Theoretical mechanics in glaciology and ice engineering

Cold Regions Science and Technology, 6 (1983) 181-184

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Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

EDITORIAL

Over the past few decades, understanding of the mechanics of ice and frozen soils has developed by piecemeal investigation of material properties, and by ad hoc application of conventional engineering mechanics to particular problemz Now, with increased motivations, resources and basic knowledge, the time may be ripe for development of more general and rigorous theory, and for greater professional involvement by applied mathematicians and theoretical mechanicianz Although a number of deliberate steps have been taken towards these goals, there is still a certain polarization between applied mathematicians interested in the application of theoretical mechanics and people following more traditional approaches to problem-solving. LooMng one way, the traditionalist may seem a barbarian stumbling blindly through confusion and inconsistency. Looking the other way, the abstract theoretician might look like an unworldly theologian, concerned more with the purity of arcane concepts than with practical needs. While these mutual suspicions may not be entirely unfounded, they have the potential to be harmful, as can sometimes be seen in peer reviews of papers and research proposals. In the hope of clearing the air a little, we have asked a member of the Editorial Board to give his views on the subject, and we invite readers to share their ideas and peeves through the correspondence columns of the journal. M.M.

THEORETICAL MECHANICS IN GLACIOLOGY AND ICE ENGINEERING L.W. Morland' School o f Mathematics and Physics, University o f East Anglia, Norwich NR4 7TJ (Gt. Britain)

Basic continuum mechanics has been the fundamental ingredient of theoretical treatments of glacier flow over the past thirty years. Constitutive equations for the macroscopic mechanical and thermomechanical response of ice to stress and heating have been deduced from laboratory and field data, and used in conjunction with the conservation equations of mass, momentum and energy, to analyse idealised glacier flow problems. The pioneering theoretical research has been conducted primarily by physicists, material scientists, engineers and geologists, and it is only in recent years that applied mathematicians whose main discipline is theoretical mechanics have made a serious entry into this field, lee engineering has many more mechanics participants than glaciology viewed as the study of long-time flows of glaciers, ice-sheets, and ice-shelves, but-both areas have developed without major contributions from non-linear theoretical

mechanics. There is, therefore, a new perspective slowly emerging in glaciology and ice mechanics, though not recognised very widely at present. While the non-linear viscoelastic response of ice to stress at constant temperature is qualitatively well established by uni-axial compression tests, there is far from sufficient data at present to determine fully even a uni-axial viscoelastic description at constant temperature. Constant stress tests exhibit primary decelerating creep, a minimum strainrate and tertiary accelerating creep for the small strain range covered (up to a few percent), and there is an associated non-monotonic stress response at constant strain.rate. These are not features of rhe01ogy models developed for other materials, and further investigations of differential and integral relations have been necessary to construct possible alternative models compatible with such features. These viscoelastic models may be of fluid

182 or solid type, with the choice depending on the properties most significant in the application, but a relation of solid type incorporating strain from a fixed reference configuration is essential to describe anisotropy. The conjecture that constant stress response and constant strain-rate response each determine the constitutive relation, being different expressions of the same material properties as in the case of a linear viscoelastic material, is true for some single integral representations but not for the lowest order differential relations compatible with the qualitative response. The two types of response reflect some common property in the fluid relation but also contain some independent information, while they are fully independent in the solid relation. More complete sets of constant stress and constant strain-rate data are needed to construct explicit relations and test the merits of the different forms of model. Of course one-dimensional data cannot determine constitutive relations required to describe the response to a general stress configuration in a three-dimensional world. A valid constitutive relation must satisfy the two fundamental invarlance principles of physics: co-ordinate invariance and frame indifference. It must therefore be expressible as a relation between frame indifferent (or frame invariant) tensors, and the uni-axial models mentioned above are derived from such relations involving suitable stress, stress-rate, strain, strain-rate and strain acceleration tensors as appropriate. Even with the usual incompressibility approximation and isotropy assumption, determination of the response coefficients of the differential relations or kernels of the integral relations requires two independent data functions associated with two independent stress components. Conventional tri-axial tests with independent axial stress and radial stress in the transverse plane (which would be more accurately described as transversely isotropic stress) do not provide two independent data functions (unless incompressibility or isotropy fails, when the model is inadequate). However, true bi-axial stress tests, with two independent stresses along two normal axes and zero stress along the third normal axis, do provide two independent relations. Combined shear and uni-axial stress tests also provide the necessary data, but appear to be experimentally less attractive. Alternative stress

