Further Extensions

The purpose of this chapter is to give attention to three directions along which more ideas could be attached, building on the preceding chapters as base. Labels or titles for the three directions are: unsteady behavior and elastic effects; the factor f; anisotropy. Throughout the preceding chapters, a highly artificial practice has been followed: attention has been focused on states where processes are occurring in the steadiest possible manner. The purpose of this chapter is to consider the question: if processes are less steady, can we still describe them concisely and predict their evolution? If we can, presumably it is by adding some terms to the descriptive equations, and we consider briefly what kinds of terms might be needed. As with turning from a single cylindrical inclusion to a granular aggregate, there is an immediate change to a vast field of complexities. The purpose of the chapter is to give just a preliminary view of how one might begin to identify possibilities. The purpose is to enquire how unsteady or transient effects might occur in a system that is capable of steady behavior. For this purpose, something simpler than a chemical nonhydrostatic system can be used, as shown in Figure 20.1. The simpler of the systems illustrated, Figure 20.1a, consists of a weight that is supported by two elements P and Q. The elements are known as dashpots; each is imagined to consist of a cylinder and piston; each cylinder is full of oil both above and below the piston and each piston has a hole. In consequence, when the element is pulled it can change length as fast as oil can slip through the piston's hole, and ideally the rate of elongation is proportional to the force pulling on the element. The fact that ultimately the piston comes to the end of the cylinder is ignored; we imagine P and Q to have as much length as we need. In Figure 20.1a, the system is such that the two elements have to elongate at equal rates; but they are dissimilar and we imagine a system where, to achieve equal rates, the force pulling P needs to be three times the force pulling Q.

Deformation and Diffusion: Quantitative Relations

The purpose of this chapter is to put the ideas of Chapter 11 into quantitative form. The first step is to link L0 to N and K; L0 is the arc-length of the imaginary quarter-cylinders in Figure 11. 5b, N is the material's viscosity (Pa-sec), and K is its coefficient for pressure-driven self-diffusion (m2/Pa-sec). The point emphasized in Chapter 11 is that if two migration paths exist, one curved and one straight, but both having the same length and the same variation of normal-stress components along their length, migration will be equally vigorous along the two paths. Further, the shortening rates at the source-ends of the two paths will be equal. The procedure used to find the relation of L0 to (NK)1/2 is to write the two shortening rates and equate them.

Change of Shape and Change of Volume

In earlier chapters we first defined a material's chemical potential, and then went on to enquire how the material responds. And similarly with a state of nonhydrostatic stress: having reviewed what it is, we consider how a material might respond. For the sake of simplicity, we imagine an extensive sample, such as a cubic meter, and suppose that the stress state is the same in every cubic centimeter; that is to say, there are no gradients in stress from point to point. Thus we do not enquire yet how a material responds to a spatial stress gradient; that comes later. We first enquire how it responds to a homogeneous but nonhydrostatic stress. Inside the material, close to the point of interest, we define a small length l by means of the material particles at its two ends. If, at a later moment, we find the distance between the particles to be l — δl, then we envisage the limit of the ratio δl/l as l goes to zero, give the limit the symbol ε, and name it the linear strain at the point of interest in the direction of l, positive when δl is positive, i.e., for a shortening and negative for an elongation. Another mental operation that can be performed in the neighborhood of the point of interest is to define a small sphere by means of the material particles that form its surface. At a later moment the particles will form the surface of an ellipsoid. (For a large sphere and an inhomogeneous situation, the new shape can be something more complicated; but as the imagined original sphere approaches zero diameter, the shape of its deformed counter-part can only approach an ellipsoid). The axes of the ellipsoid are principal directions of strain, and the magnitudes of the strains along them are named ε1, ε2, and ε3, with ε1 the largest. In an isotropic material, the principal axes of stress and strain coincide, with ε1 lying along the direction of σ1 and correspondingly; see Figure 7.la. As with stresses, the three values of ε themselves define an ellipsoid if they are all positive—see Figure 7.1b.

Disequilibrium 3: Internal Variables

In Chapters 2, 3, and 4, the usefulness of the concept chemical potential has been explored for describing and predicting movement of material from point to point in space—from a location where a component's potential is high to a location where its potential is lower. But chemical potential influences another type of material behavior as well, as in the example at the end of Chapter 2, the polymerization of vinyl chloride. The polymerization is a process that runs at a certain rate, like diffusion of salt, and the rate depends on the potential difference between the starting state and the end state; but unlike diffusion of salt, there is no overall movement from one location to a new location—the vinyl chloride simply polymerizes where it is. There are movements, of course, on the scale of the interatomic distances, but nothing corresponding to the 4 m of travel that appears in the discussion of the dike. If no travel is involved, it is not so easy to calculate a potential gradient along the travel path and go on to predict a rate of response. Yet there definitely is a rate of response, even with PVC polymerizing. The purpose of this chapter is to consider this matter; we shall then be equipped to begin considering nonhydrostatic conditions. The essential idea is to represent all possible degrees of polymerization along an axis, as in Figure 5.1. The figure is drawn to represent a condition where the chemical potential per kilogram is greater in the monomer form than in the dimer form, i.e., a condition where the material polymerizes spontaneously. Suppose we know the chemical potential per kilogram for all degrees of polymerization and also, at some temperature, the rates at which 2 forms from 1, 3 forms from 2, etc. (per kg of the starting form in a pure state). Then we arbitrarily pick a distance on the horizontal axis to separate point 1 from point 2.

