Some features of flexural-gravity wave propagation in the presence of a crack in sheet ice

The effect of a crack in an ice sheet on the propagation of surface flexural-gravity waves in a basin of constant depth is analyzed. The ice sheet is simulated by two floating semi-infinite fragments of a thin elastic isotropic plate. As the boundary-contact conditions on the line of contact between the parts of the plate the conditions of continuity of displacements for arbitrary slopes simulating one ice-floe overlying on another and free-edge conditions (crack) are considered. The dependence of the amplitude characteristics of the perturbations on the thickness of the ice, its degree of compression, the incident wave frequency, the depth of the basin, and the form of the boundary-contact conditions is investigated. Problems of wave diffraction on inhomogeneities of an elastic plate were solved in [1, 2], and on a crack in the ice sheet in [3, 4].Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–150, March–April, 1996.     

Swell wave propagation in an inhomogeneous ice sheet

The propagation of surface waves beneath a periodically inhomogeneous ice sheet is considered. Areas of broken ice and hummock ridges are considered as irregularities. It is shown that waves with frequencies corresponding to wind and swell waves are strongly scattered by the irregularities and are damped exponentially as they propagate beneath the ice.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 162–169, September–October, 1996.     

Theory of two-dimensional nonlinear waves in liquid covered by ice

The aim is to develop a method of Hamiltonian formalism for the waves in the liquid beneath an ice sheet and on that basis to construct a systematic nonlinear theory. Attention is concentrated on the investigation of the essentially two-dimensional effects whose properties depend to a large extent on the stresses in the ice.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–133, July–August, 1991.     

Unsteady gravity-elastic and gravity-capillary ship waves

The development of three-dimensional waves generated by a region of pressures moving uniformly and rectilinearly over the surface of a thin elastic isotropic plate covering an ideal fluid layer of finite depth is investigated. The pressures act starting at a certain instant. A qualitative similarity between the waves occurring and gravity-capillary waves is noted. The calculations are made for an ice cover. This model problem permits examining a number of properties of the oscillations of the ice cover occurring when hauling freight over ice roads, landing and takeoff of aircraft from ice fields, etc. [1]. The development of ship waves in a fluid of finite depth in the absence of a floating plate was investigated in [2, 3] and gravity-capillary waves were studied in [4–6]. Certain properties of steady three-dimensional waves occurring during movement of a load over the surface of a floating elastic plate were established in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–32, September–October, 1978.     

Natural vibrations of a hummock ridge in an elastic ice sheet floating on the surface of an infinitely deep fluid

The properties of the natural vibrations of a hummock ridge in an elastic ice sheet are investigated. Typical shapes of the dispersion curves for symmetric and antisymmetric boundary waves which propagate along the hummock and damp exponentially with distance from the latter are obtained. It is shown that natural vibrations can initiate failure of the ice sheet at a certain distance from the hummock. Under compression this process leads to the formation of a parallel hummock ridge.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–105, November–December, 1995.     

Instability of Waveguides in a Liquid with Surface Effects

Long surface capillary-gravity waves and waves beneath an elastic plate simulating an ice sheet are considered for a liquid of finite depth. These waves are described by a generalized Kadomtsev-Petviashvili equation containing higher (as compared with the ordinary Kadomtsev-Petviashvili equation) space derivatives. The generalized Kadomtsev-Petviashvili equation has waveguide solutions (waveguides) corresponding to traveling waves which are periodic in the direction of propagation and localized in the transverse direction. These waves result from the instability of uniform (carrier) periodic waves with respect to transverse perturbations. The stability of the waveguides with respect to longitudinal longwave perturbations is studied. The behavior of these perturbations depends on the wavenumber of the carrier periodic wave. Three intervals of wavenumbers corresponding to all the possible types of governing equations are considered.     