configurations may offer technical advantages, and collaboration between experimentalist and theoretician in the design of tests and measurement programmes could improve the yield of useful information. Analytic and numerical methods for non-linear viscoelastic stress analysis of boundary-value problems are still to be formulated, and it probably requires the stimulus of an established model to generate the considerable mathematical effort needed. Laboratory tests are limited in duration, and for the long time response appropriate to glacier flow it is inferred that ice behaves as a non-linear incompressible viscous fluid; that is, the deviatoric stess is a function of the strain-rate relative to the current configuration. Observations of ice-sheet and ice-shelf flows support this assumption, and ice-sheet data has been used to estimate the coefficient in an assumed power law relation. However, the stress vs. strain-rate relation is typically deduced from the minimum strain-rate in uni-axial constant stress tests, since the supposed long time limit is not reached, which requires a further conjecture that the minimum strain-rate and longtime (steady) strain-rate are not much different at the low deviatoric stresses arising typically in gravity-driven natural ice flows. A power law is the simplest formula providing adequate correlation over a wide range, but implies infinite viscosity at zero stress. There are associated apparent singularities in some flow analyses, which can be treated rigorously but at additional complexity, and since the interpretation of data near zero stress is open to doubt, a power law is not the most advantageous model. In fact, the detailed data used for the original power law deduction is more closely correlated by a polynomial with three terms, which exhibits finite viscosity at zero stress. However, a wide range of laboratory data presented over twenty years does not produce a consistent viscous relation, since measured strain-rates at practical stress levels vary by a factor of three, and an even greater difference results from the ice-shelf estimates. Field data is clearly the more realistic guide to long-time viscous response, but will be available only at a limited set of stress levels and also hinges on a boundary-value problem solution for interpretation. Uniformity of the longitudinal strainrate through the thickness of an ice-shelf is a val-

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uable feature in such interpretation, but there is also a significant temperature variation with depth, and ice response to stress is strongly temperature dependent, particularly near the melting point. Interpretation of data therefore requires solution of the coupled thermomechanical problem, which in turn requires the form of temperature dependence specified. Solution of a general coupled problem has not yet been accomplished, and previous parameter estimates from shelf data rely on inconsistent approximations. It is generally assumed that temperature appears only through a temperature dependent rate factor in the viscous relation. This simple concept that temperature changes only the process rate at a given stress can be applied directly to any viscoelastic relation and appeared first in early linear viscoelastic descriptions of polymers. It is equivalent to a universal mechanical relation on a reduced time scale which depends on the temperature history of the material element. However, different sets of laboratory data for fixed stress and varying temperatures do not lead to a common rate factor, and more controlled testing is necessary to establish, or reject, this model of temperature dependence. Once again, at constant temperature, a viscous shear relation requires two independent data functions for determination, so uni-axial data has been supplemented by an assumption that the deviatoric stress is parallel to the strain-rate. Apart from early limited combined shear and compression tests, which did not confirm this assumption, no further multi.axial data to determine the structure of the viscous relation have been obtained. Theoretical analyses of natural ice flow problems must therefore rest for the present on the restricted thermomechanical viscous relation accepted for thirty years, though inconsistencies between data should raise doubts and stimulate further testing. Natural large-scale ice flows are primarily the gravity-driven flows of bounded valley glaciers, grounded ice-sheets and floating ice-shelves. They are free surface flows, so the flow domain is not prescribed but is an element of the solution. In most situations acceleration terms are extremely small so momentum balance requires a stress field in equilibrium with gravity. For glaciers and icesheets there is a significant basal shear traction governed by a basal sliding law, while for a floating

