## Abstract

This paper uses the special Cosserat theory of hyperelastic shells satisfying Kirchoff’s hypothesis and irrotational flow theory to model the interaction between a heavy thin elastic sheet and an infinite ocean beneath it. From a general discussion of three-dimensional motions, involving an Eulerian description of the flow and a Lagrangian description of the elastic sheet, a special case of two-dimensional travelling waves with two wave speed parameters, one for the sheet and another for the fluid, is developed only in terms of Eulerian coordinates.

## 1. Nonlinear hydroelastic waves

Our goal is to derive nonlinear equations that govern the motion of an ideal incompressible liquid under an elastic sheet. For example, we would like to describe the propagation of waves of finite amplitude on the surface of an ocean under ice, regarding the ice sheet as an elastic shell. It is assumed throughout that there are no frictional forces between the sheet and the fluid beneath.

### (a) Fluid motion

We restrict attention to flows of infinite depth that at time *t* occupy a domain in , the boundary of which is a surface *Σ*_{t} that lies in a horizontal slab of finite height. Let points of be identified by coordinates ** x**=(

*x*

^{1},

*x*

^{2},

*x*

^{3}) and let Δ and ∇ denote the usual Laplacian and gradient operators with respect to these variables. We assume that the flow is incompressible, irrotational, satisfies the usual kinematic condition at

*Σ*

_{t}and vanishes at infinity. In particular, its velocity at a point

**at time**

*x**t*is given by ∇

*φ*(

**,**

*x**t*) where the velocity potential

*φ*satisfies 1.1 Here, is the unit normal vector to

*Σ*

_{t}outward from and 1.2 when locally

*Σ*={(

**,**

*x**t*):

*F*(

**,**

*x**t*)=0} and

*Σ*

_{t}={

**:**

*x**F*(

**,**

*x**t*)=0}. So the kinematic condition is equivalent to ∂

_{t}

*F*(

**,**

*x**t*)+∇

*F*(

**,**

*x**t*)⋅∇

*φ*(

**,**

*x**t*)=0 on

*Σ*

_{t}. The fluid pressure is After changing coordinates, the function

*C*(

*t*) can be taken to be zero.

### (b) Elastic shell dynamics

There are many approaches to elastic shell dynamics (see [1–3] for discussions and references). One possibility [4–7] is to reduce a three-dimensional theory, using physically reasonable assumptions on the kinematics and careful asymptotic analysis, to a two-dimensional model. The conclusions of that approach will be used here to specify realistic stored elastic energy functions when we regard the sheet on the ocean surface as a thin elastic shell that can bend, twist and stretch, but cannot shear or change its thickness. To be precise, we adopt the special Cosserat theory of shells, with one director satisfying Kirchoff’s hypothesis (see [1], ch. 17.8), to describe the surface sheet.

We assume that there is an *unloaded rest state* in which the thin sheet occupies the plane *x*^{3}=0 and has constant density (mass per unit area) *ρ*_{0}>0. We regard this as the reference plane and the coordinates ** X**=(

*X*

^{1},

*X*

^{2}) of its material points are the Lagrangian coordinates of points of the sheet after deformation. In Cosserat theory ([1] gives a thorough mathematical account), the elastic state of a shell, considered as an oriented surface in , is described by the position of its material points in Lagrangian coordinates together with a family of vector fields

*d*^{k}, which are called directors, on the surface.

To proceed, we recall some basic geometric notation. Suppose a smooth oriented surface in is given in local coordinates (*Ω*,** r**) by

**=**

*x***(**

*r***), , where**

*X***∈**

*r**C*

^{3}(

*Ω*) and the partial derivatives of

**are denoted by When**

*r*

*r*_{,1}and

*r*_{,2}are linearly independent at every point of

*Ω*, they form a basis for the tangent space at

**(**

*r***) and the normal vector**

*X***(**

*n***) at**

*X***(**

*r***) is a**

*X**C*

^{2}vector field given by

Let the corresponding coefficients *a*_{ij}, *b*_{ij} of the first and second fundamental forms be
1.3
Then the surface area element is
Let *A*, *B* denote the matrices (*a*_{ij}), (*b*_{ij}) and let *A*^{−1}=(*a*^{ij}), *B*^{−1}=(*b*^{ij}). Then the principal curvatures are the eigenvalues *k*_{1} and *k*_{2} of the Weingarten matrix *W*=*A*^{−1}*B* ([8], ch. 8) and the mean curvature *H* and Gauss curvature *K* are, with the usual summation convention,
where .

