Interaction of hydro-elastic waves with a vertical wall

The linear two-dimensional problem of hydroelastic waves reflected by a vertical wall is analysed. The fluid is of finite depth and is covered by an ice sheet. The fluid is assumed incompressible and inviscid. The ice sheet is assumed thin compared with both the water depth and wavelength of the incident wave. The deflection of the ice sheet is described by linear elastic plate theory, and the fluid flow by using the potential-flow model. The ice sheet extends infinitely and is clamped to the vertical-walled structure. The incident hydroelastic wave is regular. An analytic solution is found by integral-transform methods. The ice deflection, the vertical and horizontal forces acting on the wall and the bending stresses in the ice caused by the incident wave are determined. The forces on the wall are analysed in detail, and relevant physical parameters are varied for comparison. The phase shift between the incident and reflected wave amplitudes is found as part of the complete solution. It is shown that the ice clamping condition leads to a specific effect on the ice deflection.     

各向异性弹性层中SH波传播的边界元方法

本文利用坐标变换下的映象法求出了底部为刚性边界的各向异性弹性层中SH波传播问题的格林函数,并由此建立了求解分层介质问题的边界元方法。文中给出了在稳态线荷载作用下带有圆孔的弹性层中反应的计算实例。     

The Wiener–Hopf and residue calculus solutions for a submerged semi-infinite elastic plate

We present a solution for the interaction of normally incident linear waves with a submerged elastic plate of semi-infinite extent, where the water has finite depth. While the problem has been solved previously by the eigenfunction-matching method, the present study shows that this problem is also amenable to the more analytical, and extremely efficient, Wiener–Hopf (WH) and residue calculus (RC) methods. We also show that the WH and RC solutions are actually equivalent for problems of this type, a result which applies to many other problems in linear wave theory. (e.g., the much-studied floating elastic plate scattering problem, or acoustic wave propagation in a duct where one wall has an abrupt change in properties.) We present numerical results and a detailed convergence study, and discuss as well the scattering by a submerged rigid dock, particularly the radiation condition beneath the dock.     

Quarter-plane problem of a floating elastic plate

An investigation is made into the hydro-elastic behavior of a floating elastic plate, which occupies a quarter plane to infinity and is excited by water waves. A boundary-integral equation based on the Green function for this problem is shown for the case of finite water depth, as well as for the case of shallow water. The solution of the quarter-plane problem is composed of the corner effect and the solution of the half-plane problem. The corner effect is divided into two parts. The first part is the end effect of the forcing term of the integral equation, which is analytically estimated and its asymptotic form is derived. The second part is the local contribution whose asymptotic form is also obtained. The asymptotic form of the corner effect is confirmed by a numerical evaluation.     

Hydroelastic behaviour of compound floating plate in waves

The paper deals with the plane problem of the hydroelastic behaviour of floating plates under the influence of periodic surface water waves. Analysis of this problem is based on hydroelasticity, in which the coupled hydrodynamics and structural dynamics problems are solved simultaneously. The plate is modeled by an Euler beam. The method of numerical solution of the floating-beam problem is based on expansions of the hydrodynamic pressure and the beam deflection with respect to different basic functions. This makes it possible to simplify the treatment of the hydrodynamic part of the problem and at the same time to satisfy accurately the beam boundary conditions. Two approaches aimed to reduce the beam vibrations are described. In the first approach, an auxiliary floating plate is added to the main structure. The size of the auxiliary plate and its elastic characteristics can be chosen in such a way that deflections of the main structure for a given frequency of incident wave are reduced. Within the second approach the floating beam is connected to the sea bottom with a spring, the rigidity of which can be selected in such a way that deflections in the main part of the floating beam are very small. The effect of the vibration reduction is quite pronounced and can be utilized at the design stage.     

Flexural gravity wave scattering by a nearly vertical porous wall

Two-dimensional linear flexural gravity wave scattering by a nearly vertical porous wall is analyzed through a simplified perturbational analysis. A continuous semi-infinite ice sheet of uniform thickness is assumed to be floating over water of infinite depth. The ice sheet, with inclusion or exclusion of compressive stress, has either a free edge or a clamped edge at the porous wall. The first-order correction to the reflected flexural gravity wave amplitude is obtained by two different methods. The first method involves an application of Green’s theorem, and the second method involves a first kind integral equation. The integral equation method proves to be robust as it provides a complete solution in all cases of the problem, whereas the first method fails to produce the same when the ice sheet with a free edge is under compressive stress. The strain in the ice sheet and shear force along the ice sheet are computed and explained graphically for suitable parameters and a particular wall shape function.     

