Brodberg K.B. Cracks and fracture



Brodberg K.B. Cracks and fracture
23.11.2009 в 04:16 6.21 МБ djvu 75 раз

1999. , 752 pages

From Introduction:

What is fracture? The simplest answer would be: "the process of breaking" or "the condition of being broken". Actually, both fracture and break can be traced to the same Indo-European root, bhreg (to break). Each term can be either a noun or a verb.

Common-day use of the word "fracture" is not well-defined. It may refer to something that is being (or has been) broken apart, or it may indicate the presence of a crack. In the present work, "fracture" will be reserved for unstable crack growth, either all the way through a structural part or so far that the strength or stiffness of the part becomes considerably reduced.

A crack may be defined as a material separation by opening or sliding, with the separation distance substantially smaller than the separation extent - the crack length. The separation distance is often comparable to certain micro-structural length dimensions, for instance the distance between larger inhomogeneities in the material, such as inclusions. In extreme cases, the separation distance may be of the order of the atomic distance, and the crack length, while still large compared to this distance, may be smaller than some larger micro-structural dimensions, for instance a grain size. It is then appropriate to talk about a micro-crack. Micro-cracks play an important part in fracture processes, but so also do other types of material separation on a microscopic level, particularly internal voids (holes). The general term for a material separation on a microscopic level is a micro-separation.

From a practical point of view, many cracks may be considered as harmless, i.e. not leading to fracture. A big structure, such as a tanker, contains probably several thousand macroscopic cracks and several million micro-cracks. Essentially only those cracks that are situated in highly strained regions should be regarded as potential fracture initiators, and this only if they are larger than a certain size. It is a major objective of fracture mechanics to find out which cracks constitute an obvious risk for fracture and which do not.

Crack growth depends on loading conditions and environmental conditions. It may be extremely fast, over 1000 m/s, and it may be extremely slow, less than 1 mm/year.

Loading conditions include many distinct types, static, dynamic, load controlled, grip controlled, etc. An important distinction should be made between monotone loading, i.e. monotonically increasing load until either a certain level is reached or mechanical failure occurs, and repeated loading, cyclic or non-cyclic. Very few structures, such as objects loaded by their own weight, are actually exposed to monotone loading. Some structures are exposed to considerably more than ten thousand load applications. If fracture occurs after so many load applications it is referred to as fatigue, or, more clearly, high-cycle fatigue. By low-cycle fatigue is meant fatigue after relatively few load applications, usually fewer than about ten thousand!

Environmental conditions such as temperature and corrosive atmosphere influence crack growth. At a high temperature, usually several hundred degrees Celsius, metals show crack growth through creep, i.e. slow crack growth even at constant load.

The basic framework presented in this book is to a large extent common to most types of crack growth and fracture. High-cycle fatigue, creep crack growth and crack growth under corrosion will not, however, be explicitly covered.

Fracture is only one way by which mechanical failure can occur. Other types of processes leading to mechanical failure are corrosion and wear. These mechanisms do not belong to the scope of the present book. Very closely related to fracture, however, is plastic collapse. In a ductile tensile test piece, for instance, plastic instability precedes crack growth (at least on a macro-scale), and the fact that the final rupture occurs through fracture is uninteresting from a practical point of view: quite obviously the plastic instability should be given the blame for the failure. In other cases, the opposite order of events occurs, for instance at failure caused by bending of a ductile beam: crack growth may reduce the beam stiffness so much that plastic collapse takes over, but the failure may already be a fact when this happens, so the occurrence of plastic collapse is fairly uninteresting from a practical point of view.
In engineering structures, crack growth occurs generally through opening of a gap between the crack surfaces. This mechanism is conventionally referred to as the opening mode or mode I. In other cases, mainly in earthquakes and related events, crack growth occurs through sliding between the crack surfaces. This mode is called the sliding mode or shearing mode, and there are two varieties, mode II and mode III, depending on whether the sliding direction is normal to or parallel with the crack edge. Frequently both of these modes occur together (so called mixed mode growth), but mode I does not appear to mix readily with the other two.