Parametric excitation of flexural-gravity edge waves in the fluid beneath an elastic ice sheet with a crack

The properties of elastic-gravity oscillations of deep water beneath a thin elastic plate with a crack are investigated in the paper. The dependence of the reflection and transition coefficients of the waves through the crack on wave frequency and incident angle are found. The shape of the fluid surface deformed by edge waves, propagating along the crack and decreasing exponentially away from the crack, is investigated in the vicinity of the crack. The asymptotic equations describing the parametric excitation of counterpropagating edge waves by flexural-gravity waves which hit the crack at normal incidence are derived.     

Impingement of surface waves on the edge of compressed ice

The impingement of small-amplitude surface waves on the edge of a solid compressed ice sheet in a basin of finite constant depth is considered. The influence of the cylindrical rigidity and the value of the compressing force on the dependence of the amplitude coefficients of reflection and transmission on the incident wave period is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 121–126, May–June, 1995.     

Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.     

Diffraction of acoustic waves on the leading edge of a flat plate in a supersonic flow

A scheme is proposed for calculating the intensity of the acoustic wave field generated by diffraction of a beam of acoustic waves on a sharp leading edge of a flat plate in a supersonic flow. This wave field is shown to be a functional of the mass-flow amplitude distribution in the acoustic field at the level of the plate surface upstream of the latter. This distribution can be found on the basis of measurements. The discontinuity of the normal-to-plate component of the velocity perturbation on the plate edge plays an important role in determining mass-flow fluctuations along the plate. At large distances from the leading edge of the plate, where the diffraction wave on the boundary-layer edge degenerates into longitudinal acoustic waves, the amplitude of mass-flow fluctuations decreases with increasing distance from the leading edge and depends on wave orientation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 64–70, March–April, 2005.     

Dynamic mode I perturbation solution for a moving crack unsteadily

Rice et al. (Journal of Mechanics and Physics of Solids42, 813–843) analyze the propagation of a planar crack with a nominally straight front in a model elastic solid with a single displacement component. Using the form of Willis et al. (Journal of the Mechanics and Physics of Solids43, 319–341), of dynamic mode I weight functions for a moving crack, we address that problem solved by Rice et al. in the 3D context of elastodynamic theory. Oscillatory crack tip motion results from constructive-destructive interference of stress intensity waves. Those waves, including system of the dilatational, shear and Rayleigh waves, interact on each other and with moving edge of crack, can lead to continuing fluctuations of the crack front and propagation velocity.     

Motion of an External Load over a Semi-Infinite Ice Sheet in the Subcritical Regime

The three-dimensional problem of steady-state forced vibrations of fluid and semiinfinite ice sheet under the action of a local external load traveling along the rectilinear sheet edge at a constant velocity is considered. Two cases are analyzed. In the first case the fluid surface outside the ice sheet is free and in the second the fluid is confined by a rigid vertical wall and the ice sheet edge adjacent to the wall can be both clamped and free. The ice sheet is simulated by a thin elastic isotropic plate floating on the surface of fluid of finite depth. The load traveling velocity is assumed to be not higher than the minimum phase velocity of the flexural-gravity waves (subcritical regime). The solution to the linear problem is obtained by means of the integral Fourier transform and matching the expansions of the velocity potential in the vertical eigenfunctions. Examples of the numerical investigation of the ice sheet and fluid displacements are given.     

Plotnikov——Toland板模型中水弹性孤立波的迎撞

通过奇异摄动方法研究了在薄冰层覆盖的不可压缩理想流体表面上传播的两个水弹性孤立波之间的迎面碰撞.借助特殊的 Cosserat 超弹性壳 理论以及Kirchhoff--Love 板理论,冰层由 Plotnikov--Toland板模型描述.流体运动采用浅水假设和Boussinesq 近似. 应用Poincaré--Lighthill--Kuo 方法进行坐标变形,进而渐近求解控制方程及边界条件, 给出了三阶解的显式表达. 可以观察到碰撞后的孤立波不会改变它们的形状和振幅. 波浪轮廓在碰撞之前是对称的, 而在碰撞之后变成不对称的并且在波传播方向上向后倾斜. 弹性板和流体表面张力减小了波幅. 图示比 较了本文与已有结果可知线性板模型可作为本文的一个特例.      