In general Cosserat theory, there may be many directors, *d*^{1},…,*d*^{k}, and the elastic state of the shell is given by However, in the special Cosserat theory adopted here, there is only one director which, by Kirchoff’s hypothesis, is the unit vector normal to the surface. Hence the time-dependent state of the elastic shell is given in local coordinates by a mapping
in which is the position at time *t* of a material point with Lagrangian coordinates ** X**, and

**, the unit normal to the surface at**

*n**r*(

**,**

*X**t*), is the only director.

Since we assume that the shell has zero thickness, we take and *φ*=0 in ch. 17, eqns (1.11), (1.12) and (2.1c), and hence ** h**=

**=**

*l***0**and

*I*=0 in eqns (2.13b,c) and (2.5c–e), in Antman [1]. Under these assumptions, the impulse–momentum law, which follows from the principle of virtual power, can be written in weak form as (take

**=**

*h***=**

*l***0**in Antman [1], ch. 17, (8.44)–(8.46) where, since

*ρ*

_{0}is its mass per unit surface area, we have taken 2

*hρ*=

*ρ*

_{0}) for any , a smooth, compactly supported, vector-valued test function, and

**given by**

*ω*Here ** f** is a given external force, and

*N*^{α}and

*M*^{α}are functions of

*r*_{,i},

*n*_{,j}and

**which will be determined by constitutive hypotheses on the elasticity of the shell. An integration by parts leads to 1.4 In lemma 1.1 below, we derive this law from a geometric variational principle.**

*n*### (c) Hyperelasticity

We consider a special Cosserat shell with director ** n**, which in addition is

*hyperelastic*([1], ch. 17, eqn (8.34)). In other words, there exists an elastic energy density function

*W*(

*r*_{,1},

*r*_{,2},

*n*_{,1},

*n*_{,2}) with where, to ensure that the stored energy is frame-indifferent (see [1], ch. 17, eqns (8.7) and (8.34)), we suppose that 1.5 In other words,

*W*is a function of the coefficients of the first and second fundamental forms (1.3) of the sheet in local coordinates (

*Ω*,

**). In addition, we note that the kinetic energy density of the sheet is At this point, we introduce notation for variations of functions due to variations of the surface**

*r**Σ*in §1

*a*. Suppose

*G*is a function of

**and its derivatives with respect to**

*r***and**

*X**t*. For an arbitrary and consider the one-parameter family

*Σ*

^{τ}of perturbed surfaces given by Let

*g*(

**,**

*X**t*;

*τ*) denote the value of

*G*at

*r*^{τ}(

**,**

*X**t*) and let The corresponding variations of the energy densities are:

*δ*Ke=

*ρ*

_{0}

*r*_{,t}⋅

*δ*

*r*_{,t}and 1.6 The following lemma gives a formula for the energy variation of a hyperelastic shell that leads to the impulse momentum law (1.4).

### Lemma 1.1

*When W is given by equation (1.5), for* *,
*
*In particular, equation (1.4) can be written*
1.7

### Proof.

Since (** n**×

**)⋅(**

*m***×**

*n**δ*

**)=**

*n***⋅**

*m**δ*

**for any**

*n***, because |**

*m***|=1 and**

*n***⊥**

*n**δ*

**, and since it is easily checked that**

*n***=**

*ω***×**

*n**δ*

**, equation (1.6) gives since**

*n*

*n*_{,α}×∂

*W*/∂

*n*_{,α}+

*r*_{,α}×

*N*^{α}=

*n*_{,α}×∂

*W*/∂

*n*_{,α}+

*r*_{,α}×∂

*W*/∂

*r*_{,α}=0 because equation (1.5) holds, The result follows.

The function Ke−*W* is the density of an action integral and equation (1.7) is the minimum action principle for this hyperelastic material.

