Ice Plate Research Articles

Diffraction of flexural-gravity waves by a vertical cylinder of non-circular cross section

The linear three-dimensional problem of flexural-gravity wave (hydro-elastic wave) diffraction by a vertical cylinder of an arbitrary smooth cross section is studied using an asymptotic approach combined with the vertical mode method for water of finite depth. The surface of the water is covered by an infinite, continuous elastic ice plate. The rigid cylinder extends from the sea bottom to the ice surface. The ice plate is frozen to the cylinder. The ice deflection is described by the equation of a thin elastic plate of constant thickness with clamped edge conditions at the cylinder. The flow under the ice is described by the linear theory of potential flows. The coupled problem of wave diffraction is solved in two steps. First, the problem is solved without evanescent waves similar to the problem of water waves diffracted by a vertical cylinder. This solution does not satisfy the edge conditions. Second, a radiation problem with a prescribed motion of the ice plate edge is solved by the vertical mode method. The sum of these two solutions solve the original problem. Both solutions are obtained by an asymptotic method with a small parameter quantifying a small deviation of the cylinder cross section from a circular one. Third-order asymptotic solutions are obtained by solving a set of two-dimensional boundary problems for Helmholtz equations in the exterior of a circle. Strains along the edge, where the ice plate is frozen to the cylinder, are investigated for nearly square and elliptic cross sections of the vertical cylinders depending on the characteristics of ice and incident wave. The strains are shown to be highest in the places of high curvatures of the cross sections. The derived asymptotic formulae can be used in design of vertical columns in ice. They directly relate the strains in ice plate to the shape of the column.

Numerical investigation of ice breaking by a high-pressure bubble based on a coupled BEM-PD model

In this paper, by combining the boundary element method (BEM) and peridynamics (PD), a bubble-ice interaction model is established, which can investigate the dynamic interactions between a high-pressure bubble and an ice plate with particular focus on the mechanical behaviors of ice breaking. The bubble dynamics are solved by BEM based on the potential flow theory. Ice cracks initiation and propagation are simulated by the bond-based peridynamics which is validated by a three-point bending test. The fluid–structure interaction (FSI) is achieved by matching the normal velocity and hydrodynamic loads at the fluid–structure interface. To validate the proposed FSI model, an experiment is carried out in which an oscillating bubble is generated under an ice plate by underwater discharge system. The whole interaction process is captured by a Phantom V711 high-speed camera. Qualitative agreements are achieved between the numerical and experimental results. The underlying mechanism of cracks initiation, propagation, branching, and coalescence of the ice plate is found to highly depend on three parameters, i.e., bubble–ice distance, ice thickness and bubble size. The present study is expected to provide further assists in the understanding of ice breaking problems.

Crack formation and breakout of shore fast sea ice in Mordvinova Bay, south-east Sakhalin Island

Recognizing that cyclical flexural fluctuations in sea ice sheets stimulated by waves can potentially initiate and perpetuate cracks that may ultimately lead to the destruction of an ice cover, we discuss some winter observations collected beneath the fast ice cover of Mordvinova Bay in the south-east of Sakhalin Island during 2018–2019 when a large crack formed that subsequently detached the offshore sea ice from the inshore ice plate by a wide lead. Energy density spectra show the ramifications of surface-gravity waves from the open ocean—primarily at swell frequencies—penetrating far into the ice from offshore, together with comparatively long, resonant leaky wave modes that gradually dissipate as they propagate alongshore but present a diminishing standing wave envelope of antinodes perpendicular to the coast. By evaluating amplitudes over different ranges of period, three swell wave events are assessed to determine whether they alone could have induced enough bending in the sea ice plate to cause breakage, acknowledging that this would normally occur much closer to the peripheral ice edge where the stresses are largest or as the swells run ashore. The foreshortened seaward standing curvature profile of leaky wave antinodes is also considered as a prospective source of cracking. Whilst neither pathway is found conclusively to be independently responsible for breaking the ice in this case, it is conjectured that the combined influence of swell and leaky waves acting together could cause the observed crack to form. In this case, fracture would occur at a distance of less than about four or so kilometres from the shore, where the antinodal curvatures induced by the standing leaky modes are sufficient. Conceding that the unrelenting flexing of the ice plate by waves and swell could affect the longevity of the sea ice as well, fatigue-precipitated weakening is explored. By determining the value of the mechanical stress at which an ensuing rupture of the fast ice could happen, it is shown that ice destruction can eventuate approximately 17 h after the swell first arrives at the ice edge from the Sea of Okhotsk, appreciating that breakup will depend on prevailing and prior meteorological conditions, the form and state of the sea ice including any preexisting defects, and the specific periods of swell waves that infiltrate the Mordvinova Bay ice sheet.

