Modern civilization would not be possible without iron and steel. Steel is an essential component of all machinery used for manufacture of all our goods. The words iron and steel have come to suggest strength as evident in the following terms: iron willed, iron fisted, iron clad, iron curtain, and pumping iron. A steely glance is a stern look. A heart of steel implies a very hard demeanor. The Russian dictator Joseph Stalin (which means “steel” in Russian) chose that name to invoke fear in his subordinates.

This book is intended both as a resource for engineers and as an introduction to the layman to our most important metal system. The first few chapters cover the history and refining of iron and steel; the rest of the book covers physical properties and physical metallurgy.

I have drawn heavily on material from Physical Metallurgy of Steels by W. C. Leslie and Steel Metallurgy for the Nonmetallurgist by J. D. Verhoeven. However, this book includes material not covered in either of those.

Professors Robert Pehlke, Ronald Gibala, John Keough, and Paul Trojan were very helpful. Kathy Hayrynen supplied a number of micrographs.

The reader is assumed to have had a course in materials science and to be familiar with phase diagrams, Fick‘s laws of diffusion, and the concept of free energy.

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.

Introduction

The main methods for treating the defect elasticity problems considered in this book are introduced. The chapter begins by reviewing the requirements that any solution for a defect elasticity problem must satisfy. A basic differential equation for the displacements whose solutions automatically satisfy these requirements is then formulated, and useful methods of solving it are described. These include its direct solution and various formalisms that expedite its solution under different conditions, including the Fourier transform approach, the Green's function method, and the sextic and integral formalisms.

Following this, the transformation strain method, which is applicable for many defect problems, is described. Here, the defect is introduced in the form of an appropriate stress-free transformation strain. The resulting elastic field is then found by methods involving the use of Green's functions or Fourier transforms.

Next, the stress function method for solving problems is introduced, and the Airy stress function method, which is applicable when plane strain conditions prevail, is described. Finally, the problem of finding solutions for defects in finite homogeneous regions bounded by interfaces, rather than in infinite regions, is outlined. The methods by which the boundary conditions at the interfaces, which exist in such cases, can be satisfied by the method of images and use of appropriate Green’s functions are described.

Introduction

In Chapter 10, the force multipole and small inhomogeneous models for point defects in infinite homogeneous regions were described. The interaction of inhomogeneous inclusions with various types of stresses has been treated in Chapters 7 and 8. Consequently, there is no need in this chapter to devote further attention to interactions between point defects and stress in terms of the small inhomogeneous inclusion model. Attention is therefore focused on these interactions in terms of the force multipole model.

Section 11.2 includes a treatment of the interaction between a single point defect (represented by a force multipole) and a general internal or applied stress. In Section 11.3, the force multipole model is used to investigate the volume change due to a single point defect in a finite body possessing a traction-free surface, where the defect image stress can play an important role. Then, with Section 11.3 in hand, Section 11.4 takes up the particularly interesting problem of the behavior of a finite traction-free body filled with a statistically uniform distribution of point defects, which may, for example, be vacancies in thermal equilibrium or solute atoms dispersed throughout a solid solution. Analyses are given of the volume changes, macroscopic shape changes and lattice parameter changes (as measured by X-ray diffraction) produced by the defects. A demonstration is given of the intuitive result that a uniform concentration of point defects in a finite body with a traction-free surface produces a uniform average strain throughout the body. If the centers are spherically symmetric and act as centers of pure dilatation, the macroscopic body either expands or contracts uniformly throughout the body (depending upon whether the centers possess positive or negative strengths) with no change in body shape. If the centers possess lower symmetry, the body again expands or contracts uniformly but undergoes a macroscopic shape change, reflecting the symmetry of the defects.