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The response of ice to applied stress on ice-sheet flow timescales is commonly described by a non-linear incompressible viscous fluid, for which the deviatoric stress has a quadratic relation in the strain rate with two response... more
The response of ice to applied stress on ice-sheet flow timescales is commonly described by a non-linear incompressible viscous fluid, for which the deviatoric stress has a quadratic relation in the strain rate with two response coefficient functions depending on two principal strain-rate invariants I2 and I3. Commonly, a coaxial (linear) relation between the deviatoric stress and strain rate, with dependence on one strain-rate invariant I2 in a stress formulation, equivalently dependence on one deviatoric stress invariant in a strain-rate formulation, is adopted. Glen's uni-axial stress experiments determined such a coaxial law for a strain-rate formulation. The criterion for both uni-axial and shear data to determine the same relation is determined. Here, we apply Steinemann's uni-axial stress and torsion data to determine the two stress response coefficients in a quadratic relation with dependence on a single invariant I2. There is a non-negligible quadratic term for some...
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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... more
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.
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This chapter is concerned with the behaviour of polar ice on geophysical time scales and its analysis by applying a micro-mechanical approach. Based on some assumptions regarding the anisotropic properties of an individual ice crystal and... more
This chapter is concerned with the behaviour of polar ice on geophysical time scales and its analysis by applying a micro-mechanical approach. Based on some assumptions regarding the anisotropic properties of an individual ice crystal and its microscopic deformation, frame-indifferent constitutive laws for creep response of the crystal are formulated. By applying homogenization methods, the microscopic laws are then used to derive the macroscopic constitutive relations for polycrystalline ice. These relations are employed to simulate the creep behaviour of ice in simple flow configurations in order to correlate parameters in the macroscopic flow laws with the observed anisotropic behaviour of polar ice. The chapter concludes with the analysis of the mechanism of dynamic (migration) recrystallization of polycrystalline ice. Three alternative dynamic recrystallization models are formulated, which are subsequently used in the simulations for simple flows to investigate the effect of th...
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The behaviour of sea ice on geophysical length scales is considered. Hence, the motion and deformation of a large ice pack consisting of a multitude of interacting ice floes and driven by wind drag and ocean current stresses is... more
The behaviour of sea ice on geophysical length scales is considered. Hence, the motion and deformation of a large ice pack consisting of a multitude of interacting ice floes and driven by wind drag and ocean current stresses is investigated. The behaviour of the pack is analysed by treating the ice cover as a two-dimensional continuum of horizontal dimensions of the order of tens to hundreds kilometres, with local properties defined by the ice thickness and the ice area concentration. The equations governing the macroscopic behaviour of the ice pack are solved in the material coordinates by applying two discrete methods: a finite-element method and a smoothed particle hydrodynamics method. The results of numerical simulations, carried out for several constitutive models describing the large-scale rheology of sea ice, are presented to illustrate the evolution of the pack under the action of wind, including variation in the ice thickness and ice concentration.
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Abstract 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... more
Abstract 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.
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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... more
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...
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A typical sea-ice pack is a complex thermodynamic system comprising a multitude of floes of different size and geometry, driven by wind and water drag stresses, and subject to surface and basal freezing and melting in response to current... more
A typical sea-ice pack is a complex thermodynamic system comprising a multitude of floes of different size and geometry, driven by wind and water drag stresses, and subject to surface and basal freezing and melting in response to current local mechanical and thermal forcing. As individual floes move about and interact, in either ductile or brittle manner, they break, merge and override one another, giving rise to large variations in local ice thickness and ice area fraction (concentration). Since broken ice cover cannot carry tensile stresses, the mechanical behaviour of an ice pack in converging flow is remarkably different from that in diverging flow. The consequence of this is the development and subsequent propagation of interfaces that separate converging and diverging regions in sea ice, often leading to the fragmentation of an initially coherent pack domain. An important feature is also a significant change of the planar geometry of a domain occupied by the ice pack, associat...
The properties of ice and its mechanical behaviour are discussed. First, the basic facts concerning various forms of ice are presented, and relevant physical parameters are given. Then, the crystalline microstructure of ice is described,... more
The properties of ice and its mechanical behaviour are discussed. First, the basic facts concerning various forms of ice are presented, and relevant physical parameters are given. Then, the crystalline microstructure of ice is described, with an emphasis on the anisotropic properties of a single ice crystal and their effect on various types of its microscopic deformation. This is followed by the presentation of the macroscopic properties of polycrystalline ice and its behaviour in various stress and deformation regimes. Thus, the elastic, viscoelastic, viscous creep and brittle behaviour of the material is discussed, and examples of constitutive equations describing all these types of the response of ice to stress are given.
