Among the purposes of the present volume is that of recognizing Win Means’ contributions to our science, and among his contributions is his emphasis on the "global classroom." The perpetual student, the teacher who is also an eager learner, has been with us since classical times; nowadays, with electronic help, geologists can converse regardless of distance; the classroom in which we learn from each other is global indeed. I think Win would agree that it is more a seminar room than a lecture hall: avoiding the dogmatic, a proper use of global facilities is to put forward exploratory ideas so as to prompt colleagues to work them over.

These prefatory remarks are supposed to excuse the fact that the following proposals are only incomplete and tentative. I would like to have been able to link all the threads and reach conclusions that were incontrovertible, but have not been able. Readers will please exercise their own insight, and honor Win by treating the classroom as one where progress is cooperative. I hope others will continue to pursue the threads here taken in hand.

Stresses as a topic of study by themselves are of only limited interest, and the same is true of strains; it is in the interplay, when a stress causes a strain, that the topics come alive.

The link between stresses and strains is the material’s rheology. To get started in the simplest manner, one can imagine a Newtonian material, one whose viscosity remains the same no matter what stress magnitude develops. But the outcrop geologist is soon forced to note a defect of Newtonian models: they make no allowance for diffusive mass transfer, whereas outcrops are full of evidence of such effects; differentiated cleavage zones and the gap-fillings between boudins are just two from a long list of manifestations.

Diffusion effects were incorporated in abstract general formulations of mechanics in the 1960s, but the first results for specified realistic geometries were produced by Green (1970, 1980) and by Fletcher (1982). Fletcher described diffusion in three situations:

(i) across a homogeneous bar when bent;

(ii) along an inhomogeneous bar when compressed across its width;

(iii) in an almost-planar layer at the onset of folding.

The second situation was explored further by van der Molen (1985) and examined independently by Stephenson (1988). Except for Fletcher’s problem (iii), all of these can be called "one-dimensional" examples: the materials are taken to fill space in three dimensions, but it is in only one direction that a gradient exists driving a diffusive flux. Computer-chip design has prompted more one-dimensional studies (e.g. Greer 1995; Daruka et al. 1996) but in the present work we seek an example to extend Fletcher’s exploration in two and three dimensions.

Definitions: in subsequent paragraphs the following two effects are considered to be separate. Let a large sample be imagined as divided into many small elements; each element is defined by the atoms that sit in its boundary, as if one could, for example, paint them. In "deformation without diffusive mass transfer" or "deformation at constant volume," the elements all change their shape but no atom migrates out of one element across a boundary into another element; the material contents of each element do not change. By contrast, "diffusive mass transfer" here means the migration of a few atoms through the main mass of non-migrators, and includes the migration of a few atoms across boundaries. Time is considered divisible into small intervals, so that in any interval only a small fraction of the total atom population migrates or diffuses; the vast majority remain in a coherent mass, so that the element boundaries remain well defined, though in need of constant touching up. But averaged over time, all atoms behave alike; every atom spends most of its time being coherent with its neighbors and has only brief spasms of action as a diffuser. In subsequent paragraphs, "viscous deformation" and "creep" refer to the first process, to deformation with no change of element contents, and "diffusion" is used for only the second process, in place of the more cumbersome "diffusive mass transfer."

In other contexts, of course, one might distinguish creep that occurs "by diffusion" from creep that occurs for example by glide on glide-planes, while both are envisaged as processes at constant volume. It is for the present paper only that "diffusion" is used in the special sense noted.

A simple assembly with two-dimensional cross-section is shown in Figure 1. A cylindrical inclusion with high viscosity is embedded in an extensive matrix of less viscous material; at all points remote from the inclusion, the stress field is uniform --- a north-south compression M+S and a smaller east-west compression M-S; the situation extends uniformly perpendicular to the plane of the diagram so that, for all particles in the plane of the diagram, their velocities lie in the plane of the diagram too. Regions of greater compression exist around X,X and regions of less compression exist around Y,Y; if diffusive fluxes exist at all, they will carry material away from X and toward Y. In the simplest formulation, we assume that each of the materials, --- (i) the inclusion and (ii) the matrix, --- has a constant isotropic viscosity, a constant coefficient of self-diffusion and constant density. For materials that show no self-diffusion at all, equations were developed over a century ago that accurately describe the resulting instantaneous strain rates and velocity field, but for materials with self-diffusion, the behavior is still not properly known.

As far as I know, the most extensive study of this problem yet made is the one by Finley (1994, 1996). Kenkmann and Dresen (1998) cover many aspects but exclude diffusion. Ideally, one might specify material properties and then try to discover the stress field that would exist around the inclusion without any preconceptions. However, a powerful exploratory approach is to make some assumptions about the stress field and ask, "What material properties would allow a stress field of this particular form to exist?" Using this approach, Finley shows a suite of possible stress fields for different degrees of contrast between inclusion and matrix, but all depend on the material being anisotropic; not only anisotropic but anisotropic to just the right extent and with the right variation from point to point to allow the stress field to be of the form assumed. The results constitute a valuable first attack on the problem and are highly instructive, but they prompt the thought, "Let us approach this problem again with particular attention to the matter of isotropy. If we insist on the materials’ being isotropic or close to it, in what way does this guide us as regards possible stress fields? Can we progress on and describe stress states that could exist in materials with less pronounced anisotropy?"

(Fletcher, in this volume, considers the same geometry but treats only transport at the interface whereas Finley’s work and the present paper treat transport through the body of the materials.)

Preview of conclusions Regrettably, I think this problem has no elegant solution; stress fields such as Finley described, using a small number of intelligible terms, are perhaps not found in ideally isotropic materials; even this problem, selected to be the simplest possible that admits diffusion in two dimensions, perhaps suffers from intractable awkwardness. However, the second approach, emphasizing the material’s isotropy, brings some points of interest to the fore. I will therefore run through them, and hope that someone using Finley’s insights as well as the present points succeeds in making a fruitful attack on this resistant problem.

Aside from the matter of isotropy, two more features of the present work are:

(1) attention to the condition of plane strain, which is less simple in presence of diffusion than in its absence, and

(2) attention to the possibility that when stress drives diffusion, the loss or gain in a material element may not be by the same amount in all directions: diffusive loss may turn a spherical element into a smaller ellipsoid, not necessarily a smaller sphere.

In fact, part of the purpose of the present piece is to bring the second idea forward and to explore it.