### (d) Governing equations for a shell

We now use equation (1.7) to derive the governing equation for the motion of an elastic shell. Note that {*r*_{,1},*r*_{,2},** n**} spans a natural coordinate system at points of a surface defined by

**=**

*x***(**

*r***), . We consider perturbations of the form where the normal variation**

*X**ϕ*

^{n}and tangent variations

*ϕ*

^{α},

*α*=1,2, are arbitrary functions. The following lemma gives formulae for normal and tangential variations of geometrical characteristics of the surface.

### Lemma 1.2

(i) *If δ*** r**(

**)=**

*X**ϕ*

^{n}(

**)**

*X***(**

*n*

*X**) where*

*then*1.8

*where*

*is a covariant derivative of the co-vector field*

*in which the Christoffel symbols are*

*.*

(ii) *If δ*** r**(

**)=**

*X**ϕ*

^{α}(

**)**

*X*

*r*_{,α}(

*X**), where*

*, α=1,2, then*

*and*

### Proof.

Relations (1.8) are well known in differential geometry (see [9], §5.8). The formulae for tangential variations follow by similar arguments.

It follows from lemma 1.2 and the identity *δW*(*a*_{ij},*b*_{ij})=(∂*W*/∂*b*_{ij})*δb*_{ij}+(∂*W*/∂*a*_{ij})*δa*_{ij} that
Then integration by parts gives
1.9
where
1.10a
the asterisk denotes the *L*^{2}(d*x*) adjoint and
1.10b
Since *ϕ*^{n}=*δ*** r**⋅

**and**

*n**ϕ*

^{α}=

*a*

^{αj}

*r*_{,j}⋅

*δ*

**, equation (1.9) can be written 1.11 and since equations (1.11) and (1.7) yield the system governing the motion of the shell 1.12 where**

*r**Ψ*

_{n}and

*Ψ*

_{α}are defined by equations (1.10

*a*,

*b*). System (1.12) consists of three nonlinear partial differential equations for the vector-valued function

**(**

*r***,**

*X**t*). Henceforth, we consider physically reasonable hypotheses under which system (1.12) can be simplified.

Notice that the elastic energy is not a geometric invariant, in the sense that it depends not only on the shape of the surface *Σ*_{t}, but also on a specific parametrization ** r**(⋅,

*t*) associated with the reference frame. It is common to decompose the elastic energy density

*W*into two parts, the

*membrane*energy

*W*

_{m}and the

*bending*energy

*W*

_{b}. The membrane energy is associated with stretching and depends only on the first derivatives of

**. However, the energy owing to bending depends both on the first derivatives of**

*r***and on the curvatures of**

*r**Σ*. When combined with the requirements of frame indifference, this leads to the following special class of stored energies in terms of the fundamental forms of

**:**

*r*

### Remark 1.3

A possible justification of this splitting is the observation that its terms arise at different orders in an asymptotic expansion, in terms of a small thickness parameter, of a three-dimensional elastic-plate energy function. A function *W* of the form *W*(*a*_{ij},*b*_{ij}) is a geometric invariant if, for any diffeomorphism and ,
where and are the coefficients of the first and second fundamental form in local coordinates . There is only one first-order geometric invariant, , and the only second-order geometric invariants are *H* and *K*, the mean and Gauss curvatures, respectively. An elastic energy of the form
in which the membrane energy is simply the surface area, is geometrically invariant. The first term is consistent with the expression for bending energy of a shell that emerges from rigorous asymptotic analysis [4] under hypotheses that the three-dimensional elastic energy density is an isotropic function *W*_{3D}(∇** v**) of the gradient of the displacement vector

**, displacements satisfy the appropriate boundary conditions on ∂**

*v**Ω*(for example, periodic when

*Ω*is a rectangular), the plate has a smooth middle surface and its thickness

*h*is a small positive parameter. In Friesecke

*et al.*[4], it was shown that the functional

*h*

^{−3}

*E*( ,

*h*), where

*E*( ,

*h*) is the total elastic energy of a plate of thickness

*h*,

*Γ*-converges to where and the constants

*λ*,

*μ*are related to

*W*at

**=Id by If and**

*v**Σ*converges asymptotically to a horizontal plane, or if

**(**

*r***) is periodic on , then, by the Gauss–Bonnet theorem, the integral of the Gauss curvature**

*X**K*over

*Σ*depends only on the topological type of

*Σ*. Since |

*b*|

^{2}=4

*H*

^{2}−2

*K*, we may take the total bending energy to be Since , we conclude that the asymptotic theory in the case of pure bending yields . In other words, the bending energy, up to multiplicative constant, coincides with the Willmore functional [9].