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Summary A radially symmetric, gravity-driven, steady flow of a grounded polar ice sheet with a prescribed temperature field is considered. The ice is modelled as an incompressible, non-linearly viscous and anisotropic fluid with evolving... more
Summary A radially symmetric, gravity-driven, steady flow of a grounded polar ice sheet with a prescribed temperature field is considered. The ice is modelled as an incompressible, non-linearly viscous and anisotropic fluid with evolving orthotropic fabric. To describe the evolution of the fabric as an initially isotropic free surface ice descends to depth in an ice sheet, a constitutive law relating the deviatoric stress to the strain-rate and strain is applied. The solution 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 longitudinal direction to those in the thickness direction. Results of calculations show the effects of a bed topography on the ice sheet thickness profile and the velocity components. Additionally, the influence of the temperature field and the free surface snow accumulation rates on the flow is illustrated. EXTENDED SUMMARY Ice cores retr...
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This chapter deals with the analysis of the behaviour of polar ice by applying a phenomenological approach, in which the macroscopic creep response of ice is determined solely in terms of the macroscopic stress, strain-rate, and... more
This chapter deals with the analysis of the behaviour of polar ice by applying a phenomenological approach, in which the macroscopic creep response of ice is determined solely in terms of the macroscopic stress, strain-rate, and deformation. The microscopic mechanism of re-orientation of individual crystals during the deformation of ice is also accounted for in order to model the evolution of the internal structure of the material. General forms of frame-indifferent constitutive flow laws, which express either the stress in terms of the strain-rate, or the strain-rate in terms of the stress, are derived on the assumption that the type of anisotropy which develops in polar ice sheets is close to orthotropy. The parameters in the derived constitutive models are determined by the correlation of model predictions with available experimental data. The phenomenological approach is also applied to model the mechanism of the dynamic recrystallization of polar ice. All constitutive models de...
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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... more
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.
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... more
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 wa...
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In this chapter, the flow of grounded ice sheets on geophysical scales is investigated. Two ice flow configurations are considered: plane and radially-symmetric. Assuming that ice viscosities depend on local temperature, strain-rate and... more
In this chapter, the flow of grounded ice sheets on geophysical scales is investigated. Two ice flow configurations are considered: plane and radially-symmetric. Assuming that ice viscosities depend on local temperature, strain-rate and current strength of anisotropy of the material, computational models have been developed to solve the system of equations governing the flow of a large, gravity-driven, polythermal polar ice sheet. The plane flow problem is solved by a finite-element method, whereas the radially-symmetric problem is solved by applying a method of asymptotic expansions in a small parameter defining the ratio of an ice sheet’s thickness to its lateral span. The results of numerical simulations illustrate the effect of ice anisotropy on both the free-surface profile and the velocity field in a polar ice sheet. In addition, the influence of the bed topography features on the overall flow of an ice sheet is examined. The chapter is complemented with the presentation of re...
The paper is concerned with the plane strain problem of Love wave propagation through a water-saturated sand layer overlying an elastic half-space. Dynamic loads, induced by the wave passage, generate irreversible strains in the soil... more
The paper is concerned with the plane strain problem of Love wave propagation through a water-saturated sand layer overlying an elastic half-space. Dynamic loads, induced by the wave passage, generate irreversible strains in the soil matrix, giving rise to the development of excess pore water pressures and subsequent reduction of the soil effective stresses, which in an extreme case can lead to the soil liquefaction. The process of pore pressure generation is analysed within the framework of compaction theory of saturated granular media. Results of numerical calculations, carried out by means of the finite element method, illustrate the evolution of pore pressures and the development of liquefaction in the subsoil, as well as the changes in the free surface displacements.
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The processes of formation of natural ice masses on Earth are described and characteristics of the most common types of ice on our planet are presented. First, mechanisms involved in the formation of sea ice from water are discussed, with... more
The processes of formation of natural ice masses on Earth are described and characteristics of the most common types of ice on our planet are presented. First, mechanisms involved in the formation of sea ice from water are discussed, with a focus on the influence of environmental conditions prevailing during the development of ice on its structure and mechanical properties. Then, mechanisms that are involved in the transformation of snow into glacier ice in Antarctica and Greenland are described, and the processes which continuously change ice properties as ice particles move through a polar ice sheet from its surface to depth over geophysical time scales are outlined. Some aspects of the formation and evolution of macroscopic anisotropy of ice are also briefly discussed (more details on these topics are provided in Chapter 6 of the book). The chapter is concluded with a short presentation of the most important features of polar ice shelves and icebergs.
