Until now we have studied problems of homogeneous spin tunneling in small particles, in which we assumed the magnetic moment (or moments of sublattices) to be independent of spatial coordinates. In that case, as we have seen, the tunneling exponent is always proportional to the total tunneling spin. The particle, therefore, must be sufficiently small to provide a significant tunneling probability. When we switch to considering tunneling in bulk magnets, the same kind of argument can be made that suggests that magnetic tunneling can only occur in a small volume comparable to that in single-domain particles. A significant difference, however, is that a local tunneling event in a bulk magnet can trigger instability on a much greater scale, leading to really macroscopic consequences. Examples are quantum nucleation of magnetic domains and quantum depinning of domain walls, which are studied below. At the end of this chapter we will also briefly discuss tunneling of magnetic flux lines in superconductors.

Quantum nucleation of magnetic domains

Nucleation of magnetic bubbles in thin ferromagnetic films is an interesting fundamental problem that may have great potential for experimental studies. Consider a thin film of ferromagnetic material with strong perpendicular anisotropy, which is uniformly magnetized in the positive Z-direction, Fig. 4.1. The magnetic field is applied in the negative Z-direction, which favors the reversal of the magnetization. The reversal occurs via the nucleation of a critical bubble that then grows in size until the magnetization of the whole film has become aligned with the direction of the field.

To ensure the use of mechanically reliable ceramic components in technological applications, it is critical to establish an approach that can be incorporated into the engineering design process. In this chapter, the emphasis will be on the use of strength data in designing reliable ceramic components. After briefly describing strength measurement techniques, the use of failure statistics will be introduced. Finally, the time dependence of strength will be considered. As seen in the previous chapter, cracks can grow sub-critically in ceramics, causing strength to decrease with service time.

Strength testing

A common technique for the strength determination of a ceramic component is the bend (flexure) test. This approach has been popular as it involves simple specimen shapes. This is particularly useful when the specimen is machined from larger production units. The loading configuration is usually either three- or four-point bending and American Society for Testing and Materials (ASTM) standards are available for both approaches (ASTM C 1161, 1990). Assuming the material fails in tension, the bend strength is determined from the maximum applied tensile stress, using Eq. (4.6). The bending configuration is statically indeterminate and, thus, the stress equations assume that the material is linearly elastic. Four-point is often preferred over three-point bending as the specimen has a larger region under the maximum stress.

It is important to be able to solve elasticity problems for a variety of different loading geometries. The equations of elasticity do not possess unique solutions unless residual stresses are ignored but fortunately the principle of superposition allows one to deal with these as a separate issue. This principle can also be used to build up solutions of complicated problems by superposing a set of simpler problems. In this chapter, some simple elastic stress distributions for linearly elastic isotropic bodies will be shown and discussed. In these solutions there will be simplifying features that allows the general equations of elasticity to be bypassed.

An immediate difficulty in studying loaded bodies is that external forces are often applied to the body in complicated ways, e.g., through bolts, pins, collars, etc. This means that stresses can be complex in the loading region; they may vary sharply, especially if there are changes in the contours of the body. Fortunately, these stresses are generally localized to the region of contact and will not change the stress and strain distribution away from the contact region. This effect is expressed by St Venant's Principle, ‘If the forces acting on a small part of a body are replaced by a statically equivalent set of forces (i.e., with the same resultant force and couple) acting on the same area, the stress state will be changed negligibly at large distances compared to this area.’

The aim of this book is to provide a text for a senior undergraduate course on the mechanical behavior of ceramics. There are, however, some advanced sections that would allow the book to be used at the graduate level (marked ††). The format of the book owes much to the text, Mechanical Properties of Matter, by A. H. Cottrell, which helped me through graduate school. In teaching a course in this area, it has always been frustrating that there are so few texts aimed primarily at ceramics. There is often the concern of discerning whether ideas applied to other materials could also be used to understand ceramic materials. I have also been fortunate in being involved in the field of structural ceramics at a time it has undergone remarkable developments and I have tried to incorporate my interpretation of these recent advances into the text.

I would be amiss in not acknowledging the support I have received in undertaking this project. I owe much to Pat Nicholson, Dave Embury and Dick Hoagland, who patiently introduced me to this field of research and to Tom Wheat, who taught me about the processing of ceramics. I am particularly grateful to Fred Lange, who took a chance on me and became my mentor.

