Introduction

Consider a body of nominally elastic material that contains a crack. For the time being, the idealized crack is assumed to have no thickness, that is, in the absence of applied loads the two faces of the crack coincide with the same surface in space. The edge of the crack is a smooth simple space curve, either a closed curve for an internal crack or an open curve intersecting the boundary of the body at two points for an edge or surface crack.

Under the action of applied loads on the boundary of the body or on the crack faces, the crack edge is a potential site for stress concentration. If the rate at which loads are applied is sufficiently small, in some sense, then the internal stress field is essentially an equilibrium field. The body of knowledge that has been developed for describing the relationships between crack tip fields and the loads applied to a solid of specified configuration is Linear Elastic Fracture Mechanics (LEFM). This a well-developed branch of engineering science which forms the basis for results to be discussed in this chapter and the next.

If loads are rapidly applied to a cracked solid, on the other hand, the internal stress field is not, in general, an equilibrium field and inertial effects must be taken into account. There is no unambiguous criterion for deciding whether or not loads are “rapidly” applied in a particular situation.

This book is an outgrowth of my involvement in the field of dynamic fracture mechanics over a period of nearly twenty years. This subbranch of fracture mechanics has been wonderfully rich in scope and diversity, attracting the attention of both researchers and practitioners with backgrounds in the mechanics of solids, applied mathematics, structural engineering, materials science, and earth science. A wide range of analytical, experimental, and computational methods have been brought to bear on the area. Overall, the field of dynamic fracture is highly interdisciplinary, it provides a wealth of challenging fundamental issues for study, and new results have the potential for immediate practical application. In my view, this combination of characteristics accounts for its continued vitality.

I have written this book in an effort to summarize the current state of the mechanics of dynamic fracture. The emphasis is on fundamental concepts, the development of mathematical models of phenomena which are dominated by mechanical features, and the analysis of these models. Mathematical problems which are representative of the problem classes that comprise the area are stated formally, and they are also described in common language in an effort to make their features clear. These problems are solved using mathematical methods that are developed to the degree required to make the presentation more or less self-contained. Experimental and computational approaches have been of central importance in this field, and relevant results are cited in the course of discussion.

Introduction

The field of fracture mechanics is concerned with the quantitative description of the mechanical state of a deformable body containing a crack or cracks, with a view toward characterizing and measuring the resistance of materials to crack growth. The process of describing the mechanical state of a particular system is tantamount to devising a mathematical model of it, and then drawing inferences from the model by applying methods of mathematical or numerical analysis. The mathematical model typically consists of an idealized description of the geometrical configuration of the deforriiable body, an empirical relationship between internal stress and deformation, and the pertinent balance laws of physics dealing with mechanical quantities. For a given physical system, modeling can usually be done at different levels of sophistication and detail. For example, a particular material may be idealized as being elastic for some purposes but elastic-plastic for other purposes, or a particular body may be idealized as a one-dimensional structure in one case but as a three-dimensional structure in another case. It should be noted that the results of most significance for the field have not always been derived from the most sophisticated and detailed models.

A question of central importance in the development of a fracture mechanics theory is the following. Is there any particular feature of the mechanical state of a cracked solid that can be interpreted as a “driving force” acting on the crack, that is, an effect that is correlated with a tendency for the crack to extend?

Introduction

The focus in this chapter, as well as in the following chapter, is on analytical models of crack growth phenomena based on nominally elastic material response. The analysis of Chapter 4 provides information on the nature of crack tip fields during rapid crack growth for several categories of material response, and parameters that characterize the strength or intensity of these fields are also identified. A main purpose in formulating and solving boundary value problems concerned with crack propagation is to determine the dependence of the crack tip field characterizing parameters on the applied loading and on the configuration of the body.

For any particular crack growth process, it is often the case that its analytical modeling can be pursued at several scales of observation. For example, consider the case of crack growth in an engineering structure. The full structure can be analyzed to determine the stress resultants that are imposed at the fracturing section. Under suitable circumstances, these stress resultants can, in turn, be viewed as the applied loads in a two-dimensional plane stress or plane strain problem from which the dynamic stress intensity factor can be determined. At a finer scale of observation, the elastic stress intensity factor represents the applied loading in an analysis of nonlinear processes very close to the crack edge. By focusing on dominant effects at adjacent levels of observation in this way, various aspects of dynamic fracture phenomena can be analyzed.