Conventional methods for the determination of water-wave i nduced stresses in seabedscomposedofgranularsoilsarebasedonBiot-typemodels,inwhichthesoil skeleton is treated as an elastic medium. Such methods predict effective stresses in the... more
Conventional methods for the determination of water-wave i nduced stresses in seabedscomposedofgranularsoilsarebasedonBiot-typemodels,inwhichthesoil 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 paperthegranularsoilis assumedtobehaveasanelastic-ideallyplasticmaterial, with the Coulomb-Mohr yield criterion adopted to bound admissible stress states intheseabed. Thegoverningequationsaresolvednumericallybyafinitedifference 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 plasticstate.Inparticular,theeffectsontheseabedbehaviourofsuchparameters asthedegreeofporewatersaturation,thesoilpermeability,andthe...
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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... more
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 metho...
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... more
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.
In the paper the viscous creep behaviour of polycrystalline materials composed of transversely isotropic hexagonal crystals is discussed, with the main focus on the analysis of the strain-induced evolution of the macroscopic anisotropy of... more
In the paper the viscous creep behaviour of polycrystalline materials composed of transversely isotropic hexagonal crystals is discussed, with the main focus on the analysis of the strain-induced evolution of the macroscopic anisotropy of polycrystals. First, general properties of the viscous response of polycrystalline materials are inferred without assuming any specific constitutive law for constituent crystals. Then, on an example of polycrystalline ice, a model is presented which describes the evolution of the anisotropic texture of ice and its macroscopic viscous properties, resulting from the mechanisms of crystal lattice rotation and migration recrystallization. Although the results obtained are directly applicable to polar ice sheet flow analyses, the proposed general theoretical framework can be also used to describe the evolving anisotropy of other polycrystalline materials, such as certain metals, minerals and ceramics.
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Summary This paper presents the results of laboratory measurements of suspended sediment movement induced by regular non-linear water waves propagating over a mildly sloping sandy seabed covered with ripples. The measurements conducted in... more
Summary This paper presents the results of laboratory measurements of suspended sediment movement induced by regular non-linear water waves propagating over a mildly sloping sandy seabed covered with ripples. The measurements conducted in a water flume were carried out by applying the technique of particle image velocimetry (PIV). The aim of those experiments was to investigate near-bed velocities of sediment particles under controlled surface wave conditions. In particular, horizontal and vertical profiles of sand grain velocities were measured, and some comparisons between the measured and theoretically-predicted quantities were carried out. A number of selected wave cases were examined, for which the Ursell number ranged from 18 to 39, and the sediment grain mobility numbers varied between 12 and 26. For these flow conditions, the near-bed layer of intense sediment grain movements had a thickness of about 2–3 ripple heights. The maximum horizontal sediment velocities measured over ripple crests were about twice as large as those over ripple troughs. Vertical sediment velocities above ripple crests and troughs were similar, amounting to about 1/4 to 1/3 of horizontal velocities over ripple crests. The detailed quantitative results obtained in the flume can help validate other experimental techniques and can be useful in testing numerical models for simulating surface wave-induced sediment dynamics.
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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... more
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 b...
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... more
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...
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... more
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.
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The phenomenon of water wave breaking and the link between the wave breaking and the velocities of water particles on the free surface are investigated. Results of experiments carried out in a laboratory flume are presented, and then... more
The phenomenon of water wave breaking and the link between the wave breaking and the velocities of water particles on the free surface are investigated. Results of experiments carried out in a laboratory flume are presented, and then compared with predictions of a theoretical model that has also been outlined in the paper. The experiments have been conducted in a
Research Interests: Numerical Simulation, Water Waves, Dispersion Relation, Wave propagation, Free Surface, and 12 moreWater Channel, Finite Element Model, Equation of Motion, Boundary Condition, Experimental Measurement, Wave Breaking, Experimental Data, Theoretical Model, Water Depth, Direct Method, Continuity Equation, and Variational Calculus
ABSTRACT
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A plane, gravity-driven, steady flow of a polar ice sheet over a horizontal bedrock, with no-slip basal conditions, is considered. The ice is modelled as a linearly viscous, incompressible and anisotropic fluid, with evolving orthotropic... more
A plane, gravity-driven, steady flow of a polar ice sheet over a horizontal bedrock, with no-slip basal conditions, is considered. The ice is modelled as a linearly viscous, incompressible and anisotropic fluid, with evolving orthotropic fabric depending on local strain rates and deformations. For prescribed free-surface elevation and non-uniform temperature field, the ice velocities required to maintain the assumed geometry are calculated by using the finite-element method. The focus is on the mechanism of dynamic (migration) recrystallization occurring near the base ofan ice sheet and leading to significant weakening, and ultimately to the complete loss, of the anisotropic fabric developed in the upper part of the ice cap. The weakening of the fabric with increasing strain rate and temperature is modelled by means of a scaling function depending continuously on a single effective strain-rate invariant. The results of numerical calculations demonstrate the effect of the recrystalli...