Advances in Mathematical Physics

Volume 2015 (2015), Article ID 308318, 11 pages

http://dx.doi.org/10.1155/2015/308318

## The Nonlinear Hydroelastic Response of a Semi-Infinite Elastic Plate Floating on a Fluid due to Incident Progressive Waves

^{1}School of Mathematics and Physics, Qingdao University of Science and Technology, 99 Songling Road, Qingdao 266061, China^{2}Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 149 Yanchang Road, Shanghai 200072, China

Received 26 October 2014; Revised 25 February 2015; Accepted 4 March 2015

Academic Editor: Klaus Kirsten

Copyright © 2015 Ping Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The nonlinear hydroelastic response of very large floating structures (VLFSs) or an ice sheet floating on the surface of deep water, idealized as a semi-infinite thin elastic plate, is investigated analytically in the case of nonlinear incident waves. Assuming that the fluid is inviscid and incompressible and the motion is irrotational, we consider incident progressive waves with a given angular frequency within the framework of potential flow theory. With the aid of the homotopy analysis method (HAM), the convergent analytical series solutions are derived by solving the simultaneous equations in which we apply a convergence-control parameter to obtain convergent solutions with relatively few terms. The clear calculation results are represented to show nonlinear wave-plate interaction. The effects of different physical parameters, including incident wave amplitude, Young’s modulus, the thickness and density of the plate on the wave scattering, and the hydroelastic response of the floating plate, are considered. We find that the variations of the plate stiffness, thickness, and density greatly change amount of wave energy which is reflected into the open water region and is transmitted into the plate-covered region. Further, the hydroelastic response of the plate also can be affected by the amplitude of incident wave.

#### 1. Introduction

In recent decades, there have been intensive researches on hydroelastic interaction between water waves and elastic plates in the area of ocean engineering and polar engineering in view of their academic research and applications. The very large floating structures (VLFSs), which are expected to serve as floating airports, artificial floating islands, ultralarge ships, or even mobile offshore bases, are mathematically assumed as elastic plates floating on a fluid for their strong flexibility and huge horizontal scales compared to vertical scales and characteristic wavelength. Accordingly, the hydroelastic deformation in the response of the structure to water waves is a predominant factor to design and maintain a safe and economic VLFS. Early studies in this field were mainly focused on hydroelastic interaction between the water waves and the large ice sheets which were also idealized as floating elastic plates [1], and the elastic plates are usually assumed to be semi-infinitely long in comparison with the wavelength of the incident wave [2, 3]. A comprehensive summary of the early work can be found in some review papers such as [2–5]. Unfortunately, mathematical methods on solving nonlinear problems are not well developed, so most of the present analytical researches on the hydroelastic interaction between water waves and an elastic plate still are in the scope of linear theory, which only can describe small-amplitude waves. Taking a semi-infinite elastic plate by using the eigenfunction expansion method, for example, Fox and Squire [4] investigated the reflection and transmission of ocean waves at the margin of an ice sheet which is idealized as a semi-infinite elastic plate and then determined the expansion coefficients via matching the velocity potential and the pressure at the interface of the open water region and the plate-covered region and enforcing the zero shear and bending moment at the free end. However, Fox and Squire [4] also stated that the beat Lagrange’s multiplier is very difficult to choose and the mathematical calculations become impossible for the unwise choice of the multipliers. Fox and Squire [3] subsequently improved their original method by removing the free end boundary conditions from the error function and then reducing the number of the Lagrange’s multiplier from three to one and analyzed the hydroelastic response of a semi-infinite elastic with a free edge due to the obliquely incident waves. Following the framework of Fox and Squire [3], Sahoo et al. [6] developed a new inner product based on the Fourier analysis within which the original eigenfunctions for the plate-covered region were orthogonal. It is observed that the hydrodynamic behaviors of the floating semi-infinite elastic plate with different edge conditions, including a free edge, a simply supported edge, and built-in edge condition, depend on the wave conditions, the geometrical settings, and the edge conditions. Meanwhile, Teng et al. [7] optimized Fox and Squire’s method [3] by removing the Lagrange multipliers from the error function completely and improved the method of Sahoo et al. [6] through employing the eigenfunctions in the open water region and the plate-covered region, respectively. It is demonstrated with examples that the solver for the associated linear system is simplified. In Xu and Lu [8], the eigenfunction expansion method was optimized by using the orthogonality property of eigenfunctions in the open water region. And numerical analysis showed that this modified method was effective and had higher convergence than the previous results in Sahoo et al. [6] and Teng et al. [7]. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, Lin and Lu [9] extended the study of Fox and Squire [3] to the case that a semi-infinite elastic plate floating on a two-layer fluid of finite depth is subjected to the obliquely incident waves and found a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves.

More importantly, it is inevitable that large-amplitude waves will occur more often in the future with the growing frequency of extreme weather events on a warmer and warmer Earth. Thus, some scholars investigated nonlinear hydroelastic interaction between semi-infinite elastic plates and water waves by using the well-known perturbation method. Forbes [10] studied the nonlinear interaction between two-dimensional periodic waves with a constant speed and an elastic ice sheet floating on a fluid of infinite depth by using the perturbation expansion in the half-wave height and found that approximate solutions for the periodic waves had certain features in common with capillary-gravity waves. Forbes [11] subsequently improved their original perturbation method by introducing Newton-Raphson techniques to approximate the Fourier coefficients and confirmed the existence of multiple solutions for the nonlinear equations describing the shape of nonlinear periodic waves, and extremely large-amplitude waves were also found to exist. With the perturbation method in a similar way, Vanden-Broeck and Părău [12] extended the study of Forbes [11] to the case of large-amplitude periodic waves and unfortunately found that the perturbation method cannot give any information for very steep waves because this method has to depend on small physical parameters. Further, Milewski et al. [13] applied the asymptotic and numerical methods to analyze the hydroelastic solitary waves propagating under a semi-infinite elastic plate. It is found that wave packet solitary waves bifurcate from nonlinear periodic waves of minimum speed for the unforced problem, and when the problem is forced by a moving load, steady responses are possible at all subcritical speeds for small-amplitude forcing, while there was a transcritical range of forcing speeds for which there are no steady solutions for larger loads. Finally, it is also noted that these elastic models do not have a clear conservation form for the elastic potential energy.

It is well known that the traditional perturbation and asymptotic techniques depend on the small physical parameters and approximations of nonlinear problems usually break down when the nonlinearity becomes strong. Therefore, they are only valid for weakly nonlinear problems. In this paper, we apply the homotopy analysis method (HAM) developed by Liao [14], a powerful analytic method for highly nonlinear problems, to consider the nonlinear hydroelastic response of a semi-infinite elastic plate floating on a fluid due to incident progressive waves. Further, we investigate the dynamic influences of some important physical parameters, including incident wave amplitude, Young’s modulus, and the density and the thickness of the plate, on nonlinear hydroelastic response of the floating plate and the wave scattering.

#### 2. Mathematical Formulation

We consider the nonlinear hydroelastic interaction between incident progressive waves and a semi-infinite elastic plate floating on infinitely deep water for the two-dimensional case, as shown in Figure 1. Cartesian coordinates are chosen such that represents the undisturbed water surface. The -axis points horizontally rightward and the -axis points vertically upward. The semi-infinite elastic plate, which floating on a fluid, expends rightward from to infinity along the -axis without draft. Then the whole fluid domain is divided into two regions: an open water region () and a plate-covered region (). Under the assumptions that the fluid is inviscid and incompressible and the motion is irrotational, the velocity potential satisfies the Laplace equation based on the potential flow theory: