Leslie Morland

Thermomechanical balances of ice sheet flows

Abstract The flow of large natural ice masses under gravity is described by the mass, momentum, and energy balances of an incompressible, homogeneous, heat conducting, non-linearly viscous fluid in which the shear response includes a strongly temperature-dependent rate factor. Dimensionless analysis and co-ordinate stretching reflecting the long aspect ratio show that series expansions in a small parameter which determines the surface slope magnitude are uniformly valid even when temperature variation induces a strongly non-uniform mechanical response. The normalised energy balance shows that both horizontal and vertical advection are significant in thin and thick grounded sheets and in floating shelves, and that viscous dissipation can be significant in basal regions of a grounded sheet, and hence there is strong thermomechanical coupling. Moreover, though a thermal basal boundary layer may arise in a thick sheet, it would only give rise to significantly enhanced temperature and strain-rate gradients in ...

Unconfined ice shelf flow

The spreading of an unconfined ice shelf in two horizontal directions involves the variation of the two horizontal velocity components and the thickness in both directions. Exploiting the slow variation of physical quantities in both horizontal directions compared to vertical variation allows simple solution of the vertical momentum balance and the derivation of plane stress equilibrium equations for integrals of the horizontal stresses through the thickness, together with integrated traction conditions on a front contour defining the boundary of smooth flow. This contour, however, is not prescribed, but is part of the solution. Equilibrium of the region between this smooth contour and the sea margin determines the integrated front tractions in terms of the sea water pressure provided that restrictions on stresses in the margin region can be made. The resulting two-dimensional system of integropartial differential equations on the unknown domain is a complex problem.

Viscous relations for the steady creep of polycrystalline ice

Abstract Various published data from constant-stress creep tests on ice, relating minimum strain-rate to applied stress at different temperatures, are presented and compared. A temperature dependent rate factor is constructed from the Mellor and Testa (1969a) uni-axial compression data at uni-axial stress 1.18 × 10 6 N m −2 over the temperature range 212.15 K–273.15 K. This factor is used to normalise the different sets of data at different temperatures to a common temperature for comparison, but normalised strain-rates at a fixed stress still vary by a factor of 3. Furthermore, it is shown that no alternative single rate factor will adequately correlate the data at two different temperatures. A least-squares method is used to express the strain-rate as an odd polynomial in the stress; distinct polynomials are found for different sets of data. Good matches are generally obtained over a uni-axial stress range 0–10 6 N m −2 by three terms: first, third and fifth powers of stress; but less satisfactory non-monotonic polynomials involving negative coefficients are obtained in most cases if the seventh power is also included. Expressing the stress as an odd polynomial in the strain-rate, however, is not satisfactory, which is a reflection of the shape of the response at higher strain-rates. Inverse sinh function expansions failed in general, but inverse tan function expansions give good agreement to some data.

Flow of viscous fluids through a porous deformable matrix

A self-contained account of mixture theory is presented as a framework for describing the flow of fluids, liquids and gases, through a porous deformable matrix, incorporating both mechanical and thermal effects. The theory comprises the conservation laws of mass, momentum and energy for each constituent and the mixture properties which describe the interactions between constituents. Mass transfer between constituents which arises during phase change and chemical reactions influences both conservation laws and mixture properties. An analysis of discontinuity conditions at a singular surface is presented, which would be needed, for example, to describe an advancing phase-change front. Details are presented for the flow of viscous fluids through a thermoelastic matrix undergoing infinitesimal deformation, a common model for underground reservoirs. The interactions of immiscible and miscible fluids are discussed. An essential ingredient is the relation between partial physical variables defined as mean values over mixture elements, and intrinsic variables defined with respect to the constituent elements.

The Propagation of Plane Irrotational Waves through an Elastoplastic Medium

This paper is an attempt at a systematic investigation of wave propagation in a metal, treating interactions between elastic and plastic waves, and the formation and propagation of shock waves, in the general case of motions with unidirectional strain arising from an initial smooth loading unloading pulse. A stress-strain relation with linear elastic paths and concave-upward plastic paths (where compression is measured as positive) is derived and used so that the elastic wave velocity is uniform, and the plastic wave velocity an increasing function of stress. The analysis is in terms of engineering stress and strain with a Lagrangian co-ordinate system. Analytic solutions to the interactions between different types of continuous waves are developed incorporating an expression for the motion of the elastic-plastic boundary. An analysis of the breakdown of a smooth plastic compression wave into a shock wave is presented, and the propaga tion conditions derived. It is shown that the heat dissipated is proportional to the cube of the strain jump, its low value for moderate shock strength suggests that the shock does not appreciably affect the stress-strain relation, an assumption from which a solution for the unloading of a plastic compression front by an overtaking elastic wave, while shock formation is taking place, is derived. A numerical illustration of this solution for a particular pulse in aluminum is given

Dynamic plastic deformations of simply-supported square plates

Abstract I n this paper an analysis is given within the framework of thin plate theory of the problem of a simply-supported, square plate subjected to a uniformly-distributed rectangular pressure pulse. All effects due to elastic strain, work-hardening and strain-rate are neglected, although some approximate account of the two latter may be made. To simplify the analysis further, Johansens yield criterion is adopted as an approximation to Trescas. The most important results concern the maximum displacement and the total time of motion. Errors in these quantities from approximating Trescas yield criterion are estimated to be about five per cent.