Wave interaction with a floating and submerged elastic plate system

Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal mode-coupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems.     

Hydroelastic wave diffraction by a vertical cylinder

A linear three-dimensional problem of hydroelastic wave diffraction by a bottom-mounted circular cylinder is analysed. The fluid is of finite depth and is covered by an ice sheet, which is clamped to the cylinder surface. The ice stretches from the cylinder to infinity in all lateral directions. The hydroelastic behaviour of the ice sheet is described by linear elastic plate theory, and the fluid flow by a potential flow model. The two-dimensional incident wave is regular and has small amplitude. An analytical solution of the coupled problem of hydroelasticity is found by using a Weber transform. We determine the ice deflection and the vertical and horizontal forces acting on the cylinder and analyse the strain in the ice sheet caused by the incident wave.     

Three-dimensional waves beneath an ice sheet due to a steadily moving pressure

Solutions of the nonlinear water wave equations under an ice sheet are computed using a boundary integral equation method. The ice sheet is modelled as a thin elastic plate and the fluid equations are nonlinear. Depending on the velocity of the moving disturbance generating the flow, different types of responses of the floating ice sheet are discussed.     

刚性基底上弹性约束矩形板的屈曲行为分析

对刚性基底上非加载边弹性约束矩形板在固支边均匀面内压力作用下的屈曲行为进行了分析研究。用里兹能量变分法建立确定板屈曲强度的本征值问题,利用满足边界条件的屈曲位移函数导出板屈曲系数计算公式。将该方法用于矩形钢管混凝土钢板局部屈曲的分析和计算,提出了目标板边界约束系数计算公式。试验结果表明,方形钢管混凝土柱钢板局部屈曲强度计算值与试验值吻合良好,钢板边界弹性约束的处理和约束系数的计算方法是有效的。     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

含半圆形衬砌凸起弹性半空间问题的Green函数解

采用复变函数法研究了含半圆形衬砌凸起的弹性半空间中水平界面承受出平面线源荷载时的Green 函数解。该问题采用“契合”的思想求解,首先,将整个求解区域分割成两部分来处理,其一为含半圆形凹陷的弹性半空间,其二为圆形衬砌区域;其次,构造满足含半圆形凹陷半空间水平界面应力自由的散射波,构造满足圆形衬砌半圆形凸起应力自由的驻波;最后,在两个区域的“公共边界”上实施“契合”,满足公共边界处位移和应力的连续性条件,同时满足圆形衬砌内边界应力自由边界条件,建立起求解该问题的无穷代数方程组,并给出了水平表面位移幅值的具体算例和数值结果,并对其进行了讨论。     

Three-dimensional absolute nodal co-ordinate formulation: plate problem

In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no infinitesimal or finite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate coordinates. Using this interpretation of the plate coordinates the new method does not require the use of co-ordinate transformation to define the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The stiffness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental finite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain. © 1997 John Wiley & Sons, Ltd.     

A numerical study of a plate with a hole for a new class of elastic bodies

It has been shown recently that the class of elastic bodies is much larger than the classical Cauchy and Green elastic bodies, if by an elastic body one means a body incapable of dissipation (converting working into heat). In this paper, we study the boundary value problem of a hole in a finite nonlinear elastic plate that belongs to a subset of this class of the generalization of elastic bodies, subject to a uniaxial state of traction at the boundary (see Fig. 1). We consider several different specific models, including one that exhibits limiting strain. As the plate is finite, we have to solve the problem numerically, and we use the finite element method to solve the problem. In marked contrast to the results for the classical linearized elastic body, we find that the strains grow far slower than the stress.     

