyle{
= -k_0^3e^{-k_0x} \frac{\sin(k_0z+H)}{\sin(k_0H)} - \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}e^{-k_n|x-x^\prime|}
\left[k_n^7[\phi_z] + sgn(x-x^\prime)k_n^6[\partial_{x}\phi_z] \right]
}[/math]
At [math]\displaystyle{ x=x^\prime }[/math] and z=0, the first edge conditions gives
[math]\displaystyle{
k_0^2e^{-k_0x^\prime} + \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}
\left[sgn(x-x^\prime)k_n^6[\phi_z] + k_n^5[\partial_{x}\phi_z] \right] = 0
}[/math]
and the second edge condition gives
[math]\displaystyle{
-k_0^3e^{-k_0x^\prime} - \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}
\left[k_n^7[\phi_z] + sgn(x-x^\prime)k_n^6[\partial_{x}\phi_z] \right] = 0
}[/math]
The jump conditions [math]\displaystyle{ [\phi_z] }[/math] and [math]\displaystyle{ [\partial_{x}\phi_z] }[/math] can be solved by solving the edge conditions simultaneously.
The reflection and transmission coefficients, [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] can be found by taking the limit of [math]\displaystyle{ \phi_z }[/math] as
[math]\displaystyle{ x\rightarrow\pm\infty }[/math] ie
[math]\displaystyle{
R = \lim\limits_{x\rightarrow-\infty} \left( \phi_{z}^{\mathrm{In}}(x) -\psi_a [\phi_z] + \psi_s [\partial_{x}\phi_z]\right)
}[/math]
[math]\displaystyle{
= - \frac{i\beta\sin^2(k_0h)}{2\alpha C(k_0)}\left[k_0^4[\phi_z] - k_0^3[\partial_{x}\phi_z]\right]
}[/math]
and
[math]\displaystyle{
T = \lim\limits_{x\rightarrow\infty} \left(\phi_{z}^{\mathrm{In}}(x) -\psi_a [\phi_z] + \psi_s [\partial_{x}\phi_z]\right)
}[/math]
[math]\displaystyle{
=1 + \frac{i\beta\sin^2(k_0h)}{2\alpha C(k_0)}\left[k_0^4[\phi_z] + k_0^3[\partial_{x}\phi_z]\right]
}[/math]
More complicated boundary conditions
More complicated boundary conditions can be treated using this formulation.
Previously, we considered plates with free edges ie with a zero bending moment and zero shear force at each edge.
Here, we consider that each plate be connected by a series of flexural rotational springs (stiffness denoted by [math]\displaystyle{ S_r }[/math]) and vertical linear springs (stiffness denoted by [math]\displaystyle{ S_l }[/math]).
The edge conditions become:
[math]\displaystyle{
\partial_x^3\phi_z^+(x^\prime) = -\frac{S_l}{\beta}\left( \phi_z^+(x^\prime) - \phi_z^-(x^\prime)\right) = -\frac{S_l}{\beta}[\phi_z]
}[/math]
[math]\displaystyle{
\partial_x^3\phi_z^-(x^\prime) = -\frac{S_l}{\beta}\left( (\phi_z^+(x^\prime) - \phi_z^-(x^\prime)\right) = -\frac{S_l}{\beta}[\phi_z]
}[/math]
[math]\displaystyle{
\partial_x^2\phi_z^+(x^\prime) = \frac{S_r}{\beta} \left( \partial_x\phi_z^+(x^\prime) - \partial_x\phi_z^-(x^\prime)\right) = \frac{S_r}{\beta} [\partial_x\phi_z]
}[/math]
[math]\displaystyle{
\partial_x^2\phi_z^-(x^\prime) = \frac{S_r}{\beta}\left( \partial_x\phi_z^+(x^\prime) - \partial_x\phi_z^-(x^\prime) \right) = \frac{S_r}{\beta} [\partial_x\phi_z]
}[/math]
where [math]\displaystyle{ \phi^+ }[/math] is the left edge of the right plate and [math]\displaystyle{ \phi^- }[/math] is the right edge of the left plate.
These edge conditions imply [math]\displaystyle{ \frac{\partial^2\phi_n^+(x^\prime)}{\partial x^2} = \frac{\partial^2\phi_n^-(x^\prime)}{\partial x^2} }[/math] and [math]\displaystyle{ \frac{\partial^3\phi_n^+(x^\prime)}{\partial x^3} = \frac{\partial^3\phi_n^-(x^\prime)}{\partial x^3} }[/math] which imply [math]\displaystyle{ [\partial_x^2\phi_n] = 0 }[/math] and [math]\displaystyle{ [\partial_x^3\phi_n] = 0 }[/math].