Green function of steady forward speed with ice cover

The Green function representing the potential flow generated by a source with unit strength advancing beneath an ice cover along a straight path at a constant speed is studied in the paper. The ice is modelled as elastic plate. The Green function satisfying the linearized covered-surface condition is derived and expressed in the form of a double Fourier integral with respect to wavenumber k and polar angle θ on the Fourier plane. The computing method of Green function is then proposed, the polar integral is first performed by using the theorem of residues, the resultant wavenumber integral is reformulated by removing the weak singularities and evaluated numerically. The Green function is shown to be suitable for the prediction of ice plate deflection. The given solution is analysed as a function of speed and ice thickness.

Behaviour of the Ice Cover under the Conditions of Flexural-Gravity Resonance with the Pair Movement of the Load

The nature of ice cover failure technology implemented with amphibian hovercrafts (AHC) is represents generation of a wave system in an ice cover under the effect of moving load on the ice cover. If gravity wave in water accompanies flexural wave in an ice plate, this combination of waves will be called flexural gravity wave (FGW), the speed of moving load υc equal to the speed of FGW propagation will be called critical. However, during the implementation of ice breaking works by single (AHC), AHC’s parameters may appear insufficient for failure of the ice cover of specified thickness at specific ice conditions. In these cases, efficiency of ice breaking works may be increased by simultaneous use of several hovercrafts, i.e. by interference of FGWs generated by hovercrafts. This work is dedicated to research of the ice cover deformation during the motion of several (paired) loads over it. a comparison of the theoretical and experimental data obtained confirmed the operability of the selected criterion for the failure of the ice cover, which allows it to be used to develop practical recommendations for use FGWs generated by several loads to failure the ice cover. Original results obtained.

Numerical simulation of level ice impact on landing craft bow considering the transverse isotropy of Baltic Sea ice based on XFEM

Ice bending is a major failure mechanism of level ice when ships and marine structures interact with level ice. This paper aims to investigate the ice bending and ice load when level ice collides on ships and marine structures using numerical simulation method, and compare the numerical results with field test. The fracture of ice is simulated with extended finite element method (XFEM), and cohesive zone concept is used to describe the crack propagation. In order to consider the characteristics of S2 columnar ice, a transversely isotropic elastic material model is used for the ice bulk elements, and a transversely isotropic Tsai-Wu failure criterion is adopted to predict the initiation of cracks. A well-controlled field test of a landing craft bow colliding with level ice in Baltic Sea is simulated to verify the numerical scheme. The ice plate's continuous deformation, crack initiation and crack propagation at different impact velocities and angles are simulated and the results are discussed. In the simulation, the bending crack emerges at the midline of the top surface of ice plate, then propagates towards free boundary, and finally a circumferential crack forms. It is found that with the impact velocity increases, the bending load increases and the fracture size (perpendicular distance from the crack to the contact edge) decreases. And as the angle between the landing craft bow and vertical direction increases, the bending load and the fracture size decrease. The simulated results corresponds well with the field test. The competition between the circumferential crack and radial crack is also found in the simulation and will be discussed in this paper. The results show that this method well simulates the bending of level ice and predict the ice load, and provides a good approach for investigating the mechanism of different forms of level ice fracture.