In describing the mechanical properties of materials, one is interested in understanding the response of the materials to force. For example, consider the forces that are exerted on materials as we walk around. The forces arise because our bodies are being acted on by gravity and this force is acting on each particle of our body. A force that acts on every particle of a body, animate or not, is known as a body force. The force produced by our bodies is then transmitted to the floor through our feet. As the force is being transmitted via a surface, it is known as a surface force. Now, let us consider what is happening to the floor as we transmit this force. In general, we do not notice much of a reaction but, following Newton's Third Law, we know for every action there is an equal and opposite reaction. In a way this is rather remarkable, as it indicates the floor is pressing back on our feet with exactly the same force as that caused by our weight. If the reactive force was less, we would sink and if it was too high, we would rise. To understand the mechanical properties of materials, it is important to understand how this reaction arises and as materials scientists we are interested in determining whether this reaction can be controlled.

In the last chapter, the formal description of linear elasticity was introduced. It was shown that knowledge of the elastic constants for a particular material allows one to describe the strains produced by any arbitrary state of stress. In materials science one is often interested in ‘controlling’ a material property and, thus, this chapter is concerned with the influence of structure on the elastic constants. At the most basic level, the elastic constants reflect the ease of deformation of the atomic bonds but it will be shown that other levels of structure can be very important, especially with the use of composite materials.

Relationship of elastic constants to interatomic potential

In Section 2.1, linear elasticity was considered from the perspective of a single atomic bond. It was shown that the elastic constant was related to the shape of the interatomic potential, notably the curvature of the potential in the vicinity of the equilibrium spacing. For example, consider the interatomic potentials shown in Fig. 3.1 that are typical of the extremes for (a) ionic and (b) covalent ceramics. The strong directional bonding associated with covalency leads to a deep potential well. For atomic interactions acting over similar distances this would lead to a sharper curvature at the potential minimum, compared to the purely ionic case.

Inelastic deformation can occur in crystalline materials by plastic ‘flow’. This behavior can lead to large permanent strains, in some cases, at rapid strain rates. In spite of the large strains, the materials retain crystallinity during the deformation process. Surface observations on single crystals often show the presence of lines and steps, such that it appears one portion of the crystal has slipped over another, as shown schematically in Fig. 6.1 (a). The slip occurs on specific crystallographic planes in well-defined directions. Clearly, it is important to understand the mechanisms involved in such deformations and identify structural means to control this process. Permanent deformation can also be accomplished by twinning (Fig. 6.1(b)) but the emphasis in this book will be on plastic deformation by glide (slip).

Theoretical shear strength

Figure 6.2 shows one possible way in which crystal glide could occur, with one plane of atoms being sheared past an adjacent plane. In the perfect crystal, the atoms are assumed to lie directly above each other with a planar spacing d. Clearly, as the atoms are displaced, the stress will rise and pass through a maximum. Once the displacement u reaches a value of b/2, i.e., at the mid-shear position, the atoms would be equally as likely to complete the displacement (u=b) as to return to their original positions.

In the last 25 years there has been a strong movement to use ceramics in new technological applications and a key facet of this work has been directed at understanding the mechanical behavior of these materials. First, let us consider the various technological functions of ceramics as shown in Table 1.1. The diverse properties of ceramics are not always appreciated. For structural functions, adequate mechanical properties are of prime importance. Ceramic materials that are considered for these applications are termed structural ceramics. In some cases, such as engine parts, the choice is based on their high-temperature stability and corrosion resistance. These factors imply the engine temperature could be raised, making the overall performance more efficient. Unfortunately, ceramics can be brittle, failing in a sudden and catastrophic manner. Consequently, there has been a strong emphasis on understanding the mechanical properties of ceramics and on improving their strength, toughness and contact-damage resistance. Indeed, it is appropriate to state that there has been a revolution in the understanding of these properties and the associated research has led to the discovery of new classes of structural ceramic materials.

It is important to realize that mechanical properties can also be critical in non-structural applications. For example, in the design of the thermal protection system of the space shuttle, highly porous, fibrous silica tiles are used.