For geometrically invariant energy functions, the total elastic energy depends only on the shape of *Σ*_{t}, the tangent variations of the total energy are zero and
The formula for variation of mean curvature ([9], five lines below (5.55) where a factor 1/2 is missing on the right) is
where
Hence, the function *Ψ*_{n} has a straightforward representation in the important case when and *W*_{b}∈*C*^{3}:
and system (1.12) has the form
where is the density of the deformed elastic sheet at the point ** r**.

## 2. Equations for hydroelastic waves

The equations governing the elastic shell on the water surface follow from system (1.1) for the fluid dynamics and system (1.12) for the shell dynamics, if we assume that the liquid surface coincides with the elastic shell (no cavities) and the force term in the equation for the shell’s dynamics is owing to water pressure and gravity:
This leads to the free boundary problem of finding a mapping and a velocity potential , where is bounded by , satisfying the equations
2.1a
2.1b
2.1c
where (*n*^{t},** n**) is defined by equation (1.2) and the functions

*Ψ*

_{n},

*Ψ*

_{α}are defined in terms of

**by equations (1.10**

*r**a*,

*b*). Equations (2.1

*a*–

*c*) may be supplemented by initial conditions and by hypotheses determined by the behaviour of the flow at infinity.

There is an essential difference between this problem with general *W* and free boundary problems such as the familiar water wave problem that arise in hydrodynamics. A solution of equations (2.1*a*–*c*) involves not only a free boundary *Σ*_{t} but also a particular parametrization ** x**=

**(**

*r***,**

*X**t*) of it. In contrast, capillary, capillary–gravity and gravity water wave problems are geometrically invariant in that their solutions are independent of the parametrization of the free surface. When the elastic energy density function is (pure bending), equation (2.1a) becomes where is the actual density (mass per unit area) of the deformed shell. If we neglect the inertial forces by taking

*ϱ*=0 and assume that

*Σ*

_{t}is the graph

*x*

^{3}=

*η*(

*x*

^{1},

*x*

^{2},

*t*) of a smooth function

*η*, the system becomes geometrically invariant with coefficients of first and second fundamental forms where

*α*=det(

*α*

_{ij})=1+|∇

*η*|

^{2},

*δ*

_{ij}is the Kronecker

*δ*, and the governing equations are 2.2a where , and 2.2b 2.2c where , and

### (a) Travelling waves: two speeds of propagation

Returning to equations (2.1*a*–*c*) we consider waves of permanent form travelling with speed *c* on a steady irrotational flow which is at rest at infinite depth beneath an elastic sheet. To be precise, suppose the position at time *t* of the point with Lagrangian coordinates ** X** is
where , and . This describes a surface of fixed shape moving with constant speed

*c*and the point with Lagrangian coordinate

**moves relative to a frame moving with speed**

*X**d*. We call

*c*the wave speed,

*d*the drift velocity of the surface sheet and the mean velocity of the surface sheet relative to the wave. Thus, we have two velocities of wave propagation, one for the fluid and another for the elastic shell. If the fluid velocity field is supposed stationary with respect to a frame moving with speed

*c*, the fluid pressure

*p*and the flow potential

*φ*are functions of the variable . After this change of variables and after introducing the modified potential and pressure

*P*=

*p*−

*c*

^{2}/2, equations (2.1

*a*–

*c*) give the stationary equations and boundary conditions for functions and in a fixed domain bounded by . If, to simplify notation, we henceforth write

**,**

*r***and**

*X***instead of , and , the governing equations for steady travelling hydroelastic waves in a moving frame become 2.3a 2.3b 2.3c In the geometrically invariant case, when and**

*x**ρ*

_{0}=0, equations (2.3

*a*–

*c*) become where

*H*and

**Δ**are defined by equations (2.2

*a*–

*c*). These equations should be supplemented with the conditions at infinity

### (b) Steady two-dimensional flows

As remarked earlier, an important difference between hydroelastic problems and many other free boundary problems in mathematical physics is that here we are looking, not only for the shape of the free surface, but also for a special parametrization ** x**=