ОСОБЕННОСТИ КОНТАКТНОГО РАЗРУШЕНИЯ ЛЬДА

The formation of an intermediate layer under hydrostatic compression at a shear appearing due to the action of converging and diverging fronts of stress momentums (pulses) is considered. Continuous monitoring of deformational changes in the structure of ice was carried out using acoustic methods. The features of contact ice breaking in the diverging fronts of stress pulses are considered by the example of the slow impact of a rigid spherical indenter on an ice plate simulating half-space. Using the piezoelectric accelerometer, an oscillogram of the impact was recorded and a generalized dependence of the reduced stress on the reduced instantaneous velocity of the impact (semi-cubic parabola) was obtained. It is established that under conditions of the experiment (smooth convex indenter surface and icy half-space) a thin intermediate layer is formed, the properties of which determine the physical similarity in the family of curves «instantaneous force-instantaneous velocity». A rheological model with due regard for the change in the microstructure of ice during the impact is proposed. Quantitative determinations of the deformation changes in structure of solid ice samples were performed under intensive plastic deformation in a matrix with a profile similar to the Laval nozzle. The deformations created by the piston caused forced vibrations in the ice. The working surface of the piston in the form of an ellipsoid together with the smooth walls of the matrix and the reverse cone created conditions for parametric resonance and the formation of fronts of highfrequency stress pulses. Under influence of these pulses, zones with a superplastic fine-crystalline structure of ice (cumulative effect) were formed in ice. In the outlet cylindrical channel, a flow around an obstacle of the ice with the structure of an intermediate layer (dynamic viscosity 20 MPa s) and the distribution of velocities of motion over the channel cross section were studied. The obtained results can be used to simulate the processes of contact destruction of deep rocks by a support or an ice-resistant platform loaded with an ice field.

Physical Parameters of Solitary Wave Packets in Shallow Basins under Ice Cover

We determine the velocities and lengths of solitary envelope waves whose velocity is located in a left half-neighborhood of the phase velocity minimum in the dispersion relation for shallow basins under ice cover. The ice cover is modeled as an elastic Kirchhoff-Love ice plate. The Euler equation for the liquid layer (water) includes an additional pressure from the plate, which Boats freely on the liquid surface. We consider the case of weakly nonlinear waves in the limit of long wavelengths and small amplitudes where the initial dimensionless stress in the ice cover does not exceed one third. These waves are described by a fifth-order Kawahara equation. We then compare the obtained results with the parameters found using a strongly nonlinear description. The comparison yields very good results for shallow depths of the considered basin. This phenomenon is explained by the properties of the lowest nonlinearity coefficient in the equations describing the solitary envelope waves branching from the phase velocity minimum on the dispersion curve. We discuss possible applications of the obtained results to experimental wave measurements under an ice cover.

Micro‐Seismic Monitoring of a Shear Fault Within a Floating Ice Plate

AbstractThe deformation of a circular fault in a thin floating ice plate imposed by a slow rotational displacement is investigated. Temporal changes in shear strength, as a proxy for the resistance of the fault as a whole, are monitored by the torque required to impose a constant displacement rate. Micro‐seismic monitoring is used to study the relationship between fault average resistance (torque) and micro‐ruptures. The size distribution of ruptures follows a power‐law scaling characterized by an unusually high exponent (b≃3), characteristic of a deformation driven by small ruptures. In strong contrast to the typical brittle dynamics of crustal faults, an 'apparently aseismic' deformation regime is observed in which small undetected seismic ruptures, below the detection level of the monitoring system, control the slip budget. Most (≃71%) of the detected ruptures are organized in bursts with highly similar waveforms, suggesting that these ruptures are only a passive by‐product of apparently aseismic slip events. The seismic signature of this deformation regime has strong similarities with crustal faulting in settings characterized by high temperature and with non‐volcanic tremors.

Measurement and evaluation of natural frequencies of bulk ice plate using Scanning Laser Doppler Vibrometer

The structure of water in its solid state (ice) is complex. A number of process parameters and the chemistry of the freezing aqueous solution influence ice crystal growth and ice characteristics such as crystalline structure, impurity inclusions, and appearance. The resulting variation in ice properties makes determining mechanical properties challenging. This study examines the performance of a Scanning Laser Doppler Vibrometer (SLDV) used to measure the natural frequency of an ice sample and consequently evaluate its modulus of elasticity. Analytical and numerical methods were used, and the results have been compared to values found in the literature. The modulus of elasticity for the measured ice plate, with a porosity of 8.8%, was found to be 6.7 GPa, which is 2.7% lower than previously published values.

Simulation of the deflection of the ice cover

The paper proposes a mathematical model describing behavior of ice cover under loading. Not taking into account the chemical characteristics, and considering it a solid continuous medium on an elastic base, the role of which is played by water, to simulate the deflection of the ice cover under loading, it is possible to use the theory of thin shells and plates. Thus, the proposed mathematical model is based on the partial differential equation of the bending of a thin plate on an elastic base. Boundary conditions are set depending on the method of fixing the edges. Delivered the boundary value problem is solved in the class of almost-periodic functions by means of a generalized discrete Fourier transform, which allows us to move from the partial differential equation to the ordinary differential equation. The graph of the function of deflection of the ice plate is constructed.