Fracture mechanics is a rather young discipline. Even though the interest in fracture prediction probably is older than our civilization, the systematic approach to problems concerning growth of pre-existing macroscopic cracks, which is what fracture mechanics is about, is typically a 20th century concern. Basic mathematical tools were created by Kolosov (1909) in his doctoral thesis at University of Dorpat (present Tartu), Estonia. Inglis (1913), obviously independently, also solved a basic crack problem, and, in a discussion of Inglis' paper, B. Hopkinson (1913) suggested that nonlinear phenomena near the crack edge should be taken into account. This was finally done by Griffith (1920), but, by using energy considerations and the concept of surface energy, he avoided an analysis of the crack edge neighbourhood. Griffith's experiments with thin glass rods prompted Weibull (1939a,b) to establish a statistical theory of fracture. Orowan (1952) extended Griffith's approach to all cases of small scale yielding (in which plastic flow is confined to a small region near the crack edge) by inclusion of all dissipative energy, essentially the surface energy and plastic work. Irwin (1957) introduced new and expedient concepts such as the stress intensity factor (originally the crack driving force) and the energy release rate. The critical stress intensity factor or, equivalently, the fracture toughness, became concepts that laid the foundation of the linear (elastic) fracture mechanics (LEFM). Barenblatt (1959a,b) introduced the concept of autonomy of the field near the crack edge, and a linearized model of the crack edge vicinity lead to his concept of cohesion modulus. In fact, all the different concepts used in LEFM are developed explicitly or tacitly under the assumption of autonomy, which thus provides the very basis for LEFM. Briefly expressed, autonomy implies that the processes near a crack edge are always the same in each material, regardless of body and loading geometry, under certain specified general conditions.

After the 1950s the development in the fields of crack mechanics and fracture mechanics has been quite impressive, quantitatively and qualitatively. It is not possible to describe this development in a rather limited space, but a few names will be mentioned. First to mind comes J.R. Rice, who has made outstanding contributions to virtually all fields in crack and fracture mechanics, from the mid 60s, including the introduction of the J-integral concept, a path-independent integral, for crack analysis (Rice 1968a), which laid the foundation for the nonlinear fracture mechanics, to recent contributions concerning three-dimensional dynamic crack propagation (Geubelle and Rice, 1995, Cochard and Rice, 1997, Morrisey and Rice, 1998). His impact on the whole field has been singular and enormous. In the dynamic field, the significant and pioneering contributions by B.V. Kostrov and L.B. Freund deserve particular mention. Kostrov solved several problems of importance for earthquake source physics and dynamic crack propagation in general. He was the first to solve a problem of nonconstant crack expansion (Kostrov, 1966). Among numerous contributions by Freund may be mentioned a series of four papers on crack propagation with nonconstant velocity and other dynamic problems, such as stress wave interaction with cracks (Freund 1972a,b, 1973, 1974a). Finally, T. Yokobori should be mentioned, both for his outstanding merits as a scientist and for his organizational talents: he is the Founder President of the International Congress of Fracture, which started in 1965 and has had a profound importance as a forum for exchange of ideas and experiences through their quadrennial conferences. His book on fracture mechanics, first published in Japanese 1955 and translated into English ten years later (Yokobori 1965), appears to be the first monograph in the field.

There exists a fairly large number of books and overviews related to fracture mechanics. A summing up of the state of the art at the end of the 1960s is given in a seven volume large treatise on fracture, edited by H. Liebowitz (1968-1972). Other books were written by Anderson (1995), Broek (1982), Cherepanov (1979), Freund (1990), Hahn (1976), Hellan (1984), Herzberg (1983), Kanninen and Popelar (1985), Karihaloo (1995), Knott (1979), Lawn (1993) and Yokobori (1965). There are also several books on specialized subjects, such as computer methods in fracture mechanics, fatigue crack propagation and creep crack growth. Some of these, for instance the book by Riedel (1987) on fracture at high temperatures, also give account for general properties of cracks and fracture.

This volume builds on research work in various subfields of crack and fracture mechanics from all over the world. The selection naturally reflects my own interests and experiences. The two first chapters deal with the physical processes in the vicinity of the crack edge and the development of fracture. Chapter 3 develops general basic concepts and relations in crack mechanics, such as path-independent integrals, stress intensity factors and energy flux into the process region. Chapters 4-7 deal with the analysis of elastostatic cracks, stationary or slowly moving elastic-plastic cracks, elastodynamic crack processes and elastic-plastic crack dynamics. In Chapter 8, physical and engineering aspects of the processes leading to fracture are considered, and Chapter 9 deals with dynamic fracture mechanics. The appendices include general formulae, the basic theory of analytic functions, introduction to Laplace, Mellin and Hankel transforms, and description of certain basic relations, for instance for stress waves in solids. There is an extensive bibliography, covering references to both classical and recent work.

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