The paper deals with the problem of sea-ice pack motion and deformation under the action of wind and water drag forces. Differential equations describing the behaviour of ice, with its very distinct material responses in converging and diverging flows, express the mass and linear momentum balances on a horizontal plane (the free surface of the ocean). The thermodynamic effects (ice melting and lead water freezing) are accounted for by adding source terms to the equations describing the evolution of the ice thickness and area fraction (concentration). These thermodynamic source terms are described by means of a single function that idealizes typical ice growth-rates observed in winter in the Arctic. The equations governing the problem are solved by a fully Lagrangian method of the smoothed particle hydrodynamics (SPH). Assuming that the ice behaviour can be approximated by a non-linearly viscous rheology, the proposed SPH model was used to simulate the flow of a sea-ice pack driven by wind drag stresses and varying seasonal temperatures. The results of numerical simulations illustrate the evolution of an ice pack, including distributions of ice thickness and ice area fraction in space and time for assumed temperature distributions.
The paper is concerned with the problem of sea-ice pack motion and deformation under the action of wind and water currents. Differential equations describing the dynamics of ice, with its very distinct mateFfigrial responses in converging and diverging flows, express the mass and linear momentum balances on the horizontal plane (the free surface of the ocean). These equations are solved by the fully Lagrangian method of smoothed particle hydrodynamics (SPH). Assuming that the ice behaviour can be approximated by a non-linearly viscous rheology, the proposed SPH model has been used to simulate the evolution of a sea-ice pack driven by wind drag stresses. The results of numerical simulations illustrate the evolution of an ice pack, including variations in ice thickness and ice area fraction in space and time. The effects of different initial ice pack configurations and of different conditions assumed at the coast–ice interface are examined. In particular, the SPH model is applied to a pack flow driven by a vortex wind to demonstrate how well the Lagrangian formulation can capture large deformations and displacements of sea ice.
The behaviour of a water-saturated sand deposit subjected to dynamic loads induced by the propagation of Rayleigh surface waves is analysed. Cyclic shearing of the saturated sand matrix due to ground motions results in the development of excess pore pressures in the soil and its subsequent liquefaction. The phenomena of pore pressure generation and soil liquefaction are investigated within the framework of a compaction theory for saturated granular media. The results of calculations, carried out by a finite-element method, illustrate the evolution of pore pressures and the development of liquefaction zones in the soil, and show the variation of surface wave parameters with the progressive degradation of the strength of the subsoil.
In this paper the problem of transient gravitational wave propagation in a viscous incompressible fluid is considered, with a focus on flows with fast-moving free surfaces. The governing equations of the problem are solved by the smoothed particle hydrodynamics method (SPH). In order to impose the incompressibility constraint on the fluid motion, the so-called projection method is applied in which the discrete SPH equations are integrated in time by using a fractional-step technique. Numerical performance of the proposed model has been assessed by comparing its results with experimental data and with results obtained by a standard (weakly compressible) version of the SPH approach. For this purpose, a plane dam-break flow problem is simulated, in order to investigate the formation and propagation of a wave generated by a sudden collapse of a water column initially contained in a rectangular tank, as well as the impact of such a wave on a rigid vertical wall. The results of simulations show the evolution of the free surface of water, the variation of velocity and pressure fields in the fluid, and the time history of pressures exerted by an impacting wave on a wall.
The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is solved numerically by applying an element-free Galerkin method. First, the proposed model is validated by comparing its predictions with experimental data for the plane flow in water of uniform depth. Then, as illustrations, results of numerical simulations performed for plane gravity waves propagating through a region with a sloping bed are presented. These results show the evolution of the free-surface elevation, displaying progressive steepening of the wave over the sloping bed, followed by its attenuation in a region of uniform depth. In addition, some of the results of the present model are compared with those obtained earlier by using the conventional finite element method.