**(**

*r***,**

*X**t*) of it relative to a prescribed reference frame. In particular, equations (2.1

*a*–

*c*) and (2.3

*a*–

*c*) involve both a Lagrangian variable

**and an Eulerian variable**

*X***. Because of the first term in equation (2.3a), we cannot in general eliminate the Lagrangian part and write the equation in Eulerian coordinates, even when the stored energy density is geometrically invariant and the elastic energy depends only on the shape of the sheet. A remarkable property of two-dimensional travelling-wave flows is that the governing equations admit a scalar integral and the stretches can be expressed algebraically as a function of the wave elevation. Consequently, the whole system can be described in terms of one type of coordinates. There follows a brief survey of these results.**

*x*In equations (2.3*a*–*c*), assume that ** r**(

**)=(**

*X**r*

_{1}(

*X*

_{1}),

*X*

_{2},

*r*

_{3}(

*X*

_{1})) and that the velocity potential of the flow beneath

*Σ*is independent of

*x*

^{2}. So the steady waves have surface profiles

*Σ*given by their cross sections . With

*a*:=

*r*

_{1}′

^{2}+

*r*

_{3}′

^{2}, the only non-zero coefficients of the first and second fundamental forms of

*Σ*are

*a*

_{11}=

*a*, and

*a*

_{22}=1. Following Antman, set and note that

*σ*:=2

*H*=

*μ*/

*ν*is the curvature of the curve , where

*H*is the mean curvature of the surface

*Σ*.

We assume that the material is hyperelastic with frame-indifferent stored energy density *W*(*a*_{11},*b*_{11}) in the form and let
In this notation,
and equations (1.10a,*b*) become
Let the so-called pseudo elastic stress and moment functions, *N*(*ν*,*μ*) and *M*(*ν*,*μ*), be defined by
2.4
where the *pseudo elastic energy* is . Let , let ** t**=

**′/**

*r**ν*be the unit tangent vector to the curve and note that

**′′⋅**

*r***=**

*n**νμ*and

**′′⋅**

*r***=**

*t**ν*′. Then, relative to the moving frame, the wave propagation in the sheet is governed by a two-dimensional version of the steady system (2.3a), which in this notation is a system of ordinary differential equations for the functions

*μ*(

*X*) and

*ν*(

*X*): 2.5 where ′ denotes the derivative with respect to

*X*,

*M*′+

*νR*=0 and

**=(0,1) (**

*j**N*and

*R*are tangential and normal stresses and

*M*is a moment). Remarkably, the first of equations (2.5) has a first integral independent of the pressure

*P*, 2.6 where

*η*(

*X*)=

**(**

*r**X*)⋅

**is the wave elevation and**

*j**γ*is a constant. So we can simplify equation (2.5) by substituting for

*R*and using the identity (2.6). To study the resulting system, it is convenient and natural [10–12] to let

*ϕ*=1/

*ν*and put 2.7 Notice that is the ratio of the deformed elastic material density

*ϱ*at point

**(**

*r**X*) and the elastic material density

*ρ*

_{0}in the unloaded state. With this density ratio

*ϕ*and the curvature

*σ*of the elastic sheet as new unknowns, we have 2.8 and the governing equations (2.5) for the elastic sheet can be rewritten as 2.9a 2.9b where

*η*=

**⋅**

*r***and**

*j**P*is the pressure in the flow at the free surface and

*γ*is a constant. For hydroelastic waves, equations (2.9

*a*,

*b*) should be supplemented with the hydrodynamical equations that determine

*P*. Before doing so we make the following important observation.