Introduction

In Section 2.1, it was argued that the fields at an interior point on the edge of a stationary crack in an elastic solid are asymptotically two-dimensional. The convention of resolving the local deformation field into the in-plane opening mode (mode I), the in-plane shearing mode (mode II), and the antiplane shearing mode (mode III) was adopted. It was pointed out that the components of stress have an inverse square root dependence on normal distance from the crack edge and a characteristic variation with angular position around the edge. This variation is specified through the functions Σij(·) for each mode in Section 2.1. These general features are common to all configurations and all loading conditions. The influence of configuration and loading are included in the asymptotic description of stress only through scalar multipliers, one for each mode, which are the elastic stress intensity factors. The role of the stress intensity factor as a crack tip field characterizing parameter has been discussed in Sections 2.1 and 3.6.

The existence of similar universal fields for growing cracks is considered next. The same convention for categorizing local deformation modes will be followed. Except where noted explicitly to the contrary, the asymptotic crack tip fields will be determined for variable crack tip speed. Mode III will be considered first, because it is the simplest case to analyze, and the in-plane modes will be analyzed subsequently.

Introduction

Analytical methods based on the work done by applied loads and the changes in the energy of a system that accompany a real or virtual crack advance have been of central importance in the development of fracture mechanics. These methods have provided a degree of unification of seemingly diverse ideas in fracture mechanics, and they have led to procedures of enormous practical significance for the characterization of the fracture behavior of materials. In addition, some of the most elegant theoretical analyses in the field have been those associated with energy methods. In this chapter, energy concepts that are particularly relevant to the study of dynamic fracture processes are considered.

The importance of the variation of energy measures during crack growth was recognized by Griffith (1920) in his pioneering discussion of brittle fracture, as outlined in Section 1.1.2. The extension of a crack requires the formation of new surface, he reasoned, with its associated surface energy. Consequently, a crack in a brittle solid should advance when the reduction of the total potential energy of the body during a small amount of crack advance equals the surface energy of the new surface thereby created. For an elastic body containing a crack, the negative of the rate of change of total potential energy with respect to crack dimension is called the energy release rate. This quantity, which is usually denoted by the symbol G, is a function of crack size, in general.

Introduction

The preceding chapters address interactions between colloidal particles dispersed in pure liquid or electrolyte solution. The hydrodynamic and dispersion forces depend only on the bulk properties of the individual phases, i.e. the viscosity and the dielectric permittivities. Electrostatic forces arising from the surface charges, however, are accompanied by free electrolyte. The associated electric fields distribute these additional species non-uniformly in the surrounding fluid, thereby producing a spatially varying osmotic pressure. Electrostatic interactions between particles alter these ion distributions, affecting the electric and pressure fields and generating an interparticle force.

We now consider another component commonly present in colloidal systems, soluble polymer. In many ways, the phenomena and the theoretical treatment resemble those for electrostatics. The interactions between polymer and particle generate non-uniform distributions of polymer throughout the solution. Particle–particle interactions alter this equilibrium distribution, producing a force whose sign and magnitude depend on the nature of the particle–polymer interaction. The major difference from the ionic solutions lies in the internal degrees of freedom of the polymer, which necessitate detailed consideration of the solution thermodynamics.

The reasons for adding soluble polymer to colloidal dispersions are several. The earliest known role, as stabilizer, aids or preserves the dispersion through adsorption of the macromolecule onto the surfaces of the particles to produce a strongly repulsive interaction. Homopolymers achieve this by adsorbing to particles non-specifically at multiple points along their backbone, while block or graft copolymers adsorb irreversibly at one end with the other remaining in solution (Napper, 1983).

Introduction

In the preceding chapters we have examined the response of colloidal particles to interactions with one another in a quiescent fluid, to interactions with large collectors while being convected by the fluid, and to imposed forces due to electric fields, gravity, or concentration gradients. In each case, equilibrium or non-equilibrium, static or dynamic, the interparticle forces and the resulting suspension microstructure play key roles. Now we consider the stresses and the non-equilibrium microstructure generated in a flowing suspension when the velocity varies spatially on a scale large with respect to the size of the particles.