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The paper is devoted to numerical modelling of solitary wave propagation phenomena in shallow water of uniform depth. The problem governing equations are solved by applying a corrected smoothed particle hydrodynamics (SPH) method in which standard smoothing kernel functions are modified in such a way that so-called linear reproducing conditions for kernel approximations and their first-order spatial derivatives are satisfied. Numerical performance of the proposed SPH model has been verified by comparing its predictions with analytical results for a solitary wave travelling over the horizontal bottom. Also, the results obtained by applying the corrected SPH method and those given by the standard SPH method, with no kernel correction, are compared. Further, an impact of the solitary wave on a vertical rigid wall is investigated, and finally an interaction of two colliding solitary waves is considered.
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A multi-grain model for a migration recrystallization process in polar ice is presented. The model is based on the Sachs-Reuss approximation of the stress homogeneity in a polycrystalline aggregate. An individual crystal of ice is treated as a transversely isotropic and incompressible medium which deforms by viscous creep. The highly anisotropic viscous behaviour of the ice crystal is described by a constitutive law expressing microscopic strain-rate in terms of the deviatoric stress and three fluidity parameters that define different viscous resistances of the crystal in different glide directions. It is assumed that the recrystallization occurs in those crystals in the aggregate which are most slowly deforming, and new crystals are nucleated at orientations which favour the crystal deformation by basal glide. The model predictions are illustrated by results of numerical simulations of simple flows, showing the evolution of the microscopic structure of ice and the variation of macroscopic viscosities with increasing deformations.
The paper deals with numerical modelling of water flow which is generated by the break of a dam. The problem is solved by applying a smoothed particle hydrodynamics (SPH) method in which standard smoothing kernel functions are corrected in such a way that so-called linear reproducing conditions for kernel approximations and their gradients are satisfied. The proposed SPH model has been used to simulate a two-dimensional problem of the collapse of a water column inside a rectangular tank. The simulations illustrate the formation and subsequent propagation of a wave over the horizontal plane. It is shown that the model predictions of the changes of the advancing wave-front position, and of the changes of the free surface elevation of water, compare well with experimental data. Also, the results obtained with the corrected SPH method are compared with those given by the standard SPH method with no kernel correction. In addition, an impact of the surging wave against a vertical rigid wall is illustrated.
The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is formulated in the Lagrangian description, and the ensuing equations are solved numerically by a finite element method. In computations a convecting mesh that follows the material fluid particles is used. As illustrations, results of numerical simulations carried out for plane gravity waves propagating over bottoms of simple geometry are presented. For parameters typical of a laboratory flume, the transformation of a transient wave, generated by a single movement of a piston-like wave maker, is investigated. The results show the evolution of the free-surface elevation, displaying steepening of the wave over sloping beds and its gradual attenuation in regions of uniform depth.
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Conventional methods for the determination of water-wave induced stresses in seabeds composed of granular soils are based on Biot-type models, in which the soil skeleton is treated as an elastic medium. Such methods predict effective stresses in the soil that are unacceptable from the physical point of view, as they permit tensile stresses to occur near the upper surface of the seabed. Therefore, in this paper the granular soil is assumed to behave as an elastic-ideally plastic material, with the Coulomb-Mohr yield criterion adopted to bound admissible stress states in the seabed. The governing equations are solved numerically by a finite difference method. The results of simulations, carried out for the case of time-harmonic water waves, illustrate the depth distributions of the excess pore pressures and the effective stresses in the seabed, and show the shapes of zones of soil in the plastic state. In particular, the effects on the seabed behaviour of such parameters as the degree of pore water saturation, the soil permeability, and the earth pressure coefficient, are illustrated.
In the paper, the problem of dynamic impact of a floating ice sheet at an off-shore structure is considered. It is assumed that during an interaction event the dominant mechanism is the brittle fracture of ice at the ice--structure interface, that is, elastic and creep effects in ice are ignored. Since in natural conditions the edge of floating ice is usually irregular, the contact between a floe and an engineering object is imperfect. Thus, at any one time, the failure of ice occurs only in a number of small zones along a structure wall, leading to a highly irregular variation of forces exerted on the structure during the impact process. It is supposed in the analysis that the successive small-scale fracture events at the contact surface occur at random, and all these small-scale events take place independently of each other. An off-shore structure is modelled as a fixed and rigid circular cylinder with vertical walls. For an adopted geometry of the ice sheet, its initial horizontal velocity, and the variety of parameters describing the limit failure stresses in ice, the history of total loads sustained by the structure and the floe velocity variation are illustrated for a typical impact event. Furthermore, probability distributions for maximum impact forces exerted on the structure, depending on the floe size, its thickness and initial velocity, are determined.