Equations (2.9*a*,*b*) are written in Lagrangian coordinates and ′ denotes the derivative with respect to the Lagrangian variable *X*. However, from the definition of *ϕ*, we have ds=d*X*/*ϕ*(*X*), where *s* is arc length along . Using this relation, we can rewrite the governing equations (2.9*a*,*b*) in Eulerian coordinates. To illustrate this observation, we focus on the natural case when the density *W* of the stored elastic energy is a sum of bending and membrane energies. Moreover, in accordance with the asymptotic theory, we assume that the bending energy is a geometric invariant, which leads to the representation
because *H*=*σ*/2 is the mean curvature of *Σ* and *a*_{11}=*a*=*ν*^{2}. The changes of variables (2.4) and (2.7) yield that
where *F*_{b}(*σ*)=*W*_{b}(*σ*/2) and *F*_{m}(*ϕ*)=*W*_{m}(1/*ϕ*^{2}). Now assume that the elastic sheet is represented in Eulerian coordinate as a graph, *x*^{3}=*η*(*x*^{1}), and note that along the elastic sheet
Therefore, after the change of variables *X*→*x*^{1}, we arrive at a system of equations and boundary conditions (in physical variables) for the wave profile *η*(*x*^{1}), the normalized density *ϕ*(*x*^{1}), and the stream function *ψ*(*x*^{1},*x*^{3}), which determine the propagation of two-dimensional hydroelastic travelling waves. The system consists of the elasticity equations
2.10a
2.10b
and the equations from hydrodynamics,
2.10c
2.10d
2.10e
2.10f
Here *ψ* and the velocity potential *Φ* satisfy the Cauchy–Riemann equations
Systems (2.10*a*–*f*) are coupled via the pressure function *P* which appears in both the elastic equation (2.10a) and the hydrodynamic equation (2.10c). In the case of pure bending, *F*_{m}=0 and we can calculate *ϕ* (which for physical reasons should be non-negative) from equation (2.10b),
and equation (2.10a) can be replaced by
Finally, we note from the conclusions of Friesecke *et al.* [4] that in many applications it often suffices to take *F*(*σ*)=const. *σ*^{2}.

## 3. Summary

The main outcome of this paper is a mathematical model (2.10*a*–*f*) in physical coordinates for the propagation of two-dimensional nonlinear hydroelastic travelling waves. It was derived from a general treatment of three-dimensional flows beneath an elastic sheet that involved both Lagrangian and Eulerian coordinates. We finish by mentioning some particular features of these systems.

*Singular waves*. Singularities owing to the self-intersection of a three-dimensional elastic sheet are, of course, physically impossible, but the governing equations themselves do not guarantee the absence of self-intersections. It follows from work of Li & Yau [13] that if the surface *Σ*_{t} in three-dimensional space tends asymptotically to a plane at infinity and if
then *Σ*_{t} does not intersect itself. The left-hand side of this inequality, which is the Willmore functional (its critical points are called Willmore surfaces), can be regarded as the simplest example of the bending energy. The expectation is that a Willmore surface can develop singularities if the value of the Willmore functional exceeds 8*π*. Hence, if this happens in our problem, we might expect the formations of singularities in the elastic sheet similar to the formation of singularities in an Euler elastica.

*Jumps.* Consider the two-dimensional case when the membrane energy is not zero. The natural assumption is that the function *F*_{m}(*ϕ*) is convex and tends to as *ϕ*→0 or to . Even then, the left-hand side of equation (2.10b) is not a monotone function of *ϕ* for sufficiently large values of the mean velocity . In this case equation (2.10b), for given *γ* and *η*, can have more than one solution *ϕ*. This means that systems (2.10*a*–*f*) may have a solution with discontinuous *ϕ*. Plotnikov & Toland [14] found the following results on the structure of such solutions.

If the function *F* admits the representation *F*_{b}(*σ*)+*F*_{m}(*ϕ*) in which *F*_{b} is convex and *F*_{b}, *F*_{m} satisfy certain growth condition, then the problems (2.10*a*–*f*) admit periodic solutions that minimize the energy per period. For such solutions the functions *ϕ* may have jumps, but the functions *N*(*σ*,*ϕ*) and *M*(*σ*,*ϕ*) in equation (2.8) are continuous everywhere. The continuity of *M* is equivalent to continuity of *σ*, but continuity of *N* yields a Hugoniot type condition relating the values of *ϕ* on both sides of the jump where it was shown that certain stability conditions hold.

## Acknowledgements

J.F.T. acknowledges receipt of a Royal Society/Wolfson Research Award.

## Footnotes

One contribution of 13 to a Theme Issue ‘The mathematical challenges and modelling of hydroelasticity’.

- This journal is © 2011 The Royal Society