Plane Inertial Problems

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. 

References to the relevant literature will be noted at appropriate places. Problems of material response to applied load have been treated by Morland (1963), Chu (1965) and Martincek (1979). 

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Our method of attacking plane, non-inertial problems will be, in the first instance, to reduce (2.8.9) to a Hilbert problem, in precisely the manner developed by Muskhelishvili (1963), and then to handle the specifically viscoelastic aspects, essentially by the methods outlined in Sects. 2.4-6. We remark that an alternative way of approaching the first stage is the dual integral equation method originally used in this context by Sneddon (1951) but with a long history of mathematical development summarized by Gladwell (1980). 

Some standard terminology will occasionally be used. If the stresses on the crack face are purely normal, the crack is said to be subject to opening mode or Mode I displacement, or it is simply referred to as a Mode I crack. If the stresses are purely shear, the crack is subject to sliding mode or Mode II displacement, while if the stresses are perpendicular to the plane, we have tearing mode or Mode III displacement [Irwin (1960), Sih and Liebowitz (1968), Sneddon and Lowengrub (1969) for example]. In this Chapter, we consider mainly Mode I displacement and, to a certain extent. Mode II. Tearing mode cracks, which are typically the simplest to analyze, are considered briefly in Chap. 7, in the context of inertial problems. 

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. 

Figure 18. The Oxy plane of linear configurations (above) and the Oyz plane of isoceles configurations in the A-IA2A3 problem. The potential function is deeper for darker gray. The lines cross at the center and at the equilibrium points. The white dashed line is the reaction path, if there were no inertial or nondiagonal kinetic energy effects.

Let us consider now a double-sided plane indentation for a clamped beam (Fig. 17.4). To simplify the problem, we will study a slender beam, neglecting shear and rotatory inertial terms. Let d(t) and P(t) express the time dependence of both the displacement of the indentor and the applied load. Furthermore la is the length of contact, and I is the half-length of the beam. The origin of coordinates will be taken at the center of the beam. Owing to the symmetry of the problem, only the solution for x > 0 will be considered. 

We consider first a body of negligible mass moving around a body of finite mass m in an elliptic orbit. It can be proved (Murray and Dermott, 1999) that the action-angle variables of the two-body problem in the inertial frame, in the plane, are the Delaunay variables defined by... 

There are nine components of (unknown) stresses at any point in a stressed body, and they generally vary from point to point within the body. These stresses must be in equilibrium with each other and with other body forces (such as gravitational and inertial forces). For elastostatic problems, the body forces are typically assumed to be zero, and are not considered further. For simplicity, therefore, the equilibrium of an element dx, dy, 1) under plane stress (a22 = = xz = = 0) is... 

We now proceed to derive the Dirac states for a freely moving electron of mass Me- Note that the charge of the fermion does not enter the Dirac equation for this fermion being at rest or moving freely with constant velocity v. The solutions in Eqs. (5.72) and (5.73) may now be subjected to a general Lorentz boost as given by Eq. (3.81) into an inertial frame of reference moving relatively to the previous one, in which the fermion is at rest, with velocity (— ). This option, namely that the solutions for a complicated kinematic problem can be obtained from those of a simple kinematic problem in a suitably chosen frame of reference by a Lorentz transformation, cannot be overemphasized from a conceptual point of view. However, instead of this Lorentz transformation a direct solution of the Dirac Eq. (5.23) is easier. For this purpose we choose an ansatz of plane waves,... 

IX. Causality. The requirement of Causality, namely that the current situation can be influenced only by past and contemporaneous events, may be shown to impose a constraint on the analytic structure of the complex moduli in the complex (JD plane, and also on combinations of the moduli multiplying Green s functions in the solution of non-inertial boundary value problems. These quantities can have no singularities in the lower half-plane. In a restricted sense, this can be shown directly for certain combinations of complex moduli using properties of the individual complex moduli. 

We consider, in the present chapter, problems involving moving or varying loads acting on the boundaries of viscoelastic materials, where plane strain (see Sect. 2.8) conditions prevail. Inertial effects are neglected. 

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