In this paper the problem of transient gravitational wave propagation in a viscous compressible fluid is investigated. The problem is formulated in the Lagrangian description and is solved numerically by a finite element method. In computations either fixed in space or moving meshes that follow the material fluid particles are used with the purpose to compare their numerical performance. As illustrations, results of numerical simulations carried out for plane flows in a domain of simple geometry are presented. First, the finite element results are compared with available experimental data for the case of small-amplitude waves in order to validate the numerical model. Then, the problem of large-amplitude transient water wave propagation over a horizontal bottom, involving the wave reflection at a rigid wall, is considered. For the flow parameters typical of a laboratory flume, the evolution of the free-surface elevation and the time variations of the surface displacements at chosen locations are shown for a range of different moving wall amplitudes and excitation times.
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An axially symmetric, gravity driven, steady flow of a grounded polar ice sheet with a prescribed temperature field is considered. The ice is treated as an incompressible, non-linearly viscous, anisotropic fluid, the internal structure (fabric) of which evolves as ice descends from the free surface to depth in an ice sheet. The evolution of the ice fabric is described by an orthotropic constitutive law which relates the deviatoric stress to the strain-rate, strain, and three structure tensors based on the current (rotating) principal stretch axes. The solution of the problem is constructed as a leading-order approximation derived from asymptotic expansions in a small parameter that reflects the small ratio of stress and velocity gradients in the lateral direction of the ice sheet to those in the thickness direction. Numerical simulations of the flow problem have been carried out for various sets of rheological parameters defining the limit strength of the anisotropic fabric in ice. The results of calculations illustrate the influence of the ice anisotropy, basal melt conditions and temperature field in ice on the glacier thickness and lateral span, and on the depth profiles of the flow velocity.
In this paper the problem of interaction between a coherent floating ice field and a fixed, rigid, vertically-walled circular cylinder is investigated. The ice cover, of horizontal dimensions significantly larger than the characteristic size of the structure, is assumed to be driven against the cylinder by wind drag forces. The ice is treated as a viscous-plastic material, in which the permissible stress states in the horizontal plane are bound by an elliptic yield curve. By using an associated flow rule, a constitutive law, involving two parameters defining the ice strength in compression and much smaller strength in extension, is derived in order to describe the behaviour of the material. The law predicts distinct responses during yield (occurring at high strain-rates) and during the flow when the yield condition does not apply (at lower strain-rates). The results of numerical calculations performed by a finite difference method illustrate, for chosen ice rheological parameters, the distribution of contact stresses at the ice - structure interface. Two forms of boundary conditions at the cylinder wall, free-slip and no-slip, are considered, and their effects on the horizontal loads sustained by the structure are examined. In addition, the results for the viscous-plastic rheology of ice are compared with those obtained on the assumption of a purely viscous behaviour of ice.
The paper is concerned with the problem of interaction between a coherent floating ice cover and a fixed, rigid, vertically-walled circular cylinder. The ice cover, of horizontal dimensions considerably larger than the size of the structure, is assumed to be driven against the structure by wind and water current drag stresses. The floating ice cover is modelled as a plate that is subject to the action of horizontal forces and transverse bending due to the reaction of underlying water. During an interaction event, of a quasi-static character, the ice is modelled as a creeping material the behaviour of which is described by a viscous flow law with two, bulk and shear, viscosities. The viscosities change dramatically in their magnitudes during a transition from converging to diverging deformation of the material to reflect the fact that floating ice offers much less resistance to tensile rather than compressive stresses. By numerical simulations carried out by a finite difference method, the influence of the ice rheological parameters on the distribution of contact stresses at the ice - structure interface is investigated. Two types of boundary conditions at the interface, free-slip and no-slip, are considered, and their effects on the loads sustained by the structure are compared. In addition, creep buckling of the ice sheet near the structure is analysed to determine the critical time at which ice starts to fail due to exceeding its flexural strength at given loading conditions.