Disequilibrium 1: Potential Gradients and Flows

As in Chapter 2, so again here the intention is to review ideas that are already familiar, rather than to introduce the unfamiliar; to build a springboard, but not yet to leap off into space. The familiar idea is of flow down a gradient—water running downhill. Parallels are electric current in a wire, salt diffusing inland from the sea, heat flowing from the fevered brow into the cool windowpane, and helium diffusing through the membrane of a helium balloon. For any of these, we can imagine a linear relation: . . . Flow rate across a unit area = (conductivity) x (driving gradient) . . . where the conductivity retains a constant value, and if the other two quantities change, they do so in a strictly proportional way. Real life is not always so simple, but this relation serves to introduce the right quantities, some suitable units and some orders of magnitude. For present purposes, the second and fourth of the examples listed are the most relevant. To make comparison easier we imagine a barrier through which salt can diffuse and through which water can percolate, but we imagine circumstances such that only one process occurs at a time. Specifically, imagine a lagoon separated from the ocean by a manmade dike of gravel and sand 4 m thick, as in Figure 3.1. If the lagoon is full of seawater but the water levels on the two sides of the dike are unequal, water will percolate through the dike, whereas if the levels are the same and the dike is saturated but the lagoon is fresh water, salt will diffuse through but there will be no bulk flow of water. (More correctly, because seawater and fresh water have different densities, and because of other complications, the condition of no net water flow would be achieved in circumstances a little different from what was just stated. For present purposes all we need is the idea that conditions exist where water does not percolate but salt does diffuse.) For flow of water driven by a pressure gradient, suitable units are shown in the upper part of Table 3.1 and for diffusion of salt driven by a concentration gradient, suitable units are shown in the lower part.

Cylindrical Inclusions

The purpose of this chapter is to extend the ideas in Chapter 16 to the situation where one material occurs as an inclusion in the other. In Chapters 13 through 15, most of the discussion centered on conditions that varied along one direction, x, but not along orthogonal directions. At several points, a cylindrical tube with fixed radius was imagined, but this was only a handy visualization of the condition where all velocities are zero in planes normal to x. The use of the equations in Chapters 13 through 16 is to describe conditions close to a planar interface of large extent. If the ratio (distance from interface)/(breadth of a planar portion of interface) is small, behavior is as if the interface were infinitely extensive, and it is to this condition that the equations apply. As a step toward understanding behavior around an inclusion, we now consider a long cylinder of one material embedded in an unlimited extent of a second material. The axis of the cylinder is taken as the y-direction and we continue to assume that everything is uniform in this direction: all properties and all behaviors are uniform along y and all velocities along y are zero. But in xz planes we now see a circular cross-section as in Figure 18.1 instead of just a planar boundary. As regards stress state, let this be uniform throughout the host material except insofar as the inclusion causes variation; let the remote stress state have principal compressive stresses σxx and σzz with σzz larger. To start, we make the same assumptions as in Chapter 13, namely that the two materials are uniform and of the same chemical composition, differing only in viscosity; and let the material of the inclusion be stiffer. Because of σzz being larger, the entire assembly will at any moment be shortening along z and elongating along x, and if the cross-section is circular at the moment we inspect it, it will be elliptical at all later times. An impression of how deformation proceeds is given in Figure 18.2, which shows how a grid would look at a later time if it had been a square grid at the moment when the inclusion was circular.

Compounds of the Type (A, B)X

Compounds such as gallium aluminum arsenide are of interest for two reasons. First, they have practical use, so that benefit comes from understanding their properties and behavior; but second, they can be regarded as mixtures of just two components, GaAs and AlAs, and so serve as a fresh set of examples of the ideas in Chapter 14. A difficulty in that chapter arises from the fact that at some points we need to imagine a mechanical continuum, while at other points we need to imagine particles traveling independently. In this chapter we need to do the same two things but the conflict in our concepts is not as acute; we can use eqn. (14.9) with more confidence and escape from the sense of an internal contradiction. The basis for the discussion is the idea of perfect stoichiometry; in a compound of type (A, B)X it is assumed that although the abundance-ratio of A to B is variable, the ratio of (A + B) to X is always exactly 1. Departures from stoichiometry are, of course, of great importance but constitute a later topic.

