A constitutive law for the viscous and tertiary creep responses of ice to applied stress

Given the initial (secondary creep) viscous response of ice to applied stress, the subsequent tertiary creep is described by an orthotropic fabric evolution relation motivated by crystal rotation arguments. That is, the ice is described as a non-simple anisotropic fluid with dependence on the evolving deformation. Extension and modification of previous formulations are proposed, in which a general orthotropic flow law for stress includes terms which are quadratic functions of the strain-rate tensor, compared to previously analysed relations in which only linear in the strain-rate tensor terms were considered. Ice response functions in the extended law are constructed in such a way that the validity equalities and inequalities between the instantaneous directional viscosities at each stage of the tertiary creep are satisfied, and correlations with families of idealised uni-axial and simple shear tertiary creep curves for different applied stresses are possible. It is shown for a range of free parameters in the proposed orthotropic model how accurately the assumed uni-axial and shear creep curves can be approximated by the constructed response functions

Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor

A complete theoretical presentation of the Continuum-mechanical, Anisotropic Flow model, based on an anisotropic Flow Enhancement factor (CAFFE model) is given. The CAFFE model is an application of the theory of mixtures with continuous diversity for the case of large polar ice masses in which induced anisotropy occurs. The anisotropic response of the polycrystalline ice is described by a generalization of Glen's flow law, based on a scalar anisotropic enhancement factor. The enhancement factor depends on the orientation mass density, which is closely related to the orientation distribution function and describes the distribution of grain orientations (fabric). Fabric evolution is governed by the orientation mass balance, which depends on four distinct effects, interpreted as local rigid body rotation, grain rotation, rotation recrystallization (polygonization) and grain boundary migration (migration recrystallization), respectively. It is proven that the flow law of the CAFFE model is truly anisotropic despite the collinearity between the stress deviator and stretching tensors.Comment: 22 pages, 5 figure

Incompressible SPH Model for Simulating Violent Free-Surface Fluid Flows

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

Loads exerted by floating ice on a cylindrical structure

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

Loads Exerted on a Cylindrical Structure by Floating Ice Modelled as a Viscous-Plastic Material

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

Simulation of Solitary Wave Mechanics by a Corrected Smoothed Particle Hydrodymamics Method

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

SPH Modelling of Sea-ice Pack Dynamics

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

Application of an Element-free Galerkin Method to Water Wave Propagation Problems

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

On the maximum horizontal forces exerted by floating ice on engineering structures

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