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The paper is concerned with the problem of creep buckling of a floating ice plate pressin against a rigid, vertical-walled, engineering structure of a finite lenght. The plate is modelled as a truncated wedge of a semi-infinite length and constant thickness, resting on a liquid base and subjected to transverse bending due to the elastic reation of the base and in-plane axial compression due to wind and water drag forces. The ice is treated as a viscous materail, with the viscosity varying with the depth of the ice cover. The results of numerical calculations, carried out by the finite element method, show the evolution of creep buckles in the plate, and also ilustrate the behaviour of the ice cover at different levels of the in-plane axial loading, at different temperatures across the ice, and for different geometries of he wedge-shaped plate.
In this paper the problem of interaction between a coherent floating ice cover and a rigid engineering structure is considered. It is assumed that the ice cover, of horizontal dimensions considerably larger than the dimensions of the structure, is driven by wind and water current drag forces. During the interaction process of a quasi-static character, ice is assumed to behave as a creeping material, with a rheology described by the viscous fluid flow law. The ice cover is treated as a plate which sustains both bending due to the vertical reaction of the underlying water and the action of horizontal forces, which gives rise to the development of creep buckles in the plate and subsequently leads to the flexural failure of ice. An approximate solution to the problem is constructed by employing the finite element method. The results of numerical simulations illustrate the magnitudes of the forces exerted on the structure and their dependence on the wind direction and the structure geometry. In addition, the ice plate deflection in the vicinity of the structure is illustrated, and the values of the critical time at which the plate starts to fail by creep buckling are determined to show their dependence on the ice thickness, temperature, and type.
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A discrete-grain model accounting for the induced anisotropy of polycrystalline ice is formulated. An individual ice crystal is supposed to be a transversely isotropic medium whose behaviour is linearly viscous. For such a crystal a frame-indifferent constitutive law involving three microscopic rheological parameters is derived. Assuming that each crystal undergoes a homogeneous deformation of the polycrystalline aggregate (the Taylor approximation), the macroscopic viscous behaviour of the material is determined. The considerations are illustrated by the results of numerical simulations of simple flows, showing the evolution of the oriented structure of the material and the variation of macroscopic viscosities with increasing strains. In addition, the influence of the parameters describing the single crystal anisotropy on the overall behaviour of the aggregate is investigated.
The paper concerns the problem of calculation of the maximum horizontal forces that a floating ice cover can exert on isolated, vertical-walled, engineering structures. The analysis is carried out on the assumption that the largest possible force which can occur in a floating ice plate is determined by the elastic buckling failure mechanism. Hence, the buckling loads of a semi-infinite, wedge-shaped in-plane, thin elastic plate resting on a liquid base and pressing against a rigid structure of a limited width, are evaluated. The problem is solved by applying the finite-element method. The results of numerical calculations illustrate the variation of the buckling force with the thickness of ice, the width of the structure, the angle defining the in-plane shape of the plate, and the type of boundary conditions at the ice-structure contact zone. The comparison of the results obtained in this work with those given by approximate analytic estimates available in literature, has shown that the latter considerably overestimate the bearing capacity of ice, therefore new relations are proposed in this paper.
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As polycrystaline ice undergoes creep deformation over long time-periods, it develops a fabric (oriented structure) and associated, strain-induced anisotropy. In the paper, a frame-indifferent orthotropic constitutive model for secondary creep of ice is formulated, in which the strain-rate is expressed in terms of the deviatoric stress, strain, and three structure tensors based on the principal deformation axes. As an illustration, the model is used to determine the evolution of the creep response of ice to continued uniaxial compression and simple shearing.
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