Chemical Potential Under Nonhydrostatic Stress

The purpose of this chapter is to continue the unification that was begun in Chapter 8. There, first and second derivatives of normal stress with respect to orientation were used; we now examine the idea that the chemical potential of a component at a point can be a multivalued direction-dependent scalar like the normal-stress magnitude, and that it too can have a gradient with respect to orientation. The essence of a nonhydrostatic stress is that different planes through a point are subject to different normal compressive stresses: σn varies with the orientation of the plane considered. Let us focus on a plane i across which the normal compressive stress is σi: then we put forward the assertion that an equilibrium state that can be associated with plane i is a hydrostatic state whose pressure has the same magnitude as σi. For illustration, see Figure 9.1. (For the present, we take a cautious stance: each hydrostatic state in the figure is certainly an equilibrium state, and each is certainly associated with a plane, but is it the associated equilibrium state that properly belongs with that plane according to the precepts of, for example, de Groot (1951, p. 11)? For now, we make no attempt to prove that it is so: we simply use the assertion and explore its consequences. Fortunately its consequences include large amounts of classical mechanics so that it counts as a "successful assertion" on those grounds, but at least for now it lacks any underpinnings.) An immediate consequence of the assertion illustrated in Figure 9.1 is the relation in Figure 9.2.

Nonhydrostatic Stress

As in the chapters on chemical potential, it will again be assumed that the reader has thought about the topic before, so that our task is to select rather than to build. The interior of a continuous sample contains many small volumes and small areas, on any of which attention can be focused. A small internal area has the property that, across it, the material on one side exerts a normal force and a tangential force on the material on the other side. Let the normal force be F and the area A; then the ratio F/A approaches a limit as the size of A approaches zero. Thus we define the magnitude of the normal stress at a point across an infinitesimal area of a particular orientation. If we set up Cartesian coordinates so that the orientation of the area can be specified by the direction of its normal then, at a point, for every direction vector there is a normal-stress magnitude. The stress may be compressive or tensile, and in this text we treat compressions as positive. It is possible to imagine a universe where space itself has an attribute of left-handedness or right-handedness, or where space does not but materials do. But if we set these possibilities aside and use ordinary ideas about symmetry, it follows that at any point where stresses exist inside a continuum, there are three orthogonal planes across which the tangential stress is zero; these planes suffer only normal stresses. The planes themselves are principal planes, their normals are the three principal directions at the point and the normal-stress magnitudes are the principal stress magnitudes. The largest, intermediate, and smallest normal compressions will be designated σ 1, σ 2 and σ 3, respectively; for most of what follows we shall designate the directions along which these compressions act as x1, x2, and x3 (so that the plane compressed by stress σ 1 has x1 for its normal), and we shall use x1, x2, and x3 as axes for a local Cartesian system with which other planes and directions at the point can be specified. In particular, for any direction through the point, a unit vector can be imagined (magnitude = 1 unit of length); its components along the three axes will be called n1, n2, and n3, combining to give the unit vector n.

Summary

The purpose of this chapter is to consolidate. No new ideas are introduced; instead we try to sort the main thread from the side issues, and the parts that are reasonably clear and firm from the parts that are still fuzzy. The core of the chapter is a set of seventeen statements, seventeen vertebrae that form the backbone of the book, but there are also a preface and a postscript. The preface provides the setting for the seventeen-part core and the postscript takes up the question of where to go next. The purpose of the book was given at the start of Chapter 1. Even at that early point, a stressed cylinder was used as an example. The purpose is to make headway with the question: if a state of chemical equilibrium exists under hydrostatic stress and is disturbed by making the stress nonhydrostatic, what processes begin to run, and what quantitative relations should we expect to be followed? Before the seventeen-part "answer" it is to be noted that there are two alternative ways of dividing the subject matter into two parts. The division scheme is displayed in Figure 17.1a and separates eight types of change. (A somewhat similar diagram on page 111, distinguished eight circumstances in which change might be observed—a different system of divisions that is of no use here.) Of the eight boxes set up, four have been discussed, as shown in Figure 17.1b. The two ways of dividing this four-box group are by a horizontal cut or by a vertical cut that separates stars from superscript a’s. (A vertical cut separating the N-box from the rest is of no help; it would be contrary to our theme.) The horizontal cut separates stress-driven effects below from composition-driven effects above. It is in fact the traditional division between mechanics and chemistry; enormous amounts of science fall clearly above the cut or clearly below it and cause no confusion at all. This cut was used as a guide in the early chapters, especially in the flow diagram or organization chart, Figure 8.1. By contrast, the second cut appeared as late as Chapter 15, but deserves emphasis; it is at least as instructive and helpful as the first, and perhaps more helpful.