A more careful statement of the same idea is as follows. Suppose that at some point in the material, the deformation at some moment can be described using a strain-rate tensor: then if the material is both viscous with viscosity N and self-diffusing under stress with coefficient K, the total strain-rate tensor can be partitioned into a first part controlled by N and the stress state at the point, and a second part controlled by K and the variation in stress around the point. The main idea in view is that the latter part of the strain-rate tensor need not be isotropic and in general will not be. By contrast, a different idea also considered in the body of the paper is that the K-related second part is necessarily isotropic, i. e. gain or loss of material by diffusive mass transfer can change a spherical element only to a larger or smaller sphere.

In parallel with the two ideas about the mass-transfer part of the strain-rate tensor, there are two ideas about the driving gradient. One can consider the gradient through space of just the mean stress (which has a single value at any point considered) or one can consider the gradients through space of several separate stress components. The version using mean stress goes with the idea that a sphere changes only to another sphere. I have tried to show that one can make a certain amount of progress using either version, mean stress plus isotropic strain by mass transfer OR full stress state plus anisotropic strain by mass transfer; Figure 4 and Appendix 1 are based on the first version, and Figures 8 and 9 and Appendix 3 on the second.

The two versions just discussed both treat diffusive mass transfer through the body of the material, but there is also the option of considering diffusive mass transfer only at bounding surfaces or interfaces (see Fletcher, this volume). When applied to a fold or a boudin, this approach is quite different from considering the interiors of rock units in the manner of the present paper; but if one considers the grain interfaces inside an extensive body of granular rock and then averages over many grains, the resulting equations have much in common with those for the interior of a continuum. Fletcher (1982) took this approach and, in course of averaging, took the mean-stress/isotropic-strain-rate option for gains and losses by diffusive mass transfer (1982, p. 278,279). I believe that no theory has yet taken the parallel path, combining attention to interfaces with the anisotropic-strain-rate option, despite the fact that rock thin-sections contain abundant features prompting thoughts in that direction e. g. intergranular seams of insoluble material that appear to be residues.

It might seem that a continuum theory is basically different from any theory built by treating a rock unit as a mass of grains separated by interfaces, but the difference is not as great as at first appears. Macroscopic experiments designed to give estimates of N and K ignore whatever microstructure a real material may have, but as discussed in the main text and in Appendix 2, any pair of experimental values for N and K defines a characteristic length L for the material. I believe this length L, of the order of nanometers or micrometers, arises from the real material’s microstructure; then if, in a theory, we postulate that a continuum has properties N and K, we implicitly suggest that the continuum has some kind of microstructure. The difference is that in the "continuum theory" we suggest nothing about what the microstructure is, and make no distinctions such as that between interfaces and grain interiors. But in Fletcher’s approach, after individual grains have been considered, the averaging step smoothes over the geometrical details of the interfaces and solid grains. So in the granular treatment, the microstructure is specified but smoothed over, whereas in the continuum treatment, a microstructure is not specified but is implied. The two approaches are complementary and illuminate each other.

On the other hand, a difference between two basic ideas remains. Whenever there is diffusive mass transfer from a high-compression source toward a low-compression sink, we suppose that the flux is linked to some kind of stress gradient. One version holds that at any point in such a gradient, the relevant quantity is a single stress magnitude; the other version holds that in general all three principal stress magnitudes are relevant and that it is only when attention is confined to diffusion along a surface that a single stress magnitude per point suffices. To insist on a single value at each point or to admit a suite of values at each point are two fundamentally different ways of proceeding.

The same two options are current regarding chemical potential. The idea that a component’s chemical potential can have only one value at a point is of rather long standing; the second idea, that in a stressed material the potential has a suite of values at a single point, was proposed by Ramberg (1959; for the same proposal in a more accessible journal, see Ramberg 1963). Independently Bowen (1967, 1976) proposed a chemical potential tensor, with principal values conforming to Ramberg’s definitions. A strong endorsement of this approach is given by Grinfeld (1991, p. 2 and 132). Most interestingly, Green (1986) uses Bowen’s tensor (Green’s symbol G_ij, p. 202) but states that " it would be misleading to call G_ij the Free Enthalpy tensor because the Free Enthalpy is a scalar quantity." Having recognized the tensor, he directs attention strictly to an interface and uses only a single component from it.

Conclusions We take pure shear of a highly viscous cylinder in a less viscous matrix as a sample problem where diffusive mass transfer may occur. Fletcher (this volume) approaches the problem assuming diffusion only along the interface, while Finley (1994,1996) approaches it assuming only volume-diffusion. There is no incompatibility here; in a real situation, diffusion is likely to run both at the interface and through the volume, and the separate treatments are useful steps toward something more comprehensive.

In the present paper, two more treatments are offered, both emphasizing volume diffusion, both incomplete. In the first, we assume that diffusive mass transfer is driven by a gradient in the field of mean-stress magnitudes; in the second, we assume that gradients in several separate stress components need to be considered for a full analysis of diffusion effects. The conclusion is that both approaches deserve attention and need more work. (A fifth approach (Bayly and Minkel, in press) uses finite elements and explores further details.) The problem turns out to be quite intricate but a benefit is that it encourages attention to a number of behaviors that will reappear in other geometrical configurations. My personal expectation is that for volume diffusion, using several separate stress components will gain acceptance as being fundamentally correct, but that in many instances, using just the mean stress will be a wholly satisfactory approximation. Also diffusion at interfaces governed by the interface normal stress will in many situations be more important.

Facts so far ignored are that any real inclusion differs from its matrix in both composition and density. Consequences of a density contrast are explored by Green (1986) and consequences of variable composition by Bayly (1992, chapters 15 and 16). Consequences of the high mobility of cations compared with components of the Al-Si-O substrate are noted by Bayly (1987 p. 577, 578). A corollary is that hydrogen ions (protons) will tend to diffuse toward high-compression sites and mobilize oxygen atoms there by detaching them from the substrate. Overall, much remains to be done; there are many avenues to explore.

In concluding I revert to the fact that in this volume we honor Win Means’ contributions. His demonstrations of what can be learned from bench-top analogs are a continuing source of insights and stimuli. A photographic record of some augen growing may soon be available to guide the construction of relevant theories and to strengthen the link to behaviors in real rocks.

Acknowledgments I am indebted to Enrique Merino who made helpful comments on an early draft and to Ray Fletcher for a very careful review. I have learned a great deal over many years from association with Sharon Finley, Ray Fletcher, Don Drew and Don Minkel; and, irreplaceably, from Win.

During preparation of this paper, Hans Ramberg died, scholar and friend.