Though it would be preferable, for application to natural rocks, if the inclusion were spherical or ellipsoidal in shape, I give here an approximate solution for the simpler problem of an inclusion of cylindrical form with circular cross-section. The flow is plane, with the axis of the inclusion normal to the plane of flow. I expect that consideration of a cylinder, rather than a sphere, would provide insight into the natural phenomenon. This supposition is similar to that taken in the modeling of polycrystalline rock by an analog material consisting of a thin sheet of crystals about one grain thick (Means, 1977, 1980, 1989).

Let the far-field deformation be pure shear in plane flow at rate D_xx⁰,

where

Let the inclusion have a viscosity h*. The x-axis is in the direction of extension, so D_xx⁰ > 0.

A distribution of dissolution of the host over a portion of the interface and precipitation over the other portion may be written in a way that is formally equivalent to a homogeneous deformation of the inclusion (see also Bayly, this volume). Let D_xx^T* and D_zz^T* be the components of a rate of stress-free transformation of the inclusion that is not associated with its viscous deformation. Here, I take the rate of dilation to be zero, to correspond to a net rate of dissolution equal to that of precipitation. The associated radial velocity at the surface of the inclusion is

where a is the radius of the inclusion. A tangential component

has no equivalence in the actual process, although it is formally accounted for in the solution. In this respect, the present model is an approximation.

Associating v_r^T* with diffusional transport along the interface, we have

Separately, associating v_r^T* with dissolution and precipitation along the interface

The formulation used in the previous example is repeated here, with J_s the interfacial flux, m the chemical potential of the pressure shadow component in the fluid film, and s_rr* the radial, or normal, component of stress at the circular interface. The arclength is s = aq, where q is the angle of the radius vector measured counterclockwise from the positive x-axis. Further,

where s_xx* and s_zz* are the non-zero components of the uniform stress in the inclusion referred to the Cartesian axes.

The relations between the stresses in the inclusion, which are homogenous, those of the far field, and the rate of deformation corresponding to the stress-free transformation have been obtained by an analysis too lengthy for inclusion here. They were derived from a more general solution for an elliptical cylindrical cavity in an orthotropic anisotropic elastic solid (Lekhnitskii, 1963) and the method used in the analysis of an ellipsoidal inclusion in an otherwise uniformly stressed elastic solid (Eshelby, 1957). I found

Choice of the form

and its substitution into the equations (10) together with (11) and (12) yield expressions for the desired quantities. In particular

where

and

A comparison of B’ and W’ with B and W of the boudin gap model will show why I have made the present identifications.

Here, the ratio of rates of deformation is between that associated with the pressure shadow and the far field rate. If B’ = 0, of course, no pressure solution occurs, and a pressure shadow does not form. If B’ = 1, the equivalent rate of deformation from formation of the pressure shadow is just equal to the far-field value. In essence, interfacial dissolution, transport and precipitation have served to remove the effect of the rigid inclusion from the local field of deformation. However, if the inclusion is not rigid, or nearly so, the strain computed from the pressure shadow exceeds the far field value. For B’ > 1, this is always true. The maximum rate of deformation associated with the growth of a pressure shadow, when B’ ® ¥ , is twice the far- field value. The inclusion now looks like a cavity filled with an inviscid fluid at a pressure equal to the negative of the far-field mean stress. The stress difference in the inclusion is

This becomes reduced as B’ increases, the limiting value being zero. These results suggest that the use of pressure shadows to estimate strain, and perhaps "strain history," need to be carefully examined.

I speculate that if pressure solution acts to produce a pressure shadow about an inclusion that is much larger than the host grains, operating at the grain scale, it would lead to a very substantial rate of deformation within the host. The same argument would apply to the filling of a boudin gap, where, moreover, the disparity between the scale of the boudin and that of the grains is very much greater. One might then wonder how the "driving force" in terms of a variation in normal stress large enough to drive pressure solution at the much larger scale would come about. The known result for the boudin, that the gap being filled is a crack, suggests that pressure shadows around stiff inclusions may also be bound by an interfacial crack on one side. The mechanics of the process would then be somewhat different from that considered here.