As can be seen in Table 9.1, many fabrication routes are now available by which a reinforcement can be incorporated into a metal matrix. It is important to note from the outset that making the right choice of fabrication procedure is just as important in terms of the microstructure and performance of a component, as it is for its commercial viability. However, before looking in detail at the various processing options, it is worthwhile dwelling for a moment on selection of the reinforcement. Clearly, the size, shape and strength of the reinforcing particles or fibres is of central importance. Often the choice between the continuous and discontinuous options is relatively straightforward, both in terms of performance and processing cost. However, within each category there exist wide variations in reinforcement size and morphology.

Measurement of Young's modulus

Basic principle and capability

The aim is to characterise the relationship between an applied load and a material's elastic (reversible) variation in strain. The available experimental approaches can be split into two broad classes; mechanical or static methods, and ultrasonic or dynamic methods. Their measurement capability in terms of accuracy is similar (±0.5 GPa).

Of all the different means of characterising stiffness, the following measures are perhaps the most popular.

1. (1) the tangent to the initial stress-strain slope

2. (2) the tangent, subsequent to prestraining, of the initial slope on reloading

3. (3) the tangent, subsequent to prestraining, of the initial slope of the unloading curve

4. (4) the tangent to the reloading curve, subsequent to prestraining and low stress amplitude cycling about zero until stress/strain hysteresis is negligible

5. (5) the speed of ultrasonic waves through the medium

6. (6) the frequency of resonant standing vibrations, which are related to the dimensions of the specimen as well as to the stiffness of the material.

Internal stresses are commonplace in almost any material which is mechanically inhomogeneous. Typically, their magnitude varies according to the degree of inhomogeneity: for an externally loaded polycrystalline cubic metal, differently oriented crystallites will be stressed to different extents, but these differences are usually quite small. For a composite, consisting of two distinct constituents with different stiffnesses, these disparities in stress will commonly be much larger. Internal stresses arise as a result of some kind of misfit between the shapes of the constituents (matrix and reinforcement, i.e. fibre, whisker or particle). Such a misfit could arise from a temperature change, but a closely related situation is created during mechanical loading - when a stiff inclusion tends to deform less than the surrounding matrix.

Failure processes in long fibre MMCs

Failure of laminae

Under an arbitrary stress state, a lamina (a sheet containing unidirectionally aligned long fibres) can fail in one of three ways, as depicted in Fig. 7.1.

We turn now to a special kind of fracture, that produced by the contact of a hard indenter on a brittle surface. Indentation fracture, so-called, is of historical as well as practical interest. It dates back to 1880 with the celebrated studies of Hertz (see Hertz 1896) on conical fractures at elastic contacts between curved glass surfaces. The fully developed Hertzian cone crack is the prototypical stable fracture in brittle solids. Shortly after Hertz, Auerbach (1891) showed empirically that the critical load to initiate cone fractures in flat specimens is proportional to the radius of the indenting sphere, Pc ∝ r. For 75 years, ‘Auerbach's law’ remained one of the great paradoxes in fracture theory: the notion that fracture should initiate when the maximum tensile stress in the Hertzian field just equals the bulk strength of the material implies an alternative relation, Pc ∝ r2. Resolution of the paradox awaited the advent of modern-day fracture mechanics (Frank & Lawn 1967). More recently, radial–median cracks produced in elastic–plastic fields by diamond pyramid indenters have assumed centre stage. The radial crack system is now arguably the most widely used of all fracture testing methodologies in the mechanical evaluation of brittle materials.

Irwin's generalisation of the Griffith concept, as outlined in the previous chapter, provides us with a powerful tool for handling the mechanics of fracture. In particular, by balancing mechanical energy released against surface energy gained we have a thermodynamically sound criterion for predicting when an ideally brittle crack will extend. But the Irwin mechanics tell us nothing as to how that crack extends. Whenever we have had occasion to refer to events at the tip the description has proved totally inadequate. The singularities in the continuum linear elastic solutions simply cannot be reconciled with any physically realistic local rupture process. Assuming the first law of thermodynamics to be beyond question, it is clear that some vital element is missing in the fracture mechanics: it is not the Griffith energy balance that is at issue here, but rather the mechanism by which this balance is effected.

In this chapter we question the adequacy of the linear continuum representation of matter. Real solids tend to a maximum in the intrinsic stress–strain characteristic. This is true even of perfectly brittle solids which fracture by bond rupture. In addition, crack growth can be accompanied by deformation processes in the near-tip field. Such deformation can be highly dissipative. If there is to be any proper basis for understanding the behaviour of ‘tough’ ceramics and other brittle materials for structural applications we must include provision for nonlinear and irreversible elements in the equilibrium fracture mechanics.

All discussions on the strength of brittle solids in the preceding chapters are predicated on the existence of flaws. But what are the underlying origin and nature of such flaws? What flaw geometry and material parameters govern the micromechanics of initiation into the ultimate well-developed crack? How do persistent nucleation forces influence the stability of flaws during initiation and subsequent development?

