Conservation of particle identity

The essence of Lagrangian fluid dynamics is fluid particle identity acting as an independent variable. The identifier or label may be the particle position at some time, but could for example be a triple of the thermodyamic properties of the particle at some time. Time after labeling is the other independent variable. The fluid particle may not actually have been released into the flow at the time of labeling, but merely labeled with position or with some other properties at that time. Nevertheless, “time of release” will be used interchangeably with “labeling time.” The subsequent position of the particle is a dependent variable, even though it may coincide with the independently chosen position of an Eulerian observer at the subsequent time. The Eulerian observer also employs time, after some convenient initial instant, as the other independent variable. Of course, a particle path can be calculated in the Eulerian framework by integrating velocity on the path, with respect to time. Indeed, the suppression or implicitness of this detailed path information is the basis of the relative simplicity of the Eulerian formulation. On the other hand, fluid velocity is readily calculated from the particle position in the Lagrangian framework by the local operation of particle differentiation with respect to time after labeling.

Conservation of particle identity is not an immediately compelling consideration in the Eulerian framework, but is fundamental in the Lagrangian.

Viscous stresses and heat conduction

It has been assumed to this point that there are no viscous stresses, nor any heat conduction. Thus, the dynamics of the ideal fluid of the preceding chapters are compatible with an isotropic distribution of molecular velocities. In fact, anisotropy is always present in a real assembly of molecules, owing to the walls of the fluid container, fields of force or sources of heat. The Navier-Stokes equations for a real fluid may be derived from Boltzmann's equation for a dilute gas using the Chapman-Enksog expansion (Chapman and Cowling, 1970), which assumes a molecular velocity distribution close to an isotropic equilibirum. A simpler derivation, requiring less physical insight, follows from the general principles of continuum mechanics by adopting Newton's and Fourier's laws as the constitutive relations. The essential aspect of these constitutive relations is that they are local in the Eulerian framework: the viscous stress tensor is proportional to the Eulerian rate of strain tensor, while the heat flux is proportional to the Eulerian temperature gradient. The Navier-Stokes equations are accordingly expressed naturally in Eulerian form, while the Lagrangian form can only be derived by “cheating.” That is, it cannot be derived from Boltzmann's equation. Cheating can be minimized (see Aside in Section 3.2), but in the interest of moving forward, let us cheat in full.

The previous three chapters have laid out some of the basic phenomenological features of plastic deformation in crystals and have developed a mathematical constitutive framework for analyzing crystalline deformation. It is not the purpose herein to provide an exhaustive treatment of particular case studies, in particular through the review of various numerical studies that have been performed, as this is the subject of a rather different volume. We do, however, explore some of the phenomenological implications of the mechanisms and theory developed above vis-à-vis the nature of crystalline deformation. In particular, we will explore the natural tendency of plastic deformation to become highly nonuniform and in fact localized into patterns that can, inter alia, evolve into bands of intensely localized slip, kinking patterns, and the sort of heterogeneous patterns of slip on different systems that were referred to as “patchy slip” in Chapter 27. These examples of localized deformation are important because they often lead to material failure, as well as to the evolution of internal substructure that, in turn, directly influences evolving material response. On the other hand, the analysis of these deformation patterns serves to highlight some rather fundamental aspects of the process of crystalline deformation via the process of slip. This serves to reveal and, in part explain, some of the basic implications of the type of theory we have outlined herein.

The problem considered here has found application to a legion of physical applications including, inter alia, the theory of solid state phase transformations where the transformation (arising from second phase precipitation, allotropic transition, or uptake of solutes, or changes in chemical stoichiometry) causes a change in size and/or shape of the transformed, included, region; differences in thermal expansion of an included region and its surrounding matrix, which in turn causes incompatible thermal strains between the two; and, perhaps surprisingly, the concentrated stress and strain fields that develop around included regions that have different elastic modulus from those of their surrounding matrices. For the reason that the results of this analysis have application to such a wide variety of problem areas, and because the solution approach we adopt has heuristic value, we devote this chapter to the inclusion problem.

The Problem

In an infinitely extended elastic medium, a region – the “inclusion” – undergoes what would have been a stress free strain. Call this strain the “transformation strain,”eT. Due to the elastic constraint of the medium, i.e., the matrix, there are internal stresses and elastic strains. What is this resulting elastic field and what are its characteristics? In particular, can an exact solution be found for this involved elastic field? The region of interest is shown in Fig. 20.1 and is denoted as VI; the outward pointing unit normal to VI is n.

The breakdown of an initially flat, or smooth, surface into one characterized by surface roughness is an important type of phenomena occurring, inter alia, during the growth of thin films or at surfaces of solids subject to remotely applied stress in environments that induce mass removal or transport. In the case of thin films, stresses arise due to lattice mismatch and/or differences in coefficients of thermal expansion. The sources of stress are, indeed, legion but the effect can be to induce roughness, and surface restructuring, that may be either deleterious, or in some cases desirable, if the patterning can be controlled. The phenomena was first studied by Asaro and Tiller (1972) and has since been pursued by others. Our purpose is to develop some of the guiding principles, but we note that the topic is far from being thoroughly worked out. In particular, we make many simplifying assumptions, one being the assumption of surface isotropy. We also ignore some important physical attributes of surfaces, such as surface stress, which have recently been added to the description of surface patterning (Freund and Suresh, 2003).

Stressed Surface Problem

We consider here the phenomena of the breakdown of planar interfaces subject to stress into interfaces characterized by undulated topology. The phenomena is governed by those same driving forces that lead to crack growth and the growth of defects, such as inclusions, that cause internal stresses.