Introduction

An interface can experience a number of different types of force, including chemical forces (e.g., due to compositional differences across the interface), curvature forces (due to interface curvature) and mechanical forces (Sutton and Balluffi, 2006; Asaro and Lubarda, 2006). In view of the focus of this book, I consider only mechanical forces.

In general, an interface experiences a mechanical force when it lies between two adjoining regions containing different elastic fields and therefore different strain energy densities and elastic displacement fields. In such cases movement of the interface, in which one region grows at the expense of the other, can produce a decrease in the overall energy of the body and thereby give rise to a force on the interface, expressed by Eq. (5.38). Such a force can occur under a variety of circumstances. For example, during the recrystallization of a plastically deformed crystalline body, relatively strain-free crystals form and then grow into the surrounding plastically deformed and dislocated matrix. Here, the reduction in energy that occurs as the strain-free crystals grow at the expense of the dislocated matrix produces outward forces on the interfaces bounding the strain-free crystals. In other situations, elastic fields that differ across interfaces, and therefore generate a mechanical interface force, often occur in polycrystalline materials in the form of compatibility stresses, arising as a result of elastic anisotropy, anisotropic thermal expansion, and differing modes of plastic deformation in the crystals adjoining the interfaces (e.g., Sutton and Balluffi, 2006).

Contents of book

An introduction to the use of anisotropic linear elasticity in determining the static elastic properties of defects in crystals is presented. The defects possess different dimensionalities and span the defect spectrum. They include:

• Point defects (vacancies, self-interstitials, solute atoms, and small clusters of these species),

• Line defects (dislocations),

• Planar defects (homophase and heterophase interfaces),

• Volume defects (inhomogeneities and inclusions).

To avoid confusion, an inclusion is defined as a misfitting region embedded within a larger constraining matrix body, and, therefore, acts as a source of stress. It may be either homogeneous (if it possesses the same elastic properties as the matrix) or inhomogeneous (if its elastic properties differ). On the other hand, an inhomogeneity is simply an embedded region with different elastic constants but no misfit.

Introduction

The basic elements of anisotropic linear elasticity are presented in concise form. First, the deformation of an elastically strained body is described in terms of the local displacements, strains, and rotations that occur throughout the body. Requirements on the strains that ensure compatibility of the medium are then described. Next, the forces acting throughout the body are described in terms of surface tractions, body forces, and stresses. Conditions for mechanical equilibrium are derived. The stresses and strains are then linearly coupled via elastic constants, and various stress–strain relationships are derived. Finally, the energy stored in an elastically strained medium is formulated. Elements of the theory for the special case when the medium is elastically isotropic are included, along with several formulations of additional elastic quantities required for treating crystal defects.

References include: Love (1944); Sokolnikoff (1946); Muskhelishvili (1953); Nye (1957); Lekhnitskii (1963); Bacon, Barnett and Scattergood (1979b); Soutas-Little (1999); Hetnarski and Ignaczak (2004) and Asaro and Lubarda (2006).

Introduction

To formulate interactions between defects and stress it is useful to classify the various types of defect and stress that will be of concern to us. Of the defects considered in this book, inclusions, point defects, dislocations, and various interfaces containing discrete intrinsic dislocations are sources of stress, and they therefore interact elastically with imposed stress, which may be internal stress due, for example, to the presence of other defects, or applied stress, due to forces applied to the body.

On the other hand, when a defect source of stress lies in a finite region bounded by interfaces, an image stress is generated, as described in Section 3.8, which then interacts with the defect. Thus, the interface acts, in a sense, as the source of a stress that interacts with the defect. In addition, an inhomogeneity, which by itself is not a source of stress, causes a perturbation of an imposed stress field, which, in turn, interacts with the imposed stress. An inhomogeneity may therefore be regarded as the indirect source of a stress that interacts with an imposed stress.

Introduction

The fundamentals necessary to analyze the interaction energies and forces thatcan occur between the various defects treated in this book are now in place. Theelastic fields produced by the defects, and the interactions between the defectsand various elastic fields, have been formulated in previous chapters.

The general procedure for determining the interaction between two defects is toobtain the elastic field due to one defect at the location of the other and thento determine the interaction of the latter defect with this field. Since thenumber of possible defect–defect interactions is far too large toconsider comprehensively, this final chapter is restricted to a limited numberof representative interactions, which serve to demonstrate basic methods.

Point defect–point defect interactions

First, a general formulation is given of the interaction energy between two pointdefects that are represented by force multipoles, as in Chapters 10 and 11.Following this, the interaction between two point defects, each possessing cubicdefect symmetry, is analyzed for the case of an isotropic system.

General formulation

The interaction energy between a single point defect and a general elastic fieldhas been formulated in Chapter 11 (see Eqs. (11.1)–(11.4)), and the displacementfield due to a force multipole is given by Eq. (10.10). The interaction energybetween two point defects, i.e., D1 and D2, can therefore be obtained byimagining that D2 is created at the origin in the presence of D1 at theposition, x, and using the above results for the interaction energy and requiredelastic field.