Basic Concepts of Mechanics

In this section we develop the basic ideas of classical rheological thought. We presume elastomers to deform as continuous media and to be subject to the formalism of continuum mechanics [TIO]. We begin by developing the idea of the nature of applied forces and the stress tensor. 

The idea of the stress tensor in a material arises from the necessity of representing the influence of applied forces on deformation. The applied forces F acting on a body may be represented as the sum of contact forces acting on the surface and body forces f, such as gravitation, which act directly on the elements of mass. We may write [TIO] (Fig. 1) 

The idea of the stress tensor comes from relating t to the unit normal vector n to the surface through 

The concepts of the stress vector and stress tensor were developed during the 1820s by Cauchy. The direction of the stress component is /.When the direction of the stress component i is perpendicular to the plane (oiy/ = ), the stress is called the normal stress. When the direction i is tangent to the plane j ipiji the stress is called the shear stress. 

The complete dynamics of a deforming body requires including the contact forces, the body (gravitational) forces pf with inertial forces. For a macroscopic mass M 

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In this chapter we shall first outline the basic concepts of the various mechanisms for energy redistribution, followed by a very brief overview of collisional intennoleciilar energy transfer in chemical reaction systems. The main part of this chapter deals with true intramolecular energy transfer in polyatomic molecules, which is a topic of particular current importance. Stress is placed on basic ideas and concepts. It is not the aim of this chapter to review in detail the vast literature on this topic we refer to some of the key reviews and books [U, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32] and the literature cited therein. These cover a variety of aspects of tire topic and fiirther, more detailed references will be given tliroiighoiit this review. We should mention here the energy transfer processes, which are of fiindamental importance but are beyond the scope of this review, such as electronic energy transfer by mechanisms of the Forster type [33, 34] and related processes. 

The main features of the chemical bonding formed by electron pairs were captured in the early days of quantum mechanics by Heitler and London. Their model, which came to be known, as the valence bond (VB) model in its later versions, will serve as our basic tool for developing potential surfaces for molecules undergoing chemical reactions. Here we will review the basic concepts of VB theory and give examples of potential surfaces for bond-breaking processes. 

Over the past decade there has been an explosion of knowledge on the mechanisms of both coagulation and fibrinolysis that has contributed to our appreciation of the basic concepts of these two pathways. 

In this chapter we give a brief review of some of the basic concepts of quantum mechanics with emphasis on salient points of this theory relevant to the central theme of the book. We focus particularly on the electron density because it is the basis of the theory of atoms in molecules (AIM), which is discussed in Chapter 6. The Pauli exclusion principle is also given special attention in view of its role in the VSEPR and LCP models (Chapters 4 and 5). We first revisit the perhaps most characteristic feature of quantum mechanics, which differentiates it from classical mechanics its probabilistic character. For that purpose we go back to the origins of quantum mechanics, a theory that has its roots in attempts to explain the nature of light and its interactions with atoms and molecules. References to more complete and more advanced treatments of quantum mechanics are given at the end of the chapter. 

In references 38 and 39, Comba and Hambley introduce their topic in three parts (1) basic concepts of molecular mechanics, (2) applications of the techniques and difficulties encountered, and (3) a guide to molecular modeling of a new system. Only the introductory section of reference 38 is summarized here. 

The basic concepts of the instrumentation for liquid chromatography are described here, with the mechanism of their operation and their influence on the separation of analytes. 

The basic concepts of nucleophilic substitution reactions appeared in the first semester of organic chemistry. These reactions follow or Sp 2 mechanisms. (In aromatic nucleophilic substitution mechanism, we use the designation Sp Ar.) In Sfjl and Sp 2 mechanisms, a nucleophile attacks the organic species and substitutes for a leaving group. In aromatic systems, the same concepts remain applicable, but with some differences that result from the inherent stability of aromatic systems. 

The explanation of classical MD given above was meant in part to emphasize that the dynamics of atoms can be described provided that the potential energy of the atoms, U U(ru. .., r3N), is known as a function of the atomic coordinates. It has probably already occurred to you that a natural use of DFT calculations might be to perform molecular dynamics by calculating U U(r, ..., r3N) with DFT. That is, the potential energy of the system of interest can be calculated on the fly using quantum mechanics. This is the basic concept of ab initio MD. The Lagrangian for this approach can be written as... 

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