Supplemental reading:

Holton (1979), sections 9.2, 9.3

Pedlosky (1979), sections 7.1–7.3

Remarks

In Chapters 8–11 we examined the wave properties of the atmosphere under a variety of circumstances. We have also considered the interactions of waves with mean flows. In Chapter 7, we noted that eddies would have to be involved in transporting heat between the tropics and the poles. Thus far our study of waves has not provided much insight into this matter. As it turns out, vertically propagating stationary Rossby waves do carry heat poleward. This heat transport, while not insignificant, is largely a byproduct of the fact that the wave momentum flux acts to reduce shears, and geostrophic adjustment involves a concomitant reduction of meridional temperature gradients. Unfortunately, we will not have time to study these mechanisms in these notes. However, the quasi-geostrophic framework established in Chapter 12 greatly simplifies such studies. This framework will be used to study how travelling disturbances arise in the atmosphere. We will also sketch (in Chapter 14) some results which suggest that these travelling disturbances play the major role in determining the global, temperature distribution. The generation of such disturbances involves hydrodynamic instability, and before diving into this problem in a meteorological context, it will be useful to examine stability in simpler situations.

Before beginning this topic we should recall possible wave (eddy) sources considered thus far:

1. Direct forcing as produced by tidal heating, flow over mountains, or flow through quasi-stationary inhomogeneities in heating.

2. Resonant free oscillations. These are presumably preferred responses to any ‘noise'.

Supplemental reading:

Lorenz (1967)

Held and Hou (1980)

Schneider and Lindzen (1977)

Schneider (1977)

Lindzen and Hou (1988)

As we noticed in our perusal of the data, atmospheric fields are far from being zonally symmetric. Some of the deviation from symmetry is forced by the inhomogeneity of the earth's surface, and some is autonomous (travelling cyclones, for example). Nevertheless, the zonally averaged circulation has, over the centuries, been the object of special attention. Indeed, the term ‘general circulation’ is frequently taken to mean the zonally averaged behavior. This is the viewpoint of Lorenz (1967).

You are urged to read chapters 1, 3, and 4 of Lorenz. Chapter 1 is a short and especially insightful discussion of the methodology of studying the atmosphere. As is generally the case in this field, there will be views in Lorenz which are not universally agreed on, but this hardly diminishes its value.

There are several reasons for focussing on the zonally averaged circulation:

1. Significant motion systems like the tropical tradewinds are well described by zonal averages.

2. The circulation of the atmosphere is only a small perturbation on a rigidly rotating basic state which is zonally symmetric.

3. The zonally averaged circulation is a convenient subset of the total circulation.

Our approach in this chapter will be to inquire how the atmosphere would behave in the absence of eddies. It is hoped that a comparison of such results with observations will lend some insight into what maintains the observed zonally averaged state. In particular, discrepancies may point to the rôle of eddies in maintaining the zonal average.

Introduction

In the preceding chapter, the transient stress intensity factor history resulting from the application of a spatially uniform crack face traction was examined. The particular situations analyzed represent the simplest cases of transient loading of a stationary crack for each mode of crack opening. There are many similar situations that can be analyzed by the methods outlined in Chapter 2. For example, consider the plane strain situation of a plane tensile stress pulse propagating through the material toward the edge of the crack, which lies in the half plane −∞ < x < 0, y = 0. Suppose that the x and y components of the unit vector normal to the wavefront are − cos θ and − sin θ, respectively, and that the pulse front reaches the crack edge at time t = 0. Suppose further that the incident pulse carries a jump in the normal stress component from the initial value of zero to σinc. If the solid is uncracked, then the incident pulse will induce a tensile traction of magnitude σ* = σinc{1 − (1 − 2v) cos2 θ/(1 − v)} and a shear traction of magnitude τ* = σinc[1 − 2v) sin θ cos θ/(1 − v) on the plane y = 0 over the interval −Cdt/ cos θ < x 0. This particular stress wave diffraction problem was studied by de Hoop (1958) in his pioneering work on the subject.

Introduction

In this chapter, the role of material inelasticity is considered, in the form of irreversible plastic flow, dependence of the material response on the rate of deformation, or microcracking. The study of issues in dynamic fracture mechanics concerned with these effects is at an early stage. Consequently, the sections are not integrated to any significant degree. Instead, each section is intended to give an impression of the present stage of development of analytical modeling in the areas covered.

Viscoelastic crack growth

The study of crack growth in a linear viscoelastic material has been motivated primarily by interest in modeling the fracture process in relatively brittle polymeric materials, although other materials may be idealized as linear viscoelastic under some circumstances. There are numerous specific linear viscoelastic models available for stress analysis, but only some general properties are considered here for crack growth analysis. Time-dependent material response, of the kind on which the theory of viscoelasticity is based, may be of interest in the analysis of fracture phenomena at two different levels. On the one hand, the bulk properties of the body in which the crack is propagating are important in determining the way in which the effect of applied loads is transferred to the crack tip region or the way in which stress is redistributed due to crack growth.

Introduction

The restriction to constant crack speed in the early analytical work on dynamic crack propagation, as reflected in the contents of Chapter 6, was not motivated by physical considerations. Instead, the restriction was imposed in order to render the mathematical models tractable. In this chapter, progress toward relaxing this restriction for rapid crack growth in nominally elastic materials is described. Results are limited, but sufficient to provide a relatively complete conceptual basis for analysis of crack propagation under two-dimensional conditions or in simple structural elements.

The study of a problem involving crack growth at nonuniform rate proceeds in two steps. First, the underlying boundary value problem is considered for arbitrary motion of the crack tip, with a view toward obtaining a full description of the mechanical fields near the crack edge during growth. Then, an additional physical postulate in the form of a crack propagation criterion is imposed on the mechanical fields in order to determine an equation of motion for the crack tip. This approach is illustrated for antiplane shear crack growth, for plane strain crack growth under quite general loading conditions, and for one-dimensional string and beam models. The technologically important issue of crack arrest (Bluhm 1969) is an essential feature in this development; arrest is identified through the crack tip equation of motion as the point beyond which crack growth cannot be sustained.

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.