A two-dimensional model for the dynamics of sea ice

This paper develops a systematic analysis of a sea ice pack viewed as a thin layer of coherent ice floes and open water regions at the ocean surface. The pack is driven by wind stress and Coriolis force, with responsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dimensional theory for the interface between ocean and atmosphere. The theory is presented for a plane horizontal interface, but the construction is readily extended to a non-planar interface. An interacting continua framework is adopted to describe the layer mixture of ice and water, which introduces the layer thickness h and ice area fraction A as smoothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolution equation for A which ignores ridging can lead to the area fraction exceeding unity in maintained converging flow, which is physically invalid. This is a feature and weakness of current models, and is eliminated by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the excess horizontal ice flux into increasing thickness of a ridging zone of area fraction Ar, and a simple postulate for the vertical ridging flux yields an evolution equation for A which shows how A can approach unity asymptotically, but not exceed unity, in a maintained converging flow. This is a significant feature of the new model, and eliminates a serious physical and numerical flaw in existing models. The horizontal momentum “balance involves the gradients of the extra stress integrated through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stresses are zero in diverging flow and arise as a result of interactions between floes during converging flow. It is shown precisely how a mean stress in a floe is determined by such edge tractions, and in turn provides an interpretation of the local extra stress in the pack. The interpretation introduces the further model function f (A) which defines the fraction of ice-ice contact length over the boundary of a floe, describing an increase of the contact fraction as A increases. Model interaction mechanisms then suggest a qualitative law for the pack stress in terms of relative motions of the floes which define the pack-scale strain rates. A simple viscous law is presented for illustration, but it is shown that even this simple model can reflect a conventional motion of a failure criterion on the stresses in a ridging zone where the convergence greatly exceeds a threshold value. We have therefore defined precisely the two-dimensional ice pack stress arising in the momentum balance, and determined its relation to the contact forces between adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orders of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensionless parameters arise, which allows the derivation of leading-order equations defining a reduced model which describes the major balances in the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dimensional pack domain) in the flow, satisfying appropriate matching conditions to be determined as part of the complete evolution. This feature appears to have been ignored in previous treatments. Here we illustrate the evolution of a moving boundary by constructing an exact solution to a one-dimensional pack motion which describes onshore drift due to increasing, then decreasing, wind stress. During the second phase a region of diverging flow expands from the free edge. The solution demonstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constructed to determine more complex one and two-dimensional motions.

Constitutive laws for ice

Abstract A theoretical discussion of models which describe the transient and secondary creep response of polycrystalline ice is presented, including a hypothesis which incorporates temperature dependence in the rate laws by the introduction of a reduced time scale. The secondary response is described by a general non-linear incompressible viscous fluid law, and it is shown that bi-axial stress experiments are insufficient but combined shear and compression experiments are sufficient to determine, in principle, the general response functions. Transient creep can be described qualitatively by a viscoelastic fluid model, and the most simple material memory influence is given by dependence on the current creep acceleration which leads to a first order differential relation between stress and strain rate. The secondary creep response is incorporated as a steady asymptotic limit with the time scale of significant transient creep governed by the response coefficients. Quantitative tests of such transient response require data from experiments at short time intervals, and in particular the determination of an initial strain rate to complement the differential law.

Generation of Thermoelastic Stress Waves by Impulsive Electromagnetic Radiation

The penetration and absorption of electromagnetic radiation through a thin layer adjacent to the surface of an elastic half-space provides a sudden heat source distribution through the layer, and, in consequence, stress waves are generated. A solution for uniaxial motion is given in the limit situation when the pulse duration approaches zero in comparison with the wave travel time over the absorption depth, together with the neglect of thermal diffusion on this time scale. The radiation absorption is assumed to decay exponentially with depth. These simplifying features lead to stress wave profiles which clearly illustrate the effective nature of the wave propagation, so that the present solution provides a useful complement to previously obtained solutions of a more general nature. In the case when the surface is stress-free, following initial build-up (here instantaneous) of compressive stress within the absorption layer, a tensile wave propagates outward from the layer. Numerical data are presented to show that the peak tensile stress can attain significant levels within distances of a few absorption depths.

A mixture theory for a phase-changing snowpack

Abstract The continuum theory of mixtures is used as the mathematical framework for a four-constituent model of a natural snowpack. The general conservation equations in point form are derived from appropriate integral balances and constitutive requirements are identified. In particular, constitutive postulates are made for the interaction terms due to phase change in the momentum and energy equations. The conservation equations are written in terms of partial variables whereas material constitutive laws are given in terms of intrinsic variables. Generalising Morlands theory * enables these two types of variable to be related when mass transfer due to phase change is included. A reduced model is proposed which assumes linear hypo-thermoelastic response for the ice and linearly viscous fluid response for the water, water-vapour and air.