Plane Non-inertial Crack Problems

Some simple fracture mechanics problems for viscoelastic media are considered in this chapter. Much of the emphasis will be on qualitative differences in behaviour from corresponding elastic configurations, in other words specifically viscoelastic phenomena. We refer to standard texts on fracture mechanics for a treatment of the elastic problems for example, Liebowitz (1968), Sneddon and Lowengrub (1969), Lardner (1974) and Cherepanov (1979). 

There are two alternative approaches to crack problems in elastic or viscoelastic media, which might be termed the microscopic and the macroscopic approach, respectively. The microscopic approach is based on the dislocation concept [Lardner (1974), for example] while the macroscopic approach views a crack simply as a cavity in the medium. Both are finally equivalent, of course. 

The microscopic approach has the advantage of being physically intuitive and also, in some respects, mathematically convenient in that integral equations can often be written down with a minimum of theoretical development. However, the dislocation concept at times appears unnecessary and burdensome. Also, with this approach, it is difficult to bring into play the full power and elegance of the mathematical methods, based on complex variable techniques which have been developed for plane elastic (and easily adaptable to viscoelastic) problems. 

We therefore adopt the second approach, omitting the introduction and use of the dislocation concept. 

One feature of interest in crack problems is that the exact nature of the boundary conditions is not always a priori obvious, but has to be determined in the course of solving the problem. This is particularly true for viscoelastic media. 

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We consider plane contact and crack problems in this chapter, without neglecting inertial effects. Such problems are typically far more difficult than the non-inertial problems discussed in Chaps. 3 and 4, and require different techniques for their solution. This is an area still in the development stage so that it will not be possible to achieve the kind of synthesis or unification which is desirable. We confine our attention to steady-state motion at uniform velocity V in the negative x direction. We begin by deriving boundary relationships between displacement and stress. These are applied to moving contact problems in the small viscoelasticity approximation, and to Mode III crack problems without any approximation. 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

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