Plane Problem of Vibrations of an Elastic Floating Plate under Periodic External Loading

The Wiener–Hopf technique is used to construct an analytical solution of the problem of vibrations of a semiinfinite elastic floating plate under periodic external loading. The solution is obtained in explicit form ignoring draft. The dependences of the amplitudes of surface waves and iceplate deflection on the loading distribution and frequency, ice thickness, and liquid depth are studied numerically. It is established that for some types of acting load, no waves propagate in the plate and liquid and the plate vibrations are standing waves localized near the loading region. An example of such vibrations is given and a condition for the occurrence of localized vibrations is found.     

EFFECTS OF A TRANSLATING LOAD ON A FLOATING PLATE—STRUCTURAL DRAG AND PLATE DEFORMATION

The elastic deformation of a structural plate floating on water caused by a translating three-dimensional load is investigated. The problem is akin to the landing and take-off of aircraft on a structural or ice sheet. The initial-boundary-value problem is solved analytically using a free-surface condition that incorporates the flexural rigidity of the plate. The three-dimensional load is modeled as an axisymmetric, translating pressure distribution. The time-dependent analytical solution is used to obtain the unsteady drag of this moving pressure, if it exists, as well as its asymptotic behavior at large time. The behavior of the transition of the drag near a critical speed related to the minimum celerity of the free waves of the hydroelastic system is examined. Asymptotic analysis shows that the drag attains a discontinuous but finite value as the translation speed approaches the critical speed, an essential difference from some existing two-dimensional results. The growth rate of the plate slope is found to be weakly singular, like log t, for large time. Comparisons with published experimental data for plate deformation are made for the case of an ice sheet. The agreement is very favorable. Implications on the operation of floating runways are discussed.     

A direct relationship between bending waves and transition conditions of floating plates

Ocean waves travel deep into ice fields in the polar regions, both affecting the formation of sea-ice and causing its break-up. Recently, it has been shown that a relatively simple linear water and bending wave theory can predict the decay rate of the wave energy travelling through fractured ice sheets and floes at the geophysically important wave periods of 6–15 s. That work used simple free-edge conditions. A possible improvement to the current model is to better represent the effective connection due to partially frozen cracks that occur in practice. The Wiener–Hopf technique gives explicit formulae for the velocity potential and surface deflection, expressed as series expansions over the modes of the elastic plate floating on water of finite depth, with the coefficients in the expansion given as functions of four constants. These constants are determined by a system of four linear equations, represented by a 4-by-4 matrix and a four-element vector. The elements of the matrix are given as explicit functions of relationship between edge conditions. General connections between ice sheets may be interpreted as a vertical and a rotational spring providing transition conditions for the shear force and the bending moment. The reflection and the transmission of waves can then be simply calculated as direct functions of the connection conditions. Conversely, reflected and transmitted waves allow complete characterization of the effective connection conditions at a material discontinuity.     

Evolution of three-dimensional gravitationally warped waves during the movement of a pressure zone of variable intensity

Three-dimensional, unestablished, gravitationally warped waves arising due to the motion of a harmonically time-varying pressure zone over a solid, thin plate floating on the surface of a homogeneous liquid of finite depth have been studied in the linear formulation. In the absence of a plate, three-dimensional waves are generated by the movement of a region of periodic perturbations, where established waves have been studied in [1, 2], and unestablished waves have been investigated in [3–5]. The evolution of three-dimensional, gravitationally warped waves formed during the motion of a constant load over a plate has been considered in [6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 54–60, September–October, 1986.     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.     

Near crack line elastic–plastic analysis for a infinite plate loaded by two pairs of point tensile forces

The near crack line analysis method is used to investigate a center crack loaded by two pairs of point tensile forces in an infinite plate in an elastic–perfectly plastic solid, and the analytical solutions are obtained in this paper. These solutions include: the elastic–plastic stress field near the crack line, the law that the length of the plastic zone along the crack line is varied with an external loads and the bearing capacity of an infinite plate with a center crack. The results of this paper are sufficiently precise near the crack line because the assumptions of the small scale yielding theory have not been used and no other assumptions have been taken.