Modeling and simulation of ice–water interactions by coupling peridynamics with updated Lagrangian particle hydrodynamics

In this work, a coupling method between the bond-based peridynamics model for solids and the updated Lagrangian particle hydrodynamics (ULPH) model of fluids is developed to simulate interaction between ice and seawater. The ULPH (Tu and Li in J Comput Phys 348:495–513, 2017) is the latest development of using non-local differential operators to model Navier–Stokes equations of Newtonian fluids, which has the advantages to preserve the consistency with partial differential equations and second-order accuracy in spatial gradient interpolation. On the other hand, we model the ice-like materials by using the bond-based peridynamics, which formulates equations of motions in terms of integro-differential equations instead of partial differential equations. For demonstration purpose, the numerical solution of a rigid ball impacting an ice plate floating in calm water is presented. The results obtained have shown the versatility as well as the accuracy of the method.

Comparison of cooling plate configurations for automotive battery pack thermal management

A numerical simulation was developed combining micro and macro scale models, to determine the thermal state of battery packs in electric vehicles. A spatially resolved Ohm’s law model was integrated with the single particle model to resolve the electrochemistry in prismatic cells, and then coupled with a thermal transport simulation. The objective was to compare the effectiveness of two types of liquid channel cooling plate configurations. The first, known as ice plates are placed between every second cell in the battery pack. The second, known as cold plates, are placed underneath a block of cells, and absorb heat only through the small lower face of the cell, exploiting anisotropic heat transfer properties. In general, the ice plate performs better. The cold plate is less complicated and expensive to integrate into the battery pack, and has more scope for higher coolant circulation rates. This paper compares the performance of the two cooling systems, highlighting the conditions where each system works best, along with quantitative assessments obtained through numerical simulation.

Nonlinear Vibrations of a Floating Longitudinally Compressed Elastic Plate in the Interaction of Wave Harmonics of Finite Amplitude

The equations for three nonlinear approximations of flexural-gravity vibrations of a longitudinally compressed elastic plate which take into account the nonlinearity of acceleration of vertical plate displacements are obtained using the method of multiple scales. The ice plate unbounded in horizontal directions floats on the surface of a homogeneous ideal fluid. Asymptotic expansions up to the third order of smallness are constructed on the basis of the obtained equations for the plate-fluid surface elevation and the velocity potential of the liquid particles formed in the nonlinear interaction of two harmonics of progressive periodic surface waves. The amplitude-phase characteristics of the formed elevation of the fluid surface (bending of the plate) are analyzed as functions of the basin depth, the parameters of ice plate and interacting harmonics, and the nonlinearity of acceleration of vertical ice displacements.

The vertical mode method in the problems of flexural-gravity waves diffracted by a vertical cylinder

The linear three-dimensional problem of ice loads acting on a vertical circular cylinder frozen in an ice cover of infinite extent is studied. The loads are caused by an uni-directional hydroelastic wave propagating in the ice cover towards the cylinder mounted to the see bottom in water of constant depth. There are no open water surfaces in this problem. The deflection of the ice cover is described by the Bernoulli–Euler equation of a thin elastic plate of constant thickness. At the contact line between the ice cover and the surface of the cylinder, some edge conditions are imposed. In this study, the edge of the ice plate is either clamped to the cylinder or has no contact with the cylinder surface, with the plate edge being free of stresses and shear forces. The water is of finite constant depth, inviscid and incompressible. The problem is solved by both the vertical mode method and using the Weber integral transform in the radial coordinate. Each vertical mode corresponds to a root of the dispersion relation for flexural-gravity waves. It is proved that these two solutions are identical for the clamped edge conditions. This result is non-trivial because the vertical modes are non-orthogonal in a standard sense, they are linearly dependent, the roots of the dispersion relation can be double and even triple, and the set of the modes could be incomplete. A general solution of the wave-cylinder interaction problem is derived by the method of vertical modes and applied to different edge conditions on the contact line. There are three conditions of solvability in this problem. It is shown that these conditions are satisfied for any parameters of the problem.

Body motion in liquid under ice plate with snow cover

The motion of a submarine in liquid under an ice plate covered with flooded snow is considered. The ice is modelled as an elastic plate and the snow cover is modelled as a viscous layer on the top of the plate. The submarine is modelled as a slender solid of revolution with scale 1:300. The experimental and theoretical study of the influence of the viscous snow layer on deflections of the floating ice plate is conducted. The viscous layer reduces the amplitudes of flexural-gravity waves. The greatest influence of the viscous layer on the plate deflections is achieved for velocities of the submarine, where the waves of maximum amplitude are generated. Theoretical results are in good qualitative and quantitative agreement with the model experiments.