A Newtonian incompressible liquid is characterized by a linear relation between the stress tensor and the rate-of-strain tensor, with the constant of proportionality being the viscosity. Polymeric liquids are well known for their non-Newtonian behavior including shear-rate-dependent viscosities, elasticity manifested in recoil upon the cessation of flow, solid-like fracture during extrusion, and a variety of secondary flow phenomena. Colloidal suspensions also depart from Newtonian behavior. They often behave as solids requiring a finite stress, the yield stress, before deforming continuously as a liquid. The contrast with the polymeric liquids reflects the fundamentally different microstructures. Both microstructures deform under stress, but macromolecular systems can recover from strains of several hundred per cent because the restoring force increases with the degree of deformation. The interparticle forces governing the microstructure in colloidal dispersions generally have a short range and the magnitude decreases with increasing separation, providing no mechanism for recovery beyond strains of a few per cent.

Introduction

Microscopic observations of colloidal particles in the nineteenth century revealed their tendency to form persistent aggregates through collisions induced by Brownian motion, clearly indicating an attractive interparticle force. Identification of its origin, however, awaited the quantitative descriptions of van der Waals forces between molecules developed in the 1920s (Israelachvili, 1985). This development prompted Kallman & Willstätter (1932) and Bradley (1932) to realize the summation over pairs of molecules in interacting particles would yield a long-range attraction.

Subsequently, de Boer (1936) and Hamaker (1937) performed explicit calculations of dispersion forces between colloidal particles by assuming the intermolecular forces to be strictly pairwise additive. Although approximate, this theory captures the essence of the phenomenon. The attraction arises because local fluctuations in the polarization within one particle induce, via the propagation of electromagnetic waves, a correlated response in the other. The associated free energy decreases with decreasing separation. Phase shifts introduced at large separations by the finite velocity of propagation reduce the degree of correlation, and, therefore, the magnitude of the attraction. Although the intermolecular potential decays rapidly on the molecular scale, the cumulative effect is a long-range interparticle potential that scales on the particle size.

Introduction

In the preceding chapters, fundamental aspects of colloid behavior have been emphasized. Now we are ready to apply this knowledge to processes involving suspensions. Here we investigate the capture of small particles by stationary collector units, one aspect of filtration technology.

Elementary considerations show that a strong attractive force is necessary if freely suspended particles are to come together, because at close separations viscous resistance increases dramatically. Since the interparticle force derives from the combination of electrostatic and dispersion forces, capture is particularly sensitive to the balance between colloidal and hydrodynamic forces. Several mechanisms contribute to particle capture and retention. Inertia is the dominant factor when fast-moving particles impact on a stationary object, whereas geometry and proximity govern the interception of slow-moving particles. The capture of submicron particles is influenced enormously by interparticle forces and Brownian motion. All these aspects are treated here, but technological issues are ignored. For example, a persistent problem encountered in the filtration of small particles is buildup of a deposit. Our treatment deals with the behavior of clean collector units to emphasize basic colloidal phenomena.

Aerosols have received the most study by a wide margin and many comprehensive reviews exist, e.g. Hidy & Brock (1970), Davies (1973), Friedlander (1977), and Kirsch & Stechkina (1978). Ives (1975) and Tien & Payatakes (1979) present broad reviews of liquid filtration; Spielman (1977) concentrates on small-scale processes in liquids.

Introduction

The sedimentation of colloidal particles is important both in technology and in the laboratory. Gravity settlers, thickeners, or clarifiers commonly remove particles from waste streams issuing from a variety of processes. These generally operate as continuous processes that split the feed into two product streams, one the clear fluid and the other a sludge. Successful design requires knowledge of the sedimentation velocity of the particles over the relevant range of volume fractions and the role of interparticle forces in determining the structure of the dense sludge. Centrifugation provides a means of enhancing the driving force for commercial-scale operations, as well as concentrating or analyzing dispersions in the laboratory.

Despite their longstanding use, much remains to be understood about the details of processes which convert dilute dispersions into dense sediments. The key issues appear to be

1. (i) the variation of the settling velocity with volume fraction and interparticle potential,

2. (ii) the role of forces transmitted by interparticle potentials, and

3. (iii) the formulation of macroscopic models to predict the evolution of volume fraction as a function of position and time.

As with other colloidal phenomena, the complexity arises from the importance of a variety of interparticle forces and the fact that many systems of interest tend to be flocculated.