Inertial problems

Except in Chap. 7, the non-inertial approximation is adopted throughout this work. Even for the very simple inertial problems considered in Chap. 7, it is apparent that there are significant difficulties associated with the retention of inertial terms. 

In this chapter, we discuss methods of solution of viscoelastic boundary value problems in general terms, together with certain relevant theorems. The main emphasis is on non-inertial problems. Also, most of the discussion is confined to the isothermal case. 

The Classical Correspondence Principle was enunciated in reasonably general form by Read (1950) and Lee (1955) among others and discussed rigorously by Sternberg (1964) for the more general non-isothermal case. Sternberg (1964) reviews the older literature in some detail. Tao (1966) discusses correspondences between elastic and viscoelastic inertial problems in terms of Laplace transforms, essentially generalizing the work of Lee (1955). 

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. 

Inertial normal contact problems in three dimensions have been considered by Sabin (1975, 1987). By means of an integral transform technique, he reduces the problem to a set of dual Integral equations, which are in turn reduced to a single Volterra integral equation. This is solved numerically. He obtains the interesting result that the contact pressure is not significantly different from that in the non-inertial problem. A similar observation had been made earlier, in connection with the elastic problem, by Tsai (1971). 

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. 

These are in fact the same as for the non-inertial problem. The results (7.3.11 -13) for the stress ahead of the crack and the stress intensity factor have been given by Willis (1%7), for a standard linear solid, and Walton (1982) for a general material, using quite different methods to the one outlined here. 

During the seventies, the work on non-inertial problems was consolidated. The main purpose of the present volume is to present a coherent, unified development of this topic, in particular of those problem classes which are not covered by the Classical Correspondence Principle. There has also been some progress on inertial problems. Typically however, to make progress on such problems it is necessary either to confine one s attention to the most idealized configurations or to introduce some approximation. Also, the mathematical techniques used have been generally rather sophisticated. We briefly discuss this work in the last chapter, and derive certain results by comparatively elementary methods. 

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. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The Reynolds number Re = vl/v, where v and l are the characteristic velocity and length for the problem, respectively, gauges the relative importance of inertial and viscous forces in the system. Insight into the nature of the Reynolds number for a spherical particle with radius l in a flow with velocity v may be obtained by expressing it in terms of the Stokes time, t5 = i/v, and the kinematic time, xv = l2/v. We have Re = xv/xs. The Stokes time measures the time it takes a particle to move a distance equal to its radius while the kinematic time measures the time it takes momentum to diffuse over... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

The problem relates directly to the constancy of c, which implies that the velocity of light is independent of both the motion of its source and the direction of propagation, a condition that cannot hold in more than one Newtonian inertial frame if the Galilean principle of relativity applies. Since there is no evidence that the laws of physics are not identical in all inertial frames of reference the only conclusion is that the prescription for Galilean transformations needs modification to be consistent, not only with simple mechanics, but also with electromagnetic effects. 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

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