[math]\displaystyle{ \phi_n }[/math] can now be re-expressed by
[math]\displaystyle{
\phi_{n}\left( x\right)
= \phi_{n}^{\mathrm{In}}\left( x\right) +
\psi_a [\phi_n] - \psi_s [\partial_{x^\prime}\phi_n]
}[/math]
We now have two unknowns which can be solved simultaneously using the following two edge conditions
[math]\displaystyle{
[\phi_z] = -s_l \left( \frac{\partial^3\phi_z(x)}{\partial x^3} \right)
}[/math]
[math]\displaystyle{
= -s_l \left(k_0^2e^{-k_0x^\prime} + \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}
\left(sgn(x-x^\prime)k_n^6[\phi_z] + k_n^5[\partial_{x}\phi_z] \right) \right)
}[/math]
and
[math]\displaystyle{
[\partial_x\phi_z] = s_r \left( \frac{\partial^2\phi_z(x)}{\partial x^2} \right)
}[/math]
[math]\displaystyle{
= s_r \left( -k_0^3e^{-k_0x^\prime} - \frac{i\beta}{2\alpha}\sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)}
\left(k_n^7[\phi_z] + sgn(x-x^\prime)k_n^6[\partial_{x}\phi_z] \right)\right)
}[/math]
where [math]\displaystyle{ s_l = \beta/S_l }[/math] and [math]\displaystyle{ s_r = \beta/S_r }[/math].
A Set of Floating Elastic Plates of Identical Properties
We now present the solution for the case of a set of cracks with waves incident from normal.
The solution is similiar to the case for one crack except that we solve for a set of cracks where [math]\displaystyle{ x^\prime_j }[/math] is the position of the [math]\displaystyle{ j^{th} }[/math] crack so that
[math]\displaystyle{
\phi_{z} = \phi_{z}^{\mathrm{In}} + \sum_{j=1}^N\left( - \psi_a(x-x^\prime_j) [\phi_z]_j + \psi_s(x-x^\prime_j) [\partial_{x}\phi_z]_j
- \chi_a(x-x^\prime_j) [\partial_{x}^2\phi_z]_j + \chi_s(x-x^\prime_j) [\partial_{x}^3\phi_z]_j \right)
}[/math]
where there are [math]\displaystyle{ N }[/math] cracks and [] is the jump at the [math]\displaystyle{ j^{th} }[/math] crack.
If we consider the standard edge conditions of [math]\displaystyle{ \partial_x^2\phi = 0 }[/math] and [math]\displaystyle{ \partial_x^3\phi = 0 }[/math], our edge conditions become
[math]\displaystyle{
k_0^2e^{-k_0x_r} + \frac{i\beta}{2\alpha}\sum_{j=1}^N \left( \sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)} e^{k_n(x_r-x_j^\prime)}
\left[sgn(x_r-x_j^\prime)k_n^6[\phi_z]_j + k_n^5[\partial_{x}\phi_z]_j \right]\right) = 0
}[/math]
[math]\displaystyle{
-k_0^3e^{-k_0x_r} - \frac{i\beta}{2\alpha}\sum_{j=1}^N \left( \sum_{n=-2}^\infty \frac{\sin^2{(k_n H)}}{ C(k_n)} e^{k_n(x_r-x_j^\prime)}
\left[k_n^7[\phi_z]_j + sgn(x_r-x_j^\prime)k_n^6[\partial_{x}\phi_z]_j \right]\right) = 0
}[/math]
where we consider the jump across [math]\displaystyle{ x_r }[/math] for each [math]\displaystyle{ r =1,2, ..., N }[/math]
The reflection and transmission coefficients, [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] can now be expressed as
[math]\displaystyle{
R = - \frac{i\beta\sin^2(k_0h)}{2\alpha C(k_0)}\sum_{j=1}^N \left[k_0^4[\phi_z]_j - k_0^3[\partial_{x}\phi_z]_j\right]
}[/math]
and
[math]\displaystyle{
=1 + \frac{i\beta\sin^2(k_0h)}{2\alpha C(k_0)}\sum_{j=1}^N \left[k_0^4[\phi_z]_j + k_0^3[\partial_{x}\phi_z]_j\right]
}[/math]