shelf the basal shear traction is zero, and in consequence the respective stress and velocity patterns are quite different. A common feature, however, is the very small surface slope relative to the mean base plane, including the more complex transition zone between grounded sheet and floating shelf. While a direct small surface slope approximation has led in early two-dimensional treatments of both sheets and shelves to simple expressions for the respective stress components in terms of the surface profile, only the shelf flow analysis was completed successfully, and that only for uniform density and temperature-independent ice response in plane flow. More recent dimensionless variable analyses of sheets and shelves display a natural small parameter in the equations, and co-ordinate stretching necessary to achieve the correct gradient balances for equilbrium lead to rational series expansions in a small parameter which defines the magnitude of the surface slope. The lead order terms confirm the earlier approximate stress expressions and the special plane flow shelf analysis, but demonstrate the inconsistencies of further approximations made to incorporate non-uniform density and lateral expansion of shelves and to complete the plane or axi-symmetric solution for a grounded sheet. Full solutions so far have been presented only for steady flow with the assumption of temperature-independent ice response or isothermal conditions, both physically unrealistic. However, the dimensionless analysis, co-ordinate stretching and series expansions approach applies equally to the coupled thermomechanical equations, and the lead order approximation for steady flow satisfies a purely parabolic system of partial differential equations in contrast to the elliptic nature of the original stress and velocity system. There are attendant advantages for numerical methods of solution. In the isothermal two-dimensional case the problem reduces to a standard non.linear second order ordinary differential equation for the surface profile with straightforward numerical solution, and the coupled thermomechanical parabolic system offers a more attractive approach than direct numerical treatment of the full unscaled equations. It is still a non-linear free (unknown) surface problem, so not trivial. This approach demonstrates that the simple stress expressions still hold for the

184 thermomechanical problem (though the surface profile will differ from the isothermal solution) and for unsteady flows, but more important it shows the relative significance of different terms in the physical balance equations, and in turn the errors of various direct physical approximations. In particular, in the energy balance both horizontal and vertical advections are significant, viscous dissipation is significant in thick ice-sheets and is not negligible for moderate sheets and shelves, and a weak thermal boundary layer may arise in thick sheets but not in moderate sheets nor in shelves. There is also an inference that annual variation of surface conditions about a long-time steady mean induces unsteady conditions only in a thin surface layer. This form of mathematical analysis must be a useful approach to a variety of ice flow problems - for example, the sheet-shelf transition, lateral shelf expansion, unsteady surface conditions - in order to determine the correct physical significance of different terms in the balances and to obtain a rational simplifying approximation of the system of equations. Solutions of idealised problems can also serve to test the accuracy of numerical algorithms constructed for more realistic conditions. It should be clear that the recent mathematical treatments of ice flow problems are not attempts to displace good physical models by rigorous but artificial analysis but, on the contrary, aim to construct valid approximations of the fundamental physical balances. First attempts naturally oversimplify, omitting important physical ingredients, but are useful to develop the technique, and in fact can provide better understanding of environmental influence than more general treatments which fail to satisfy the physical balances. They also highlight the errors which can arise by carrying direct physical assumptions too far. Empirical and

mathematical approaches may appear complementary, or even disjoint, but it is much better if empirical evidence is digested in a proper formulation of a flow problem which is then treated rigorously. While two distinct philosophies exist there will be strong differences of opinion about the merits of research on both sides, which will influence reviews of papers submitted for publication. Use is still made of simple formulae or approximations long established in the literature to interpret data or model flow configurations, and unsuspecting authors are naturally surprised to receive a review pointing out that their theoretical base is invalid. An editor cannot be expected to publish results based on invalid theory because it has been accepted in the past. On the other hand, very theoretical expositions are criticised (and rejected) for their (apparent) lack of physical application and their length devoted to mathematical analysis. A strategy I favour is to include sufficient mathematical detail for a non-mathematical reader to seek assurance from a colleague with theoretical mechanics experience that the methods and results are sensible and, of course, to demonstrate how the methods and results relate to physical application. Glaciology has benefited by being multidisciplinary throughout its development, and must surely benefit further with the interest and participation of applied mathematicians. Collaboration between the applied mathematicians and practically oriented glaciologists is essential to develop mutual understanding and yield more constructive theoretical advances, but does require extra effort, patience, and tolerance of different perspectives. I would like to close this very personal view of a science which has attracted and held my interest for several years now by making a plea for collaboration.