The decades following 1920 witnessed a preoccupation of silicate-glass researchers with the search for ‘Griffith’ flaws. But observing such flaws by any direct means proved elusive. Typically, characteristic flaw dimensions in these and other homogeneous covalent–ionic solids like monocrystalline silicon, sapphire and quartz range from 1 nm (pristine fibres and whiskers) to 1 μm (aged, as-handled solids), and usually occur at the surface. As we indicated in sect. 1.6, planar defects on this scale are likely to lie below the limit of detectability by optical means. More recently, with the advent of modern heterogeneous polycrystalline ceramics, we have seen a shift in focus to microstructural fabrication flaws, the existence and character of which can be all too obvious. These latter range from 1 μm (high-density, fine-grained, polished materials) to 1 mm and above (refractories, concrete), and occur in both the surface and the bulk.

The small scale of typical flaws highlights the sensitivity of brittle solids to seemingly innocuous extraneous events and treatments (recall the one-hundred-fold reduction in strength on aging glass fibres, sect. 1.6).

Until now we have approached crack propagation from the continuum viewpoint. Nonetheless, repeated allusions have been made in chapters 3 and 5 to the fundamental limitations of any such approach that disregards the atomic structure of solids. There we argued for the incorporation of a lattice-plane range parameter as a critical scaling dimension in the brittle crack description. We noted that the Barenblatt cohesion-zone model avoids reference to the atomic structure by resorting to the Irwin slit description of cracks; yet estimates of the critical crack-opening dimensions using this same model confirm that the intrinsic separation process indeed operates at the atomic level. The Elliot lattice half-space model of sect. 3.3.2 represents one attempt to incorporate an essential element of discreteness. The phenomenological kinetic models of sect. 5.5, with their presumption of energy barriers, represent another. However, those models are at the very least quasi-continuous. In brittle fracture, as in any thermodynamic process, the final answers must be sought at the atomic or molecular level.

On the other hand, while an atomistic approach provides greater physical insight into the crack problem, it inevitably involves greater mathematical complexity. Classically, solids may be represented as manybody assemblages of point masses (atoms) linked by springs (bonds). We will see that the mass–spring representation can lead us to a deeper understanding of brittle cracks. But even this representation is oversimplistic. In some cases, particularly when the crack interacts with environmental species, it is necessary to consider atoms as elastic spheres rather than point masses, to allow properly for molecular size effects.

We have thus far dealt with the resistance to crack propagation at opposite extremes of material representation, continuum solid and atomic lattice. It is now appropriate to investigate the problem at an intermediate level, that of the microstructure. By ‘microstructure’ we mean the compositional configuration of discrete structural ‘defects’: voids, inclusions, secondphase particles (volume defects); secondary crack surfaces, grain boundaries, stacking faults, twin or phase boundaries (surface defects); dislocations (line defects). It is principally at this intermediate level that significant improvements in the mechanical properties of traditional brittle polycrystalline ceramics (cf. table 3.1) may be realised. By tailoring the microstructure it is possible to introduce an interactive defect structure that acts as an effective restraint on crack propagation and thus enhances the material toughness.

In this chapter we examine some of these ‘toughening’ interactions. We identify two classes of restraint. The first involves purely geometrical processes, deflections along or across weak interfaces, etc. The responsible microstructural elements may be regarded as ‘transitory obstacles’, in the sense that their impeding influence lasts only for the duration of crackfront intersection. Because of their ephemeral nature such interactions are relatively ineffective as sources of toughening, accounting at very most for increases of a factor of four in crack-resistance energy R or, equivalently, a factor of two in toughness T.

The second class of restraint comprises shielding processes. The critical interactions occur away from the tip, within a ‘frontal zone’ ahead or at a ‘bridged interface’ behind.

Thus far we have considered only static crack systems. Now, if an unbalanced force acts on any volume elements within a cracked body, that element will be accelerated, and will thereby acquire kinetic energy. The system is then a dynamic one, and the static equilibrium conditions of Griffith and Irwin–Orowan no longer apply. In certain instances, such as when stable cracks are made to grow slowly in controlled fracture surface energy tests, the kinetic energy component may be relatively insignificant in comparison with the system mechanical energy. The system may then be regarded as quasi-static, insofar as the static solutions describe the critical requirements for crack extension to sufficient accuracy.

There are two ways in which a crack system may become dynamic. The first arises when a crack reaches a point of instability in its length: the system acquires kinetic energy by virtue of the inertia of the material surrounding the rapidly separating crack walls. Such a dynamical state may be realised even under fixed loading conditions. A ‘running’ crack is typified by a rapid acceleration toward a terminal velocity governed by the speed of elastic waves. In most (but not all) test specimens the running crack divides the material into two or more fragments. The second type of dynamical state arises when the applied loading is subject to a rapid time variation, as in impact loading. In this case response may be limited by the characteristic duration of the loading pulse.