Expansion formulae for wave structure interaction problems with applications in hydroelasticity

Alternate derivations of the expansion formulae for wave structure interaction problems are obtained in case of water of infinite depth and utilized to analyze the hydroelastic behavior of large floating structures. Considering the boundary value problem associated with Laplace equation having higher order boundary condition on the horizontal boundary and a Dirichlet type boundary condition on the vertical boundary in a quarter plane, Fourier sine transform is applied in the horizontal direction to convert the problem to a Sturm-Liouville type boundary value problem associated with non-homogeneous ordinary differential equation (ODE) in the transformed variable. Finally, inverting the transformed functions and applying the regularity criterion of the transformed function, the required expansion formula is derived. The expansion formula thus derived is extended to deal with similar boundary value problems having Neumann type boundary condition. The expansion formulae are applied to (i) analyze oblique scattering of flexural gravity waves by an articulated floating elastic plate and (ii) study the effect of compression on the oblique scattering of flexural gravity waves by a line discontinuity in a large floating ice sheet in water of infinite depth, which find applications in marine technology and arctic engineering, respectively. The present derivations of the expansion formulae are very simple and straightforward and can be easily used to study a large class of problems in the area of fluids and structures in mathematical physics and engineering.     

Crack arresting low-density porous materials with periodic microstructure

The problem of brittle fracture of a 2D periodic porous material is considered. For the considered relative density range the cellular material approximation with beam elements becomes invalid and a novel analysis technique based on the discrete Fourier transform is suggested. Mode I fracture toughness is derived from the continuum analysis of the stress state in a perforated elastic plane with a macroscopic crack composed of a number of microcracks between the neighboring voids. The crack length is proved to be sufficiently large such that the K-field of the linear elastic fracture mechanics takes place. The solution is obtained in the form of superposition of weighted Green functions every one of which corresponds to a unit displacement jump at a single microcrack. The Green function solutions are found by a combination of the representative cell technique and the finite element method. As a result the initial problem for infinite plane is solved by multiple analysis of the repetitive periodicity cell. An optimization of the voids shape for fixed relative density is performed in order to design the material with improved fracture toughness. Several types of voids arrangement are considered. In the limiting case of low density the results are compared with known data for cellular materials.     

Vibration isolation using open or filled trenches

The problem of isolating structures from surface waves by open or filled trenches under conditions of plane strain is numerically studied. The soil is assumed to be an isotropic, linear elastic or viscoelastic nonhomogeneous (layered) half-space medium. Waves generated by the harmonic motion of a rigid surface machine foundatin are considered. The formulation and solution of the problem are accomplished by the frequency domain boundary element method. The Green's function of Kausel-Peek-Hull for a thin layered half-space is employed and this essentially requires only a discretization of the trench perimeter and the soil-foundation interface. The proposed methodology is used for the solution of a number of vibration isolation problems and the effect of soil inhomogeneity on the wave screening effectiveness of trenches is discussed.     

Bending of a thin plate containing a rigid inclusion and a crack

The bending of a thin infinite plate with a line crack and an arbitrarily shaped rigid inclusion is analyzed. The superposition principle is used to reduce the original formulation to two subsidiary problems. A distribution of dislocation is assumed along the crack line. The solution is obtained in an integral form by using the Green function of a point dislocation. The stress functions for both subsidiary problems are obtained by employing the rational mapping function technique. The stress intensity factors are obtained in terms of the dislocation density function. Numerical results are demonstrated for the plate containing a square rigid inclusion and a line crack.     

弹性浮板在周期集中载荷作用下的水弹性响应

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf方法,研究了水面上浮动弹性平板受周期集中载荷作用下的水弹性响应。流体被假设为不可压理想且等深,流动有势。首先分析求解了两种区域(自由水面下区域和浮板下区域)内等深流场的导波问题。而后将原来混合边值问题的求解归结为求解无穷线性代数方程组的问题。其次,采用该方法研究了三种周期集中载荷激励下浮板上挠度和动弯矩幅值的分布情况;最后,对浮板上挠度幅值分布与不同周期载荷的作用点的关系进行了研究。     

水面上弹性浮板在外载荷作用下的动力学特性研究

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf方法,研究了水面上浮动弹性平板局部受动载荷(周期分布载荷)作用下的动力学特性。首先将分析计算结果与采用其他方法得到的计算结果进行了对比和分析;其次,采用本文方法研究了三种周期载荷激励下浮板上挠度和动弯矩幅值的分布情况;最后,对浮板挠度幅值分布与不同周期载荷的中心位置及其分布宽度的关系进行了研究。