Moving of a submarine under an ice cover in fluid of finite depth

The influence of a basin depth on ice cover deflections, when a submarine moves under it in fluid, is experimentally and theoretically analyzed. For a theoretical research the ice cover is modelled by an elastic floating plate, and the submarine is modelled by a source-sink system in a fluid flow. The problem is solved by means of asymptotic analysis and integral transformations. The experimental study is conducted with submarine models in the pool with the sizes L × B × H = 10 m × 3 m × 1 m . The ice cover for experiments is modelled in two ways. The first way is the thin ice plate received by freezing of the surface layer of water at low temperatures. In the second case the rubber sheet floating on a reservoir surface is used. The special velocity (the velocity at which maximum values of the wave height are reached) is analyzed. It is obtained that the decrease of the basin depth leads to decrease of the special speed. For small speeds (less than the special speed) there is the fact that the basin depth decrease leads to the increase of the height of the flexural-gravity wave.

The ice response to an oscillating load moving along a frozen channel

Unsteady response of an ice cover to an oscillating load moving along a frozen rectangular channel is studied for large times. The channel is filled with ideal incompressible fluid. The ice cover is modelled by a thin elastic plate. The flow caused by the deflection of the ice cover is potential. The problem is formulated within the linear theory of hydroelasticity. External load is modelled by a smooth localized pressure distribution. The load has periodic magnitude and moves along the channel with constant speed. Joint system of equations for the ice plate and the flow potential is closed by initial and boundary conditions: the ice plate is frozen to the walls of the channel, the flow velocity potential satisfies the impermeability condition at the rigid walls of the channel and linearized kinematic and dynamic conditions at the ice-liquid interface; at the initial time the load is stationary, the fluid in the channel is at rest and the stationary ice deflection is determined from the plate equation for the initial magnitude of the load. The problem is solved with the help of the Fourier transform along the channel. The ice deflection profile across the channel is sought in the form of the series of the eigenmodes of the ice cover oscillations in a channel. The solution of the problem is obtained in quadratures and consists of three parts: (1) symmetric with respect to the load deflection corresponding to the stationary load; (2) deflection corresponding to steady waves propagating at the load speed; (3) deflection corresponding to waves propagating from the load and caused by the oscillations of the load. The number of the last waves, depending on the parameters, can not exceed four for each eigenmode. In this article the results of the analytical and numerical analysis of the considered problem is presented.

Floating ice plate failure due to its thermal expansion at the surface

The problem of a floating ice sheet failure caused by stresses induced in ice by temperature changes at its top surface is investigated. The ice cover is modelled as a plate of uniform thickness, which is laterally constrained at its edges by rigid walls, and is assumed to deform, and ultimately fail, by the mechanism of creep buckling. The floating plate is subjected to in-plane compressive stresses developing in ice to prevent its lateral expansion due to heating, and is transversely (vertically) bent by the forces acting at its base and caused by the reaction of underlying water. The sea ice is treated as a material whose elastic and viscous properties depend on its porosity and current temperature, and therefore vary with the depth of ice. The results of simulations, carried out for a variety of ice plate spans, thicknesses and temperature-change scenarios, illustrate the evolution of creep buckles in the plate prior to its failure, and show the time variation of the magnitudes of forces exerted by ice on the constraining walls.

2D numerical modeling of icebreaker advancing in ice-covered water

This paper presents 2D numerical modeling to calculate ship–ice interactions that occur when an icebreaker advances into ice-covered water. The numerical model calculates repeated icebreaking of an ice plate and removal of small ice floes. The icebreaking of the ice plate is calculated using a ship–ice contact detection technique and fluid–structural interaction of ice plate bending behavior. The ship–ice interactions in small ice floes are calculated using a physically based modeling with 3DOF rigid body equations. The ice plate is broken in crushing, bending, and splitting mode. The ice floes drift by wind or current and by the force induced by the ship–ice interaction. The time history of ice force and ice floe distribution when an icebreaker advances into the ice-covered water are obtained numerically. Numerical results demonstrate that the time history of ice force and distribution of ice floes (ice channel width) depend on the ice floe size, ship motion and ice drifting by wind or current. It is shown that the numerical model of ship maneuvering in realistic ice conditions is necessary to obtain precise information about the ship in ice-covered water. The proposed numerical model can be useful to provide data of a ship operating in ice-covered water.