Most materials show a tendency to fracture when stressed beyond some critical level. This fact was appreciated well enough by nineteenth century structural engineers, and to them it must have seemed reasonable to suppose strength to be a material property. After all, it had long been established that the stress response of materials within the elastic limit could be specified completely in terms of characteristic elastic constants. Thus arose the premise of a ‘critical applied stress’, and this provided the basis of the first theories of fracture. The idea of a well-defined stress limit was (and remains) particularly attractive in engineering design; one simply had to ensure that the maximum stress level in a given structural component did not exceed this limit.

However, as knowledge from structural failures accumulated, the universal validity of the critical applied stress thesis became more suspect. The fracture strength of a given material was not, in general, highly reproducible, in the more brittle materials fluctuating by as much as an order of magnitude. Changes in test conditions, e.g. temperature, chemical environment, load rate, etc., resulted in further, systematic variations in strengths. Moreover, different material types appeared to fracture in radically different ways: for instance, glasses behaved elastically up to the critical point, there to fail suddenly under the action of a tensile stress component, while many metallic solids deformed extensively by plastic flow prior to rupture under shear.

In developing the Griffith–Irwin fracture mechanics of chapters 2 and 3 we presumed that equilibrium brittle fracture properties are governed by invacuo surface energies. In practice, most fractures take place in a chemically interactive environment. The effects of environment on crack propagation can be strongly detrimental. One of the most distinctive manifestations is a rate-dependent growth, even at sustained applied stresses well below the ‘inert strength’, with velocities sufficiently high as to be clearly measurable but too low as to be considered inertial. We use the term kinetics (typical velocity range ≈ m s-1 down to and below nm s-1) to distinguish from true dynamics (velocity range ≈ m s-1 to km s-1). Kinetic crack propagation (alternatively referred to as ‘slow’ or ‘subcritical’ crack growth) is notable for its extreme sensitivity to applied load, specifically to G and K. It tends also to depend on concentration of environmental species, temperature, and other extraneous variables.

How does kinetic behaviour reconcile with the Griffith concept? A crack growing at constant velocity at a specific driving force implies a condition of steady state, whereas Griffith deals explicitly with equilibrium states. Experimentally, it is found that velocity diminishes with decreasing G or K until, at some threshold, motion ceases. On unloading the system still further the crack closes up and, under favourable conditions, heals, even in the presence of environmental species.

This book is a restructured version of a first edition published in 1975. As before, the objective is a text for higher degree students in materials science and researchers concerned with the strength and toughness of brittle solids. More specifically, the aim is to present fracture mechanics in the context of the ‘materials revolution’, particularly in ceramics, that is now upon us. Thus whereas some chapters from the original are barely changed, most are drastically rewritten, and still others are entirely new.

Our focus, therefore, is ‘brittle ceramics’. By brittle, we mean cracks of atomic sharpness that propagate essentially by bond rupture. By ceramics, we mean covalent–ionic materials of various persuasions, including glasses, polycrystalline aggregates, minerals, and even composites. Since 1975, our knowledge of structural ceramics has equalled, some would insist surpassed, that of metals and polymers. But it is brittleness that remains the singular limiting factor in the design of ceramic components. If one is to overcome this limitation, it is necessary first to understand the underlying mechanics and micromechanics of crack initiation and propagation. Prominent among improvements in this understanding have been a continuing evolution in the theories of continuum fracture mechanics and new conceptions of fundamental crack-tip laws. Most significant, however, is the advent of ‘microstructural shielding’ processes, as manifested in the so-called crack-resistance- or toughness-curve, with far-reaching consequences in relation to strength and toughness. This developing area promises to revolutionise traditional attitudes toward properties design and processing strategies for ceramics.

In turning to engineering aspects of brittle fracture the focus shifts from toughness to strength. However, structural design is concerned not just with strength but also with reliability. How may we guarantee the strength of a brittle component? Or, more realistically, how well may we guarantee the strength and for how long? Much of our current methodology for quantifying the reliability of intrinsically brittle materials can be traced back to the endeavours of Evans, Wiederhorn, Davidge, Ritter and others in the early 1970s to address such questions in relation to fracture mechanics, specifically in the context of failure from Griffith flaws.

Reliability inevitably embodies a probabilistic element. Designers reconcile themselves to the notion of a ‘risk’ of failure over a ‘lifetime’, acceptable values of these quantities depending on specific applications. The classical form in which risk is expressed for any mechanical structure (including, interestingly, the human body) is the ‘bathtub’ curve of fig. 10.1, representing the ‘hazard’ (mortality) rate as a function of time. Such a curve is certainly representative of the strength characteristics of ceramic components: component failure is most frequent during manufacture and initial screening or after prolonged wear and tear in stringent service environments.

It is in the recognition that variability in strength and lifetime is unavoidable that ‘flaw statistics’ enters as a central element of reliability analysis in brittle materials. One regards individual flaws as members of some determinable distribution.